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Introduction to the Theory of COMPUTATION
THIRD EDITION
MICHAEL SIPSER
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To Ina, Rachel, and Aaron
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CONTENTS
Preface to the First Edition xi
Tothestudent……………………… xi Totheeducator …………………….. xii The first edition . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Feedbacktotheauthor …………………. xiii Acknowledgments……………………. xiv
Preface to the Second Edition xvii
Preface to the Third Edition xxi
0 Introduction 1
0.1 Automata,Computability,andComplexity . . . . . . . . . . . . . 1 Complexitytheory……………………. 2 Computabilitytheory ………………….. 3 Automatatheory…………………….. 3
0.2 MathematicalNotionsandTerminology . . . . . . . . . . . . . . 3 Sets…………………………… 3 Sequencesandtuples ………………….. 6 Functionsandrelations …………………. 7 Graphs…………………………. 10 Stringsandlanguages ………………….. 13 Booleanlogic………………………. 14 Summaryofmathematicalterms …………….. 16
0.3 Definitions,Theorems,andProofs …………….. 17 Findingproofs……………………… 17 0.4 TypesofProof ………………………. 21 Proofbyconstruction………………….. 21 Proofbycontradiction………………….. 21 Proofbyinduction……………………. 22 Exercises,Problems,andSolutions ………………. 25
v
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vi CONTENTS
Part One: Automata and Languages 29
1 Regular Languages 31
1.1 FiniteAutomata ……………………… 31 Formaldefinitionofafiniteautomaton . . . . . . . . . . . . . 35 Examplesoffiniteautomata……………….. 37 Formaldefinitionofcomputation ……………. 40 Designingfiniteautomata………………… 41 Theregularoperations …………………. 44
1.2 Nondeterminism……………………… 47 Formal definition of a nondeterministic finite automaton . . . . 53 EquivalenceofNFAsandDFAs …………….. 54 Closureundertheregularoperations. . . . . . . . . . . . . . . 58
1.3 RegularExpressions ……………………. 63 Formaldefinitionofaregularexpression . . . . . . . . . . . . 64 Equivalencewithfiniteautomata…………….. 66
1.4 NonregularLanguages…………………… 77 Thepumpinglemmaforregularlanguages . . . . . . . . . . . 77 Exercises,Problems,andSolutions ………………. 82
2 Context-Free Languages 101
2.1 Context-FreeGrammars…………………..102 Formal definition of a context-free grammar . . . . . . . . . . 104 Examplesofcontext-freegrammars ……………105 Designingcontext-freegrammars …………….106 Ambiguity ………………………..107 Chomskynormalform ………………….108
2.2 PushdownAutomata…………………….111 Formaldefinitionofapushdownautomaton. . . . . . . . . . . 113 Examples of pushdown automata . . . . . . . . . . . . . . . . . 114 Equivalencewithcontext-freegrammars. . . . . . . . . . . . . 117
2.3 Non-Context-FreeLanguages………………..125 The pumping lemma for context-free languages . . . . . . . . . 125
2.4 DeterministicContext-FreeLanguages. . . . . . . . . . . . . . . 130 PropertiesofDCFLs …………………..133 Deterministiccontext-freegrammars . . . . . . . . . . . . . . 135 Relationship of DPDAs and DCFGs . . . . . . . . . . . . . . . 146 Parsing and LR(k) Grammars . . . . . . . . . . . . . . . . . . . 151 Exercises,Problems,andSolutions ……………….154
Part Two: Computability Theory 163
3 The Church–Turing Thesis 165
3.1 TuringMachines………………………165 FormaldefinitionofaTuringmachine . . . . . . . . . . . . . . 167
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Examples of Turing machines . . . . . . . . . . . . . . . . . . . 170
3.2 VariantsofTuringMachines…………………176 MultitapeTuringmachines ………………..176 NondeterministicTuringmachines. . . . . . . . . . . . . . . . 178 Enumerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Equivalence with other models . . . . . . . . . . . . . . . . . . 181
3.3 TheDefinitionofAlgorithm ………………..182 Hilbert’s problems . . . . . . . . . . . . . . . . . . . . . . . . . 182 Terminology for describing Turing machines . . . . . . . . . . 184 Exercises,Problems,andSolutions ……………….187
4 Decidability 193
4.1 DecidableLanguages…………………….194 Decidable problems concerning regular languages . . . . . . . 194 Decidable problems concerning context-free languages . . . . . 198
4.2 Undecidability ……………………….201 Thediagonalizationmethod ……………….202 Anundecidablelanguage …………………207 A Turing-unrecognizable language . . . . . . . . . . . . . . . . 209
Exercises,Problems,andSolutions ……………….210
5 Reducibility 215
5.1 Undecidable Problems from Language Theory . . . . . . . . . . 216 Reductionsviacomputationhistories. . . . . . . . . . . . . . . 220
5.2 ASimpleUndecidableProblem……………….227
5.3 MappingReducibility ……………………234 Computable functions . . . . . . . . . . . . . . . . . . . . . . . 234 Formaldefinitionofmappingreducibility . . . . . . . . . . . . 235 Exercises,Problems,andSolutions ……………….239
6 Advanced Topics in Computability Theory 245
6.1 TheRecursionTheorem…………………..245 Self-reference ………………………246 Terminologyfortherecursiontheorem . . . . . . . . . . . . . 249 Applications ……………………….250
6.2 Decidabilityoflogicaltheories ……………….252 A decidable theory . . . . . . . . . . . . . . . . . . . . . . . . . 255 An undecidable theory . . . . . . . . . . . . . . . . . . . . . . . 257
6.3 TuringReducibility……………………..260
6.4 ADefinitionofInformation…………………261 Minimallengthdescriptions ……………….262 Optimality of the definition . . . . . . . . . . . . . . . . . . . . 266 Incompressiblestringsandrandomness . . . . . . . . . . . . . 267 Exercises,Problems,andSolutions ……………….270
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CONTENTS vii
viii CONTENTS
Part Three: Complexity Theory 273
7 Time Complexity 275
7.1 MeasuringComplexity……………………275 Big-O and small-o notation . . . . . . . . . . . . . . . . . . . . 276 Analyzingalgorithms …………………..279 Complexityrelationshipsamongmodels . . . . . . . . . . . . . 282
7.2 TheClassP…………………………284 Polynomial time . . . . . . . . . . . . . . . . . . . . . . . . . . 284 ExamplesofproblemsinP ………………..286
7.3 TheClassNP………………………..292 ExamplesofproblemsinNP ……………….295 ThePversusNPquestion ………………..297
7.4 NP-completeness………………………299 Polynomialtimereducibility ……………….300 Definition of NP-completeness . . . . . . . . . . . . . . . . . . 304 The Cook–Levin Theorem . . . . . . . . . . . . . . . . . . . . 304
7.5 AdditionalNP-completeProblems ……………..311 The vertex cover problem . . . . . . . . . . . . . . . . . . . . . 312 TheHamiltonianpathproblem ……………..314 Thesubsetsumproblem …………………319
Exercises,Problems,andSolutions ……………….322
8 Space Complexity 331
8.1 Savitch’sTheorem ……………………..333
8.2 TheClassPSPACE …………………….336
8.3 PSPACE-completeness …………………..337
The TQBF problem . . . . . . . . . . . . . . . . . . . . . . . . 338 Winning strategies for games . . . . . . . . . . . . . . . . . . . 341 Generalizedgeography ………………….343
8.4 TheClassesLandNL……………………348
8.5 NL-completeness ……………………..351 Searchingingraphs ……………………353
8.6 NLequalscoNL………………………354 Exercises,Problems,andSolutions ……………….356
9 Intractability 363
9.1 HierarchyTheorems…………………….364 Exponentialspacecompleteness ……………..371
9.2 Relativization………………………..376 Limitsofthediagonalizationmethod . . . . . . . . . . . . . . 377
9.3 CircuitComplexity……………………..379 Exercises,Problems,andSolutions ……………….388
10 Advanced Topics in Complexity Theory 393
10.1ApproximationAlgorithms …………………393
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10.2 Probabilistic Algorithms . . . . . . . . . . . . . . . . . . . . . . . 396 TheclassBPP ………………………396 Primality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Read-once branching programs . . . . . . . . . . . . . . . . . . 404
10.3Alternation …………………………408 Alternatingtimeandspace ………………..410 The Polynomial time hierarchy . . . . . . . . . . . . . . . . . . 414
10.4InteractiveProofSystems ………………….415 Graph nonisomorphism . . . . . . . . . . . . . . . . . . . . . . 415 Definitionofthemodel ………………….416 IP=PSPACE ………………………418
10.5ParallelComputation ……………………427 Uniform Boolean circuits . . . . . . . . . . . . . . . . . . . . . 428 TheclassNC ………………………430 P-completeness ……………………..432
10.6 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Secret keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Public-keycryptosystems …………………435 One-way functions . . . . . . . . . . . . . . . . . . . . . . . . . 435 Trapdoorfunctions ……………………437
Exercises,Problems,andSolutions ……………….439 Selected Bibliography 443 Index 448
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CONTENTS ix
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PREFACE TO THE FIRST EDITION
TO THE STUDENT
Welcome!
You are about to embark on the study of a fascinating and important subject: the theory of computation. It comprises the fundamental mathematical proper- ties of computer hardware, software, and certain applications thereof. In study- ing this subject, we seek to determine what can and cannot be computed, how quickly, with how much memory, and on which type of computational model. The subject has obvious connections with engineering practice, and, as in many sciences, it also has purely philosophical aspects.
I know that many of you are looking forward to studying this material but some may not be here out of choice. You may want to obtain a degree in com- puter science or engineering, and a course in theory is required—God knows why. After all, isn’t theory arcane, boring, and worst of all, irrelevant?
To see that theory is neither arcane nor boring, but instead quite understand- able and even interesting, read on. Theoretical computer science does have many fascinating big ideas, but it also has many small and sometimes dull details that can be tiresome. Learning any new subject is hard work, but it becomes easier and more enjoyable if the subject is properly presented. My primary ob- jective in writing this book is to expose you to the genuinely exciting aspects of computer theory, without getting bogged down in the drudgery. Of course, the only way to determine whether theory interests you is to try learning it.
xi
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xii PREFACE TO THE FIRST EDITION
Theory is relevant to practice. It provides conceptual tools that practition- ers use in computer engineering. Designing a new programming language for a specialized application? What you learned about grammars in this course comes in handy. Dealing with string searching and pattern matching? Remember finite automata and regular expressions. Confronted with a problem that seems to re- quire more computer time than you can afford? Think back to what you learned about NP-completeness. Various application areas, such as modern cryptographic protocols, rely on theoretical principles that you will learn here.
Theory also is relevant to you because it shows you a new, simpler, and more elegant side of computers, which we normally consider to be complicated ma- chines. The best computer designs and applications are conceived with elegance in mind. A theoretical course can heighten your aesthetic sense and help you build more beautiful systems.
Finally, theory is good for you because studying it expands your mind. Com- puter technology changes quickly. Specific technical knowledge, though useful today, becomes outdated in just a few years. Consider instead the abilities to think, to express yourself clearly and precisely, to solve problems, and to know when you haven’t solved a problem. These abilities have lasting value. Studying theory trains you in these areas.
Practical considerations aside, nearly everyone working with computers is cu- rious about these amazing creations, their capabilities, and their limitations. A whole new branch of mathematics has grown up in the past 30 years to answer certain basic questions. Here’s a big one that remains unsolved: If I give you a large number—say, with 500 digits—can you find its factors (the numbers that divide it evenly) in a reasonable amount of time? Even using a supercomputer, no one presently knows how to do that in all cases within the lifetime of the universe! The factoring problem is connected to certain secret codes in modern cryptosys- tems. Find a fast way to factor, and fame is yours!
TO THE EDUCATOR
This book is intended as an upper-level undergraduate or introductory gradu- ate text in computer science theory. It contains a mathematical treatment of the subject, designed around theorems and proofs. I have made some effort to accommodate students with little prior experience in proving theorems, though more experienced students will have an easier time.
My primary goal in presenting the material has been to make it clear and interesting. In so doing, I have emphasized intuition and “the big picture” in the subject over some lower level details.
For example, even though I present the method of proof by induction in Chapter 0 along with other mathematical preliminaries, it doesn’t play an im- portant role subsequently. Generally, I do not present the usual induction proofs of the correctness of various constructions concerning automata. If presented clearly, these constructions convince and do not need further argument. An in- duction may confuse rather than enlighten because induction itself is a rather sophisticated technique that many find mysterious. Belaboring the obvious with
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PREFACE TO THE FIRST EDITION xiii
an induction risks teaching students that a mathematical proof is a formal ma- nipulation instead of teaching them what is and what is not a cogent argument.
AsecondexampleoccursinPartsTwoandThree,whereIdescribealgorithms in prose instead of pseudocode. I don’t spend much time programming Turing machines (or any other formal model). Students today come with a program- ming background and find the Church–Turing thesis to be self-evident. Hence I don’t present lengthy simulations of one model by another to establish their equivalence.
Besides giving extra intuition and suppressing some details, I give what might be called a classical presentation of the subject material. Most theorists will find the choice of material, terminology, and order of presentation consistent with that of other widely used textbooks. I have introduced original terminology in only a few places, when I found the standard terminology particularly obscure or confusing. For example, I introduce the term mapping reducibility instead of many–one reducibility.
Practice through solving problems is essential to learning any mathemati- cal subject. In this book, the problems are organized into two main categories called Exercises and Problems. The Exercises review definitions and concepts. The Problems require some ingenuity. Problems marked with a star are more difficult. I have tried to make the Exercises and Problems interesting challenges.
THE FIRST EDITION
Introduction to the Theory of Computation first appeared as a Preliminary Edition in paperback. The first edition differs from the Preliminary Edition in several substantial ways. The final three chapters are new: Chapter 8 on space complex- ity; Chapter 9 on provable intractability; and Chapter 10 on advanced topics in complexity theory. Chapter 6 was expanded to include several advanced topics in computability theory. Other chapters were improved through the inclusion of additional examples and exercises.
Comments from instructors and students who used the Preliminary Edition were helpful in polishing Chapters 0–7. Of course, the errors they reported have been corrected in this edition.
Chapters 6 and 10 give a survey of several more advanced topics in com- putability and complexity theories. They are not intended to comprise a cohesive unit in the way that the remaining chapters are. These chapters are included to allow the instructor to select optional topics that may be of interest to the serious student. The topics themselves range widely. Some, such as Turing reducibility and alternation, are direct extensions of other concepts in the book. Others, such as decidable logical theories and cryptography, are brief introductions to large fields.
FEEDBACK TO THE AUTHOR
The internet provides new opportunities for interaction between authors and readers. I have received much e-mail offering suggestions, praise, and criticism, and reporting errors for the Preliminary Edition. Please continue to correspond!
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xiv PREFACE TO THE FIRST EDITION
I try to respond to each message personally, as time permits. The e-mail address
for correspondence related to this book is sipserbook@math.mit.edu .
A web site that contains a list of errata is maintained. Other material may be added to that site to assist instructors and students. Let me know what you would like to see there. The location for that site is
http://math.mit.edu/~sipser/book.html . ACKNOWLEDGMENTS
I could not have written this book without the help of many friends, colleagues, and my family.
I wish to thank the teachers who helped shape my scientific viewpoint and educational style. Five of them stand out. My thesis advisor, Manuel Blum, is due a special note for his unique way of inspiring students through clarity of thought, enthusiasm, and caring. He is a model for me and for many others. I am grateful to Richard Karp for introducing me to complexity theory, to John Addison for teaching me logic and assigning those wonderful homework sets, to Juris Hartmanis for introducing me to the theory of computation, and to my father for introducing me to mathematics, computers, and the art of teaching.
This book grew out of notes from a course that I have taught at MIT for the past 15 years. Students in my classes took these notes from my lectures. I hope they will forgive me for not listing them all. My teaching assistants over the years—Avrim Blum, Thang Bui, Benny Chor, Andrew Chou, Stavros Cos- madakis, Aditi Dhagat, Wayne Goddard, Parry Husbands, Dina Kravets, Jakov Kucˇan, Brian O’Neill, Ioana Popescu, and Alex Russell—helped me to edit and expand these notes and provided some of the homework problems.
Nearly three years ago, Tom Leighton persuaded me to write a textbook on the theory of computation. I had been thinking of doing so for some time, but it took Tom’s persuasion to turn theory into practice. I appreciate his generous advice on book writing and on many other things.
I wish to thank Eric Bach, Peter Beebee, Cris Calude, Marek Chrobak, Anna Chefter, Guang-Ien Cheng, Elias Dahlhaus, Michael Fischer, Steve Fisk, Lance Fortnow, Henry J. Friedman, Jack Fu, Seymour Ginsburg, Oded Goldreich, Brian Grossman, David Harel, Micha Hofri, Dung T. Huynh, Neil Jones, H. Chad Lane, Kevin Lin, Michael Loui, Silvio Micali, Tadao Murata, Chris- tos Papadimitriou, Vaughan Pratt, Daniel Rosenband, Brian Scassellati, Ashish Sharma, Nir Shavit, Alexander Shen, Ilya Shlyakhter, Matt Stallmann, Perry Susskind, Y. C. Tay, Joseph Traub, Osamu Watanabe, Peter Widmayer, David Williamson, Derick Wood, and Charles Yang for comments, suggestions, and assistance as the writing progressed.
The following people provided additional comments that have improved this book: Isam M. Abdelhameed, Eric Allender, Shay Artzi, Michelle Ather- ton, Rolfe Blodgett, Al Briggs, Brian E. Brooks, Jonathan Buss, Jin Yi Cai,
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PREFACE TO THE FIRST EDITION xv
Steve Chapel, David Chow, Michael Ehrlich, Yaakov Eisenberg, Farzan Fallah, Shaun Flisakowski, Hjalmtyr Hafsteinsson, C. R. Hale, Maurice Herlihy, Vegard Holmedahl, Sandy Irani, Kevin Jiang, Rhys Price Jones, James M. Jowdy, David M. Martin Jr., Manrique Mata-Montero, Ryota Matsuura, Thomas Minka, Farooq Mohammed, Tadao Murata, Jason Murray, Hideo Nagahashi, Kazuo Ohta, Constantine Papageorgiou, Joseph Raj, Rick Regan, Rhonda A. Reumann, Michael Rintzler, Arnold L. Rosenberg, Larry Roske, Max Rozenoer, Walter L. Ruzzo, Sanatan Sahgal, Leonard Schulman, Steve Seiden, Joel Seiferas, Ambuj Singh, David J. Stucki, Jayram S. Thathachar, H. Venkateswaran, Tom Whaley, Christopher Van Wyk, Kyle Young, and Kyoung Hwan Yun.
Robert Sloan used an early version of the manuscript for this book in a class that he taught and provided me with invaluable commentary and ideas from his experience with it. Mark Herschberg, Kazuo Ohta, and Latanya Sweeney read over parts of the manuscript and suggested extensive improvements. Shafi Goldwasser helped me with material in Chapter 10.
I received expert technical support from William Baxter at Superscript, who wrote the LATEX macro package implementing the interior design, and from Larry Nolan at the MIT mathematics department, who keeps things running.
It has been a pleasure to work with the folks at PWS Publishing in creat- ing the final product. I mention Michael Sugarman, David Dietz, Elise Kaiser, Monique Calello, Susan Garland and Tanja Brull because I have had the most contact with them, but I know that many others have been involved, too. Thanks to Jerry Moore for the copy editing, to Diane Levy for the cover design, and to Catherine Hawkes for the interior design.
I am grateful to the National Science Foundation for support provided under grant CCR-9503322.
My father, Kenneth Sipser, and sister, Laura Sipser, converted the book di- agrams into electronic form. My other sister, Karen Fisch, saved us in various computer emergencies, and my mother, Justine Sipser, helped out with motherly advice. I thank them for contributing under difficult circumstances, including insane deadlines and recalcitrant software.
Finally, my love goes to my wife, Ina, and my daughter, Rachel. Thanks for putting up with all of this.
Cambridge, Massachusetts Michael Sipser October, 1996
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PREFACE TO THE SECOND EDITION
Judging from the email communications that I’ve received from so many of you, the biggest deficiency of the first edition is that it provides no sample solutions to any of the problems. So here they are. Every chapter now contains a new Selected Solutions section that gives answers to a representative cross-section of that chapter’s exercises and problems. To make up for the loss of the solved problems as interesting homework challenges, I’ve also added a variety of new problems. Instructors may request an Instructor’s Manual that contains addi- tional solutions by contacting the sales representative for their region designated at www.course.com .
A number of readers would have liked more coverage of certain “standard” topics, particularly the Myhill–Nerode Theorem and Rice’s Theorem. I’ve par- tially accommodated these readers by developing these topics in the solved prob- lems. I did not include the Myhill–Nerode Theorem in the main body of the text because I believe that this course should provide only an introduction to finite automata and not a deep investigation. In my view, the role of finite automata here is for students to explore a simple formal model of computation as a prelude to more powerful models, and to provide convenient examples for subsequent topics. Of course, some people would prefer a more thorough treatment, while others feel that I ought to omit all references to (or at least dependence on) finite automata. I did not include Rice’s Theorem in the main body of the text because, though it can be a useful “tool” for proving undecidability, some students might
xvii
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xviii PREFACE TO THE SECOND EDITION
use it mechanically without really understanding what is going on. Using reduc- tions instead, for proving undecidability, gives more valuable preparation for the reductions that appear in complexity theory.
I am indebted to my teaching assistants—Ilya Baran, Sergi Elizalde, Rui Fan, Jonathan Feldman, Venkatesan Guruswami, Prahladh Harsha, Christos Kapout- sis, Julia Khodor, Adam Klivans, Kevin Matulef, Ioana Popescu, April Rasala, Sofya Raskhodnikova, and Iuliu Vasilescu—who helped me to craft some of the new problems and solutions. Ching Law, Edmond Kayi Lee, and Zulfikar Ramzan also contributed to the solutions. I thank Victor Shoup for coming up with a simple way to repair the gap in the analysis of the probabilistic primality algorithm that appears in the first edition.
I appreciate the efforts of the people at Course Technology in pushing me and the other parts of this project along, especially Alyssa Pratt and Aimee Poirier. Many thanks to Gerald Eisman, Weizhen Mao, Rupak Majumdar, Chris Umans, and Christopher Wilson for their reviews. I’m indebted to Jerry Moore for his superb job copy editing and to Laura Segel of ByteGraphics (lauras@bytegraphics.com) for her beautiful rendition of the figures.
The volume of email I’ve received has been more than I expected. Hearing from so many of you from so many places has been absolutely delightful, and I’ve tried to respond to all eventually—my apologies for those I missed. I’ve listed here the people who made suggestions that specifically affected this edition, but I thank everyone for their correspondence:
Luca Aceto, Arash Afkanpour, Rostom Aghanian, Eric Allender, Karun Bak- shi, Brad Ballinger, Ray Bartkus, Louis Barton, Arnold Beckmann, Mihir Bel- lare, Kevin Trent Bergeson, Matthew Berman, Rajesh Bhatt, Somenath Biswas, Lenore Blum, Mauro A. Bonatti, Paul Bondin, Nicholas Bone, Ian Bratt, Gene Browder, Doug Burke, Sam Buss, Vladimir Bychkovsky, Bruce Carneal, Soma Chaudhuri, Rong-Jaye Chen, Samir Chopra, Benny Chor, John Clausen, Alli- son Coates, Anne Condon, Jeffrey Considine, John J. Crashell, Claude Crepeau, Shaun Cutts, Susheel M. Daswani, Geoff Davis, Scott Dexter, Peter Drake, Jeff Edmonds, Yaakov Eisenberg, Kurtcebe Eroglu, Georg Essl, Alexander T. Fader, Farzan Fallah, Faith Fich, Joseph E. Fitzgerald, Perry Fizzano, David Ford, Jeannie Fromer, Kevin Fu, Atsushi Fujioka, Michel Galley, K. Gane- san, Simson Garfinkel, Travis Gebhardt, Peymann Gohari, Ganesh Gopalakr- ishnan, Steven Greenberg, Larry Griffith, Jerry Grossman, Rudolf de Haan, Michael Halper, Nick Harvey, Mack Hendricks, Laurie Hiyakumoto, Steve Hockema, Michael Hoehle, Shahadat Hossain, Dave Isecke, Ghaith Issa, Raj D. Iyer, Christian Jacobi, Thomas Janzen, Mike D. Jones, Max Kanovitch, Aaron Kaufman, Roger Khazan, Sarfraz Khurshid, Kevin Killourhy, Seungjoo Kim, Victor Kuncak, Kanata Kuroda, Thomas Lasko, Suk Y. Lee, Edward D. Leg- enski, Li-Wei Lehman, Kong Lei, Zsolt Lengvarszky, Jeffrey Levetin, Baekjun Lim, Karen Livescu, Stephen Louie, TzerHung Low, Wolfgang Maass, Arash Madani, Michael Manapat, Wojciech Marchewka, David M. Martin Jr., Anders Martinson, Lyle McGeoch, Alberto Medina, Kurt Mehlhorn, Nihar Mehta, Al- bert R. Meyer, Thomas Minka, Mariya Minkova, Daichi Mizuguchi, G. Allen Morris III, Damon Mosk-Aoyama, Xiaolong Mou, Paul Muir, German Muller,
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PREFACE TO THE SECOND EDITION xix
Donald Nelson, Gabriel Nivasch, Mary Obelnicki, Kazuo Ohta, Thomas M. Oleson, Jr., Curtis Oliver, Owen Ozier, Rene Peralta, Alexander Perlis, Holger Petersen, Detlef Plump, Robert Prince, David Pritchard, Bina Reed, Nicholas Riley, Ronald Rivest, Robert Robinson, Christi Rockwell, Phil Rogaway, Max Rozenoer, John Rupf, Teodor Rus, Larry Ruzzo, Brian Sanders, Cem Say, Kim Schioett, Joel Seiferas, Joao Carlos Setubal, Geoff Lee Seyon, Mark Skandera, Bob Sloan, Geoff Smith, Marc L. Smith, Stephen Smith, Alex C. Snoeren, Guy St-Denis, Larry Stockmeyer, Radu Stoleru, David Stucki, Hisham M. Sueyllam, Kenneth Tam, Elizabeth Thompson, Michel Toulouse, Eric Tria, Chittaranjan Tripathy, Dan Trubow, Hiroki Ueda, Giora Unger, Kurt L. Van Etten, Jesir Vargas, Bienvenido Velez-Rivera, Kobus Vos, Alex Vrenios, Sven Waibel, Marc Waldman, Tom Whaley, Anthony Widjaja, Sean Williams, Joseph N. Wilson, Chris Van Wyk, Guangming Xing, Vee Voon Yee, Cheng Yongxi, Neal Young, Timothy Yuen, Kyle Yung, Jinghua Zhang, Lilla Zollei.
I thank Suzanne Balik, Matthew Kane, Kurt L. Van Etten, Nancy Lynch, Gregory Roberts, and Cem Say for pointing out errata in the first printing.
Most of all, I thank my family—Ina, Rachel, and Aaron—for their patience, understanding, and love as I sat for endless hours here in front of my computer screen.
Cambridge, Massachusetts Michael Sipser December, 2004
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
PREFACE TO THE THIRD EDITION
The third edition contains an entirely new section on deterministic context-free languages. I chose this topic for several reasons. First of all, it fills an obvious gap in my previous treatment of the theory of automata and languages. The older editions introduced finite automata and Turing machines in deterministic and nondeterministic variants, but covered only the nondeterministic variant of pushdown automata. Adding a discussion of deterministic pushdown automata provides a missing piece of the puzzle.
Second, the theory of deterministic context-free grammars is the basis for LR(k) grammars, an important and nontrivial application of automata theory in programming languages and compiler design. This application brings together several key concepts, including the equivalence of deterministic and nondeter- ministic finite automata, and the conversions between context-free grammars and pushdown automata, to yield an efficient and beautiful method for parsing. Here we have a concrete interplay between theory and practice.
Last, this topic seems underserved in existing theory textbooks, considering its importance as a genuine application of automata theory. I studied LR(k) gram- mars years ago but without fully understanding how they work, and without seeing how nicely they fit into the theory of deterministic context-free languages. My goal in writing this section is to give an intuitive yet rigorous introduction to this area for theorists as well as practitioners, and thereby contribute to its broader appreciation. One note of caution, however: Some of the material in this section is rather challenging, so an instructor in a basic first theory course
xxi
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xxii PREFACE TO THE THIRD EDITION
may prefer to designate it as supplementary reading. Later chapters do not de- pend on this material.
Many people helped directly or indirectly in developing this edition. I’m in- debted to reviewers Christos Kapoutsis and Cem Say who read a draft of the new section and provided valuable feedback. Several individuals at Cengage Learning assisted with the production, notably Alyssa Pratt and Jennifer Feltri-George. Suzanne Huizenga copyedited the text and Laura Segel of ByteGraphics created the new figures and modified some of the older figures.
I wish to thank my teaching assistants at MIT, Victor Chen, Andy Drucker, Michael Forbes, Elena Grigorescu, Brendan Juba, Christos Kapoutsis, Jon Kel- ner, Swastik Kopparty, Kevin Matulef, Amanda Redlich, Zack Remscrim, Ben Rossman, Shubhangi Saraf, and Oren Weimann. Each of them helped me by discussing new problems and their solutions, and by providing insight into how well our students understood the course content. I’ve greatly enjoyed working with such talented and enthusiastic young people.
It has been gratifying to receive email from around the globe. Thanks to all for your suggestions, questions, and ideas. Here is a list of those correspondents whose comments affected this edition:
Djihed Afifi, Steve Aldrich, Eirik Bakke, Suzanne Balik, Victor Bandur, Paul Beame, Elazar Birnbaum, Goutam Biswas, Rob Bittner, Marina Blanton, Rod- ney Bliss, Promita Chakraborty, Lewis Collier, Jonathan Deber, Simon Dex- ter, Matt Diephouse, Peter Dillinger, Peter Drake, Zhidian Du, Peter Fe- jer, Margaret Fleck, Atsushi Fujioka, Valerio Genovese, Evangelos Georgiadis, Joshua Grochow, Jerry Grossman, Andreas Guelzow, Hjalmtyr Hafsteinsson, Arthur Hall III, Cihat Imamoglu, Chinawat Isradisaikul, Kayla Jacobs, Flem- ming Jensen, Barbara Kaiser, Matthew Kane, Christos Kapoutsis, Ali Durlov Khan, Edwin Sze Lun Khoo, Yongwook Kim, Akash Kumar, Eleazar Leal, Zsolt Lengvarszky, Cheng-Chung Li, Xiangdong Liang, Vladimir Lifschitz, Ryan Lortie, Jonathan Low, Nancy Lynch, Alexis Maciel, Kevin Matulef, Nelson Max, Hans-Rudolf Metz, Mladen Miksˆa, Sara Miner More, Rajagopal Nagara- jan, Marvin Nakayama, Jonas Nyrup, Gregory Roberts, Ryan Romero, Santhosh Samarthyam, Cem Say, Joel Seiferas, John Sieg, Marc Smith, John Steinberger, Nuri Tas ̧demir, Tamir Tassa, Mark Testa, Jesse Tjang, John Trammell, Hi- roki Ueda, Jeroen Vaelen, Kurt L. Van Etten, Guillermo Va ́zquez, Phanisekhar Botlaguduru Venkata, Benjamin Bing-Yi Wang, Lutz Warnke, David Warren, Thomas Watson, Joseph Wilson, David Wittenberg, Brian Wongchaowart, Kis- han Yerubandi, Dai Yi.
Above all, I thank my family—my wife, Ina, and our children, Rachel and Aaron. Time is finite and fleeting. Your love is everything.
Cambridge, Massachusetts Michael Sipser April, 2012
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0
INTRODUCTION
We begin with an overview of those areas in the theory of computation that we present in this course. Following that, you’ll have a chance to learn and/or review some mathematical concepts that you will need later.
0.1
AUTOMATA, COMPUTABILITY, AND COMPLEXITY
This book focuses on three traditionally central areas of the theory of computa- tion: automata, computability, and complexity. They are linked by the question:
What are the fundamental capabilities and limitations of computers?
This question goes back to the 1930s when mathematical logicians first began to explore the meaning of computation. Technological advances since that time have greatly increased our ability to compute and have brought this question out of the realm of theory into the world of practical concern.
In each of the three areas—automata, computability, and complexity—this question is interpreted differently, and the answers vary according to the in- terpretation. Following this introductory chapter, we explore each area in a
1
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2 CHAPTER 0 / INTRODUCTION
separate part of this book. Here, we introduce these parts in reverse order be- cause by starting from the end you can better understand the reason for the beginning.
COMPLEXITY THEORY
Computer problems come in different varieties; some are easy, and some are hard. For example, the sorting problem is an easy one. Say that you need to arrange a list of numbers in ascending order. Even a small computer can sort a million numbers rather quickly. Compare that to a scheduling problem. Say that you must find a schedule of classes for the entire university to satisfy some reasonable constraints, such as that no two classes take place in the same room at the same time. The scheduling problem seems to be much harder than the sorting problem. If you have just a thousand classes, finding the best schedule may require centuries, even with a supercomputer.
What makes some problems computationally hard and others easy?
This is the central question of complexity theory. Remarkably, we don’t know the answer to it, though it has been intensively researched for over 40 years. Later, we explore this fascinating question and some of its ramifications.
In one important achievement of complexity theory thus far, researchers have discovered an elegant scheme for classifying problems according to their com- putational difficulty. It is analogous to the periodic table for classifying elements according to their chemical properties. Using this scheme, we can demonstrate a method for giving evidence that certain problems are computationally hard, even if we are unable to prove that they are.
You have several options when you confront a problem that appears to be computationally hard. First, by understanding which aspect of the problem is at the root of the difficulty, you may be able to alter it so that the problem is more easily solvable. Second, you may be able to settle for less than a perfect solution to the problem. In certain cases, finding solutions that only approximate the perfect one is relatively easy. Third, some problems are hard only in the worst case situation, but easy most of the time. Depending on the application, you may be satisfied with a procedure that occasionally is slow but usually runs quickly. Finally, you may consider alternative types of computation, such as randomized computation, that can speed up certain tasks.
One applied area that has been affected directly by complexity theory is the ancient field of cryptography. In most fields, an easy computational problem is preferable to a hard one because easy ones are cheaper to solve. Cryptography is unusual because it specifically requires computational problems that are hard, rather than easy. Secret codes should be hard to break without the secret key or password. Complexity theory has pointed cryptographers in the direction of computationally hard problems around which they have designed revolutionary new codes.
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 3 COMPUTABILITY THEORY
During the first half of the twentieth century, mathematicians such as Kurt Go ̈ del, Alan Turing, and Alonzo Church discovered that certain basic problems cannot be solved by computers. One example of this phenomenon is the prob- lem of determining whether a mathematical statement is true or false. This task is the bread and butter of mathematicians. It seems like a natural for solution by computer because it lies strictly within the realm of mathematics. But no computer algorithm can perform this task.
Among the consequences of this profound result was the development of ideas concerning theoretical models of computers that eventually would help lead to the construction of actual computers.
The theories of computability and complexity are closely related. In com- plexity theory, the objective is to classify problems as easy ones and hard ones; whereas in computability theory, the classification of problems is by those that are solvable and those that are not. Computability theory introduces several of the concepts used in complexity theory.
AUTOMATA THEORY
Automata theory deals with the definitions and properties of mathematical mod- els of computation. These models play a role in several applied areas of computer science. One model, called the finite automaton, is used in text processing, com- pilers, and hardware design. Another model, called the context-free grammar, is used in programming languages and artificial intelligence.
Automata theory is an excellent place to begin the study of the theory of computation. The theories of computability and complexity require a precise definition of a computer. Automata theory allows practice with formal definitions of computation as it introduces concepts relevant to other nontheoretical areas of computer science.
0.2
MATHEMATICAL NOTIONS AND TERMINOLOGY
As in any mathematical subject, we begin with a discussion of the basic mathe- matical objects, tools, and notation that we expect to use.
SETS
A set is a group of objects represented as a unit. Sets may contain any type of object, including numbers, symbols, and even other sets. The objects in a set are called its elements or members. Sets may be described formally in several ways.
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4 CHAPTER 0 / INTRODUCTION
One way is by listing a set’s elements inside braces. Thus the set
S = {7, 21, 57}
contains the elements 7, 21, and 57. The symbols ∈ and ̸∈ denote set member- ship and nonmembership. We write 7 ∈ {7, 21, 57} and 8 ̸∈ {7, 21, 57}. For two sets A and B, we say that A is a subset of B, written A ⊆ B, if every member of A also is a member of B. We say that A is a proper subset of B, written A B, if A is a subset of B and not equal to B.
The order of describing a set doesn’t matter, nor does repetition of its mem- bers. We get the same set S by writing {57, 7, 7, 7, 21}. If we do want to take the number of occurrences of members into account, we call the group a multiset instead of a set. Thus {7} and {7, 7} are different as multisets but identical as sets. An infinite set contains infinitely many elements. We cannot write a list of all the elements of an infinite set, so we sometimes use the “. . .” notation to mean “continue the sequence forever.” Thus we write the set of natural numbers N as
{1,2,3,…}. The set of integers Z is written as
{…,−2,−1,0,1,2,…}.
The set with zero members is called the empty set and is written ∅. A set with one member is sometimes called a singleton set, and a set with two members is called an unordered pair.
When we want to describe a set containing elements according to some rule, we write {n|rule about n}. Thus {n|n = m2 for some m ∈ N} means the set of perfect squares.
If we have two sets A and B, the union of A and B, written A∪B, is the set we get by combining all the elements in A and B into a single set. The intersection of A and B, written A ∩ B, is the set of elements that are in both A and B. The complement of A, written A, is the set of all elements under consideration that are not in A.
As is often the case in mathematics, a picture helps clarify a concept. For sets, we use a type of picture called a Venn diagram. It represents sets as regions enclosed by circular lines. Let the set START-t be the set of all English words that start with the letter “t”. For example, in the figure, the circle represents the set START-t. Several members of this set are represented as points inside the circle.
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 5
FIGURE 0.1
Venn diagram for the set of English words starting with “t”
Similarly, we represent the set END-z of English words that end with “z” in the following figure.
FIGURE 0.2
Venn diagram for the set of English words ending with “z”
To represent both sets in the same Venn diagram, we must draw them so that they overlap, indicating that they share some elements, as shown in the following figure. For example, the word topaz is in both sets. The figure also contains a circle for the set START-j. It doesn’t overlap the circle for START-t because no word lies in both sets.
FIGURE 0.3
Overlapping circles indicate common elements
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6 CHAPTER 0 / INTRODUCTION
The next two Venn diagrams depict the union and intersection of sets A
and B.
FIGURE 0.4
Diagrams for (a) A ∪ B and (b) A ∩ B
SEQUENCES AND TUPLES
A sequence of objects is a list of these objects in some order. We usually designate a sequence by writing the list within parentheses. For example, the sequence 7, 21, 57 would be written
(7, 21, 57).
The order doesn’t matter in a set, but in a sequence it does. Hence (7, 21, 57) is not the same as (57, 7, 21). Similarly, repetition does matter in a sequence, but it doesn’t matter in a set. Thus (7, 7, 21, 57) is different from both of the other sequences, whereas the set {7, 21, 57} is identical to the set {7, 7, 21, 57}.
As with sets, sequences may be finite or infinite. Finite sequences often are called tuples. A sequence with k elements is a k-tuple. Thus (7, 21, 57) is a 3-tuple. A 2-tuple is also called an ordered pair.
Sets and sequences may appear as elements of other sets and sequences. For example, the power set of A is the set of all subsets of A. If A is the set {0, 1}, the power set of A is the set { ∅, {0}, {1}, {0, 1} }. The set of all ordered pairs whose elements are 0s and 1s is { (0, 0), (0, 1), (1, 0), (1, 1) }.
If A and B are two sets, the Cartesian product or cross product of A and B, written A × B, is the set of all ordered pairs wherein the first element is a member of A and the second element is a member of B.
EXAMPLE 0.5
If A = {1,2} and B = {x,y,z},
A × B = { (1, x), (1, y), (1, z), (2, x), (2, y), (2, z) }.
We can also take the Cartesian product of k sets, A1, A2, … , Ak, written A1 ×A2 ×···×Ak. Itisthesetconsistingofallk-tuples(a1,a2,…,ak)where ai ∈Ai.
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 7
EXAMPLE 0.6
If A and B are as in Example 0.5,
A × B × A = (1, x, 1), (1, x, 2), (1, y, 1), (1, y, 2), (1, z, 1), (1, z, 2), (2,x,1), (2,x,2), (2,y,1), (2,y,2), (2,z,1), (2,z,2) .
If we have the Cartesian product of a set with itself, we use the shorthand
k
k
A×A×···×A=A .
The set N 2 equals N × N . It consists of all ordered pairs of natural numbers.
EXAMPLE 0.7
We also may write it as {(i, j)| i, j ≥ 1}.
FUNCTIONS AND RELATIONS
Functions are central to mathematics. A function is an object that sets up an input–output relationship. A function takes an input and produces an output. In every function, the same input always produces the same output. If f is a function whose output value is b when the input value is a, we write
f(a) = b.
A function also is called a mapping, and, if f (a) = b, we say that f maps a to b. For example, the absolute value function abs takes a number x as input and returns x if x is positive and −x if x is negative. Thus abs(2) = abs(−2) = 2. Addition is another example of a function, written add. The input to the addition function is an ordered pair of numbers, and the output is the sum of
those numbers.
The set of possible inputs to the function is called its domain. The outputs
of a function come from a set called its range. The notation for saying that f is a function with domain D and range R is
f : D −→ R .
In the case of the function abs, if we are working with integers, the domain and the range are Z, so we write abs : Z−→Z. In the case of the addition function for integers, the domain is the set of pairs of integers Z × Z and the range is Z, so we write add : Z × Z−→Z. Note that a function may not necessarily use all the elements of the specified range. The function abs never takes on the value −1 even though −1 ∈ Z. A function that does use all the elements of the range is said to be onto the range.
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8 CHAPTER 0 / INTRODUCTION
We may describe a specific function in several ways. One way is with a pro- cedure for computing an output from a specified input. Another way is with a table that lists all possible inputs and gives the output for each input.
EXAMPLE 0.8
Consider the function f : {0, 1, 2, 3, 4}−→ {0, 1, 2, 3, 4}.
n f(n) 01 12 23 34 40
This function adds 1 to its input and then outputs the result modulo 5. A number modulo m is the remainder after division by m. For example, the minute hand on a clock face counts modulo 60. When we do modular arithmetic, we define Zm = {0, 1, 2, . . . , m − 1}. With this notation, the aforementioned function f h a s t h e f o r m f : Z 5 −→ Z 5 .
EXAMPLE 0.9
Sometimes a two-dimensional table is used if the domain of the function is the Cartesian product of two sets. Here is another function, g : Z4 × Z4 −→ Z4 . The entry at the row labeled i and the column labeled j in the table is the value of g(i,j).
g0123 00123 11230 22301 33012
The function g is the addition function modulo 4.
Whenthedomainofafunctionf isA1×···×Ak forsomesetsA1,…,Ak,the input to f is a k-tuple (a1,a2,…,ak) and we call the ai the arguments to f. A function with k arguments is called a k-ary function, and k is called the arity of the function. If k is 1, f has a single argument and f is called a unary function. If k is 2, f is a binary function. Certain familiar binary functions are written in a special infix notation, with the symbol for the function placed between its two arguments, rather than in prefix notation, with the symbol preceding. For example, the addition function add usually is written in infix notation with the + symbol between its two arguments as in a + b instead of in prefix notation add (a, b).
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 9
A predicate or property is a function whose range is {TRUE, FALSE}. For example, let even be a property that is TRUE if its input is an even number and FALSE if its input is an odd number. Thus even(4) = TRUE and even(5) = FALSE.
A property whose domain is a set of k-tuples A × · · · × A is called a relation, a k-ary relation, or a k-ary relation on A. A common case is a 2-ary relation, called a binary relation. When writing an expression involving a binary rela- tion, we customarily use infix notation. For example, “less than” is a relation usually written with the infix operation symbol <. “Equality”, written with the = symbol, is another familiar relation. If R is a binary relation, the statement aRb means that aRb = TRUE. Similarly, if R is a k-ary relation, the statement R(a1,...,ak) means that R(a1,...,ak) = TRUE.
EXAMPLE 0.10
In a children’s game called Scissors–Paper–Stone, the two players simultaneously select a member of the set {SCISSORS, PAPER, STONE} and indicate their selec- tions with hand signals. If the two selections are the same, the game starts over. If the selections differ, one player wins, according to the relation beats.
beats
SCISSORS PAPER STONE
SCISSORS FALSE FALSE TRUE
PAPER TRUE FALSE FALSE
STONE FALSE TRUE FALSE
From this table we determine that SCISSORS beats PAPER is TRUE and that PAPER beats SCISSORS is FALSE.
Sometimes describing predicates with sets instead of functions is more con- venient. The predicate P : D−→{TRUE, FALSE} may be written (D, S), where S = {a ∈ D| P(a) = TRUE}, or simply S if the domain D is obvious from the context. Hence the relation beats may be written
{(SCISSORS, PAPER), (PAPER, STONE), (STONE, SCISSORS)}.
A special type of binary relation, called an equivalence relation, captures the notion of two objects being equal in some feature. A binary relation R is an equivalence relation if R satisfies three conditions:
1. R is reflexive if for every x, xRx;
2. R is symmetric if for every x and y, xRy implies yRx; and
3. R is transitive if for every x, y, and z, xRy and yRz implies xRz.
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10 CHAPTER 0 / INTRODUCTION EXAMPLE 0.11
Define an equivalence relation on the natural numbers, written ≡7 . For i, j ∈ N , say that i ≡7 j, if i−j is a multiple of 7. This is an equivalence relation because it satisfies the three conditions. First, it is reflexive, as i − i = 0, which is a multiple of 7. Second, it is symmetric, as i − j is a multiple of 7 if j − i is a multiple of 7. Third, it is transitive, as whenever i − j is a multiple of 7 and j − k is a multiple of 7, then i − k = (i − j) + (j − k) is the sum of two multiples of 7 and hence a multiple of 7, too.
GRAPHS
An undirected graph, or simply a graph, is a set of points with lines connecting some of the points. The points are called nodes or vertices, and the lines are called edges, as shown in the following figure.
FIGURE 0.12 Examples of graphs
The number of edges at a particular node is the degree of that node. In Figure 0.12(a), all the nodes have degree 2. In Figure 0.12(b), all the nodes have degree 3. No more than one edge is allowed between any two nodes. We may allow an edge from a node to itself, called a self-loop, depending on the situation.
In a graph G that contains nodes i and j, the pair (i, j) represents the edge that connects i and j. The order of i and j doesn’t matter in an undirected graph, so the pairs (i,j) and (j,i) represent the same edge. Sometimes we describe undirected edges with unordered pairs using set notation as in {i, j}. If V is the set of nodes of G and E is the set of edges, we say G = (V, E). We can describe a graph with a diagram or more formally by specifying V and E. For example, a formal description of the graph in Figure 0.12(a) is
{1, 2, 3, 4, 5}, {(1, 2), (2, 3), (3, 4), (4, 5), (5, 1)},
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 11
and a formal description of the graph in Figure 0.12(b) is
{1, 2, 3, 4}, {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}.
Graphs frequently are used to represent data. Nodes might be cities and edges the connecting highways, or nodes might be people and edges the friendships between them. Sometimes, for convenience, we label the nodes and/or edges of a graph, which then is called a labeled graph. Figure 0.13 depicts a graph whose nodes are cities and whose edges are labeled with the dollar cost of the cheapest nonstop airfare for travel between those cities if flying nonstop between them is possible.
FIGURE 0.13
Cheapest nonstop airfares between various cities
We say that graph G is a subgraph of graph H if the nodes of G are a subset of the nodes of H, and the edges of G are the edges of H on the corresponding nodes. The following figure shows a graph H and a subgraph G.
FIGURE 0.14
Graph G (shown darker) is a subgraph of H
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12 CHAPTER 0 / INTRODUCTION
A path in a graph is a sequence of nodes connected by edges. A simple path is a path that doesn’t repeat any nodes. A graph is connected if every two nodes have a path between them. A path is a cycle if it starts and ends in the same node. A simple cycle is one that contains at least three nodes and repeats only the first and last nodes. A graph is a tree if it is connected and has no simple cycles, as shown in Figure 0.15. A tree may contain a specially designated node called the root. The nodes of degree 1 in a tree, other than the root, are called the leaves of the tree.
FIGURE 0.15
(a) A path in a graph, (b) a cycle in a graph, and (c) a tree
A directed graph has arrows instead of lines, as shown in the following figure. The number of arrows pointing from a particular node is the outdegree of that node, and the number of arrows pointing to a particular node is the indegree.
FIGURE 0.16 A directed graph
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 13
In a directed graph, we represent an edge from i to j as a pair (i,j). The formal description of a directed graph G is (V, E), where V is the set of nodes and E is the set of edges. The formal description of the graph in Figure 0.16 is
{1,2,3,4,5,6}, {(1,2), (1,5), (2,1), (2,4), (5,4), (5,6), (6,1), (6,3)}.
A path in which all the arrows point in the same direction as its steps is called a directed path. A directed graph is strongly connected if a directed path connects every two nodes. Directed graphs are a handy way of depicting binary relations. If R is a binary relation whose domain is D × D, a labeled graph G = (D, E) represents R, where E = {(x, y)| xRy}.
EXAMPLE 0.17
The directed graph shown here represents the relation given in Example 0.10.
FIGURE 0.18
The graph of the relation beats
STRINGS AND LANGUAGES
Strings of characters are fundamental building blocks in computer science. The alphabet over which the strings are defined may vary with the application. For our purposes, we define an alphabet to be any nonempty finite set. The members of the alphabet are the symbols of the alphabet. We generally use capital Greek letters Σ and Γ to designate alphabets and a typewriter font for symbols from an alphabet. The following are a few examples of alphabets.
Σ1 = {0,1}
Σ2 = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}
Γ = {0,1,x,y,z}
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14 CHAPTER 0 / INTRODUCTION
A string over an alphabet is a finite sequence of symbols from that alphabet, usually written next to one another and not separated by commas. If Σ1 = {0,1}, then 01001 is a string over Σ1. If Σ2 = {a,b,c,...,z}, then abracadabra is a string over Σ2. If w is a string over Σ, the length of w, written |w|, is the number of symbols that it contains. The string of length zero is called the empty string and is written ε. The empty string plays the role of 0 in a number system. If w has length n, we can write w = w1w2 ···wn where each wi ∈ Σ. The reverse of w, written wR, is the string obtained by writing w in the opposite order (i.e., wnwn−1 · · · w1). String z is a substring of w if z appears consecutively within w. For example, cad is a substring of abracadabra.
If we have string x of length m and string y of length n, the concatenation of x and y, written xy, is the string obtained by appending y to the end of x, as in x1 · · · xmy1 · · · yn. To concatenate a string with itself many times, we use the superscript notation xk to mean
k
xx···x.
The lexicographic order of strings is the same as the familiar dictionary order. We’ll occasionally use a modified lexicographic order, called shortlex order or simply string order, that is identical to lexicographic order, except that shorter strings precede longer strings. Thus the string ordering of all strings over the alphabet {0,1} is
(ε,0,1,00,01,10,11,000,...).
Say that string x is a prefix of string y if a string z exists where xz = y, and that xisaproperprefixofyifinadditionx̸=y. Alanguageisasetofstrings. A language is prefix-free if no member is a proper prefix of another member.
BOOLEAN LOGIC
Boolean logic is a mathematical system built around the two values TRUE and FALSE. Though originally conceived of as pure mathematics, this system is now considered to be the foundation of digital electronics and computer design. The values TRUE and FALSE are called the Boolean values and are often represented by the values 1 and 0. We use Boolean values in situations with two possibilities, such as a wire that may have a high or a low voltage, a proposition that may be true or false, or a question that may be answered yes or no.
We can manipulate Boolean values with the Boolean operations. The sim- plest Boolean operation is the negation or NOT operation, designated with the symbol ¬. The negation of a Boolean value is the opposite value. Thus ¬0 = 1 and ¬1 = 0. We designate the conjunction or AND operation with the sym- bol ∧. The conjunction of two Boolean values is 1 if both of those values are 1. The disjunction or OR operation is designated with the symbol ∨. The disjunc-
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0.2 MATHEMATICAL NOTIONS AND TERMINOLOGY 15 tion of two Boolean values is 1 if either of those values is 1. We summarize this
information as follows.
0 ∧ 0 = 0 0 ∨ 0 = 0 ¬0 = 1 0 ∧ 1 = 0 0 ∨ 1 = 1 ¬1 = 0 1∧0=0 1∨0=1
1∧1=1 1∨1=1
We use Boolean operations for combining simple statements into more com- plex Boolean expressions, just as we use the arithmetic operations + and × to construct complex arithmetic expressions. For example, if P is the Boolean value representing the truth of the statement “the sun is shining” and Q represents the truth of the statement “today is Monday”, we may write P ∧ Q to represent the truth value of the statement “the sun is shining and today is Monday” and sim- ilarly for P ∨ Q with and replaced by or. The values P and Q are called the operands of the operation.
Several other Boolean operations occasionally appear. The exclusive or, or XOR, operation is designated by the ⊕ symbol and is 1 if either but not both of its two operands is 1. The equality operation, written with the symbol ↔, is 1 if both of its operands have the same value. Finally, the implication operation is designated by the symbol → and is 0 if its first operand is 1 and its second operand is 0; otherwise, → is 1. We summarize this information as follows.
0⊕0=0 0↔0=1 0→0=1 0⊕1=1 0↔1=0 0→1=1 1⊕0=1 1↔0=0 1→0=0 1⊕1=0 1↔1=1 1→1=1
We can establish various relationships among these operations. In fact, we can express all Boolean operations in terms of the AND and NOT operations, as the following identities show. The two expressions in each row are equivalent. Each row expresses the operation in the left-hand column in terms of operations above it and AND and NOT.
P ∨ Q P → Q P ↔Q P ⊕ Q
¬(¬P ∧ ¬Q)
¬P ∨ Q
(P →Q)∧(Q→P) ¬(P ↔ Q)
The distributive law for AND and OR comes in handy when we manipulate Boolean expressions. It is similar to the distributive law for addition and multi- plication, which states that a × (b + c) = (a × b) + (a × c). The Boolean version comes in two forms:
• P ∧(Q∨R)equals(P ∧Q)∨(P ∧R),anditsdual • P ∨(Q∧R)equals(P ∨Q)∧(P ∨R).
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16 CHAPTER 0 / INTRODUCTION
SUMMARY OF MATHEMATICAL TERMS
Alphabet Argument
Binary relation Boolean operation Boolean value Cartesian product
Complement Concatenation Conjunction Connected graph Cycle
Directed graph Disjunction Domain
Edge
Element
Empty set
Empty string Equivalencerelation Function
Graph Intersection k-tuple Language Member Node Ordered pair Path Predicate Property Range Relation Sequence Set
Simple path Singleton set String Symbol
Tree
Union Unordered pair Vertex
A finite, nonempty set of objects called symbols An input to a function
A relation whose domain is a set of pairs
An operation on Boolean values
The values TRUE or FALSE, often represented by 1 or 0
An operation on sets forming a set of all tuples of elements from
respective sets
An operation on a set, forming the set of all elements not present An operation that joins strings together
Boolean AND operation
A graph with paths connecting every two nodes
A path that starts and ends in the same node
A collection of points and arrows connecting some pairs of points Boolean OR operation
The set of possible inputs to a function
A line in a graph
An object in a set
The set with no members
The string of length zero Abinaryrelationthatisreflexive,symmetric,andtransitive
An operation that translates inputs into outputs
A collection of points and lines connecting some pairs of points An operation on sets forming the set of common elements
A list of k objects
A set of strings
An object in a set
A point in a graph
A list of two elements
A sequence of nodes in a graph connected by edges
A function whose range is {TRUE, FALSE}
A predicate
The set from which outputs of a function are drawn
A predicate, most typically when the domain is a set of k-tuples A list of objects
A group of objects
A path without repetition
A set with one member
A finite list of symbols from an alphabet
A member of an alphabet
A connected graph without simple cycles
An operation on sets combining all elements into a single set
A set with two members
A point in a graph
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0.3 DEFINITIONS, THEOREMS, AND PROOFS 17
0.3
DEFINITIONS, THEOREMS, AND PROOFS
Theorems and proofs are the heart and soul of mathematics and definitions are its spirit. These three entities are central to every mathematical subject, includ- ing ours.
Definitions describe the objects and notions that we use. A definition may be simple, as in the definition of set given earlier in this chapter, or complex as in the definition of security in a cryptographic system. Precision is essential to any mathematical definition. When defining some object, we must make clear what constitutes that object and what does not.
After we have defined various objects and notions, we usually make math- ematical statements about them. Typically, a statement expresses that some object has a certain property. The statement may or may not be true; but like a definition, it must be precise. No ambiguity about its meaning is allowed.
A proof is a convincing logical argument that a statement is true. In mathe- matics, an argument must be airtight; that is, convincing in an absolute sense. In everyday life or in the law, the standard of proof is lower. A murder trial demands proof “beyond any reasonable doubt.” The weight of evidence may compel the jury to accept the innocence or guilt of the suspect. However, evidence plays no role in a mathematical proof. A mathematician demands proof beyond any doubt.
A theorem is a mathematical statement proved true. Generally we reserve the use of that word for statements of special interest. Occasionally we prove state- ments that are interesting only because they assist in the proof of another, more significant statement. Such statements are called lemmas. Occasionally a theo- rem or its proof may allow us to conclude easily that other, related statements are true. These statements are called corollaries of the theorem.
FINDING PROOFS
The only way to determine the truth or falsity of a mathematical statement is with a mathematical proof. Unfortunately, finding proofs isn’t always easy. It can’t be reduced to a simple set of rules or processes. During this course, you will be asked to present proofs of various statements. Don’t despair at the prospect! Even though no one has a recipe for producing proofs, some helpful general strategies are available.
First, carefully read the statement you want to prove. Do you understand all the notation? Rewrite the statement in your own words. Break it down and consider each part separately.
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18 CHAPTER 0 / INTRODUCTION
Sometimes the parts of a multipart statement are not immediately evident. One frequently occurring type of multipart statement has the form “P if and only if Q”, often written “P iff Q”, where both P and Q are mathematical state- ments. This notation is shorthand for a two-part statement. The first part is “P only if Q,” which means: If P is true, then Q is true, written P ⇒ Q. The second is “P if Q,” which means: If Q is true, then P is true, written P ⇐ Q. The first of these parts is the forward direction of the original statement and the second is the reverse direction. We write “P if and only if Q” as P ⇐⇒ Q. To prove a statement of this form, you must prove each of the two directions. Often, one of these directions is easier to prove than the other.
Another type of multipart statement states that two sets A and B are equal. The first part states that A is a subset of B, and the second part states that B is a subset of A. Thus one common way to prove that A = B is to prove that every member of A also is a member of B, and that every member of B also is a member of A.
Next, when you want to prove a statement or part thereof, try to get an in- tuitive, “gut” feeling of why it should be true. Experimenting with examples is especially helpful. Thus if the statement says that all objects of a certain type have a particular property, pick a few objects of that type and observe that they actually do have that property. After doing so, try to find an object that fails to have the property, called a counterexample. If the statement actually is true, you will not be able to find a counterexample. Seeing where you run into difficulty when you attempt to find a counterexample can help you understand why the statement is true.
EXAMPLE 0.19
Suppose that you want to prove the statement for every graph G, the sum of the degrees of all the nodes in G is an even number.
First, pick a few graphs and observe this statement in action. Here are two examples.
Next, try to find a counterexample; that is, a graph in which the sum is an odd number.
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0.3 DEFINITIONS, THEOREMS, AND PROOFS 19
Can you now begin to see why the statement is true and how to prove it?
If you are still stuck trying to prove a statement, try something easier. Attempt to prove a special case of the statement. For example, if you are trying to prove that some property is true for every k > 0, first try to prove it for k = 1. If you succeed, try it for k = 2, and so on until you can understand the more general case. If a special case is hard to prove, try a different special case or perhaps a special case of the special case.
Finally, when you believe that you have found the proof, you must write it up properly. A well-written proof is a sequence of statements, wherein each one follows by simple reasoning from previous statements in the sequence. Carefully writing a proof is important, both to enable a reader to understand it, and for you to be sure that it is free from errors.
The following are a few tips for producing a proof.
• Be patient. Finding proofs takes time. If you don’t see how to do it right away, don’t worry. Researchers sometimes work for weeks or even years to find a single proof.
• Come back to it. Look over the statement you want to prove, think about it a bit, leave it, and then return a few minutes or hours later. Let the unconscious, intuitive part of your mind have a chance to work.
• Be neat. When you are building your intuition for the statement you are trying to prove, use simple, clear pictures and/or text. You are trying to develop your insight into the statement, and sloppiness gets in the way of insight. Furthermore, when you are writing a solution for another person to read, neatness will help that person understand it.
• Beconcise.Brevityhelpsyouexpresshigh-levelideaswithoutgettinglostin details. Good mathematical notation is useful for expressing ideas concisely. But be sure to include enough of your reasoning when writing up a proof so that the reader can easily understand what you are trying to say.
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20 CHAPTER 0 / INTRODUCTION
For practice, let’s prove one of DeMorgan’s laws.
THEOREM 0.20
For any two sets A and B, A ∪ B = A ∩ B.
First, is the meaning of this theorem clear? If you don’t understand the mean- ing of the symbols ∪ or ∩ or the overbar, review the discussion on page 4.
To prove this theorem, we must show that the two sets A ∪ B and A ∩ B are equal. Recall that we may prove that two sets are equal by showing that every member of one set also is a member of the other and vice versa. Before looking at the following proof, consider a few examples and then try to prove it yourself.
PROOF This theorem states that two sets, A ∪ B and A ∩ B, are equal. We prove this assertion by showing that every element of one also is an element of the other and vice versa.
SupposethatxisanelementofA∪B. ThenxisnotinA∪Bfromthe definition of the complement of a set. Therefore, x is not in A and x is not in B, from the definition of the union of two sets. In other words, x is in A and x is in B. Hence the definition of the intersection of two sets shows that x is in A ∩ B.
For the other direction, suppose that x is in A ∩ B. Then x is in both A and B. Therefore,xisnotinAandxisnotinB,andthusnotintheunionof these two sets. Hence x is in the complement of the union of these sets; in other words, x is in A ∪ B, which completes the proof of the theorem.
Let’s now prove the statement in Example 0.19.
THEOREM 0.21
For every graph G, the sum of the degrees of all the nodes in G is an even
number.
PROOF Every edge in G is connected to two nodes. Each edge contributes 1 to the degree of each node to which it is connected. Therefore, each edge con- tributes 2 to the sum of the degrees of all the nodes. Hence, if G contains e edges, then the sum of the degrees of all the nodes of G is 2e, which is an even number.
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0.4
TYPES OF PROOF
Several types of arguments arise frequently in mathematical proofs. Here, we describe a few that often occur in the theory of computation. Note that a proof may contain more than one type of argument because the proof may contain within it several different subproofs.
PROOF BY CONSTRUCTION
Many theorems state that a particular type of object exists. One way to prove such a theorem is by demonstrating how to construct the object. This technique is a proof by construction.
Let’s use a proof by construction to prove the following theorem. We define a graph to be k-regular if every node in the graph has degree k.
THEOREM 0.22
For each even number n greater than 2, there exists a 3-regular graph with n
nodes.
PROOF Let n be an even number greater than 2. Construct graph G = (V, E) withnnodesasfollows.ThesetofnodesofGisV ={0,1,…,n−1},andthe set of edges of G is the set
E={{i,i+1}| for0≤i≤n−2}∪{{n−1,0}} ∪ {{i,i+n/2}| for0≤i≤n/2−1}.
Picture the nodes of this graph written consecutively around the circumference of a circle. In that case, the edges described in the top line of E go between adjacent pairs around the circle. The edges described in the bottom line of E go between nodes on opposite sides of the circle. This mental picture clearly shows that every node in G has degree 3.
PROOF BY CONTRADICTION
In one common form of argument for proving a theorem, we assume that the theorem is false and then show that this assumption leads to an obviously false consequence, called a contradiction. We use this type of reasoning frequently in everyday life, as in the following example.
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0.4 TYPES OF PROOF 21
22 CHAPTER 0 / INTRODUCTION EXAMPLE 0.23
Jack sees Jill, who has just come in from outdoors. On observing that she is completely dry, he knows that it is not raining. His “proof” that it is not raining is that if it were raining (the assumption that the statement is false), Jill would be wet (the obviously false consequence). Therefore, it must not be raining.
Next, let’s prove by contradiction that the square root of 2 is an irrational number. A number is rational if it is a fraction m , where m and n are integers;
n
in other words, a rational number is the ratio of integers m and n. For example, 2 obviously is a rational number. A number is irrational if it is not rational.
3
THEOREM 0.24 √2 is irrational.
√2 = m, n
where m and n are integers. If both m and n are divisible by the same integer greater than 1, divide both by the largest such integer. Doing so doesn’t change the value of the fraction. Now, at least one of m and n must be an odd number.
We multiply both sides of the equation by n and obtain n√2 = m.
We square both sides and obtain
2n2 = m2.
Because m2 is 2 times the integer n2, we know that m2 is even. Therefore, m, too, is even, as the square of an odd number always is odd. So we can write m = 2k for some integer k. Then, substituting 2k for m, we get
2n2 = (2k)2 = 4k2.
Dividing both sides by 2, we obtain
n2 = 2k2.
But this result shows that n2 is even and hence that n is even. Thus we have established that both m and n are even. But we had earlier reduced m and n so that they were not both even—a contradiction.
PROOF BY INDUCTION
Proof by induction is an advanced method used to show that all elements of an infinite set have a specified property. For example, we may use a proof by induction to show that an arithmetic expression computes a desired quantity for
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PROOF First, we assume for the purpose of later obtaining a contradiction
that √2 is rational. Thus
every assignment to its variables, or that a program works correctly at all steps or for all inputs.
To illustrate how proof by induction works, let’s take the infinite set to be the natural numbers, N = {1, 2, 3, . . . }, and say that the property is called P . Our goal is to prove that P(k) is true for each natural number k. In other words, we want to prove that P(1) is true, as well as P(2), P(3), P(4), and so on.
Every proof by induction consists of two parts, the basis and the induction step. Each part is an individual proof on its own. The basis proves that P(1) is true. The induction step proves that for each i ≥ 1, if P(i) is true, then so is P(i + 1).
When we have proven both of these parts, the desired result follows—namely, that P(i) is true for each i. Why? First, we know that P(1) is true because the basis alone proves it. Second, we know that P(2) is true because the induction step proves that if P(1) is true then P(2) is true, and we already know that P(1) is true. Third, we know that P(3) is true because the induction step proves that if P(2) is true then P(3) is true, and we already know that P(2) is true. This process continues for all natural numbers, showing that P(4) is true, P(5) is true, and so on.
Once you understand the preceding paragraph, you can easily understand variations and generalizations of the same idea. For example, the basis doesn’t necessarily need to start with 1; it may start with any value b. In that case, the induction proof shows that P(k) is true for every k that is at least b.
In the induction step, the assumption that P(i) is true is called the induction hypothesis. Sometimes having the stronger induction hypothesis that P(j) is true for every j ≤ i is useful. The induction proof still works because when we want to prove that P(i + 1) is true, we have already proved that P(j) is true for every j ≤ i.
The format for writing down a proof by induction is as follows.
Basis: Prove that P(1) is true.
.
Induction step: For each i ≥ 1, assume that P(i) is true and use this assumption
to show that P(i + 1) is true.
.
Now, let’s prove by induction the correctness of the formula used to calculate the size of monthly payments of home mortgages. When buying a home, many people borrow some of the money needed for the purchase and repay this loan over a certain number of years. Typically, the terms of such repayments stipulate that a fixed amount of money is paid each month to cover the interest, as well as part of the original sum, so that the total is repaid in 30 years. The formula for calculating the size of the monthly payments is shrouded in mystery, but actually is quite simple. It touches many people’s lives, so you should find it interesting. We use induction to prove that it works, making it a good illustration of that technique.
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0.4 TYPES OF PROOF 23
24 CHAPTER 0 / INTRODUCTION
First, we set up the names and meanings of several variables. Let P be the principal, the amount of the original loan. Let I > 0 be the yearly interest rate of the loan, where I = 0.06 indicates a 6% rate of interest. Let Y be the monthly payment. For convenience, we use I to define another variable M , the monthly multiplier. It is the rate at which the loan changes each month because of the interest on it. Following standard banking practice, the monthly interest rate is one-twelfth of the annual rate so M = 1 + I/12, and interest is paid monthly (monthly compounding).
Two things happen each month. First, the amount of the loan tends to in- crease because of the monthly multiplier. Second, the amount tends to decrease because of the monthly payment. Let Pt be the amount of the loan outstand- ing after the tth month. Then P0 = P is the amount of the original loan, P1 = MP0 − Y is the amount of the loan after one month, P2 = MP1 − Y is the amount of the loan after two months, and so on. Now we are ready to state and prove a theorem by induction on t that gives a formula for the value of Pt.
THEOREM 0.25 For each t ≥ 0,
PROOF
Pt = P M t − Y M t − 1 . M−1
Basis: Prove that the formula is true for t = 0. If t = 0, then the formula states
that
P0 = P M 0 − Y M 0 − 1 . M−1
We can simplify the right-hand side by observing that M 0 = 1. Thus we get P0 = P,
which holds because we have defined P0 to be P . Therefore, we have proved that the basis of the induction is true.
Induction step: For each k ≥ 0, assume that the formula is true for t = k and show that it is true for t = k + 1. The induction hypothesis states that
Pk =PMk −Y Mk −1. M−1
Our objective is to prove that
Pk+1 =PMk+1 −Y Mk+1 −1. M−1
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EXERCISES 25 We do so with the following steps. First, from the definition of Pk+1 from
Pk, we know that
Pk+1 =PkM−Y.
Therefore, using the induction hypothesis to calculate Pk, Pk+1 = PMk −Y Mk −1M −Y.
M−1 Multiplying through by M and rewriting Y yields
Pk+1 = P M k+1 − Y M k+1 − M − Y M − 1 M−1 M−1
=PMk+1−Y Mk+1−1 . M−1
Thus the formula is correct for t = k + 1, which proves the theorem.
Problem 0.15 asks you to use the preceding formula to calculate actual mort- gage payments.
EXERCISES
0.1 Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description of each set.
a. {1,3,5,7, …}
b. {…,−4,−2,0,2,4, …}
c. {n|n=2mforsomeminN}
d. {n|n=2mforsomeminN,andn=3kforsomekinN} e. {w|wisastringof0sand1sandwequalsthereverseofw}
f. {n|nisanintegerandn=n+1}
0.2 Write formal descriptions of the following sets.
a. The set containing the numbers 1, 10, and 100
b. The set containing all integers that are greater than 5
c. The set containing all natural numbers that are less than 5
d. The set containing the string aba
e. The set containing the empty string
f. The set containing nothing at all
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26 CHAPTER 0 / INTRODUCTION
0.3 Let A be the set {x,y,z} and B be the set {x,y}.
a. IsAasubsetofB?
b. IsBasubsetofA?
c. WhatisA∪B?
d. WhatisA∩B?
e. WhatisA×B?
f. What is the power set of B?
0.4 If A has a elements and B has b elements, how many elements are in A × B? Explain your answer.
0.5 If C is a set with c elements, how many elements are in the power set of C? Explain your answer.
0.6 LetX betheset{1,2,3,4,5}andY betheset{6,7,8,9,10}. Theunaryfunction f: X−→Y andthebinaryfunctiong: X×Y−→Y aredescribedinthefollowing tables.
n f(n) 16 27 36 47 56
g678910 11010101010 2789106 377889 4987610 566666
a. What is the value
b. What are the range and domain of f?
c. What is the value of g(2, 10)?
d. What are the range and domain of g?
e. What is the value of g(4, f (4))?
0.7 For each part, give a relation that satisfies the condition.
a. Reflexive and symmetric but not transitive
b. Reflexive and transitive but not symmetric
c. Symmetric and transitive but not reflexive
0.8 ConsidertheundirectedgraphG=(V,E)whereV,thesetofnodes,is{1,2,3,4} and E, the set of edges, is {{1, 2}, {2, 3}, {1, 3}, {2, 4}, {1, 4}}. Draw the graph G. What are the degrees of each node? Indicate a path from node 3 to node 4 on your drawing of G.
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of
f (2)?
0.9 Write a formal description of the following graph.
PROBLEMS 27
PROBLEMS
0.10 Find the error in the following proof that 2 = 1.
Consider the equation a = b. Multiply both sides by a to obtain a2 = ab. Subtract b2 from both sides to get a2 −b2 = ab−b2. Now factor each side, (a+b)(a−b) = b(a−b), and divide each side by (a−b) to get a+b = b. Finally, let a and b equal 1, which shows that 2 = 1.
0.11 LetS(n) = 1+2+···+nbethesumofthefirstnnaturalnumbersandlet C(n) = 13 + 23 + ··· + n3 be the sum of the first n cubes. Prove the following equalities by induction on n, to arrive at the curious conclusion that C(n) = S2(n) for every n.
a. S(n)=1n(n+1). 2
b. C(n)= 1(n4 +2n3 +n2)= 1n2(n+1)2. 44
0.12 Find the error in the following proof that all horses are the same color. CLAIM: In any set of h horses, all horses are the same color.
PROOF: By induction on h.
Basis: For h = 1. In any set containing just one horse, all horses clearly are the same color.
Induction step: For k ≥ 1, assume that the claim is true for h = k and prove that it is true for h = k + 1. Take any set H of k + 1 horses. We show that all the horses in this set are the same color. Remove one horse from this set to obtain the set H1 with just k horses. By the induction hypothesis, all the horses in H1 are the same color. Now replace the removed horse and remove a different one to obtain the set H2. By the same argument, all the horses in H2 are the same color. Therefore, all the horses in H must be the same color, and the proof is complete.
0.13 Show that every graph with two or more nodes contains two nodes that have equal degrees.
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28
CHAPTER 0 / INTRODUCTION
A⋆ 0.14
A 0.15
Ramsey’s theorem. Let G be a graph. A clique in G is a subgraph in which every two nodes are connected by an edge. An anti-clique, also called an independent set, is a subgraph in which every two nodes are not connected by an edge. Show that every graph with n nodes contains either a clique or an anti-clique with at least
1 log n nodes. 22
Use Theorem 0.25 to derive a formula for calculating the size of the monthly pay- ment for a mortgage in terms of the principal P , the interest rate I , and the number of payments t. Assume that after t payments have been made, the loan amount is reduced to 0. Use the formula to calculate the dollar amount of each monthly pay- ment for a 30-year mortgage with 360 monthly payments on an initial loan amount of $100,000 with a 5% annual interest rate.
SELECTED SOLUTIONS
0.14 Make space for two piles of nodes: A and B. Then, starting with the entire graph,
repeatedly add each remaining node x to A if its degree is greater than one half the
number of remaining nodes and to B otherwise, and discard all nodes to which x
isn’t (is) connected if it was added to A (B). Continue until no nodes are left. At
most half of the nodes are discarded at each of these steps, so at least log2 n steps
will occur before the process terminates. Each step adds a node to one of the piles,
so one of the piles ends up with at least 1 log n nodes. The A pile contains the 22
nodes of a clique and the B pile contains the nodes of an anti-clique.
0.15 WeletPt=0andsolveforYtogettheformula:Y=PMt(M−1)/(Mt−1). For P = $100,000, I = 0.05, and t = 360, we have M = 1 + (0.05)/12. We use a calculator to find that Y ≈ $536.82 is the monthly payment.
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PART ONE
AUTOMATA AND LANGUAGES
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1
REGULAR LANGUAGES
The theory of computation begins with a question: What is a computer? It is perhaps a silly question, as everyone knows that this thing I type on is a com- puter. But these real computers are quite complicated—too much so to allow us to set up a manageable mathematical theory of them directly. Instead, we use an idealized computer called a computational model. As with any model in science, a computational model may be accurate in some ways but perhaps not in others. Thus we will use several different computational models, depending on the fea- tures we want to focus on. We begin with the simplest model, called the finite state machine or finite automaton.
1.1
FINITE AUTOMATA
Finite automata are good models for computers with an extremely limited amount of memory. What can a computer do with such a small memory? Many useful things! In fact, we interact with such computers all the time, as they lie at the heart of various electromechanical devices.
The controller for an automatic door is one example of such a device. Often found at supermarket entrances and exits, automatic doors swing open when the controller senses that a person is approaching. An automatic door has a pad
31
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32 CHAPTER 1 / REGULAR LANGUAGES
in front to detect the presence of a person about to walk through the doorway. Another pad is located to the rear of the doorway so that the controller can hold the door open long enough for the person to pass all the way through and also so that the door does not strike someone standing behind it as it opens. This configuration is shown in the following figure.
FIGURE 1.1
Top view of an automatic door
The controller is in either of two states: “OPEN” or “CLOSED,” representing the corresponding condition of the door. As shown in the following figures, there are four possible input conditions: “FRONT” (meaning that a person is standing on the pad in front of the doorway), “REAR” (meaning that a person is standing on the pad to the rear of the doorway), “BOTH” (meaning that people are standing on both pads), and “NEITHER” (meaning that no one is standing on either pad).
FIGURE 1.2
State diagram for an automatic door controller
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state
NEITHER CLOSED CLOSED OPEN CLOSED
1.1 FINITE AUTOMATA 33 input signal
FRONT REAR BOTH OPEN CLOSED CLOSED OPEN OPEN OPEN
FIGURE 1.3
State transition table for an automatic door controller
The controller moves from state to state, depending on the input it receives. When in the CLOSED state and receiving input NEITHER or REAR, it remains in the CLOSED state. In addition, if the input BOTH is received, it stays CLOSED because opening the door risks knocking someone over on the rear pad. But if the input FRONT arrives, it moves to the OPEN state. In the OPEN state, if input FRONT, REAR, or BOTH is received, it remains in OPEN. If input NEITHER arrives, it returns to CLOSED.
For example, a controller might start in state CLOSED and receive the series of input signals FRONT, REAR, NEITHER, FRONT, BOTH, NEITHER, REAR, and NEITHER. It then would go through the series of states CLOSED (starting), OPEN, OPEN, CLOSED, OPEN, OPEN, CLOSED, CLOSED, and CLOSED.
Thinking of an automatic door controller as a finite automaton is useful be- cause that suggests standard ways of representation as in Figures 1.2 and 1.3. This controller is a computer that has just a single bit of memory, capable of recording which of the two states the controller is in. Other common devices have controllers with somewhat larger memories. In an elevator controller, a state may represent the floor the elevator is on and the inputs might be the sig- nals received from the buttons. This computer might need several bits to keep track of this information. Controllers for various household appliances such as dishwashers and electronic thermostats, as well as parts of digital watches and calculators, are additional examples of computers with limited memories. The design of such devices requires keeping the methodology and terminology of finite automata in mind.
Finite automata and their probabilistic counterpart Markov chains are useful tools when we are attempting to recognize patterns in data. These devices are used in speech processing and in optical character recognition. Markov chains have even been used to model and predict price changes in financial markets.
We will now take a closer look at finite automata from a mathematical per- spective. We will develop a precise definition of a finite automaton, terminology for describing and manipulating finite automata, and theoretical results that de- scribe their power and limitations. Besides giving you a clearer understanding of what finite automata are and what they can and cannot do, this theoreti- cal development will allow you to practice and become more comfortable with mathematical definitions, theorems, and proofs in a relatively simple setting.
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34 CHAPTER 1 / REGULAR LANGUAGES
In beginning to describe the mathematical theory of finite automata, we do so in the abstract, without reference to any particular application. The following figure depicts a finite automaton called M1.
FIGURE 1.4
A finite automaton called M1 that has three states
Figure 1.4 is called the state diagram of M1. It has three states, labeled q1, q2, and q3. The start state, q1, is indicated by the arrow pointing at it from nowhere. The accept state, q2, is the one with a double circle. The arrows going from one state to another are called transitions.
When this automaton receives an input string such as 1101, it processes that string and produces an output. The output is either accept or reject. We will consider only this yes/no type of output for now to keep things simple. The processing begins in M1’s start state. The automaton receives the symbols from the input string one by one from left to right. After reading each symbol, M1 moves from one state to another along the transition that has that symbol as its label. When it reads the last symbol, M1 produces its output. The output is accept if M1 is now in an accept state and reject if it is not.
For example, when we feed the input string 1101 into the machine M1 in Figure 1.4, the processing proceeds as follows:
1. Start in state q1.
2. Read 1, follow transition from q1 to q2.
3. Read 1, follow transition from q2 to q2.
4. Read 0, follow transition from q2 to q3.
5. Read 1, follow transition from q3 to q2.
6. Accept because M1 is in an accept state q2 at the end of the input.
Experimenting with this machine on a variety of input strings reveals that it accepts the strings 1, 01, 11, and 0101010101. In fact, M1 accepts any string that ends with a 1, as it goes to its accept state q2 whenever it reads the symbol 1. In addition, it accepts strings 100, 0100, 110000, and 0101000000, and any string that ends with an even number of 0s following the last 1. It rejects other strings, such as 0, 10, 101000. Can you describe the language consisting of all strings that M1 accepts? We will do so shortly.
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1.1 FINITE AUTOMATA 35 FORMAL DEFINITION OF A FINITE AUTOMATON
In the preceding section, we used state diagrams to introduce finite automata. Now we define finite automata formally. Although state diagrams are easier to grasp intuitively, we need the formal definition, too, for two specific reasons.
First, a formal definition is precise. It resolves any uncertainties about what is allowed in a finite automaton. If you were uncertain about whether finite automata were allowed to have 0 accept states or whether they must have ex- actly one transition exiting every state for each possible input symbol, you could consult the formal definition and verify that the answer is yes in both cases. Sec- ond, a formal definition provides notation. Good notation helps you think and express your thoughts clearly.
The language of a formal definition is somewhat arcane, having some simi- larity to the language of a legal document. Both need to be precise, and every detail must be spelled out.
A finite automaton has several parts. It has a set of states and rules for going from one state to another, depending on the input symbol. It has an input al- phabet that indicates the allowed input symbols. It has a start state and a set of accept states. The formal definition says that a finite automaton is a list of those five objects: set of states, input alphabet, rules for moving, start state, and accept states. In mathematical language, a list of five elements is often called a 5-tuple. Hence we define a finite automaton to be a 5-tuple consisting of these five parts.
We use something called a transition function, frequently denoted δ, to de- fine the rules for moving. If the finite automaton has an arrow from a state x to a state y labeled with the input symbol 1, that means that if the automaton is in state x when it reads a 1, it then moves to state y. We can indicate the same thing with the transition function by saying that δ(x, 1) = y. This notation is a kind of mathematical shorthand. Putting it all together, we arrive at the formal definition of finite automata.
DEFINITION 1.5
A finite automaton is a 5-tuple (Q, Σ, δ, q0, F ), where
1. Q is a finite set called the states,
2. Σ is a finite set called the alphabet,
3. δ : Q × Σ−→ Q is the transition function,1 4. q0 ∈ Q is the start state, and
5. F ⊆ Q is the set of accept states.2
1Refer back to page 7 if you are uncertain about the meaning of δ : Q × Σ−→ Q. 2Accept states sometimes are called final states.
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36 CHAPTER 1 / REGULAR LANGUAGES
The formal definition precisely describes what we mean by a finite automa- ton. For example, returning to the earlier question of whether 0 accept states is allowable, you can see that setting F to be the empty set ∅ yields 0 accept states, which is allowable. Furthermore, the transition function δ specifies exactly one next state for each possible combination of a state and an input symbol. That an- swers our other question affirmatively, showing that exactly one transition arrow exits every state for each possible input symbol.
We can use the notation of the formal definition to describe individual finite automata by specifying each of the five parts listed in Definition 1.5. For exam- ple, let’s return to the finite automaton M1 we discussed earlier, redrawn here for convenience.
FIGURE 1.6
The finite automaton M1
We can describe M1 formally by writing M1 = (Q, Σ, δ, q1 , F ), where
1. Q = {q1,q2,q3}, 2. Σ = {0,1},
3. δ is described as
4. q1 is the start state, and 5. F = {q2}.
01
q1 q1 q2 q2 q3 q2 q3 q2 q2,
If A is the set of all strings that machine M accepts, we say that A is the language of machine M and write L(M ) = A. We say that M recognizes A or that M accepts A. Because the term accept has different meanings when we refer to machines accepting strings and machines accepting languages, we prefer the term recognize for languages in order to avoid confusion.
A machine may accept several strings, but it always recognizes only one lan- guage. If the machine accepts no strings, it still recognizes one language— namely, the empty language ∅.
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In our example, let
A = {w| w contains at least one 1 and
an even number of 0s follow the last 1}. Then L(M1) = A, or equivalently, M1 recognizes A.
EXAMPLES OF FINITE AUTOMATA
EXAMPLE 1.7
Here is the state diagram of finite automaton M2.
1.1 FINITE AUTOMATA 37
FIGURE 1.8
State diagram of the two-state finite automaton M2
In the formal description, M2 is {q1, q2}, {0,1}, δ, q1, {q2}. The transition function δ is
01
q1 q1 q2 q2 q1 q2.
Remember that the state diagram of M2 and the formal description of M2 contain the same information, only in different forms. You can always go from one to the other if necessary.
A good way to begin understanding any machine is to try it on some sample input strings. When you do these “experiments” to see how the machine is working, its method of functioning often becomes apparent. On the sample string 1101, the machine M2 starts in its start state q1 and proceeds first to state q2 after reading the first 1, and then to states q2, q1, and q2 after reading 1, 0, and 1. The string is accepted because q2 is an accept state. But string 110 leaves M2 in state q1, so it is rejected. After trying a few more examples, you would see that M2 accepts all strings that end in a 1. Thus L(M2) = {w| w ends in a 1}.
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38 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.9
Consider the finite automaton M3.
FIGURE 1.10
State diagram of the two-state finite automaton M3
Machine M3 is similar to M2 except for the location of the accept state. As usual, the machine accepts all strings that leave it in an accept state when it has finished reading. Note that because the start state is also an accept state, M3 accepts the empty string ε. As soon as a machine begins reading the empty string, it is at the end; so if the start state is an accept state, ε is accepted. In addition to the empty string, this machine accepts any string ending with a 0. Here,
L(M3) = {w| w is the empty string ε or ends in a 0}. EXAMPLE 1.11
The following figure shows a five-state machine M4.
FIGURE 1.12 Finite automaton M4
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Machine M4 has two accept states, q1 and r1, and operates over the alphabet Σ = {a, b}. Some experimentation shows that it accepts strings a, b, aa, bb, and bab, but not strings ab, ba, or bbba. This machine begins in state s, and after it reads the first symbol in the input, it goes either left into the q states or right into the r states. In both cases, it can never return to the start state (in contrast to the previous examples), as it has no way to get from any other state back to s. If the first symbol in the input string is a, then it goes left and accepts when the string ends with an a. Similarly, if the first symbol is a b, the machine goes right and accepts when the string ends in b. So M4 accepts all strings that start and end with a or that start and end with b. In other words, M4 accepts strings that start and end with the same symbol.
EXAMPLE 1.13
Figure 1.14 shows the three-state machine M5, which has a four-symbol input
alphabet, Σ = {⟨RESET⟩, 0, 1, 2}. We treat ⟨RESET⟩ as a single symbol.
1.1 FINITE AUTOMATA 39
FIGURE 1.14 Finite automaton M5
Machine M5 keeps a running count of the sum of the numerical input symbols it reads, modulo 3. Every time it receives the ⟨RESET⟩ symbol, it resets the count to 0. It accepts if the sum is 0 modulo 3, or in other words, if the sum is a multiple of 3.
Describing a finite automaton by state diagram is not possible in some cases. That may occur when the diagram would be too big to draw or if, as in the next example, the description depends on some unspecified parameter. In these cases, we resort to a formal description to specify the machine.
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40 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.15
Consider a generalization of Example 1.13, using the same four-symbol alpha- bet Σ. For each i ≥ 1 let Ai be the language of all strings where the sum of the numbers is a multiple of i, except that the sum is reset to 0 whenever the symbol ⟨RESET⟩ appears. For each Ai we give a finite automaton Bi, recognizing Ai. We describe the machine Bi formally as follows: Bi =(Qi,Σ,δi,q0,{q0}), where Qi is the set of i states {q0, q1, q2, . . . , qi−1}, and we design the transi- tion function δi so that for each j, if Bi is in qj, the running sum is j, modulo i. For each qj let
δi(qj,0) = qj,
δi (qj , 1) = qk , where k = j + 1 modulo i,
δi (qj , 2) = qk , where k = j + 2 modulo i, and δi(qj,⟨RESET⟩) = q0.
FORMAL DEFINITION OF COMPUTATION
So far we have described finite automata informally, using state diagrams, and with a formal definition, as a 5-tuple. The informal description is easier to grasp at first, but the formal definition is useful for making the notion precise, resolv- ing any ambiguities that may have occurred in the informal description. Next we do the same for a finite automaton’s computation. We already have an informal idea of the way it computes, and we now formalize it mathematically.
Let M = (Q,Σ,δ,q0,F) be a finite automaton and let w = w1w2 ··· wn be a string where each wi is a member of the alphabet Σ. Then M accepts w if a sequenceofstatesr0,r1,…,rn inQexistswiththreeconditions:
1. r0 = q0,
2. δ(ri,wi+1)=ri+1, fori=0,…,n−1, and 3. rn ∈ F .
Condition 1 says that the machine starts in the start state. Condition 2 says that the machine goes from state to state according to the transition function. Condition 3 says that the machine accepts its input if it ends up in an accept state. We say that M recognizes language A if A = {w| M accepts w}.
DEFINITION 1.16
A language is called a regular language if some finite automaton recognizes it.
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EXAMPLE 1.17
Take machine M5 from Example 1.13. Let w be the string
10⟨RESET⟩22⟨RESET⟩012.
Then M5 accepts w according to the formal definition of computation because
the sequence of states it enters when computing on w is q0, q1, q1, q0, q2, q1, q0, q0, q1, q0,
which satisfies the three conditions. The language of M5 is
L(M5) = {w| the sum of the symbols in w is 0 modulo 3,
except that ⟨RESET⟩ resets the count to 0}. As M5 recognizes this language, it is a regular language.
DESIGNING FINITE AUTOMATA
Whether it be of automaton or artwork, design is a creative process. As such, it cannot be reduced to a simple recipe or formula. However, you might find a particular approach helpful when designing various types of automata. That is, put yourself in the place of the machine you are trying to design and then see how you would go about performing the machine’s task. Pretending that you are the machine is a psychological trick that helps engage your whole mind in the design process.
Let’s design a finite automaton using the “reader as automaton” method just described. Suppose that you are given some language and want to design a finite automaton that recognizes it. Pretending to be the automaton, you receive an input string and must determine whether it is a member of the language the automaton is supposed to recognize. You get to see the symbols in the string one by one. After each symbol, you must decide whether the string seen so far is in the language. The reason is that you, like the machine, don’t know when the end of the string is coming, so you must always be ready with the answer.
First, in order to make these decisions, you have to figure out what you need to remember about the string as you are reading it. Why not simply remember all you have seen? Bear in mind that you are pretending to be a finite automaton and that this type of machine has only a finite number of states, which means a finite memory. Imagine that the input is extremely long—say, from here to the moon—so that you could not possibly remember the entire thing. You have a finite memory—say, a single sheet of paper—which has a limited storage ca- pacity. Fortunately, for many languages you don’t need to remember the entire input. You need to remember only certain crucial information. Exactly which information is crucial depends on the particular language considered.
For example, suppose that the alphabet is {0,1} and that the language consists of all strings with an odd number of 1s. You want to construct a finite automaton E1 to recognize this language. Pretending to be the automaton, you start getting
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1.1 FINITE AUTOMATA 41
42 CHAPTER 1 / REGULAR LANGUAGES
an input string of 0s and 1s symbol by symbol. Do you need to remember the entire string seen so far in order to determine whether the number of 1s is odd? Of course not. Simply remember whether the number of 1s seen so far is even or odd and keep track of this information as you read new symbols. If you read a 1, flip the answer; but if you read a 0, leave the answer as is.
But how does this help you design E1? Once you have determined the neces- sary information to remember about the string as it is being read, you represent this information as a finite list of possibilities. In this instance, the possibilities would be
1. even so far, and 2. odd so far.
Then you assign a state to each of the possibilities. These are the states of E1, as shown here.
FIGURE 1.18
The two states qeven and qodd
Next, you assign the transitions by seeing how to go from one possibility to another upon reading a symbol. So, if state qeven represents the even possibility and state qodd represents the odd possibility, you would set the transitions to flip state on a 1 and stay put on a 0, as shown here.
FIGURE 1.19
Transitions telling how the possibilities rearrange
Next, you set the start state to be the state corresponding to the possibility associated with having seen 0 symbols so far (the empty string ε). In this case, the start state corresponds to state qeven because 0 is an even number. Last, set the accept states to be those corresponding to possibilities where you want to accept the input string. Set qodd to be an accept state because you want to accept
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1.1 FINITE AUTOMATA 43 when you have seen an odd number of 1s. These additions are shown in the
following figure.
FIGURE 1.20
Adding the start and accept states
EXAMPLE 1.21
This example shows how to design a finite automaton E2 to recognize the regu- lar language of all strings that contain the string 001 as a substring. For example, 0010, 1001, 001, and 11111110011111 are all in the language, but 11 and 0000 are not. How would you recognize this language if you were pretending to be E2? As symbols come in, you would initially skip over all 1s. If you come to a 0, then you note that you may have just seen the first of the three symbols in the pattern 001 you are seeking. If at this point you see a 1, there were too few 0s, so you go back to skipping over 1s. But if you see a 0 at that point, you should remember that you have just seen two symbols of the pattern. Now you simply need to continue scanning until you see a 1. If you find it, remember that you succeeded in finding the pattern and continue reading the input string until you get to the end.
So there are four possibilities: You
1. haven’t just seen any symbols of the pattern, 2. have just seen a 0,
3. have just seen 00, or
4. have seen the entire pattern 001.
Assign the states q, q0, q00, and q001 to these possibilities. You can assign the transitions by observing that from q reading a 1 you stay in q, but reading a 0 you move to q0. In q0 reading a 1 you return to q, but reading a 0 you move to q00. In q00 reading a 1 you move to q001, but reading a 0 leaves you in q00. Finally, in q001 reading a 0 or a 1 leaves you in q001. The start state is q, and the only accept state is q001, as shown in Figure 1.22.
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44 CHAPTER 1 / REGULAR LANGUAGES
FIGURE 1.22
Accepts strings containing 001
THE REGULAR OPERATIONS
In the preceding two sections, we introduced and defined finite automata and regular languages. We now begin to investigate their properties. Doing so will help develop a toolbox of techniques for designing automata to recognize partic- ular languages. The toolbox also will include ways of proving that certain other languages are nonregular (i.e., beyond the capability of finite automata).
In arithmetic, the basic objects are numbers and the tools are operations for manipulating them, such as + and ×. In the theory of computation, the ob- jects are languages and the tools include operations specifically designed for manipulating them. We define three operations on languages, called the reg- ular operations, and use them to study properties of the regular languages.
DEFINITION 1.23
Let A and B be languages. We define the regular operations union,
concatenation, and star as follows:
• Union:A∪B={x|x∈Aorx∈B}.
• Concatenation:A◦B={xy|x∈Aandy∈B}. • Star:A∗ ={x1x2…xk|k≥0andeachxi ∈A}.
You are already familiar with the union operation. It simply takes all the strings in both A and B and lumps them together into one language.
The concatenation operation is a little trickier. It attaches a string from A in front of a string from B in all possible ways to get the strings in the new language.
The star operation is a bit different from the other two because it applies to a single language rather than to two different languages. That is, the star oper- ation is a unary operation instead of a binary operation. It works by attaching any number of strings in A together to get a string in the new language. Because
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1.1 FINITE AUTOMATA 45 “any number” includes 0 as a possibility, the empty string ε is always a member
of A∗, no matter what A is. EXAMPLE 1.24
Let the alphabet Σ be the standard 26 letters {a, b, . . . , z}. If A = {good, bad} and B = {boy, girl}, then
A ∪ B = {good, bad, boy, girl},
A ◦ B = {goodboy, goodgirl, badboy, badgirl}, and
A∗ = {ε, good, bad, goodgood, goodbad, badgood, badbad, goodgoodgood, goodgoodbad, goodbadgood, goodbadbad, . . . }.
Let N = {1,2,3,…} be the set of natural numbers. When we say that N is closed under multiplication, we mean that for any x and y in N, the product x × y also is in N. In contrast, N is not closed under division, as 1 and 2 are in N but 1/2 is not. Generally speaking, a collection of objects is closed under some operation if applying that operation to members of the collection returns an object still in the collection. We show that the collection of regular languages is closed under all three of the regular operations. In Section 1.3, we show that these are useful tools for manipulating regular languages and understanding the power of finite automata. We begin with the union operation.
THEOREM 1.25
The class of regular languages is closed under the union operation.
In other words, if A1 and A2 are regular languages, so is A1 ∪ A2.
PROOF IDEA We have regular languages A1 and A2 and want to show that A1 ∪ A2 also is regular. Because A1 and A2 are regular, we know that some finite automaton M1 recognizes A1 and some finite automaton M2 recognizes A2. To prove that A1 ∪ A2 is regular, we demonstrate a finite automaton, call it M , that recognizes A1 ∪ A2.
This is a proof by construction. We construct M from M1 and M2. Machine M must accept its input exactly when either M1 or M2 would accept it in order to recognize the union language. It works by simulating both M1 and M2 and accepting if either of the simulations accept.
How can we make machine M simulate M1 and M2? Perhaps it first simulates M1 on the input and then simulates M2 on the input. But we must be careful here! Once the symbols of the input have been read and used to simulate M1, we can’t “rewind the input tape” to try the simulation on M2. We need another approach.
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46 CHAPTER 1 / REGULAR LANGUAGES
Pretend that you are M . As the input symbols arrive one by one, you simulate both M1 and M2 simultaneously. That way, only one pass through the input is necessary. But can you keep track of both simulations with finite memory? All you need to remember is the state that each machine would be in if it had read up to this point in the input. Therefore, you need to remember a pair of states. How many possible pairs are there? If M1 has k1 states and M2 has k2 states, the number of pairs of states, one from M1 and the other from M2, is the product k1 × k2 . This product will be the number of states in M , one for each pair. The transitions of M go from pair to pair, updating the current state for both M1 and M2. The accept states of M are those pairs wherein either M1 or M2 is in an accept state.
PROOF
Let M1 recognize A1, where M1 = (Q1, Σ, δ1, q1, F1), and M2 recognize A2, where M2 = (Q2, Σ, δ2, q2, F2).
Construct M to recognize A1 ∪ A2, where M = (Q, Σ, δ, q0, F ).
1. Q = {(r1, r2)| r1 ∈ Q1 and r2 ∈ Q2}.
This set is the Cartesian product of sets Q1 and Q2 and is written Q1 × Q2. It is the set of all pairs of states, the first from Q1 and the second from Q2.
2. Σ, the alphabet, is the same as in M1 and M2. In this theorem and in all subsequent similar theorems, we assume for simplicity that both M1 and M2 have the same input alphabet Σ. The theorem remains true if they have different alphabets, Σ1 and Σ2. We would then modify the proof to let Σ = Σ1 ∪ Σ2.
3. δ, the transition function, is defined as follows. For each (r1, r2) ∈ Q and
each a ∈ Σ, let
δ(r1, r2), a = δ1(r1, a), δ2(r2, a).
Hence δ gets a state of M (which actually is a pair of states from M1 and M2), together with an input symbol, and returns M’s next state.
4. q0 is the pair (q1, q2).
5. F is the set of pairs in which either member is an accept state of M1 or M2.
We can write it as
This expression is the same as F = (F1 × Q2) ∪ (Q1 × F2). (Note that it is
F ={(r1,r2)|r1 ∈F1 orr2 ∈F2}.
not the same as F = F1 × F2. What would that give us instead?3)
3 This expression would define M’s accept states to be those for which both members of the pair are accept states. In this case, M would accept a string only if both M1 and M2 accept it, so the resulting language would be the intersection and not the union. In fact, this result proves that the class of regular languages is closed under intersection.
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This concludes the construction of the finite automaton M that recognizes the union of A1 and A2. This construction is fairly simple, and thus its correct- ness is evident from the strategy described in the proof idea. More complicated constructions require additional discussion to prove correctness. A formal cor- rectness proof for a construction of this type usually proceeds by induction. For an example of a construction proved correct, see the proof of Theorem 1.54. Most of the constructions that you will encounter in this course are fairly simple and so do not require a formal correctness proof.
We have just shown that the union of two regular languages is regular, thereby proving that the class of regular languages is closed under the union operation. We now turn to the concatenation operation and attempt to show that the class of regular languages is closed under that operation, too.
THEOREM 1.26
The class of regular languages is closed under the concatenation operation.
In other words, if A1 and A2 are regular languages then so is A1 ◦ A2.
To prove this theorem, let’s try something along the lines of the proof of the union case. As before, we can start with finite automata M1 and M2 recognizing the regular languages A1 and A2. But now, instead of constructing automaton M to accept its input if either M1 or M2 accept, it must accept if its input can be broken into two pieces, where M1 accepts the first piece and M2 accepts the second piece. The problem is that M doesn’t know where to break its input (i.e., where the first part ends and the second begins). To solve this problem, we introduce a new technique called nondeterminism.
1.2
NONDETERMINISM
Nondeterminism is a useful concept that has had great impact on the theory of computation. So far in our discussion, every step of a computation follows in a unique way from the preceding step. When the machine is in a given state and reads the next input symbol, we know what the next state will be—it is deter- mined. We call this deterministic computation. In a nondeterministic machine, several choices may exist for the next state at any point.
Nondeterminism is a generalization of determinism, so every deterministic finite automaton is automatically a nondeterministic finite automaton. As Fig- ure 1.27 shows, nondeterministic finite automata may have additional features.
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1.2 NONDETERMINISM 47
48 CHAPTER 1 / REGULAR LANGUAGES
FIGURE 1.27
The nondeterministic finite automaton N1
The difference between a deterministic finite automaton, abbreviated DFA, and a nondeterministic finite automaton, abbreviated NFA, is immediately ap- parent. First, every state of a DFA always has exactly one exiting transition arrow for each symbol in the alphabet. The NFA shown in Figure 1.27 violates that rule. State q1 has one exiting arrow for 0, but it has two for 1; q2 has one arrow for 0, but it has none for 1. In an NFA, a state may have zero, one, or many exiting arrows for each alphabet symbol.
Second, in a DFA, labels on the transition arrows are symbols from the alpha- bet. This NFA has an arrow with the label ε. In general, an NFA may have arrows labeled with members of the alphabet or ε. Zero, one, or many arrows may exit from each state with the label ε.
How does an NFA compute? Suppose that we are running an NFA on an input string and come to a state with multiple ways to proceed. For example, say that we are in state q1 in NFA N1 and that the next input symbol is a 1. After reading that symbol, the machine splits into multiple copies of itself and follows all the possibilities in parallel. Each copy of the machine takes one of the possible ways to proceed and continues as before. If there are subsequent choices, the machine splits again. If the next input symbol doesn’t appear on any of the arrows exiting the state occupied by a copy of the machine, that copy of the machine dies, along with the branch of the computation associated with it. Finally, if any one of these copies of the machine is in an accept state at the end of the input, the NFA accepts the input string.
If a state with an ε symbol on an exiting arrow is encountered, something similar happens. Without reading any input, the machine splits into multiple copies, one following each of the exiting ε-labeled arrows and one staying at the current state. Then the machine proceeds nondeterministically as before.
Nondeterminism may be viewed as a kind of parallel computation wherein multiple independent “processes” or “threads” can be running concurrently. When the NFA splits to follow several choices, that corresponds to a process “forking” into several children, each proceeding separately. If at least one of these processes accepts, then the entire computation accepts.
Another way to think of a nondeterministic computation is as a tree of possi- bilities. The root of the tree corresponds to the start of the computation. Every branching point in the tree corresponds to a point in the computation at which the machine has multiple choices. The machine accepts if at least one of the computation branches ends in an accept state, as shown in Figure 1.28.
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1.2 NONDETERMINISM 49
FIGURE 1.28
Deterministic and nondeterministic computations with an accepting branch
Let’s consider some sample runs of the NFA N1 shown in Figure 1.27. The computation of N1 on input 010110 is depicted in the following figure.
FIGURE 1.29
The computation of N1 on input 010110
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50 CHAPTER 1 / REGULAR LANGUAGES
On input 010110, start in the start state q1 and read the first symbol 0. From q1 there is only one place to go on a 0—namely, back to q1—so remain there. Next, read the second symbol 1. In q1 on a 1 there are two choices: either stay in q1 or move to q2. Nondeterministically, the machine splits in two to follow each choice. Keep track of the possibilities by placing a finger on each state where a machine could be. So you now have fingers on states q1 and q2. An ε arrow exits state q2 so the machine splits again; keep one finger on q2, and move the other to q3. You now have fingers on q1, q2, and q3.
When the third symbol 0 is read, take each finger in turn. Keep the finger on q1 in place, move the finger on q2 to q3, and remove the finger that has been on q3. That last finger had no 0 arrow to follow and corresponds to a process that simply “dies.” At this point, you have fingers on states q1 and q3.
When the fourth symbol 1 is read, split the finger on q1 into fingers on states q1 and q2, then further split the finger on q2 to follow the ε arrow to q3, and move the finger that was on q3 to q4. You now have a finger on each of the four states.
When the fifth symbol 1 is read, the fingers on q1 and q3 result in fingers on states q1, q2, q3, and q4, as you saw with the fourth symbol. The finger on state q2 is removed. The finger that was on q4 stays on q4. Now you have two fingers on q4, so remove one because you only need to remember that q4 is a possible state at this point, not that it is possible for multiple reasons.
When the sixth and final symbol 0 is read, keep the finger on q1 in place, move the one on q2 to q3, remove the one that was on q3, and leave the one on q4 in place. You are now at the end of the string, and you accept if some finger is on an accept state. You have fingers on states q1, q3, and q4; and as q4 is an accept state, N1 accepts this string.
What does N1 do on input 010? Start with a finger on q1. After reading the 0, you still have a finger only on q1; but after the 1 there are fingers on q1, q2, and q3 (don’t forget the ε arrow). After the third symbol 0, remove the finger on q3, move the finger on q2 to q3, and leave the finger on q1 where it is. At this point you are at the end of the input; and as no finger is on an accept state, N1 rejects this input.
By continuing to experiment in this way, you will see that N1 accepts all strings that contain either 101 or 11 as a substring.
Nondeterministic finite automata are useful in several respects. As we will show, every NFA can be converted into an equivalent DFA, and constructing NFAs is sometimes easier than directly constructing DFAs. An NFA may be much smaller than its deterministic counterpart, or its functioning may be easier to understand. Nondeterminism in finite automata is also a good introduction to nondeterminism in more powerful computational models because finite au- tomata are especially easy to understand. Now we turn to several examples of NFAs.
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EXAMPLE 1.30
Let A be the language consisting of all strings over {0,1} containing a 1 in the third position from the end (e.g., 000100 is in A but 0011 is not). The following four-state NFA N2 recognizes A.
FIGURE 1.31
The NFA N2 recognizing A
One good way to view the computation of this NFA is to say that it stays in the start state q1 until it “guesses” that it is three places from the end. At that point, if the input symbol is a 1, it branches to state q2 and uses q3 and q4 to “check” on whether its guess was correct.
As mentioned, every NFA can be converted into an equivalent DFA; but some- times that DFA may have many more states. The smallest DFA for A contains eight states. Furthermore, understanding the functioning of the NFA is much easier, as you may see by examining the following figure for the DFA.
1.2 NONDETERMINISM 51
FIGURE 1.32
A DFA recognizing A
Suppose that we added ε to the labels on the arrows going from q2 to q3 and from q3 to q4 in machine N2 in Figure 1.31. So both arrows would then have the label 0, 1, ε instead of just 0, 1. What language would N2 recognize with this modification? Try modifying the DFA in Figure 1.32 to recognize that language.
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52 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.33
The following NFA N3 has an input alphabet {0} consisting of a single symbol. An alphabet containing only one symbol is called a unary alphabet.
FIGURE 1.34 The NFA N3
This machine demonstrates the convenience of having ε arrows. It accepts all strings of the form 0k where k is a multiple of 2 or 3. (Remember that the superscript denotes repetition, not numerical exponentiation.) For example, N3 accepts the strings ε, 00, 000, 0000, and 000000, but not 0 or 00000.
Think of the machine operating by initially guessing whether to test for a multiple of 2 or a multiple of 3 by branching into either the top loop or the bot- tom loop and then checking whether its guess was correct. Of course, we could replace this machine by one that doesn’t have ε arrows or even any nondeter- minism at all, but the machine shown is the easiest one to understand for this language.
EXAMPLE 1.35
We give another example of an NFA in Figure 1.36. Practice with it to satisfy yourself that it accepts the strings ε, a, baba, and baa, but that it doesn’t ac- cept the strings b, bb, and babba. Later we use this machine to illustrate the procedure for converting NFAs to DFAs.
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1.2 NONDETERMINISM 53
FIGURE 1.36 The NFA N4
FORMAL DEFINITION OF A NONDETERMINISTIC FINITE AUTOMATON
The formal definition of a nondeterministic finite automaton is similar to that of a deterministic finite automaton. Both have states, an input alphabet, a transition function, a start state, and a collection of accept states. However, they differ in one essential way: in the type of transition function. In a DFA, the transition function takes a state and an input symbol and produces the next state. In an NFA, the transition function takes a state and an input symbol or the empty string and produces the set of possible next states. In order to write the formal definition, we need to set up some additional notation. For any set Q we write P(Q) to be the collection of all subsets of Q. Here P (Q) is called the power set of Q. For any alphabet Σ we write Σε to be Σ ∪ {ε}. Now we can write the formal description o f t h e t y p e o f t h e t r a n s i t i o n f u n c t i o n i n a n N F A a s δ : Q × Σ ε −→ P ( Q ) .
DEFINITION 1.37
A nondeterministic finite automaton is a 5-tuple (Q,Σ,δ,q0,F),
where
1. Q is a finite set of states,
2. Σ is a finite alphabet,
3. δ : Q × Σε −→ P (Q) is the transition function, 4. q0 ∈ Q is the start state, and
5. F ⊆ Q is the set of accept states.
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54 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.38
Recall the NFA N1:
The formal description of N1 is (Q, Σ, δ, q1, F ), where
1. Q = {q1,q2,q3,q4}, 2. Σ = {0,1},
3. δ is given as
4. q1 is the start state, and 5. F = {q4}.
01ε q1 {q1} {q1,q2} ∅
q2 {q3} ∅ {q3} q3 ∅ {q4} ∅ q4 {q4} {q4} ∅,
The formal definition of computation for an NFA is similar to that for a DFA. Let N = (Q,Σ,δ,q0,F) be an NFA and w a string over the alphabet Σ. Then we say that N accepts w if we can write w as w = y1y2 ···ym, where each yi is a member of Σε and a sequence of states r0,r1,…,rm exists in Q with three conditions:
1. r0 = q0,
2. ri+1 ∈δ(ri,yi+1), fori=0,…,m−1, and 3. rm ∈ F .
Condition 1 says that the machine starts out in the start state. Condition 2 says that state ri+1 is one of the allowable next states when N is in state ri and reading yi+1. Observe that δ(ri, yi+1) is the set of allowable next states and so we say that ri+1 is a member of that set. Finally, condition 3 says that the machine accepts its input if the last state is an accept state.
EQUIVALENCE OF NFAS AND DFAS
Deterministic and nondeterministic finite automata recognize the same class of languages. Such equivalence is both surprising and useful. It is surprising be- cause NFAs appear to have more power than DFAs, so we might expect that NFAs recognize more languages. It is useful because describing an NFA for a given language sometimes is much easier than describing a DFA for that language.
Say that two machines are equivalent if they recognize the same language.
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THEOREM 1.39
Every nondeterministic finite automaton has an equivalent deterministic finite
automaton.
PROOF IDEA If a language is recognized by an NFA, then we must show the existence of a DFA that also recognizes it. The idea is to convert the NFA into an equivalent DFA that simulates the NFA.
Recall the “reader as automaton” strategy for designing finite automata. How would you simulate the NFA if you were pretending to be a DFA? What do you need to keep track of as the input string is processed? In the examples of NFAs, you kept track of the various branches of the computation by placing a finger on each state that could be active at given points in the input. You updated the simulation by moving, adding, and removing fingers according to the way the NFA operates. All you needed to keep track of was the set of states having fingers on them.
If k is the number of states of the NFA, it has 2k subsets of states. Each subset corresponds to one of the possibilities that the DFA must remember, so the DFA simulating the NFA will have 2k states. Now we need to figure out which will be the start state and accept states of the DFA, and what will be its transition function. We can discuss this more easily after setting up some formal notation.
PROOF Let N = (Q, Σ, δ, q0 , F ) be the NFA recognizing some language A. WeconstructaDFAM =(Q′,Σ,δ′,q0′,F′)recognizingA.Beforedoingthefull construction, let’s first consider the easier case wherein N has no ε arrows. Later we take the ε arrows into account.
1. Q′ = P(Q).
Every state of M is a set of states of N. Recall that P(Q) is the set of subsets of Q.
2. For R ∈ Q′ and a ∈ Σ, let δ′(R, a) = {q ∈ Q| q ∈ δ(r, a) for some r ∈ R}. IfRisastateofM,itisalsoasetofstatesofN. WhenM readsasymbol a in state R, it shows where a takes each state in R. Because each state may go to a set of states, we take the union of all these sets. Another way to write this expression is
δ′(R, a) = δ(r, a).4 r∈R
3. q0′ = {q0}.
M starts in the state corresponding to the collection containing just the start state of N .
4. F′ = {R ∈ Q′| R contains an accept state of N}.
The machine M accepts if one of the possible states that N could be in at this point is an accept state.
4The notation δ(r, a) means: the union of the sets δ(r, a) for each possible r in R. r∈R
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1.2 NONDETERMINISM 55
56 CHAPTER 1 / REGULAR LANGUAGES
Now we need to consider the ε arrows. To do so, we set up an extra bit of notation. For any state R of M, we define E(R) to be the collection of states that can be reached from members of R by going only along ε arrows, including the members of R themselves. Formally, for R ⊆ Q let
E(R) = {q| q can be reached from R by traveling along 0 or more ε arrows}.
Then we modify the transition function of M to place additional fingers on all states that can be reached by going along ε arrows after every step. Replacing δ(r, a) by E(δ(r, a)) achieves this effect. Thus
δ′(R, a) = {q ∈ Q| q ∈ E(δ(r, a)) for some r ∈ R}.
Additionally, we need to modify the start state of M to move the fingers ini- tially to all possible states that can be reached from the start state of N along the ε arrows. Changing q0′ to be E({q0}) achieves this effect. We have now completed the construction of the DFA M that simulates the NFA N .
The construction of M obviously works correctly. At every step in the com- putation of M on an input, it clearly enters a state that corresponds to the subset of states that N could be in at that point. Thus our proof is complete.
Theorem 1.39 states that every NFA can be converted into an equivalent DFA. Thus nondeterministic finite automata give an alternative way of characterizing the regular languages. We state this fact as a corollary of Theorem 1.39.
COROLLARY 1.40
A language is regular if and only if some nondeterministic finite automaton rec-
ognizes it.
One direction of the “if and only if” condition states that a language is regular if some NFA recognizes it. Theorem 1.39 shows that any NFA can be converted into an equivalent DFA. Consequently, if an NFA recognizes some language, so does some DFA, and hence the language is regular. The other direction of the “if and only if” condition states that a language is regular only if some NFA rec- ognizes it. That is, if a language is regular, some NFA must be recognizing it. Obviously, this condition is true because a regular language has a DFA recogniz- ing it and any DFA is also an NFA.
EXAMPLE 1.41
Let’s illustrate the procedure we gave in the proof of Theorem 1.39 for convert- ing an NFA to a DFA by using the machine N4 that appears in Example 1.35. For clarity, we have relabeled the states of N4 to be {1, 2, 3}. Thus in the formal description of N4 = (Q, {a,b}, δ, 1, {1}), the set of states Q is {1, 2, 3} as shown in Figure 1.42.
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To construct a DFA D that is equivalent to N4, we first determine D’s states. N4 has three states, {1, 2, 3}, so we construct D with eight states, one for each subset of N4’s states. We label each of D’s states with the corresponding subset. Thus D’s state set is
∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.
1.2 NONDETERMINISM 57
FIGURE 1.42 The NFA N4
Next, we determine the start and accept states of D. The start state is E({1}), the set of states that are reachable from 1 by traveling along ε arrows, plus 1 itself. An ε arrow goes from 1 to 3, so E({1}) = {1, 3}. The new accept states are those containing N4’s accept state; thus {1}, {1,2}, {1,3}, {1,2,3}.
Finally, we determine D’s transition function. Each of D’s states goes to one place on input a and one place on input b. We illustrate the process of deter- mining the placement of D’s transition arrows with a few examples.
In D, state {2} goes to {2,3} on input a because in N4, state 2 goes to both 2 and 3 on input a and we can’t go farther from 2 or 3 along ε arrows. State {2} goes to state {3} on input b because in N4, state 2 goes only to state 3 on input b and we can’t go farther from 3 along ε arrows.
State {1} goes to ∅ on a because no a arrows exit it. It goes to {2} on b. Note that the procedure in Theorem 1.39 specifies that we follow the ε arrows after each input symbol is read. An alternative procedure based on following the ε arrows before reading each input symbol works equally well, but that method is not illustrated in this example.
State {3} goes to {1,3} on a because in N4, state 3 goes to 1 on a and 1 in turn goes to 3 with an ε arrow. State {3} on b goes to ∅.
State {1,2} on a goes to {2,3} because 1 points at no states with a arrows, 2 points at both 2 and 3 with a arrows, and neither points anywhere with ε ar- rows. State {1,2} on b goes to {2,3}. Continuing in this way, we obtain the diagram for D in Figure 1.43.
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58 CHAPTER 1 / REGULAR LANGUAGES
FIGURE 1.43
A DFA D that is equivalent to the NFA N4
We may simplify this machine by observing that no arrows point at states {1} and {1, 2}, so they may be removed without affecting the performance of the machine. Doing so yields the following figure.
FIGURE 1.44
DFA D after removing unnecessary states
CLOSURE UNDER THE REGULAR OPERATIONS
Now we return to the closure of the class of regular languages under the regular operations that we began in Section 1.1. Our aim is to prove that the union, concatenation, and star of regular languages are still regular. We abandoned the original attempt to do so when dealing with the concatenation operation was too complicated. The use of nondeterminism makes the proofs much easier.
First, let’s consider again closure under union. Earlier we proved closure under union by simulating deterministically both machines simultaneously via a Cartesian product construction. We now give a new proof to illustrate the
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1.2 NONDETERMINISM 59 technique of nondeterminism. Reviewing the first proof, appearing on page 45,
may be worthwhile to see how much easier and more intuitive the new proof is.
THEOREM 1.45
The class of regular languages is closed under the union operation.
PROOF IDEA We have regular languages A1 and A2 and want to prove that A1 ∪ A2 is regular. The idea is to take two NFAs, N1 and N2 for A1 and A2, and combine them into one new NFA, N .
Machine N must accept its input if either N1 or N2 accepts this input. The new machine has a new start state that branches to the start states of the old ma- chines with ε arrows. In this way, the new machine nondeterministically guesses which of the two machines accepts the input. If one of them accepts the input, N will accept it, too.
We represent this construction in the following figure. On the left, we in- dicate the start and accept states of machines N1 and N2 with large circles and some additional states with small circles. On the right, we show how to combine N1 and N2 into N by adding additional transition arrows.
FIGURE 1.46
Construction of an NFA N to recognize A1 ∪ A2
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60 CHAPTER 1 / REGULAR LANGUAGES PROOF
Let N1 = (Q1, Σ, δ1, q1, F1) recognize A1, and N2 = (Q2, Σ, δ2, q2, F2) recognize A2.
ConstructN =(Q,Σ,δ,q0,F)torecognizeA1 ∪A2.
1. Q = {q0} ∪ Q1 ∪ Q2.
The states of N are all the states of N1 and N2, with the addition of a new start state q0.
2. The state q0 is the start state of N .
3. The set of accept states F = F1 ∪ F2.
The accept states of N are all the accept states of N1 and N2. That way, N accepts if either N1 accepts or N2 accepts.
4 . D e fi n e δ s o t h a t f o r a n y q ∈⎧Q a n d a n y a ∈ Σ ε , ⎪⎨⎪⎩δ1(q, a) q ∈ Q1
δ(q,a) = δ2(q,a) q ∈ Q2 ⎪{q1,q2} q=q0 anda=ε
∅ q=q0 anda̸=ε.
Now we can prove closure under concatenation. Recall that earlier, without nondeterminism, completing the proof would have been difficult.
THEOREM 1.47
The class of regular languages is closed under the concatenation operation.
PROOF IDEA We have regular languages A1 and A2 and want to prove that A1 ◦ A2 is regular. The idea is to take two NFAs, N1 and N2 for A1 and A2, and combine them into a new NFA N as we did for the case of union, but this time in a different way, as shown in Figure 1.48.
Assign N’s start state to be the start state of N1. The accept states of N1 have additional ε arrows that nondeterministically allow branching to N2 whenever N1 is in an accept state, signifying that it has found an initial piece of the input that constitutes a string in A1. The accept states of N are the accept states of N2 only. Therefore, it accepts when the input can be split into two parts, the first accepted by N1 and the second by N2. We can think of N as nondeterministically guessing where to make the split.
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1.2 NONDETERMINISM 61
FIGURE 1.48
Construction of N to recognize A1 ◦ A2
PROOF
Let N1 = (Q1, Σ, δ1, q1, F1) recognize A1, and N2 = (Q2, Σ, δ2, q2, F2) recognize A2.
Construct N = (Q, Σ, δ, q1, F2) to recognize A1 ◦ A2. 1. Q = Q1 ∪ Q2.
The states of N are all the states of N1 and N2.
2. The state q1 is the same as the start state of N1.
3. The accept states F2 are the same as the accept states of N2. 4. Define δ so that for any q ∈ Q and any a ∈ Σε,
q ∈ Q1 and q ̸∈ F1 q∈F1 anda̸=ε q∈F1 anda=ε q ∈ Q2.
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⎧⎪⎨⎪⎩δ1(q, a) δ(q,a)= δ1(q,a)
⎪δ1(q,a)∪{q2} δ2(q,a)
62 CHAPTER 1 / REGULAR LANGUAGES THEOREM 1.49
The class of regular languages is closed under the star operation.
PROOF IDEA We have a regular language A1 and want to prove that A∗1 also is regular. We take an NFA N1 for A1 and modify it to recognize A∗1 , as shown in the following figure. The resulting NFA N will accept its input whenever it can be broken into several pieces and N1 accepts each piece.
We can construct N like N1 with additional ε arrows returning to the start state from the accept states. This way, when processing gets to the end of a piece that N1 accepts, the machine N has the option of jumping back to the start state to try to read another piece that N1 accepts. In addition, we must modify N so that it accepts ε, which always is a member of A∗1. One (slightly bad) idea is simply to add the start state to the set of accept states. This approach certainly adds ε to the recognized language, but it may also add other, undesired strings. Exercise 1.15 asks for an example of the failure of this idea. The way to fix it is to add a new start state, which also is an accept state, and which has an ε arrow to the old start state. This solution has the desired effect of adding ε to the language without adding anything else.
FIGURE 1.50
Construction of N to recognize A∗
PROOF Let N1 = (Q1, Σ, δ1, q1, F1) recognize A1. Construct N = (Q,Σ,δ,q0,F) to recognize A∗1.
1. Q = {q0} ∪ Q1.
The states of N are the states of N1 plus a new start state.
2. The state q0 is the new start state. 3. F = {q0} ∪ F1.
The accept states are the old accept states plus the new start state.
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1.3 REGULAR EXPRESSIONS 63 4. Define δ so that for any q ∈ Q and any a ∈ Σε,
⎧⎪δ1(q, a)
⎪⎨δ1(q, a)
δ(q, a) = ⎪δ1(q, a) ∪ {q1}
⎪⎩{q1 } ∅
q∈Q1 andq̸∈F1 q∈F1 anda̸=ε q∈F1 anda=ε q=q0 anda=ε q=q0 anda̸=ε.
1.3
REGULAR EXPRESSIONS
In arithmetic, we can use the operations + and × to build up expressions such as
(5+3)×4.
Similarly, we can use the regular operations to build up expressions describing languages, which are called regular expressions. An example is:
(0 ∪ 1)0∗.
The value of the arithmetic expression is the number 32. The value of a regular expression is a language. In this case, the value is the language consisting of all strings starting with a 0 or a 1 followed by any number of 0s. We get this result by dissecting the expression into its parts. First, the symbols 0 and 1 are shorthand for the sets {0} and {1}. So (0 ∪ 1) means ({0} ∪ {1}). The value of this part is the language {0,1}. The part 0∗ means {0}∗, and its value is the language consisting of all strings containing any number of 0s. Second, like the × symbol in algebra, the concatenation symbol ◦ often is implicit in regular expressions. Thus (0 ∪ 1)0∗ actually is shorthand for (0 ∪ 1) ◦ 0∗ . The concatenation attaches the strings from the two parts to obtain the value of the entire expression.
Regular expressions have an important role in computer science applications. In applications involving text, users may want to search for strings that satisfy certain patterns. Regular expressions provide a powerful method for describing such patterns. Utilities such as awk and grep in UNIX, modern programming languages such as Perl, and text editors all provide mechanisms for the descrip- tion of patterns by using regular expressions.
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64 CHAPTER 1 / REGULAR LANGUAGES
EXAMPLE 1.51
Another example of a regular expression is
(0 ∪ 1)∗.
It starts with the language (0 ∪ 1) and applies the ∗ operation. The value of this expression is the language consisting of all possible strings of 0s and 1s. If Σ = {0,1}, we can write Σ as shorthand for the regular expression (0 ∪ 1). More generally, if Σ is any alphabet, the regular expression Σ describes the language consisting of all strings of length 1 over this alphabet, and Σ∗ describes the lan- guage consisting of all strings over that alphabet. Similarly, Σ∗1 is the language that contains all strings that end in a 1. The language (0Σ∗) ∪ (Σ∗1) consists of all strings that start with a 0 or end with a 1.
In arithmetic, we say that × has precedence over + to mean that when there is a choice, we do the × operation first. Thus in 2+3×4, the 3×4 is done before the addition. To have the addition done first, we must add parentheses to obtain (2 + 3) × 4. In regular expressions, the star operation is done first, followed by concatenation, and finally union, unless parentheses change the usual order.
FORMAL DEFINITION OF A REGULAR EXPRESSION
DEFINITION 1.52
Say that R is a regular expression if R is
1. a for some a in the alphabet Σ,
2. ε,
3. ∅,
4. (R1 ∪ R2), where R1 and R2 are regular expressions, 5. (R1 ◦ R2), where R1 and R2 are regular expressions, or 6. (R1∗), where R1 is a regular expression.
In items 1 and 2, the regular expressions a and ε represent the languages {a} and {ε}, respectively. In item 3, the regular expres- sion ∅ represents the empty language. In items 4, 5, and 6, the expressions represent the languages obtained by taking the union or concatenation of the languages R1 and R2, or the star of the language R1, respectively.
Don’t confuse the regular expressions ε and ∅. The expression ε represents the language containing a single string—namely, the empty string—whereas ∅ represents the language that doesn’t contain any strings.
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1.3 REGULAR EXPRESSIONS 65
Seemingly, we are in danger of defining the notion of a regular expression in terms of itself. If true, we would have a circular definition, which would be invalid. However, R1 and R2 always are smaller than R. Thus we actually are defining regular expressions in terms of smaller regular expressions and thereby avoiding circularity. A definition of this type is called an inductive definition.
Parentheses in an expression may be omitted. If they are, evaluation is done in the precedence order: star, then concatenation, then union.
For convenience, we let R+ be shorthand for RR∗. In other words, whereas R∗ has all strings that are 0 or more concatenations of strings from R, the lan- guage R+ has all strings that are 1 or more concatenations of strings from R. So R+ ∪ ε = R∗. In addition, we let Rk be shorthand for the concatenation of k R’s with each other.
When we want to distinguish between a regular expression R and the lan- guage that it describes, we write L(R) to be the language of R.
EXAMPLE 1.53
In the following instances, we assume that the alphabet Σ is {0,1}.
1. 0∗10∗ = {w| w contains a single 1}.
2. Σ∗1Σ∗ = {w| w has at least one 1}.
3. Σ∗001Σ∗ = {w| w contains the string 001 as a substring}.
4. 1∗(01+)∗ = {w| every 0 in w is followed by at least one 1}.
5. (ΣΣ)∗ = {w| w is a string of even length}.5
6. (ΣΣΣ)∗ = {w| the length of w is a multiple of 3}.
7. 01∪10={01,10}.
8. 0Σ∗0 ∪ 1Σ∗1 ∪ 0 ∪ 1 = {w| w starts and ends with the same symbol}.
9. (0 ∪ ε)1∗ = 01∗ ∪ 1∗.
The expression 0 ∪ ε describes the language {0, ε}, so the concatenation operation adds either 0 or ε before every string in 1∗.
10. (0∪ε)(1∪ε)={ε,0,1,01}.
11. 1∗∅ = ∅.
Concatenating the empty set to any set yields the empty set.
12. ∅∗ = {ε}.
The star operation puts together any number of strings from the language to get a string in the result. If the language is empty, the star operation can put together 0 strings, giving only the empty string.
5The length of a string is the number of symbols that it contains.
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66 CHAPTER 1 / REGULAR LANGUAGES
If we let R be any regular expression, we have the following identities. They
are good tests of whether you understand the definition.
R ∪ ∅ = R.
Adding the empty language to any other language will not change it.
R ◦ ε = R.
Joining the empty string to any string will not change it.
However, exchanging ∅ and ε in the preceding identities may cause the equalities to fail.
R ∪ ε may not equal R.
For example, if R = 0, then L(R) = {0} but L(R ∪ ε) = {0, ε}.
R ◦ ∅ may not equal R.
For example, if R = 0, then L(R) = {0} but L(R ◦ ∅) = ∅.
Regular expressions are useful tools in the design of compilers for program- ming languages. Elemental objects in a programming language, called tokens, such as the variable names and constants, may be described with regular ex- pressions. For example, a numerical constant that may include a fractional part and/or a sign may be described as a member of the language
+∪-∪εD+ ∪D+.D∗ ∪D∗.D+
where D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is the alphabet of decimal digits. Examples of generated strings are: 72, 3.14159, +7., and -.01.
Once the syntax of a programming language has been described with a regular expression in terms of its tokens, automatic systems can generate the lexical analyzer, the part of a compiler that initially processes the input program.
EQUIVALENCE WITH FINITE AUTOMATA
Regular expressions and finite automata are equivalent in their descriptive power. This fact is surprising because finite automata and regular expressions superficially appear to be rather different. However, any regular expression can be converted into a finite automaton that recognizes the language it describes, and vice versa. Recall that a regular language is one that is recognized by some finite automaton.
THEOREM 1.54
A language is regular if and only if some regular expression describes it.
This theorem has two directions. We state and prove each direction as a separate lemma.
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1.3 REGULAR EXPRESSIONS 67
LEMMA 1.55
If a language is described by a regular expression, then it is regular.
PROOF IDEA Say that we have a regular expression R describing some lan- guage A. We show how to convert R into an NFA recognizing A. By Corol- lary 1.40, if an NFA recognizes A then A is regular.
PROOF Let’s convert R into an NFA N. We consider the six cases in the formal definition of regular expressions.
1.
R = a for some a ∈ Σ. Then L(R) = {a}, and the following NFA recog- nizes L(R).
2.
3.
4. 5. 6 .
Formally, N = {q1, q2}, Σ, δ, q1, {q2}, where we describe δ by saying thatδ(q1,a)={q2}andthatδ(r,b)=∅forr̸=q1 orb̸=a.
R = ε. Then L(R) = {ε}, and the following NFA recognizes L(R).
Note that this machine fits the definition of an NFA but not that of a DFA because it has some states with no exiting arrow for each possible input symbol. Of course, we could have presented an equivalent DFA here; but an NFA is all we need for now, and it is easier to describe.
Formally, N = {q1}, Σ, δ, q1, {q1}, where δ(r, b) = ∅ for any r and b. R = ∅. Then L(R) = ∅, and the following NFA recognizes L(R).
Formally, N = {q},Σ,δ,q,∅, where δ(r,b) = ∅ for any r and b. R = R1 ∪ R2.
R = R1 ◦ R2.
R = R 1∗ .
For the last three cases, we use the constructions given in the proofs that the class of regular languages is closed under the regular operations. In other words, we construct the NFA for R from the NFAs for R1 and R2 (or just R1 in case 6) and the appropriate closure construction.
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68 CHAPTER 1 / REGULAR LANGUAGES
That ends the first part of the proof of Theorem 1.54, giving the easier di- rection of the if and only if condition. Before going on to the other direction, let’s consider some examples whereby we use this procedure to convert a regular expression to an NFA.
EXAMPLE 1.56
We convert the regular expression (ab ∪ a)∗ to an NFA in a sequence of stages. We build up from the smallest subexpressions to larger subexpressions until we have an NFA for the original expression, as shown in the following diagram. Note that this procedure generally doesn’t give the NFA with the fewest states. In this example, the procedure gives an NFA with eight states, but the smallest equivalent NFA has only two states. Can you find it?
FIGURE 1.57
Building an NFA from the regular expression (ab ∪ a)∗
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1.3 REGULAR EXPRESSIONS 69
EXAMPLE 1.58
In Figure 1.59, we convert the regular expression (a ∪ b)∗aba to an NFA. A few
of the minor steps are not shown.
FIGURE 1.59
Building an NFA from the regular expression (a ∪ b)∗aba
Now let’s turn to the other direction of the proof of Theorem 1.54.
LEMMA 1.60
If a language is regular, then it is described by a regular expression.
PROOF IDEA We need to show that if a language A is regular, a regular expression describes it. Because A is regular, it is accepted by a DFA. We describe a procedure for converting DFAs into equivalent regular expressions.
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70 CHAPTER 1 / REGULAR LANGUAGES
We break this procedure into two parts, using a new type of finite automaton called a generalized nondeterministic finite automaton, GNFA. First we show how to convert DFAs into GNFAs, and then GNFAs into regular expressions.
Generalized nondeterministic finite automata are simply nondeterministic fi- nite automata wherein the transition arrows may have any regular expressions as labels, instead of only members of the alphabet or ε. The GNFA reads blocks of symbols from the input, not necessarily just one symbol at a time as in an ordi- nary NFA. The GNFA moves along a transition arrow connecting two states by reading a block of symbols from the input, which themselves constitute a string described by the regular expression on that arrow. A GNFA is nondeterministic and so may have several different ways to process the same input string. It ac- cepts its input if its processing can cause the GNFA to be in an accept state at the end of the input. The following figure presents an example of a GNFA.
FIGURE 1.61
A generalized nondeterministic finite automaton
For convenience, we require that GNFAs always have a special form that meets the following conditions.
• Thestartstatehastransitionarrowsgoingtoeveryotherstatebutnoarrows coming in from any other state.
• Thereisonlyasingleacceptstate,andithasarrowscominginfromevery other state but no arrows going to any other state. Furthermore, the accept state is not the same as the start state.
• Except for the start and accept states, one arrow goes from every state to every other state and also from each state to itself.
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1.3 REGULAR EXPRESSIONS 71
We can easily convert a DFA into a GNFA in the special form. We simply add a new start state with an ε arrow to the old start state and a new accept state with ε arrows from the old accept states. If any arrows have multiple labels (or if there are multiple arrows going between the same two states in the same direction), we replace each with a single arrow whose label is the union of the previous labels. Finally, we add arrows labeled ∅ between states that had no arrows. This last step won’t change the language recognized because a transition labeled with ∅ can never be used. From here on we assume that all GNFAs are in the special form.
Now we show how to convert a GNFA into a regular expression. Say that the GNFA has k states. Then, because a GNFA must have a start and an accept state and they must be different from each other, we know that k ≥ 2. If k > 2, we construct an equivalent GNFA with k − 1 states. This step can be repeated on the new GNFA until it is reduced to two states. If k = 2, the GNFA has a single arrow that goes from the start state to the accept state. The label of this arrow is the equivalent regular expression. For example, the stages in converting a DFA with three states to an equivalent regular expression are shown in the following figure.
FIGURE 1.62
Typical stages in converting a DFA to a regular expression
The crucial step is constructing an equivalent GNFA with one fewer state when k > 2. We do so by selecting a state, ripping it out of the machine, and repairing the remainder so that the same language is still recognized. Any state will do, provided that it is not the start or accept state. We are guaranteed that such a state will exist because k > 2. Let’s call the removed state qrip.
After removing qrip we repair the machine by altering the regular expressions that label each of the remaining arrows. The new labels compensate for the absence of qrip by adding back the lost computations. The new label going from a state qi to a state qj is a regular expression that describes all strings that would
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72 CHAPTER 1 / REGULAR LANGUAGES
take the machine from qi to qj either directly or via qrip. We illustrate this
approach in Figure 1.63.
FIGURE 1.63
Constructing an equivalent GNFA with one fewer state
In the old machine, if
1. qi goes to qrip with an arrow labeled R1,
2. qrip goes to itself with an arrow labeled R2, 3. qrip goes to qj with an arrow labeled R3, and 4. qi goes to qj with an arrow labeled R4,
then in the new machine, the arrow from qi to qj gets the label (R1)(R2)∗(R3) ∪ (R4).
We make this change for each arrow going from any state qi to any state qj , including the case where qi = qj. The new machine recognizes the original language.
PROOF Let’s now carry out this idea formally. First, to facilitate the proof, we formally define the new type of automaton introduced. A GNFA is similar to a nondeterministic finite automaton except for the transition function, which has the form
δ: Q − {qaccept} × Q − {qstart}−→R.
The symbol R is the collection of all regular expressions over the alphabet Σ, and qstart and qaccept are the start and accept states. If δ(qi,qj) = R, the arrow from state qi to state qj has the regular expression R as its label. The domain of the transition function is Q − {qaccept} × Q − {qstart} because an arrow connects every state to every other state, except that no arrows are coming from qaccept or going to qstart.
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1.3 REGULAR EXPRESSIONS 73
DEFINITION 1.64
A generalized nondeterministic finite automaton is a 5-tuple,
(Q, Σ, δ, qstart, qaccept), where
1. Q is the finite set of states,
3. δ: Q − {qaccept} × Q − {qstart}−→R is the transition function,
4. qstart is the start state, and
5. qaccept is the accept state.
2. Σ is the input alphabet,
A GNFA accepts a string w in Σ∗ if w = w1w2 ···wk, where each wi is in Σ∗ and a sequence of states q0, q1, . . . , qk exists such that
1. q0 = qstart is the start state,
2. qk = qaccept is the accept state, and
3. for each i, we have wi ∈ L(Ri), where Ri = δ(qi−1, qi); in other words, Ri is the expression on the arrow from qi−1 to qi.
Returning to the proof of Lemma 1.60, we let M be the DFA for language A. Then we convert M to a GNFA G by adding a new start state and a new accept state and additional transition arrows as necessary. We use the procedure CONVERT(G), which takes a GNFA and returns an equivalent regular expression. This procedure uses recursion, which means that it calls itself. An infinite loop is avoided because the procedure calls itself only to process a GNFA that has one fewer state. The case where the GNFA has two states is handled without recursion.
CONVERT(G):
1. Let k be the number of states of G.
2. If k = 2, then G must consist of a start state, an accept state, and a single arrow connecting them and labeled with a regular expression R.
Return the expression R.
3. If k > 2, we select any state qrip ∈ Q different from qstart and qaccept and let G′ be the GNFA (Q′, Σ, δ′, qstart, qaccept), where
Q′ =Q−{qrip},
and for any qi ∈ Q′ − {qaccept} and any qj ∈ Q′ − {qstart}, let
δ′(qi,qj) = (R1)(R2)∗(R3) ∪ (R4),
for R1 = δ(qi,qrip), R2 = δ(qrip,qrip), R3 = δ(qrip,qj), and R4 = δ(qi,qj).
4. Compute CONVERT(G′) and return this value.
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74 CHAPTER 1 / REGULAR LANGUAGES
Next we prove that CONVERT returns a correct value.
CLAIM 1.65
For any GNFA G, CONVERT(G) is equivalent to G.
We prove this claim by induction on k, the number of states of the GNFA.
Basis: Prove the claim true for k = 2 states. If G has only two states, it can have only a single arrow, which goes from the start state to the accept state. The regular expression label on this arrow describes all the strings that allow G to get to the accept state. Hence this expression is equivalent to G.
Induction step: Assume that the claim is true for k − 1 states and use this as- sumption to prove that the claim is true for k states. First we show that G and G′ recognize the same language. Suppose that G accepts an input w. Then in an accepting branch of the computation, G enters a sequence of states:
qstart, q1, q2, q3, . . . , qaccept.
If none of them is the removed state qrip, clearly G′ also accepts w. The reason is that each of the new regular expressions labeling the arrows of G′ contains the old regular expression as part of a union.
If qrip does appear, removing each run of consecutive qrip states forms an accepting computation for G′. The states qi and qj bracketing a run have a new regular expression on the arrow between them that describes all strings taking qi to qj via qrip on G. So G′ accepts w.
Conversely, suppose that G′ accepts an input w. As each arrow between any two states qi and qj in G′ describes the collection of strings taking qi to qj in G, either directly or via qrip, G must also accept w. Thus G and G′ are equivalent.
The induction hypothesis states that when the algorithm calls itself recur- sively on input G′, the result is a regular expression that is equivalent to G′ because G′ has k − 1 states. Hence this regular expression also is equivalent to G, and the algorithm is proved correct.
This concludes the proof of Claim 1.65, Lemma 1.60, and Theorem 1.54.
EXAMPLE 1.66
In this example, we use the preceding algorithm to convert a DFA into a regular expression. We begin with the two-state DFA in Figure 1.67(a).
In Figure 1.67(b), we make a four-state GNFA by adding a new start state and a new accept state, called s and a instead of qstart and qaccept so that we can draw them conveniently. To avoid cluttering up the figure, we do not draw the arrows
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1.3 REGULAR EXPRESSIONS 75
labeled ∅, even though they are present. Note that we replace the label a, b on the self-loop at state 2 on the DFA with the label a ∪ b at the corresponding point on the GNFA. We do so because the DFA’s label represents two transitions, one for a and the other for b, whereas the GNFA may have only a single transition going from 2 to itself.
In Figure 1.67(c), we remove state 2 and update the remaining arrow labels. In this case, the only label that changes is the one from 1 to a. In part (b) it was ∅, but in part (c) it is b(a ∪ b)∗. We obtain this result by following step 3 of the CONVERT procedure. State qi is state 1, state qj is a, and qrip is 2, so R1 = b, R2 =a∪b,R3 =ε,andR4 =∅.Therefore,thenewlabelonthearrowfrom1 to a is (b)(a ∪ b)∗(ε) ∪ ∅. We simplify this regular expression to b(a ∪ b)∗.
In Figure 1.67(d), we remove state 1 from part (c) and follow the same pro- cedure. Because only the start and accept states remain, the label on the arrow joining them is the regular expression that is equivalent to the original DFA.
FIGURE 1.67
Converting a two-state DFA to an equivalent regular expression
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76 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.68
In this example, we begin with a three-state DFA. The steps in the conversion are shown in the following figure.
FIGURE 1.69
Converting a three-state DFA to an equivalent regular expression
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1.4 NONREGULAR LANGUAGES 77
1.4
NONREGULAR LANGUAGES
To understand the power of finite automata, you must also understand their limitations. In this section, we show how to prove that certain languages cannot be recognized by any finite automaton.
Let’s take the language B = {0n1n| n ≥ 0}. If we attempt to find a DFA that recognizes B, we discover that the machine seems to need to remember how many 0s have been seen so far as it reads the input. Because the number of 0s isn’t limited, the machine will have to keep track of an unlimited number of possibilities. But it cannot do so with any finite number of states.
Next, we present a method for proving that languages such as B are not regu- lar. Doesn’t the argument already given prove nonregularity because the number of 0s is unlimited? It does not. Just because the language appears to require un- bounded memory doesn’t mean that it is necessarily so. It does happen to be true for the language B; but other languages seem to require an unlimited number of possibilities, yet actually they are regular. For example, consider two languages over the alphabet Σ = {0,1}:
C = {w| w has an equal number of 0s and 1s}, and
D = {w| w has an equal number of occurrences of 01 and 10 as substrings}.
At first glance, a recognizing machine appears to need to count in each case, and therefore neither language appears to be regular. As expected, C is not regular, but surprisingly D is regular!6 Thus our intuition can sometimes lead us astray, which is why we need mathematical proofs for certainty. In this section, we show how to prove that certain languages are not regular.
THE PUMPING LEMMA FOR REGULAR LANGUAGES
Our technique for proving nonregularity stems from a theorem about regular languages, traditionally called the pumping lemma. This theorem states that all regular languages have a special property. If we can show that a language does not have this property, we are guaranteed that it is not regular. The property states that all strings in the language can be “pumped” if they are at least as long as a certain special value, called the pumping length. That means each such string contains a section that can be repeated any number of times with the resulting string remaining in the language.
6See Problem 1.48.
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78 CHAPTER 1 / REGULAR LANGUAGES THEOREM 1.70
Pumping lemma If A is a regular language, then there is a number p (the pumping length) where if s is any string in A of length at least p, then s may be divided into three pieces, s = xyz, satisfying the following conditions:
1. for each i ≥ 0, xyiz ∈ A, 2. |y| > 0, and
3. |xy| ≤ p.
Recall the notation where |s| represents the length of string s, yi means that i copies of y are concatenated together, and y0 equals ε.
When s is divided into xyz, either x or z may be ε, but condition 2 says that y ̸= ε. Observe that without condition 2 the theorem would be trivially true. Condition 3 states that the pieces x and y together have length at most p. It is an extra technical condition that we occasionally find useful when proving certain languages to be nonregular. See Example 1.74 for an application of condition 3.
PROOF IDEA Let M = (Q, Σ, δ, q1, F ) be a DFA that recognizes A. We assign the pumping length p to be the number of states of M . We show that any string s in A of length at least p may be broken into the three pieces xyz, satisfying our three conditions. What if no strings in A are of length at least p? Then our task is even easier because the theorem becomes vacuously true: Obviously the three conditions hold for all strings of length at least p if there aren’t any such strings.
If s in A has length at least p, consider the sequence of states that M goes through when computing with input s. It starts with q1 the start state, then goes to, say, q3, then, say, q20, then q9, and so on, until it reaches the end of s in state q13. With s in A, we know that M accepts s, so q13 is an accept state.
If we let n be the length of s, the sequence of states q1,q3,q20,q9,…,q13 has lengthn+1. Becausenisatleastp,weknowthatn+1isgreaterthanp,the number of states of M. Therefore, the sequence must contain a repeated state. This result is an example of the pigeonhole principle, a fancy name for the rather obvious fact that if p pigeons are placed into fewer than p holes, some hole has to have more than one pigeon in it.
The following figure shows the string s and the sequence of states that M goes through when processing s. State q9 is the one that repeats.
FIGURE 1.71
Example showing state q9 repeating when M reads s
We now divide s into the three pieces x, y, and z. Piece x is the part of s appearing before q9, piece y is the part between the two appearances of q9, and
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1.4 NONREGULAR LANGUAGES 79
piece z is the remaining part of s, coming after the second occurrence of q9. So xtakesMfromthestateq1 toq9,ytakesMfromq9 backtoq9,andztakesM from q9 to the accept state q13, as shown in the following figure.
FIGURE 1.72
Example showing how the strings x, y, and z affect M
Let’s see why this division of s satisfies the three conditions. Suppose that we run M on input xyyz. We know that x takes M from q1 to q9, and then the first y takes it from q9 back to q9, as does the second y, and then z takes it to q13. With q13 being an accept state, M accepts input xyyz. Similarly, it will accept xyiz for any i > 0. For the case i = 0, xyiz = xz, which is accepted for similar reasons. That establishes condition 1.
Checking condition 2, we see that |y| > 0, as it was the part of s that occurred between two different occurrences of state q9.
In order to get condition 3, we make sure that q9 is the first repetition in the sequence. By the pigeonhole principle, the first p + 1 states in the sequence must contain a repetition. Therefore, |xy| ≤ p.
PROOF LetM = (Q,Σ,δ,q1,F)beaDFArecognizingAandpbethenumber of states of M.
Lets=s1s2···sn beastringinAoflengthn,wheren≥p.Letr1,…,rn+1 be the sequence of states that M enters while processing s, so ri+1 = δ(ri,si) for1≤i≤n. Thissequencehaslengthn+1,whichisatleastp+1. Among the first p + 1 elements in the sequence, two must be the same state, by the pigeonhole principle. We call the first of these rj and the second rl. Because rl occurs among the first p+1 places in a sequence starting at r1, we have l ≤ p+1. Now let x = s1 ···sj−1, y = sj ···sl−1, and z = sl ···sn.
AsxtakesMfromr1 torj,ytakesMfromrj torj,andztakesMfromrj to rn+1, which is an accept state, M must accept xyiz for i ≥ 0. We know that j ̸= l, so |y| > 0; and l ≤ p+1, so |xy| ≤ p. Thus we have satisfied all conditions of the pumping lemma.
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80 CHAPTER 1 / REGULAR LANGUAGES
To use the pumping lemma to prove that a language B is not regular, first as- sume that B is regular in order to obtain a contradiction. Then use the pumping lemma to guarantee the existence of a pumping length p such that all strings of length p or greater in B can be pumped. Next, find a string s in B that has length p or greater but that cannot be pumped. Finally, demonstrate that s cannot be pumped by considering all ways of dividing s into x, y, and z (taking condition 3 of the pumping lemma into account if convenient) and, for each such division, finding a value i where xyiz ̸∈ B. This final step often involves grouping the various ways of dividing s into several cases and analyzing them individually. The existence of s contradicts the pumping lemma if B were regular. Hence B cannot be regular.
Finding s sometimes takes a bit of creative thinking. You may need to hunt through several candidates for s before you discover one that works. Try mem- bers of B that seem to exhibit the “essence” of B’s nonregularity. We further discuss the task of finding s in some of the following examples.
EXAMPLE 1.73
Let B be the language {0n1n|n ≥ 0}. We use the pumping lemma to prove that B is not regular. The proof is by contradiction.
Assume to the contrary that B is regular. Let p be the pumping length given by the pumping lemma. Choose s to be the string 0p1p. Because s is a member of B and s has length more than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, where for any i ≥ 0 the string xyiz is in B. We consider three cases to show that this result is impossible.
1. The string y consists only of 0s. In this case, the string xyyz has more 0s than 1s and so is not a member of B, violating condition 1 of the pumping lemma. This case is a contradiction.
2. The string y consists only of 1s. This case also gives a contradiction.
3. The string y consists of both 0s and 1s. In this case, the string xyyz may have the same number of 0s and 1s, but they will be out of order with some 1s before 0s. Hence it is not a member of B, which is a contradiction.
Thus a contradiction is unavoidable if we make the assumption that B is reg- ular, so B is not regular. Note that we can simplify this argument by applying condition 3 of the pumping lemma to eliminate cases 2 and 3.
In this example, finding the string s was easy because any string in B of length p or more would work. In the next two examples, some choices for s do not work so additional care is required.
EXAMPLE 1.74
Let C = {w| w has an equal number of 0s and 1s}. We use the pumping lemma
to prove that C is not regular. The proof is by contradiction.
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1.4 NONREGULAR LANGUAGES 81
Assume to the contrary that C is regular. Let p be the pumping length given by the pumping lemma. As in Example 1.73, let s be the string 0p1p. With s being a member of C and having length more than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, where for any i ≥ 0 the string xyiz is in C. We would like to show that this outcome is impossible. But wait, it is possible! If we let x and z be the empty string and y be the string 0p1p, then xyiz always has an equal number of 0s and 1s and hence is in C. So it seems that s can be pumped.
Here condition 3 in the pumping lemma is useful. It stipulates that when pumping s, it must be divided so that |xy| ≤ p. That restriction on the way that s may be divided makes it easier to show that the string s = 0p1p we selected cannot be pumped. If |xy| ≤ p, then y must consist only of 0s, so xyyz ̸∈ C. Therefore, s cannot be pumped. That gives us the desired contradiction.
Selecting the string s in this example required more care than in Exam- ple 1.73. If we had chosen s = (01)p instead, we would have run into trouble because we need a string that cannot be pumped and that string can be pumped, even taking condition 3 into account. Can you see how to pump it? One way to do so sets x = ε, y = 01, and z = (01)p−1. Then xyiz ∈ C for every value of i. If you fail on your first attempt to find a string that cannot be pumped, don’t despair. Try another one!
An alternative method of proving that C is nonregular follows from our knowledge that B is nonregular. If C were regular, C ∩ 0∗1∗ also would be regular. The reasons are that the language 0∗1∗ is regular and that the class of regular languages is closed under intersection, which we proved in footnote 3 (page 46). But C ∩ 0∗1∗ equals B, and we know that B is nonregular from Example 1.73.
EXAMPLE 1.75
Let F = {ww| w ∈ {0,1}∗}. We show that F is nonregular, using the pumping lemma.
Assume to the contrary that F is regular. Let p be the pumping length given by the pumping lemma. Let s be the string 0p10p1. Because s is a member of F and s has length more than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, satisfying the three conditions of the lemma. We show that this outcome is impossible.
Condition 3 is once again crucial because without it we could pump s if we let x and z be the empty string. With condition 3 the proof follows because y must consist only of 0s, so xyyz ̸∈ F .
Observe that we chose s = 0p10p1 to be a string that exhibits the “essence” of the nonregularity of F, as opposed to, say, the string 0p0p. Even though 0p0p is a member of F , it fails to demonstrate a contradiction because it can be pumped.
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82 CHAPTER 1 / REGULAR LANGUAGES EXAMPLE 1.76
Here we demonstrate a nonregular unary language. Let D = {1n2 | n ≥ 0}. In other words, D contains all strings of 1s whose length is a perfect square. We use the pumping lemma to prove that D is not regular. The proof is by contradiction.
Assume to the contrary that D is regular. Let p be the pumping length given by the pumping lemma. Let s be the string 1p2 . Because s is a member of D and s has length at least p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, where for any i ≥ 0 the string xyiz is in D. As in the preceding examples, we show that this outcome is impossible. Doing so in this case requires a little thought about the sequence of perfect squares:
0,1,4,9,16,25,36,49,…
Note the growing gap between successive members of this sequence. Large members of this sequence cannot be near each other.
Now consider the two strings xyz and xy2z. These strings differ from each other by a single repetition of y, and consequently their lengths differ by the length of y. By condition 3 of the pumping lemma, |xy| ≤ p and thus |y| ≤ p. Wehave|xyz|=p2 andso|xy2z|≤p2+p.Butp2+p
EXAMPLE 1.77
Sometimes “pumping down” is useful when we apply the pumping lemma. We use the pumping lemma to show that E = {0i1j| i > j} is not regular. The proof is by contradiction.
Assume that E is regular. Let p be the pumping length for E given by the pumping lemma. Let s = 0p+11p. Then s can be split into xyz, satisfying the conditions of the pumping lemma. By condition 3, y consists only of 0s. Let’s examine the string xyyz to see whether it can be in E. Adding an extra copy of y increases the number of 0s. But, E contains all strings in 0∗1∗ that have more 0s than 1s, so increasing the number of 0s will still give a string in E. No contradiction occurs. We need to try something else.
The pumping lemma states that xyiz ∈ E even when i = 0, so let’s consider the string xy0z = xz. Removing string y decreases the number of 0s in s. Recall that s has just one more 0 than 1. Therefore, xz cannot have more 0s than 1s, so it cannot be a member of E. Thus we obtain a contradiction.
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EXERCISES
A 1.1
The following are the state diagrams of two DFAs, M1 and M2 . Answer the follow- ing questions about each of these machines.
EXERCISES 83
A 1.2 1.3
a. What is the start state?
b. What is the set of accept states?
c. What sequence of states does the machine go through on input aabb?
d. Does the machine accept the string aabb?
e. Does the machine accept the string ε?
Give the formal description of the machines M1 and M2 pictured in Exercise 1.1. The formal description of a DFA M is {q1, q2, q3, q4, q5}, {u, d}, δ, q3, {q3},
where δ is given by the following table. Give the state diagram of this machine.
ud
q1 q1 q2 q2 q1 q3 q3 q2 q4 q4 q3 q5 q5 q4 q5
Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts, Σ = {a, b}.
a. {w| w has at least three a’s and at least two b’s} Ab. {w| w has exactly two a’s and at least two b’s}
c. {w| w has an even number of a’s and one or two b’s}
Ad. {w| w has an even number of a’s and each a is followed by at least one b}
e. {w| w starts with an a and has at most one b}
f. {w| w has an odd number of a’s and ends with a b}
g. {w| w has even length and an odd number of a’s}
1.4
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84 CHAPTER 1 / REGULAR LANGUAGES
1.5 Each of the following languages is the complement of a simpler language. In each part, construct a DFA for the simpler language, then use it to give the state diagram of a DFA for the language given. In all parts, Σ = {a, b}.
Aa. {w| w does not contain the substring ab}
Ab. {w| w does not contain the substring baba}
c. {w| w contains neither the substrings ab nor ba}
d. {w| w is any string not in a∗b∗}
e. {w| w is any string not in (ab+)∗}
f. {w|wisanystringnotina∗ ∪b∗}
g. {w| w is any string that doesn’t contain exactly two a’s}
h. {w| w is any string except a and b}
1.6 Give state diagrams of DFAs recognizing the following languages. In all parts, the alphabet is {0,1}.
a. {w|wbeginswitha1andendswitha0}
b. {w| w contains at least three 1s}
c. {w| w contains the substring 0101 (i.e., w = x0101y for some x and y)}
d. {w| w has length at least 3 and its third symbol is a 0}
e. {w| w starts with 0 and has odd length, or starts with 1 and has even length}
f. {w| w doesn’t contain the substring 110}
g. {w| the length of w is at most 5}
h. {w| w is any string except 11 and 111}
i. {w| every odd position of w is a 1}
j. {w|w contains at least two 0s and at most one 1}
k. {ε, 0}
l. {w|w contains an even number of 0s, or contains exactly two 1s}
m. The empty set
n. All strings except the empty string
1.7 Give state diagrams of NFAs with the specified number of states recognizing each
of the following languages. In all parts, the alphabet is {0,1}.
Aa. The language {w| w ends with 00} with three states
b. The language of Exercise 1.6c with five states
c. The language of Exercise 1.6l with six states
d. The language {0} with two states
e. The language 0∗ 1∗ 0+ with three states
Af. The language 1∗ (001+ )∗ with three states
g. The language {ε} with one state
h. The language 0∗ with one state
1.8 Use the construction in the proof of Theorem 1.45 to give the state diagrams of NFAs recognizing the union of the languages described in
a. Exercises 1.6a and 1.6b.
b. Exercises 1.6c and 1.6f.
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1.9
1.10
A 1.11 1.12
1.13
EXERCISES 85 Use the construction in the proof of Theorem 1.47 to give the state diagrams of
NFAs recognizing the concatenation of the languages described in
a. Exercises 1.6g and 1.6i.
b. Exercises 1.6b and 1.6m.
Use the construction in the proof of Theorem 1.49 to give the state diagrams of
NFAs recognizing the star of the languages described in
a. Exercise 1.6b.
b. Exercise 1.6j.
c. Exercise 1.6m.
Prove that every NFA can be converted to an equivalent one that has a single accept state.
Let D = {w| w contains an even number of a’s and an odd number of b’s and does not contain the substring ab}. Give a DFA with five states that recognizes D and a regular expression that generates D. (Suggestion: Describe D more simply.)
Let F be the language of all strings over {0,1} that do not contain a pair of 1s that are separated by an odd number of symbols. Give the state diagram of a DFA with five states that recognizes F . (You may find it helpful first to find a 4-state NFA for the complement of F .)
1.14 a.
Show that if M is a DFA that recognizes language B, swapping the accept and nonaccept states in M yields a new DFA recognizing the complement of B. Conclude that the class of regular languages is closed under complement.
1.15
b. Show by giving an example that if M is an NFA that recognizes language C, swapping the accept and nonaccept states in M doesn’t necessarily yield a new NFA that recognizes the complement of C. Is the class of languages recognized by NFAs closed under complement? Explain your answer.
Give a counterexample to show that the following construction fails to prove The- orem 1.49, the closure of the class of regular languages under the star operation.7 Let N1 = (Q1,Σ,δ1,q1,F1) recognize A1. Construct N = (Q1,Σ,δ,q1,F) as follows. N is supposed to recognize A∗1.
a. The states of N are the states of N1.
b. The start state of N is the same as the start state of N1.
c. F={q1}∪F1.
The accept states F are the old accept states plus its start state.
d. Defineδsothatforanyq∈Q1 andanya∈Σε, δ(q,a)=δ1(q,a) q̸∈F1 ora̸=ε
δ1(q,a)∪{q1} q∈F1 anda=ε.
(Suggestion: Show this construction graphically, as in Figure 1.50.)
7In other words, you must present a finite automaton, N1, for which the constructed automaton N does not recognize the star of N1’s language.
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86
CHAPTER 1 / REGULAR LANGUAGES
1.16
Use the construction given in Theorem 1.39 to convert the following two nonde- terministic finite automata to equivalent deterministic finite automata.
1.17 a. Give an NFA recognizing the language (01 ∪ 001 ∪ 010)∗.
b. Convert this NFA to an equivalent DFA. Give only the portion of the DFA
1.18 1.19
1.20
1.21
that is reachable from the start state.
Give regular expressions generating the languages of Exercise 1.6.
Use the procedure described in Lemma 1.55 to convert the following regular ex- pressions to nondeterministic finite automata.
a. (0 ∪ 1)∗ 000(0 ∪ 1)∗
b. (((00)∗(11)) ∪ 01)∗
c. ∅∗
For each of the following languages, give two strings that are members and two strings that are not members—a total of four strings for each part. Assume the alphabet Σ = {a,b} in all parts.
a. a∗b∗
b. a(ba)∗ b
c. a∗∪b∗
d. (aaa)∗ h.
Use the procedure described in Lemma 1.60 to convert the following finite au- tomata to regular expressions.
e. Σ∗aΣ∗bΣ∗aΣ∗ f. aba ∪ bab
g. (ε∪a)b (a∪ba∪bb)Σ∗
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1.22
A 1.23 1.24
In certain programming languages, comments appear between delimiters such as /# and #/. Let C be the language of all valid delimited comment strings. A mem- ber of C must begin with /# and end with #/ but have no intervening #/. For simplicity, assume that the alphabet for C is Σ = {a, b, /, #}.
a. Give a DFA that recognizes C.
b. Give a regular expression that generates C.
Let B be any language over the alphabet Σ. Prove that B = B+ iff BB ⊆ B.
A finite state transducer (FST) is a type of deterministic finite automaton whose output is a string and not just accept or reject . The following are state diagrams of finite state transducers T1 and T2.
EXERCISES 87
1.25
1.26
Read the informal definition of the finite state transducer given in Exercise 1.24. Give a formal definition of this model, following the pattern in Definition 1.5 (page 35). Assume that an FST has an input alphabet Σ and an output alphabet Γ but not a set of accept states. Include a formal definition of the computation of an FST. (Hint: AnFSTisa5-tuple. Itstransitionfunctionisoftheformδ: Q×Σ−→Q×Γ.)
Using the solution you gave to Exercise 1.25, give a formal description of the ma- chines T1 and T2 depicted in Exercise 1.24.
Each transition of an FST is labeled with two symbols, one designating the input symbol for that transition and the other designating the output symbol. The two symbols are written with a slash, /, separating them. In T1, the transition from q1 to q2 has input symbol 2 and output symbol 1. Some transitions may have multiple input–output pairs, such as the transition in T1 from q1 to itself. When an FST computes on an input string w, it takes the input symbols w1 · · · wn one by one and, starting at the start state, follows the transitions by matching the input labels with the sequence of symbols w1 · · · wn = w. Every time it goes along a transition, it outputs the corresponding output symbol. For example, on input 2212011, machine T1 enters the sequence of states q1, q2, q2, q2, q2, q1, q1, q1 and produces output 1111000. On input abbb, T2 outputs 1011. Give the sequence of states entered and the output produced in each of the following parts.
a. T1 oninput011
b. T1 on input 211
c. T1 on input 121
d. T1 on input 0202
e. T2 oninputb
f. T2 on input bbab
g. T2 on input bbbbbb h. T2 on input ε
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88 CHAPTER 1 / REGULAR LANGUAGES
1.27 Read the informal definition of the finite state transducer given in Exercise 1.24. Give the state diagram of an FST with the following behavior. Its input and output alphabets are {0,1}. Its output string is identical to the input string on the even positions but inverted on the odd positions. For example, on input 0000111 it should output 1010010.
1.28 Convert the following regular expressions to NFAs using the procedure given in Theorem 1.54. In all parts, Σ = {a, b}.
a. a(abb)∗∪b
b. a+ ∪ (ab)+
c. (a ∪ b+)a+b+
1.29 Use the pumping lemma to show that the following languages are not regular.
Aa. A1={0n1n2n|n≥0}
b. A2 ={www|w∈{a,b}∗}
Ac. A3 ={a2n|n≥0} (Here,a2n meansastringof2n a’s.)
1.30 Describe the error in the following “proof” that 0∗1∗ is not a regular language. (An error must exist because 0∗1∗ is regular.) The proof is by contradiction. Assume that 0∗1∗ is regular. Let p be the pumping length for 0∗1∗ given by the pumping lemma. Choose s to be the string 0p1p. You know that s is a member of 0∗1∗, but Example 1.73 shows that s cannot be pumped. Thus you have a contradiction. So 0∗1∗ is not regular.
PROBLEMS
1.31 For any string w = w1w2 · · · wn, the reverse of w, written wR, is the string w in reverse order, wn · · · w2w1. For any language A, let AR = {wR| w ∈ A}.
Show that if A is regular, so is AR.
1.32 Let
Σ3 contains all size 3 columns of 0s and 1s. A string of symbols in Σ3 gives three
rows of 0s and 1s. Consider each row to be a binary number and let
B = { w ∈ Σ ∗3 | t h e b o t t o m r o w o f w i s t h e s u m o f t h e t o p t w o r o w s } .
For example,
Σ3 = 0 ,01 ,010 ,…,1 .
011 01
0 0 1 ∈B, but 0 0 ̸∈B.
100 11
Show that B is regular. (Hint: Working with BR is easier. You may assume the result claimed in Problem 1.31.)
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PROBLEMS 89 Σ2 = 0,01,10,1.
1.33 Let
Here, Σ2 contains all columns of 0s and 1s of height two. A string of symbols in
Σ2 gives two rows of 0s and 1s. Consider each row to be a binary number and let C = {w ∈ Σ∗2| the bottom row of w is three times the top row}.
For example, 0 01 1 0 ∈ C, but 01 01 10 ̸∈ C. Show that C is regular.
(You may assume the result claimed in Problem 1.31.)
1.34 Let Σ2 be the same as in Problem 1.33. Consider each row to be a binary number and let
D = {w ∈ Σ∗2| the top row of w is a larger number than is the bottom row}. For example, 0 10 1 0 ∈ D, but 0 01 1 0 ̸∈ D. Show that D is regular.
1.35 Let Σ2 be the same as in Problem 1.33. Consider the top and bottom rows to be strings of 0s and 1s, and let
E = { w ∈ Σ ∗2 | t h e b o t t o m r o w o f w i s t h e r e v e r s e o f t h e t o p r o w o f w } . Show that E is not regular.
1.36 Let Bn = {ak| k is a multiple of n}. Show that for each n ≥ 1, the language Bn is regular.
1.37 Let Cn = {x| x is a binary number that is a multiple of n}. Show that for each n ≥ 1, the language Cn is regular.
1.38 An all-NFA M is a 5-tuple (Q, Σ, δ, q0 , F ) that accepts x ∈ Σ∗ if every possible state that M could be in after reading input x is a state from F . Note, in contrast, that an ordinary NFA accepts a string if some state among these possible states is an accept state. Prove that all-NFAs recognize the class of regular languages.
1.39 The construction in Theorem 1.54 shows that every GNFA is equivalent to a GNFA with only two states. We can show that an opposite phenomenon occurs for DFAs. Prove that for every k > 1, a language Ak ⊆ {0,1}∗ exists that is recognized by a DFA with k states but not by one with only k − 1 states.
1.40 Recall that string x is a prefix of string y if a string z exists where xz = y, and that x is a proper prefix of y if in addition x ̸= y. In each of the following parts, we define an operation on a language A. Show that the class of regular languages is closed under that operation.
Aa. NOPREFIX(A) = {w ∈ A| no proper prefix of w is a member of A}.
b. NOEXTEND (A) = {w ∈ A| w is not the proper prefix of any string in A}.
1.41 For languages A and B, let the perfect shuffle of A and B be the language {w|w=a1b1···akbk, wherea1···ak ∈Aandb1···bk ∈B, eachai,bi ∈Σ}.
Show that the class of regular languages is closed under perfect shuffle.
1.42 For languages A and B, let the shuffle of A and B be the language
{w|w=a1b1···akbk, wherea1···ak ∈Aandb1···bk ∈B, eachai,bi ∈Σ∗}. Show that the class of regular languages is closed under shuffle.
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90
CHAPTER 1 / REGULAR LANGUAGES
1.43
A 1.44
⋆1.45 1.46
1.47
1.48
Let A be any language. Define DROP-OUT(A) to be the language containing all strings that can be obtained by removing one symbol from a string in A. Thus, DROP-OUT(A) = {xz| xyz ∈ A where x,z ∈ Σ∗,y ∈ Σ}. Show that the class of regular languages is closed under the DROP-OUT operation. Give both a proof by picture and a more formal proof by construction as in Theorem 1.47.
A 1.50 1.51
Show that B is a regular language.
b. LetC={1ky|y∈{0,1}∗ andycontainsatmostk1s,fork≥1}.
Show that C isn’t a regular language.
Read the informal definition of the finite state transducer given in Exercise 1.24. Prove that no FST can output wR for every input w if the input and output alpha- bets are {0,1}.
Let x and y be strings and let L be any language. We say that x and y are distin- guishable by L if some string z exists whereby exactly one of the strings xz and yz is a member of L; otherwise, for every string z, we have xz ∈ L whenever yz ∈ L and we say that x and y are indistinguishable by L. If x and y are indistinguishable by L, we write x ≡L y. Show that ≡L is an equivalence relation.
Let B and C be languages over Σ = {0, 1}. Define 1
B ← C = {w∈B| for some y∈C, strings w and y contain equal numbers of 1s}. Show that the class of regular languages is closed under the ←1 operation.
LetA/B={w|wx∈Aforsomex∈B}.ShowthatifAisregularandBisany language, then A/B is regular.
Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection, and complement.
a. {0n1m0n|m,n≥0} Ab. {0m1n| m ̸= n}
c. {w| w ∈ {0,1}∗ is not a palindrome}8 ⋆d. {wtw|w,t∈{0,1}+}
Let Σ = {1,#} and let
Y ={w|w=x1#x2#···#xk fork≥0, eachxi ∈1∗, andxi ̸=xj fori̸=j}.
Prove that Y is not regular. Let Σ = {0,1} and let
D = {w|w contains an equal number of occurrences of the substrings 01 and 10}. Thus 101 ∈ D because 101 contains a single 01 and a single 10, but 1010 ̸∈ D
because 1010 contains two 10s and one 01. Show that D is a regular language. 1.49 a. LetB={1ky|y∈{0,1}∗ andycontainsatleastk1s,fork≥1}.
8A palindrome is a string that reads the same forward and backward.
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A⋆ 1.52
Myhill–Nerode theorem. Refer to Problem 1.51. Let L be a language and let X be a set of strings. Say that X is pairwise distinguishable by L if every two distinct strings in X are distinguishable by L. Define the index of L to be the maximum number of elements in any set that is pairwise distinguishable by L. The index of L may be finite or infinite.
a. Show that if L is recognized by a DFA with k states, L has index at most k.
b. Show that if the index of L is a finite number k, it is recognized by a DFA
with k states.
c. Conclude that L is regular iff it has finite index. Moreover, its index is the
size of the smallest DFA recognizing it.
Let Σ = {0,1,+,=} and
ADD = {x=y+z| x, y, z are binary integers, and x is the sum of y and z}.
Show that ADD is not regular.
ConsiderthelanguageF ={aibjck|i,j,k≥0andifi=1thenj=k}.
a. Show that F is not regular.
b. Show that F acts like a regular language in the pumping lemma. In other words, give a pumping length p and demonstrate that F satisfies the three conditions of the pumping lemma for this value of p.
c. Explain why parts (a) and (b) do not contradict the pumping lemma.
The pumping lemma says that every regular language has a pumping length p, such that every string in the language can be pumped if it has length p or more. If p is a pumping length for language A, so is any length p′ ≥ p. The minimum pumping length for A is the smallest p that is a pumping length for A. For example, if A = 01∗, the minimum pumping length is 2. The reason is that the string s = 0 is in A and has length 1 yet s cannot be pumped; but any string in A of length 2 or more contains a 1 and hence can be pumped by dividing it so that x = 0, y = 1, and z is the rest. For each of the following languages, give the minimum pumping length and justify your answer.
1.53
1.54
1.55
⋆ 1.56
Aa. 0001∗
Ab. 0∗1∗
c. 001 ∪ 0∗1∗
Ad. 0∗1+0+1∗ ∪ 10∗1
e. (01)∗
If A is a set of natural numbers and k is a natural number greater than 1, let
Bk(A) = {w| w is the representation in base k of some number in A}.
Here, we do not allow leading 0s in the representation of a number. For example, B2 ({3, 5}) = {11, 101} and B3 ({3, 5}) = {10, 12}. Give an example of a set A for which B2(A) is regular but B3(A) is not regular. Prove that your example works.
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f. ε
g. 1∗01∗01∗ h. 10(11∗0)∗0
i. 1011 j. Σ∗
PROBLEMS 91
92
CHAPTER 1 / REGULAR LANGUAGES
⋆ 1.57
⋆ 1.58
⋆1.59
1.60
1.61 1.62
⋆ 1.63
1.64
If A is any language, let A1 − be the set of all first halves of strings in A so that 2
A1− ={x| forsomey, |x|=|y|andxy∈A}. 2
Show that if A is regular, then so is A1 −. 2
If A is any language, let A 1 − 1 be the set of all strings in A with their middle thirds
33
removed so that
A1−1 ={xz| forsomey, |x|=|y|=|z|andxyz∈A}.
33
Show that if A is regular, then A 1 − 1 is not necessarily regular. 33
Let M = (Q,Σ,δ,q0,F) be a DFA and let h be a state of M called its “home”. A synchronizing sequence for M and h is a string s ∈ Σ∗ where δ(q, s) = h for every q ∈ Q. (Here we have extended δ to strings, so that δ(q, s) equals the state where M ends up when M starts at state q and reads input s.) Say that M is synchronizable if it has a synchronizing sequence for some state h. Prove that if M is a k-state synchronizable DFA, then it has a synchronizing sequence of length at most k3. Can you improve upon this bound?
Let Σ = {a, b}. For each k ≥ 1, let Ck be the language consisting of all strings that contain an a exactly k places from the right-hand end. Thus Ck = Σ∗aΣk−1. Describe an NFA with k + 1 states that recognizes Ck in terms of both a state diagram and a formal description.
Consider the languages Ck defined in Problem 1.60. Prove that for each k, no DFA can recognize Ck with fewer than 2k states.
Let Σ = {a, b}. For each k ≥ 1, let Dk be the language consisting of all strings that have at least one a among the last k symbols. Thus Dk = Σ∗a(Σ ∪ ε)k−1. Describe a DFA with at most k + 1 states that recognizes Dk in terms of both a state diagram and a formal description.
a. Let A be an infinite regular language. Prove that A can be split into two infinite disjoint regular subsets.
b. LetBandDbetwolanguages.WriteBDifB⊆DandDcontains infinitely many strings that are not in B. Show that if B and D are two regular languages where B D, then we can find a regular language C where B C D.
Let N be an NFA with k states that recognizes some language A.
a. Show that if A is nonempty, A contains some string of length at most k.
b. Show, by giving an example, that part (a) is not necessarily true if you replace both A’s by A.
c. Show that if A is nonempty, A contains some string of length at most 2k .
d. Show that the bound given in part (c) is nearly tight; that is, for each k, demonstrate an NFA recognizing a language Ak where Ak is nonempty and where Ak’s shortest member strings are of length exponential in k. Come as close to the bound in (c) as you can.
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⋆ 1.65
1.66
Prove that for each n > 0, a language Bn exists where
a. Bn is recognizable by an NFA that has n states, and
b. ifBn =A1 ∪ ··· ∪Ak,forregularlanguagesAi,thenatleastoneoftheAi requires a DFA with exponentially many states.
A homomorphism is a function f: Σ−→Γ∗ from one alphabet to strings over another alphabet. We can extend f to operate on strings by defining f(w) = f(w1)f(w2)···f(wn), where w = w1w2 ···wn and each wi ∈ Σ. We further extend f to operate on languages by defining f(A) = {f(w)| w ∈ A}, for any language A.
a. Show, by giving a formal construction, that the class of regular languages is closed under homomorphism. In other words, given a DFA M that rec- ognizes B and a homomorphism f, construct a finite automaton M′ that recognizes f(B). Consider the machine M′ that you constructed. Is it a DFA in every case?
b. Show, by giving an example, that the class of non-regular languages is not closed under homomorphism.
Let the rotational closure of language A be RC(A) = {yx| xy ∈ A}.
a. Show that for any language A, we have RC(A) = RC(RC(A)).
b. Show that the class of regular languages is closed under rotational closure.
In the traditional method for cutting a deck of playing cards, the deck is arbitrarily split two parts, which are exchanged before reassembling the deck. In a more complex cut, called Scarne’s cut, the deck is broken into three parts and the middle part in placed first in the reassembly. We’ll take Scarne’s cut as the inspiration for an operation on languages. For a language A, let CUT(A) = {yxz| xyz ∈ A}.
a. Exhibit a language B for which CUT(B) ̸= CUT(CUT(B)).
b. Show that the class of regular languages is closed under CUT.
LetΣ={0,1}.LetWWk ={ww|w∈Σ∗ andwisoflengthk}.
a. Show that for each k, no DFA can recognize WWk with fewer than 2k states.
b. Describe a much smaller NFA for WWk, the complement of WWk.
We define the avoids operation for languages A and B to be
A avoids B = {w| w ∈ A and w doesn’t contain any string in B as a substring}.
Prove that the class of regular languages is closed under the avoids operation. Let Σ = {0,1}.
a. LetA={0ku0k|k≥1andu∈Σ∗}.ShowthatAisregular.
b. LetB={0k1u0k|k≥1andu∈Σ∗}.ShowthatBisnotregular.
Let M1 and M2 be DFAs that have k1 and k2 states, respectively, and then let U = L(M1) ∪ L(M2).
a. Show that if U ̸= ∅, then U contains some string s, where |s| < max(k1, k2).
b. Show that if U ̸= Σ∗, then U excludes some string s, where |s| < k1k2.
Let Σ = {0,1, #}. Let C = {x#xR#x| x ∈ {0,1}∗}. Show that C is a CFL.
⋆ 1.67
⋆ 1.68
1.69
1.70
1.71
1.72
1.73
PROBLEMS 93
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94 CHAPTER 1 / REGULAR LANGUAGES
SELECTED SOLUTIONS
1.1 For M1: (a) q1; (b) {q2}; (c) q1, q2, q3, q1, q1; (d) No; (e) No
For M2: (a) q1; (b) {q1, q4}; (c) q1, q1, q1, q2, q4;
(d) Yes;
ab
q1 q2 q3 q4
q2 q1 q3 q4.
(e) Yes
1.2 M1 =({q1,q2,q3},{a,b},δ1,q1,{q2}).
M2 =({q1,q2,q3,q4},{a,b},δ2,q1,{q1,q4}). The transition functions are
δ1 a b q1 q2 q1
q2 q3 q3 q3 q2 q1
δ2 q1 q2 q3 q4
1.4 (b) The following are DFAs for the two languages {w| w has exactly two a’s} and {w| w has at least two b’s}.
a a a,b bb
Combining them using the intersection construction gives the following DFA.
Though the problem doesn’t request you to simplify the DFA, certain states can be combined to give the following DFA.
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SELECTED SOLUTIONS 95
(d) These are DFAs for the two languages {w| w has an even number of a’s} and {w| each a in w is followed by at least one b}.
Combining them using the intersection construction gives the following DFA.
Though the problem doesn’t request you to simplify the DFA, certain states can be combined to give the following DFA.
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96 CHAPTER 1 / REGULAR LANGUAGES
1.5 (a) The left-hand DFA recognizes {w| w contains ab}. The right-hand DFA recog-
nizes its complement, {w| w doesn’t contain ab}.
(b) This DFA recognizes {w| w contains baba}.
This DFA recognizes {w| w does not contain baba}.
1.7 (a) (f)
1.11 Let N = (Q,Σ,δ,q0,F) be any NFA. Construct an NFA N′ with a single accept state that recognizes the same language as N. Informally, N′ is exactly like N except it has ε-transitions from the states corresponding to the accept states of N, to a new accept state, qaccept. State qaccept has no emerging transitions. More formally,N′ =(Q∪{qaccept},Σ,δ′,q0,{qaccept}),whereforeachq∈Qanda∈Σε
δ′(q, a) = δ(q, a) if a ̸= ε or q ̸∈ F δ(q,a)∪{qaccept} ifa=εandq∈F
and δ′(qaccept, a) = ∅ for each a ∈ Σε.
1.23 We prove both directions of the “iff.”
(→) Assume that B = B+ and show that BB ⊆ B.
ForeverylanguageBB⊆B+ holds,soifB=B+,thenBB⊆B.
(←) Assume that BB ⊆ B and show that B = B+.
ForeverylanguageB ⊆ B+,soweneedtoshowonlyB+ ⊆ B. Ifw ∈ B+, thenw=x1x2···xk whereeachxi ∈Bandk≥1. Becausex1,x2 ∈Band BB ⊆ B, we have x1x2 ∈ B. Similarly, because x1x2 is in B and x3 is in B, we have x1x2x3 ∈ B. Continuing in this way, x1 ···xk ∈ B. Hence w ∈ B, and so we may conclude that B+ ⊆ B.
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The latter argument may be written formally as the following proof by induction. Assume that BB ⊆ B.
Claim: For each k ≥ 1, if x1,...,xk ∈ B, then x1 ···xk ∈ B.
Basis: Prove for k = 1. This statement is obviously true.
Induction step: For each k ≥ 1, assume that the claim is true for k and prove it to be true for k + 1.
If x1,...,xk,xk+1 ∈ B, then by the induction assumption, x1 ···xk ∈ B. There- fore, x1 ···xkxk+1 ∈ BB, but BB ⊆ B, so x1 ···xk+1 ∈ B. That proves the induction step and the claim. The claim implies that if BB ⊆ B, then B+ ⊆ B.
1.29 (a) Assume that A1 = {0n1n2n| n ≥ 0} is regular. Let p be the pumping length given by the pumping lemma. Choose s to be the string 0p1p2p. Because s is a member of A1 and s is longer than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, where for any i ≥ 0 the string xyiz is in A1. Consider two possibilities:
1. The string y consists only of 0s, only of 1s, or only of 2s. In these cases, the string xyyz will not have equal numbers of 0s, 1s, and 2s. Hence xyyz is not a member of A1, a contradiction.
2. The string y consists of more than one kind of symbol. In this case, xyyz will have the 0s, 1s, or 2s out of order. Hence xyyz is not a member of A1, a contradiction.
Either way we arrive at a contradiction. Therefore, A1 is not regular.
(c) Assume that A3 = {a2n | n ≥ 0} is regular. Let p be the pumping length given by the pumping lemma. Choose s to be the string a2p . Because s is a member of A3 and s is longer than p, the pumping lemma guarantees that s can be split into three pieces, s = xyz, satisfying the three conditions of the pumping lemma.
The third condition tells us that |xy| ≤ p. Furthermore, p < 2p and so |y| < 2p. Therefore,|xyyz|=|xyz|+|y|<2p+2p =2p+1.Thesecondconditionrequires |y| > 0 so 2p < |xyyz| < 2p+1. The length of xyyz cannot be a power of 2. Hence xyyz is not a member of A3, a contradiction. Therefore, A3 is not regular.
1.40 (a) Let M = (Q,Σ,δ,q0,F) be a DFA recognizing A, where A is some regular language. Construct M′ = (Q′,Σ,δ′,q0′,F′) recognizing NOPREFIX(A) as follows:
1. Q′ =Q. {δ(r,a)} ifr∈/F 2. Forr∈Q′ anda∈Σ,define δ′(r,a)= ∅ ifr∈F.
3. q0′ =q0. 4. F′ =F.
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SELECTED SOLUTIONS 97
98
CHAPTER 1 / REGULAR LANGUAGES
1.44
Let MB = (QB,Σ,δB,qB,FB) and MC = (QC,Σ,δC,qC,FC) be DFAs recog-
nizingBandC,respectively.ConstructNFAM =(Q,Σ,δ,q0,F)thatrecognizes 11
B ← C as follows. To decide whether its input w is in B ← C, the machine M checks that w ∈ B, and in parallel nondeterministically guesses a string y that contains the same number of 1s as contained in w and checks that y ∈ C.
1. Q=QB ×QC.
2. For(q,r)∈Qanda∈Σε,define
⎧⎪⎨⎪⎩{(δB (q, 0), r)} δ((q,r),a) = {(δB(q,1),δC(r,1))}
{(q,δC(r,0))}
3. q0 = (qB,qC). 4. F = FB × FC .
1.46 (b)LetB={0m1n|m̸=n}.ObservethatB∩0∗1∗ ={0k1k|k≥0}.IfBwere regular, then B would be regular and so would B ∩ 0∗ 1∗ . But we already know that {0k1k| k ≥ 0} isn’t regular, so B cannot be regular.
Alternatively, we can prove B to be nonregular by using the pumping lemma di- rectly, though doing so is trickier. Assume that B = {0m1n| m ̸= n} is regular. Let p be the pumping length given by the pumping lemma. Observe that p! is di- visible by all integers from 1 to p, where p! = p(p − 1)(p − 2)···1. The string s = 0p1p+p! ∈ B, and |s| ≥ p. Thus the pumping lemma implies that s can be di- vided as xyz with x = 0a, y = 0b, and z = 0c1p+p!, where b ≥ 1 and a+b+c = p. Let s′ be the string xyi+1z, where i = p!/b. Then yi = 0p! so yi+1 = 0b+p!, and so s′ = 0a+b+c+p!1p+p!. That gives s′ = 0p+p!1p+p! ̸∈ B, a contradiction.
1.50 Assume to the contrary that some FST T outputs wR on input w. Consider the input strings 00 and 01. On input 00, T must output 00, and on input 01, T must output 10. In both cases, the first input bit is a 0 but the first output bits differ. Operating in this way is impossible for an FST because it produces its first output bit before it reads its second input. Hence no such FST can exist.
1.52 (a) We prove this assertion by contradiction. Let M be a k-state DFA that recog- nizes L. Suppose for a contradiction that L has index greater than k. That means some set X with more than k elements is pairwise distinguishable by L. Because M has k states, the pigeonhole principle implies that X contains two distinct strings x and y, where δ(q0, x) = δ(q0, y). Here δ(q0, x) is the state that M is in after start- ing in the start state q0 and reading input string x. Then, for any string z ∈ Σ∗, δ(q0,xz) = δ(q0,yz). Therefore, either both xz and yz are in L or neither are in L. But then x and y aren’t distinguishable by L, contradicting our assumption that X is pairwise distinguishable by L.
(b) Let X = {s1,...,sk} be pairwise distinguishable by L. We construct DFA M = (Q,Σ,δ,q0,F) with k states recognizing L. Let Q = {q1,...,qk}, and define δ(qi,a) to be qj, where sj ≡L sia (the relation ≡L is defined in Prob- lem 1.51). Note that sj ≡L sia for some sj ∈ X; otherwise, X ∪ sia would have k + 1 elements and would be pairwise distinguishable by L, which would contra- dict the assumption that L has index k. Let F = {qi| si ∈ L}. Let the start state q0 be the qi such that si ≡L ε. M is constructed so that for any state qi, {s| δ(q0, s) = qi} = {s| s ≡L si}. Hence M recognizes L.
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if a = 0 if a = 1 if a = ε.
(c) Suppose that L is regular and let k be the number of states in a DFA recognizing L. Then from part (a), L has index at most k. Conversely, if L has index k, then by part (b) it is recognized by a DFA with k states and thus is regular. To show that the index of L is the size of the smallest DFA accepting it, suppose that L’s index is exactly k. Then, by part (b), there is a k-state DFA accepting L. That is the smallest such DFA because if it were any smaller, then we could show by part (a) that the index of L is less than k.
1.55 (a) The minimum pumping length is 4. The string 000 is in the language but cannot be pumped, so 3 is not a pumping length for this language. If s has length 4 or more, it contains 1s. By dividing s into xyz, where x is 000 and y is the first 1 and z is everything afterward, we satisfy the pumping lemma’s three conditions. (b) The minimum pumping length is 1. The pumping length cannot be 0 because the string ε is in the language and it cannot be pumped. Every nonempty string in the language can be divided into xyz, where x, y, and z are ε, the first character, and the remainder, respectively. This division satisfies the three conditions.
(d) The minimum pumping length is 3. The pumping length cannot be 2 because the string 11 is in the language and it cannot be pumped. Let s be a string in the language of length at least 3. If s is generated by 0∗1+0+1∗ and s begins either 0 or 11, write s = xyz where x = ε, y is the first symbol, and z is the remainder of s. If s is generated by 0∗1+0+1∗ and s begins 10, write s = xyz where x = 10, y is the next symbol, and z is the remainder of s. Breaking s up in this way shows that it can be pumped. If s is generated by 10∗1, we can write it as xyz where x = 1, y = 0, and z is the remainder of s. This division gives a way to pump s.
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SELECTED SOLUTIONS 99
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2
CONTEXT-FREE LANGUAGES
In Chapter 1 we introduced two different, though equivalent, methods of de- scribing languages: finite automata and regular expressions. We showed that many languages can be described in this way but that some simple languages, such as {0n1n| n ≥ 0}, cannot.
In this chapter we present context-free grammars, a more powerful method of describing languages. Such grammars can describe certain features that have a recursive structure, which makes them useful in a variety of applications.
Context-free grammars were first used in the study of human languages. One way of understanding the relationship of terms such as noun, verb, and preposition and their respective phrases leads to a natural recursion because noun phrases may appear inside verb phrases and vice versa. Context-free grammars help us organize and understand these relationships.
An important application of context-free grammars occurs in the specification and compilation of programming languages. A grammar for a programming lan- guage often appears as a reference for people trying to learn the language syntax. Designers of compilers and interpreters for programming languages often start by obtaining a grammar for the language. Most compilers and interpreters con- tain a component called a parser that extracts the meaning of a program prior to generating the compiled code or performing the interpreted execution. A num- ber of methodologies facilitate the construction of a parser once a context-free grammar is available. Some tools even automatically generate the parser from the grammar.
101
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102 CHAPTER 2 / CONTEXT-FREE LANGUAGES
The collection of languages associated with context-free grammars are called the context-free languages. They include all the regular languages and many additional languages. In this chapter, we give a formal definition of context-free grammars and study the properties of context-free languages. We also introduce pushdown automata, a class of machines recognizing the context-free languages. Pushdown automata are useful because they allow us to gain additional insight into the power of context-free grammars.
2.1
CONTEXT-FREE GRAMMARS
The following is an example of a context-free grammar, which we call G1.
A → 0A1 A→B B→#
A grammar consists of a collection of substitution rules, also called produc- tions. Each rule appears as a line in the grammar, comprising a symbol and a string separated by an arrow. The symbol is called a variable. The string consists of variables and other symbols called terminals. The variable symbols often are represented by capital letters. The terminals are analogous to the in- put alphabet and often are represented by lowercase letters, numbers, or special symbols. One variable is designated as the start variable. It usually occurs on the left-hand side of the topmost rule. For example, grammar G1 contains three rules. G1’s variables are A and B, where A is the start variable. Its terminals are 0, 1, and #.
You use a grammar to describe a language by generating each string of that language in the following manner.
1. Write down the start variable. It is the variable on the left-hand side of the top rule, unless specified otherwise.
2. Find a variable that is written down and a rule that starts with that variable. Replace the written down variable with the right-hand side of that rule.
3. Repeat step 2 until no variables remain.
For example, grammar G1 generates the string 000#111. The sequence of substitutions to obtain a string is called a derivation. A derivation of string 000#111 in grammar G1 is
A ⇒ 0A1 ⇒ 00A11 ⇒ 000A111 ⇒ 000B111 ⇒ 000#111.
You may also represent the same information pictorially with a parse tree. An
example of a parse tree is shown in Figure 2.1.
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2.1 CONTEXT-FREE GRAMMARS 103
FIGURE 2.1
Parse tree for 000#111 in grammar G1
All strings generated in this way constitute the language of the grammar. We write L(G1) for the language of grammar G1. Some experimentation with the grammar G1 shows us that L(G1) is {0n#1n| n ≥ 0}. Any language that can be generated by some context-free grammar is called a context-free language (CFL). For convenience when presenting a context-free grammar, we abbreviate several rules with the same left-hand variable, such as A → 0A1 and A → B, into a single line A → 0A1 | B, using the symbol “ | ” as an “or”.
The following is a second example of a context-free grammar, called G2, which describes a fragment of the English language.
⟨SENTENCE⟩ → ⟨NOUN-PHRASE⟩⟨VERB-PHRASE⟩ ⟨NOUN-PHRASE⟩ → ⟨CMPLX-NOUN⟩ | ⟨CMPLX-NOUN⟩⟨PREP-PHRASE⟩
⟨VERB-PHRASE⟩ → ⟨CMPLX-VERB⟩ | ⟨CMPLX-VERB⟩⟨PREP-PHRASE⟩ ⟨PREP-PHRASE⟩ → ⟨PREP⟩⟨CMPLX-NOUN⟩
⟨CMPLX-NOUN⟩ → ⟨ARTICLE⟩⟨NOUN⟩ ⟨CMPLX-VERB⟩ → ⟨VERB⟩ | ⟨VERB⟩⟨NOUN-PHRASE⟩
⟨ARTICLE⟩ → a | the
⟨NOUN⟩ → boy | girl | flower
⟨VERB⟩ → touches | likes | sees ⟨PREP⟩ → with
Grammar G2 has 10 variables (the capitalized grammatical terms written in- side brackets); 27 terminals (the standard English alphabet plus a space charac- ter); and 18 rules. Strings in L(G2) include:
a boy sees
the boy sees a flower
a girl with a flower likes the boy
Each of these strings has a derivation in grammar G2. The following is a deriva- tion of the first string on this list.
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104 CHAPTER 2 / CONTEXT-FREE LANGUAGES
⟨SENTENCE⟩ ⇒ ⟨NOUN-PHRASE⟩⟨VERB-PHRASE⟩ ⇒ ⟨CMPLX-NOUN⟩⟨VERB-PHRASE⟩
⇒ ⟨ARTICLE⟩⟨NOUN⟩⟨VERB-PHRASE⟩ ⇒ a ⟨NOUN⟩⟨VERB-PHRASE⟩
⇒ a boy ⟨VERB-PHRASE⟩
⇒ a boy ⟨CMPLX-VERB⟩
⇒ a boy ⟨VERB⟩ ⇒ a boy sees
FORMAL DEFINITION OF A CONTEXT-FREE GRAMMAR
Let’s formalize our notion of a context-free grammar (CFG).
DEFINITION 2.2
A context-free grammar is a 4-tuple (V, Σ, R, S), where
1. V is a finite set called the variables,
2. Σ is a finite set, disjoint from V , called the terminals,
3. R is a finite set of rules, with each rule being a variable and a
string of variables and terminals, and 4. S ∈ V is the start variable.
If u, v, and w are strings of variables and terminals, and A → w is a rule of the grammar, we say that uAv yields uwv, written uAv ⇒ uwv. Say that u derives v, written u ⇒∗ v, if u = v or if a sequence u1,u2,...,uk exists for k ≥ 0 and
u⇒u1 ⇒u2 ⇒...⇒uk ⇒v.
The language of the grammar is {w ∈ Σ∗| S ⇒∗ w}.
In grammar G1, V = {A,B}, Σ = {0,1,#}, S = A, and R is the collection
of the three rules appearing on page 102. In grammar G2,
V =⟨SENTENCE⟩,⟨NOUN-PHRASE⟩,⟨VERB-PHRASE⟩, ⟨PREP-PHRASE⟩, ⟨CMPLX-NOUN⟩, ⟨CMPLX-VERB⟩, ⟨ARTICLE⟩,⟨NOUN⟩,⟨VERB⟩,⟨PREP⟩ ,
and Σ = {a, b, c, . . . , z, “ ”}. The symbol “ ” is the blank symbol, placed invisibly after each word (a, boy, etc.), so the words won’t run together.
Often we specify a grammar by writing down only its rules. We can identify the variables as the symbols that appear on the left-hand side of the rules and the terminals as the remaining symbols. By convention, the start variable is the variable on the left-hand side of the first rule.
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2.1 CONTEXT-FREE GRAMMARS 105 EXAMPLES OF CONTEXT-FREE GRAMMARS
EXAMPLE 2.3
Consider grammar G3 = ({S}, {a, b}, R, S). The set of rules, R, is
S → aSb | SS | ε.
This grammar generates strings such as abab, aaabbb, and aababb. You can see more easily what this language is if you think of a as a left parenthesis “(” and b as a right parenthesis “)”. Viewed in this way, L(G3) is the language of all strings of properly nested parentheses. Observe that the right-hand side of a rule may be the empty string ε.
EXAMPLE 2.4
ConsidergrammarG4 =(V,Σ,R,⟨EXPR⟩).
V is {⟨EXPR⟩, ⟨TERM⟩, ⟨FACTOR⟩} and Σ is {a, +, x, (, )}. The rules are
⟨EXPR⟩ → ⟨EXPR⟩+⟨TERM⟩ | ⟨TERM⟩ ⟨TERM⟩ → ⟨TERM⟩x⟨FACTOR⟩ | ⟨FACTOR⟩
⟨FACTOR⟩ → ( ⟨EXPR⟩ ) | a
The two strings a+axa and (a+a)xa can be generated with grammar G4.
The parse trees are shown in the following figure.
FIGURE 2.5
Parse trees for the strings a+axa and (a+a)xa
A compiler translates code written in a programming language into another form, usually one more suitable for execution. To do so, the compiler extracts
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106 CHAPTER 2 / CONTEXT-FREE LANGUAGES
the meaning of the code to be compiled in a process called parsing. One rep- resentation of this meaning is the parse tree for the code, in the context-free grammar for the programming language. We discuss an algorithm that parses context-free languages later in Theorem 7.16 and in Problem 7.45.
Grammar G4 describes a fragment of a programming language concerned with arithmetic expressions. Observe how the parse trees in Figure 2.5 “group” the operations. The tree for a+axa groups the x operator and its operands (the second two a’s) as one operand of the + operator. In the tree for (a+a)xa, the grouping is reversed. These groupings fit the standard precedence of mul- tiplication before addition and the use of parentheses to override the standard precedence. Grammar G4 is designed to capture these precedence relations.
DESIGNING CONTEXT-FREE GRAMMARS
As with the design of finite automata, discussed in Section 1.1 (page 41), the design of context-free grammars requires creativity. Indeed, context-free gram- mars are even trickier to construct than finite automata because we are more accustomed to programming a machine for specific tasks than we are to describ- ing languages with grammars. The following techniques are helpful, singly or in combination, when you’re faced with the problem of constructing a CFG.
First, many CFLs are the union of simpler CFLs. If you must construct a CFG for a CFL that you can break into simpler pieces, do so and then construct individual grammars for each piece. These individual grammars can be easily merged into a grammar for the original language by combining their rules and then adding the new rule S → S1 | S2 | ··· | Sk, where the variables Si are the start variables for the individual grammars. Solving several simpler problems is often easier than solving one complicated problem.
For example, to get a grammar for the language {0n1n|n ≥ 0}∪{1n0n|n ≥ 0}, first construct the grammar
S1 →0S11|ε for the language {0n1n| n ≥ 0} and the grammar
S2 →1S20|ε forthelanguage{1n0n|n≥0}andthenaddtheruleS→S1 |S2 togivethe
grammar
S → S1 | S2 S1 →0S11|ε S2 →1S20|ε.
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2.1 CONTEXT-FREE GRAMMARS 107
Second, constructing a CFG for a language that happens to be regular is easy if you can first construct a DFA for that language. You can convert any DFA into an equivalent CFG as follows. Make a variable Ri for each state qi of the DFA. Add the rule Ri → aRj to the CFG if δ(qi,a) = qj is a transition in the DFA. Add the rule Ri → ε if qi is an accept state of the DFA. Make R0 the start variable of the grammar, where q0 is the start state of the machine. Verify on your own that the resulting CFG generates the same language that the DFA recognizes.
Third, certain context-free languages contain strings with two substrings that are “linked” in the sense that a machine for such a language would need to re- member an unbounded amount of information about one of the substrings to verify that it corresponds properly to the other substring. This situation occurs in the language {0n1n| n ≥ 0} because a machine would need to remember the number of 0s in order to verify that it equals the number of 1s. You can construct a CFG to handle this situation by using a rule of the form R → uRv, which gen- erates strings wherein the portion containing the u’s corresponds to the portion containing the v’s.
Finally, in more complex languages, the strings may contain certain structures that appear recursively as part of other (or the same) structures. That situation occurs in the grammar that generates arithmetic expressions in Example 2.4. Any time the symbol a appears, an entire parenthesized expression might appear recursively instead. To achieve this effect, place the variable symbol generating the structure in the location of the rules corresponding to where that structure may recursively appear.
AMBIGUITY
Sometimes a grammar can generate the same string in several different ways. Such a string will have several different parse trees and thus several different meanings. This result may be undesirable for certain applications, such as pro- gramming languages, where a program should have a unique interpretation.
If a grammar generates the same string in several different ways, we say that the string is derived ambiguously in that grammar. If a grammar generates some string ambiguously, we say that the grammar is ambiguous.
For example, consider grammar G5:
⟨EXPR⟩ → ⟨EXPR⟩+⟨EXPR⟩ | ⟨EXPR⟩x⟨EXPR⟩ | ( ⟨EXPR⟩ ) | a
This grammar generates the string a+axa ambiguously. The following figure shows the two different parse trees.
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108 CHAPTER 2 / CONTEXT-FREE LANGUAGES
FIGURE 2.6
The two parse trees for the string a+axa in grammar G5
This grammar doesn’t capture the usual precedence relations and so may group the + before the × or vice versa. In contrast, grammar G4 generates exactly the same language, but every generated string has a unique parse tree. Hence G4 is unambiguous, whereas G5 is ambiguous.
Grammar G2 (page 103) is another example of an ambiguous grammar. The sentence the girl touches the boy with the flower has two different derivations. In Exercise 2.8 you are asked to give the two parse trees and observe their correspondence with the two different ways to read that sentence.
Now we formalize the notion of ambiguity. When we say that a grammar generates a string ambiguously, we mean that the string has two different parse trees, not two different derivations. Two derivations may differ merely in the order in which they replace variables yet not in their overall structure. To con- centrate on structure, we define a type of derivation that replaces variables in a fixed order. A derivation of a string w in a grammar G is a leftmost derivation if at every step the leftmost remaining variable is the one replaced. The derivation preceding Definition 2.2 (page 104) is a leftmost derivation.
Sometimes when we have an ambiguous grammar we can find an unambigu- ous grammar that generates the same language. Some context-free languages, however, can be generated only by ambiguous grammars. Such languages are called inherently ambiguous. Problem 2.29 asks you to prove that the language {aibjck| i = j or j = k} is inherently ambiguous.
CHOMSKY NORMAL FORM
When working with context-free grammars, it is often convenient to have them in simplified form. One of the simplest and most useful forms is called the
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DEFINITION 2.7
A string w is derived ambiguously in context-free grammar G if it has two or more different leftmost derivations. Grammar G is ambiguous if it generates some string ambiguously.
2.1 CONTEXT-FREE GRAMMARS 109 Chomsky normal form. Chomsky normal form is useful in giving algorithms
for working with context-free grammars, as we do in Chapters 4 and 7.
DEFINITION 2.8
A context-free grammar is in Chomsky normal form if every rule is
of the form
A → BC A→a
where a is any terminal and A, B, and C are any variables—except that B and C may not be the start variable. In addition, we permit the rule S → ε, where S is the start variable.
THEOREM 2.9
Any context-free language is generated by a context-free grammar in Chomsky
normal form.
PROOF IDEA We can convert any grammar G into Chomsky normal form. The conversion has several stages wherein rules that violate the conditions are replaced with equivalent ones that are satisfactory. First, we add a new start variable. Then, we eliminate all ε-rules of the form A → ε. We also eliminate all unit rules of the form A → B. In both cases we patch up the grammar to be sure that it still generates the same language. Finally, we convert the remaining rules into the proper form.
PROOF First, we add a new start variable S0 and the rule S0 → S, where S was the original start variable. This change guarantees that the start variable doesn’t occur on the right-hand side of a rule.
Second, we take care of all ε-rules. We remove an ε-rule A → ε, where A is not the start variable. Then for each occurrence of an A on the right-hand side of a rule, we add a new rule with that occurrence deleted. In other words, if R → uAv is a rule in which u and v are strings of variables and terminals, we add rule R → uv. We do so for each occurrence of an A, so the rule R → uAvAw causesustoaddR→uvAw,R→uAvw,andR→uvw. Ifwehavetherule R → A, we add R → ε unless we had previously removed the rule R → ε. We repeat these steps until we eliminate all ε-rules not involving the start variable.
Third, we handle all unit rules. We remove a unit rule A → B. Then, whenever a rule B → u appears, we add the rule A → u unless this was a unit rule previously removed. As before, u is a string of variables and terminals. We repeat these steps until we eliminate all unit rules.
Finally, we convert all remaining rules into the proper form. We replace each rule A → u1u2 ···uk, where k ≥ 3 and each ui is a variable or terminal symbol,
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110 CHAPTER 2 / CONTEXT-FREE LANGUAGES
with the rules A → u1A1, A1 → u2A2, A2 → u3A3, ... , and Ak−2 → uk−1uk. The Ai’s are new variables. We replace any terminal ui in the preceding rule(s) with the new variable Ui and add the rule Ui → ui.
EXAMPLE 2.10
Let G6 be the following CFG and convert it to Chomsky normal form by using the conversion procedure just given. The series of grammars presented illus- trates the steps in the conversion. Rules shown in bold have just been added. Rules shown in gray have just been removed.
1. The original CFG G6 is shown on the left. The result of applying the first step to make a new start variable appears on the right.
2.
3a.
S0 → S
S → ASA | aB | a A → B | S | ε
B → b | ε
Remove unit rules S → S, shown on the left, and S0 → S, shown on the
S0 → S
S → ASA | aB A→B|S
B→b|ε
Remove ε-rules B → ε, shown on the left, and A → ε, shown on the right.
S → ASA | aB A→B|S B→b|ε
right.
S0 →S S0 →S|ASA|aB|a|SA|AS S → ASA | aB | a | SA | AS | S S → ASA | aB | a | SA | AS A→B|S A→B|S
B→b B→b 3b. RemoveunitrulesA→BandA→S.
S0 →ASA|aB|a|SA|AS S → ASA | aB | a | SA | AS A → B | S | b
S0 →ASA|aB|a|SA|AS
S → ASA | aB | a | SA | AS
A → S | b | ASA | aB | a | SA | AS
B→b B→b
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S0 → S
S → ASA | aB | a | SA | AS | S A → B | S | ε
B → b
2.2 PUSHDOWN AUTOMATA 111
4. Convert the remaining rules into the proper form by adding additional vari- ables and rules. The final grammar in Chomsky normal form is equivalent to G6. (Actually the procedure given in Theorem 2.9 produces several variables Ui and several rules Ui → a. We simplified the resulting grammar by using a single variable U and rule U → a.)
S0 → AA1 | UB | a | SA | AS
S → AA1 | UB | a | SA | AS
A → b | AA1 | UB | a | SA | AS
A1 → SA U→a B→b
2.2
PUSHDOWN AUTOMATA
In this section we introduce a new type of computational model called pushdown automata. These automata are like nondeterministic finite automata but have an extra component called a stack. The stack provides additional memory beyond the finite amount available in the control. The stack allows pushdown automata to recognize some nonregular languages.
Pushdown automata are equivalent in power to context-free grammars. This equivalence is useful because it gives us two options for proving that a language is context free. We can give either a context-free grammar generating it or a push- down automaton recognizing it. Certain languages are more easily described in terms of generators, whereas others are more easily described by recognizers.
The following figure is a schematic representation of a finite automaton. The control represents the states and transition function, the tape contains the in- put string, and the arrow represents the input head, pointing at the next input symbol to be read.
FIGURE 2.11
Schematic of a finite automaton
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112 CHAPTER 2 / CONTEXT-FREE LANGUAGES
With the addition of a stack component we obtain a schematic representation
of a pushdown automaton, as shown in the following figure.
FIGURE 2.12
Schematic of a pushdown automaton
A pushdown automaton (PDA) can write symbols on the stack and read them back later. Writing a symbol “pushes down” all the other symbols on the stack. At any time the symbol on the top of the stack can be read and removed. The remaining symbols then move back up. Writing a symbol on the stack is of- ten referred to as pushing the symbol, and removing a symbol is referred to as popping it. Note that all access to the stack, for both reading and writing, may be done only at the top. In other words a stack is a “last in, first out” storage device. If certain information is written on the stack and additional information is written afterward, the earlier information becomes inaccessible until the later information is removed.
Plates on a cafeteria serving counter illustrate a stack. The stack of plates rests on a spring so that when a new plate is placed on top of the stack, the plates below it move down. The stack on a pushdown automaton is like a stack of plates, with each plate having a symbol written on it.
A stack is valuable because it can hold an unlimited amount of information. Recall that a finite automaton is unable to recognize the language {0n 1n | n ≥ 0} because it cannot store very large numbers in its finite memory. A PDA is able to recognize this language because it can use its stack to store the number of 0s it has seen. Thus the unlimited nature of a stack allows the PDA to store numbers of unbounded size. The following informal description shows how the automaton for this language works.
Read symbols from the input. As each 0 is read, push it onto the stack. As soon as 1s are seen, pop a 0 off the stack for each 1 read. If reading the input is finished exactly when the stack becomes empty of 0s, accept the input. If the stack becomes empty while 1s remain or if the 1s are finished while the stack still contains 0s or if any 0s appear in the input following 1s, reject the input.
As mentioned earlier, pushdown automata may be nondeterministic. Deter- ministic and nondeterministic pushdown automata are not equivalent in power.
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2.2 PUSHDOWN AUTOMATA 113
Nondeterministic pushdown automata recognize certain languages that no de- terministic pushdown automata can recognize, as we will see in Section 2.4. We give languages requiring nondeterminism in Examples 2.16 and 2.18. Recall that deterministic and nondeterministic finite automata do recognize the same class of languages, so the pushdown automata situation is different. We focus on nondeterministic pushdown automata because these automata are equivalent in power to context-free grammars.
FORMAL DEFINITION OF A PUSHDOWN AUTOMATON
The formal definition of a pushdown automaton is similar to that of a finite automaton, except for the stack. The stack is a device containing symbols drawn from some alphabet. The machine may use different alphabets for its input and its stack, so now we specify both an input alphabet Σ and a stack alphabet Γ.
At the heart of any formal definition of an automaton is the transition func- tion, which describes its behavior. Recall that Σε = Σ ∪ {ε} and Γε = Γ ∪ {ε}. The domain of the transition function is Q × Σε × Γε. Thus the current state, next input symbol read, and top symbol of the stack determine the next move of a pushdown automaton. Either symbol may be ε, causing the machine to move without reading a symbol from the input or without reading a symbol from the stack.
For the range of the transition function we need to consider what to allow the automaton to do when it is in a particular situation. It may enter some new state and possibly write a symbol on the top of the stack. The function δ can indicate this action by returning a member of Q together with a member of Γε, that is, a member of Q × Γε. Because we allow nondeterminism in this model, a situation may have several legal next moves. The transition function incorporates nondeterminism in the usual way, by returning a set of members of Q × Γε, that is, a member of P(Q × Γε). Putting it all together, our transition f u n c t i o n δ t a k e s t h e f o r m δ : Q × Σ ε × Γε −→ P ( Q × Γε ) .
DEFINITION 2.13
A pushdown automaton is a 6-tuple (Q, Σ, Γ, δ, q0, F ), where Q, Σ,
Γ, and F are all finite sets, and
1. Q is the set of states,
2. Σ is the input alphabet,
3. Γ is the stack alphabet,
4. δ: Q × Σε × Γε−→P(Q × Γε) is the transition function, 5. q0 ∈ Q is the start state, and
6. F ⊆ Q is the set of accept states.
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114 CHAPTER 2 / CONTEXT-FREE LANGUAGES
A pushdown automaton M = (Q, Σ, Γ, δ, q0, F ) computes as follows. It ac- cepts input w if w can be written as w = w1w2 ···wm, where each wi ∈ Σε and sequences of states r0,r1,...,rm ∈ Q and strings s0,s1,...,sm ∈ Γ∗ exist that satisfy the following three conditions. The strings si represent the sequence of stack contents that M has on the accepting branch of the computation.
1. r0 = q0 and s0 = ε. This condition signifies that M starts out properly, in the start state and with an empty stack.
2. For i = 0,...,m − 1, we have (ri+1,b) ∈ δ(ri,wi+1,a), where si = at and si+1 = bt for some a, b ∈ Γε and t ∈ Γ∗. This condition states that M moves properly according to the state, stack, and next input symbol.
3. rm ∈ F . This condition states that an accept state occurs at the input end. EXAMPLES OF PUSHDOWN AUTOMATA
EXAMPLE 2.14
The following is the formal description of the PDA (page 112) that recognizes
thelanguage{0n1n|n≥0}.LetM1 be(Q,Σ,Γ,δ,q1,F),where Q = {q1, q2, q3, q4},
Σ = {0,1},
Γ = {0, $},
F = {q1, q4}, and
δ is given by the following table, wherein blank entries signify ∅.
Input: 0 1 ε Stack:
0
$
ε
0
$
ε
0
$
ε
q1 q2 q3 q4
{(q2, 0)}
{(q2, $)} {(q3, ε)} {(q4, ε)}
{(q3, ε)}
We can also use a state diagram to describe a PDA, as in Figures 2.15, 2.17, and 2.19. Such diagrams are similar to the state diagrams used to describe finite automata, modified to show how the PDA uses its stack when going from state to state. We write “a,b → c” to signify that when the machine is reading an a from the input, it may replace the symbol b on the top of the stack with a c. Any of a, b, and c may be ε. If a is ε, the machine may make this transition without reading any symbol from the input. If b is ε, the machine may make this transition without reading and popping any symbol from the stack. If c is ε, the machine does not write any symbol on the stack when going along this transition.
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2.2 PUSHDOWN AUTOMATA 115
FIGURE 2.15
State diagram for the PDA M1 that recognizes {0n1n| n ≥ 0}
The formal definition of a PDA contains no explicit mechanism to allow the PDA to test for an empty stack. This PDA is able to get the same effect by initially placing a special symbol $ on the stack. Then if it ever sees the $ again, it knows that the stack effectively is empty. Subsequently, when we refer to testing for an empty stack in an informal description of a PDA, we implement the procedure in the same way.
Similarly, PDAs cannot test explicitly for having reached the end of the input string. This PDA is able to achieve that effect because the accept state takes effect only when the machine is at the end of the input. Thus from now on, we assume that PDAs can test for the end of the input, and we know that we can implement it in the same manner.
EXAMPLE 2.16
This example illustrates a pushdown automaton that recognizes the language
{aibjck| i,j,k ≥ 0 and i = j or i = k}.
Informally, the PDA for this language works by first reading and pushing the a’s. When the a’s are done, the machine has all of them on the stack so that it can match, them with either the b’s or the c’s. This maneuver is a bit tricky because the machine doesn’t know in advance whether to match the a’s with the b’s or the c’s. Nondeterminism comes in handy here.
Using its nondeterminism, the PDA can guess whether to match the a’s with the b’s or with the c’s, as shown in Figure 2.17. Think of the machine as having two branches of its nondeterminism, one for each possible guess. If either of them matches, that branch accepts and the entire machine accepts. Problem 2.57 asks you to show that nondeterminism is essential for recognizing this language with a PDA.
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116 CHAPTER 2 / CONTEXT-FREE LANGUAGES
FIGURE 2.17
State diagram for PDA M2 that recognizes {aibjck| i,j,k ≥ 0 and i = j or i = k}
EXAMPLE 2.18
In this example we give a PDA M3 recognizing the language {wwR|w ∈ {0,1}∗}. Recall that wR means w written backwards. The informal description and state diagram of the PDA follow.
Begin by pushing the symbols that are read onto the stack. At each point, nondeterministically guess that the middle of the string has been reached and then change into popping off the stack for each symbol read, checking to see that they are the same. If they were always the same symbol and the stack empties at the same time as the input is finished, accept; otherwise reject.
FIGURE 2.19
State diagram for the PDA M3 that recognizes {wwR| w ∈ {0, 1}∗}
Problem 2.58 shows that this language requires a nondeterministic PDA.
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2.2 PUSHDOWN AUTOMATA 117 EQUIVALENCE WITH CONTEXT-FREE GRAMMARS
In this section we show that context-free grammars and pushdown automata are equivalent in power. Both are capable of describing the class of context-free languages. We show how to convert any context-free grammar into a pushdown automaton that recognizes the same language and vice versa. Recalling that we defined a context-free language to be any language that can be described with a context-free grammar, our objective is the following theorem.
THEOREM 2.20
A language is context free if and only if some pushdown automaton recognizes it.
As usual for “if and only if” theorems, we have two directions to prove. In this theorem, both directions are interesting. First, we do the easier forward direction.
LEMMA 2.21
If a language is context free, then some pushdown automaton recognizes it.
PROOF IDEA Let A be a CFL. From the definition we know that A has a CFG, G, generating it. We show how to convert G into an equivalent PDA, which we call P .
The PDA P that we now describe will work by accepting its input w, if G gen- erates that input, by determining whether there is a derivation for w. Recall that a derivation is simply the sequence of substitutions made as a grammar generates a string. Each step of the derivation yields an intermediate string of variables and terminals. We design P to determine whether some series of substitutions using the rules of G can lead from the start variable to w.
One of the difficulties in testing whether there is a derivation for w is in figuring out which substitutions to make. The PDA’s nondeterminism allows it to guess the sequence of correct substitutions. At each step of the derivation, one of the rules for a particular variable is selected nondeterministically and used to substitute for that variable.
The PDA P begins by writing the start variable on its stack. It goes through a series of intermediate strings, making one substitution after another. Eventually it may arrive at a string that contains only terminal symbols, meaning that it has used the grammar to derive a string. Then P accepts if this string is identical to the string it has received as input.
Implementing this strategy on a PDA requires one additional idea. We need to see how the PDA stores the intermediate strings as it goes from one to an- other. Simply using the stack for storing each intermediate string is tempting. However, that doesn’t quite work because the PDA needs to find the variables in the intermediate string and make substitutions. The PDA can access only the top
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118 CHAPTER 2 / CONTEXT-FREE LANGUAGES
symbol on the stack and that may be a terminal symbol instead of a variable. The way around this problem is to keep only part of the intermediate string on the stack: the symbols starting with the first variable in the intermediate string. Any terminal symbols appearing before the first variable are matched immediately with symbols in the input string. The following figure shows the PDA P .
FIGURE 2.22
P representing the intermediate string 01A1A0
The following is an informal description of P .
1. Place the marker symbol $ and the start variable on the stack. 2. Repeat the following steps forever.
a. If the top of stack is a variable symbol A, nondeterministically select one of the rules for A and substitute A by the string on the right-hand side of the rule.
b. If the top of stack is a terminal symbol a, read the next symbol from the input and compare it to a. If they match, repeat. If they do not match, reject on this branch of the nondeterminism.
c. If the top of stack is the symbol $, enter the accept state. Doing so accepts the input if it has all been read.
We now give the formal details of the construction of the pushdown automaton P = (Q, Σ, Γ, δ, qstart, F ). To make the construction clearer, we use shorthand notation for the transition function. This notation provides a way to write an entire string on the stack in one step of the machine. We can simulate this action by introducing additional states to write the string one symbol at a time, as implemented in the following formal construction.
LetqandrbestatesofthePDAandletabeinΣε andsbeinΓε. Saythat we want the PDA to go from q to r when it reads a and pops s. Furthermore, we wantittopushtheentirestringu=u1···ul onthestackatthesametime.We can implement this action by introducing new states q1, . . . , ql−1 and setting the
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PROOF
transition function as follows:
2.2 PUSHDOWN AUTOMATA 119
δ(q, a, s) to contain (q1, ul), δ(q1, ε, ε) = {(q2, ul−1)}, δ(q2, ε, ε) = {(q3, ul−2)},
.
δ(ql−1, ε, ε) = {(r, u1)}.
We use the notation (r, u) ∈ δ(q, a, s) to mean that when q is the state of the automaton, a is the next input symbol, and s is the symbol on the top of the stack, the PDA may read the a and pop the s, then push the string u onto the stack and go on to the state r. The following figure shows this implementation.
as xyz
asz
x
y
FIGURE 2.23
Implementing the shorthand (r, xyz) ∈ δ(q, a, s)
The states of P are Q = {qstart, qloop, qaccept} ∪ E, where E is the set of states we need for implementing the shorthand just described. The start state is qstart. The only accept state is qaccept.
The transition function is defined as follows. We begin by initializing the stack to contain the symbols $ and S, implementing step 1 in the informal de- scription: δ(qstart, ε, ε) = {(qloop, S$)}. Then we put in transitions for the main loop of step 2.
First, we handle case (a) wherein the top of the stack contains a variable. Let δ(qloop, ε, A) = {(qloop, w)| where A → w is a rule in R}.
Second, we handle case (b) wherein the top of the stack contains a terminal. Let δ(qloop, a, a) = {(qloop, ε)}.
Finally, we handle case (c) wherein the empty stack marker $ is on the top of the stack. Let δ(qloop, ε, $) = {(qaccept, ε)}.
The state diagram is shown in Figure 2.24.
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120 CHAPTER 2 / CONTEXT-FREE LANGUAGES
FIGURE 2.24 State diagram of P
That completes the proof of Lemma 2.21.
EXAMPLE 2.25
We use the procedure developed in Lemma 2.21 to construct a PDA P1 from the
following CFG G.
S → aT b | b T→Ta|ε
The transition function is shown in the following diagram.
FIGURE 2.26 State diagram of P1
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2.2 PUSHDOWN AUTOMATA 121
Now we prove the reverse direction of Theorem 2.20. For the forward di- rection, we gave a procedure for converting a CFG into a PDA. The main idea was to design the automaton so that it simulates the grammar. Now we want to give a procedure for going the other way: converting a PDA into a CFG. We design the grammar to simulate the automaton. This task is challenging because “programming” an automaton is easier than “programming” a grammar.
LEMMA 2.27
If a pushdown automaton recognizes some language, then it is context free.
PROOF IDEA We have a PDA P , and we want to make a CFG G that generates all the strings that P accepts. In other words, G should generate a string if that string causes the PDA to go from its start state to an accept state.
To achieve this outcome, we design a grammar that does somewhat more. For each pair of states p and q in P , the grammar will have a variable Apq . This variable generates all the strings that can take P from p with an empty stack to q with an empty stack. Observe that such strings can also take P from p to q, regardless of the stack contents at p, leaving the stack at q in the same condition as it was at p.
First, we simplify our task by modifying P slightly to give it the following three features.
1. It has a single accept state, qaccept.
2. It empties its stack before accepting.
3. Each transition either pushes a symbol onto the stack (a push move) or pops one off the stack (a pop move), but it does not do both at the same time.
Giving P features 1 and 2 is easy. To give it feature 3, we replace each transition that simultaneously pops and pushes with a two transition sequence that goes through a new state, and we replace each transition that neither pops nor pushes with a two transition sequence that pushes then pops an arbitrary stack symbol.
To design G so that Apq generates all strings that take P from p to q, starting and ending with an empty stack, we must understand how P operates on these strings. For any such string x, P ’s first move on x must be a push, because every move is either a push or a pop and P can’t pop an empty stack. Similarly, the last move on x must be a pop because the stack ends up empty.
Two possibilities occur during P’s computation on x. Either the symbol popped at the end is the symbol that was pushed at the beginning, or not. If so, the stack could be empty only at the beginning and end of P ’s computation on x. If not, the initially pushed symbol must get popped at some point be- fore the end of x and thus the stack becomes empty at this point. We simulate the former possibility with the rule Apq → aArsb, where a is the input read at the first move, b is the input read at the last move, r is the state following p, and s is the state preceding q. We simulate the latter possibility with the rule Apq → AprArq, where r is the state when the stack becomes empty.
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122 CHAPTER 2 / CONTEXT-FREE LANGUAGES
PROOF Say that P = (Q, Σ, Γ, δ, q0, {qaccept}) and construct G. The variables of G are {Apq | p, q ∈ Q}. The start variable is Aq0 ,qaccept . Now we describe G’s rules in three parts.
1. For each p,q,r,s ∈ Q, u ∈ Γ, and a,b ∈ Σε, if δ(p,a,ε) contains (r,u) and δ(s, b, u) contains (q, ε), put the rule Apq → aArsb in G.
2.Foreachp,q,r∈Q,puttheruleApq →AprArq inG. 3.Finally,foreachp∈Q,puttheruleApp →εinG.
You may gain some insight for this construction from the following figures.
FIGURE 2.28
PDA computation corresponding to the rule Apq → AprArq
FIGURE 2.29
PDA computation corresponding to the rule Apq → aArsb
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2.2 PUSHDOWN AUTOMATA 123
Now we prove that this construction works by demonstrating that Apq gener- ates x if and only if (iff) x can bring P from p with empty stack to q with empty stack. We consider each direction of the iff as a separate claim.
CLAIM 2.30
If Apq generates x, then x can bring P from p with empty stack to q with empty
stack.
We prove this claim by induction on the number of steps in the derivation of
x from Apq.
Basis: The derivation has 1 step.
A derivation with a single step must use a rule whose right-hand side contains no variables. The only rules in G where no variables occur on the right-hand side are App → ε. Clearly, input ε takes P from p with empty stack to p with empty stack so the basis is proved.
Induction step: Assume true for derivations of length at most k, where k ≥ 1, and prove true for derivations of length k + 1.
Suppose that Apq ⇒∗ x with k + 1 steps. The first step in this derivation is either Apq ⇒ aArsb or Apq ⇒ AprArq. We handle these two cases separately.
In the first case, consider the portion y of x that Ars generates, so x = ayb. Because Ars ⇒∗ y with k steps, the induction hypothesis tells us that P can go from r on empty stack to s on empty stack. Because Apq → aArsb is a rule of G, δ(p, a, ε) contains (r, u) and δ(s, b, u) contains (q, ε), for some stack symbol u. Hence, if P starts at p with empty stack, after reading a it can go to state r and push u onto the stack. Then reading string y can bring it to s and leave u on the stack. Then after reading b it can go to state q and pop u off the stack. Therefore, x can bring it from p with empty stack to q with empty stack.
In the second case, consider the portions y and z of x that Apr and Arq re- spectively generate, so x = yz. Because Apr ⇒∗ y in at most k steps and Arq ⇒∗ z in at most k steps, the induction hypothesis tells us that y can bring P from p to r, and z can bring P from r to q, with empty stacks at the beginning and end. Hence x can bring it from p with empty stack to q with empty stack. This completes the induction step.
CLAIM 2.31
If x can bring P from p with empty stack to q with empty stack, Apq generates x.
We prove this claim by induction on the number of steps in the computation of P that goes from p to q with empty stacks on input x.
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124 CHAPTER 2 / CONTEXT-FREE LANGUAGES
Basis: The computation has 0 steps.
If a computation has 0 steps, it starts and ends at the same state—say, p. So we must show that App ⇒∗ x. In 0 steps, P cannot read any characters, so x = ε. By construction, G has the rule App → ε, so the basis is proved.
Induction step: Assume true for computations of length at most k, where k ≥ 0, and prove true for computations of length k + 1.
Suppose that P has a computation wherein x brings p to q with empty stacks in k + 1 steps. Either the stack is empty only at the beginning and end of this computation, or it becomes empty elsewhere, too.
In the first case, the symbol that is pushed at the first move must be the same as the symbol that is popped at the last move. Call this symbol u. Let a be the input read in the first move, b be the input read in the last move, r be the state after the first move, and s be the state before the last move. Then δ(p, a, ε) contains (r, u) and δ(s, b, u) contains (q, ε), and so rule Apq → aArsb is in G.
Letybetheportionofxwithoutaandb,sox=ayb. Inputycanbring P from r to s without touching the symbol u that is on the stack and so P can go from r with an empty stack to s with an empty stack on input y. We have removed the first and last steps of the k + 1 steps in the original computation on xsothecomputationonyhas(k+1)−2=k−1steps. Thustheinduction hypothesis tells us that Ars ⇒∗ y. Hence Apq ⇒∗ x.
In the second case, let r be a state where the stack becomes empty other than at the beginning or end of the computation on x. Then the portions of the computation from p to r and from r to q each contain at most k steps. Say that y is the input read during the first portion and z is the input read during the second portion. The induction hypothesis tells us that Apr ⇒∗ y and Arq ⇒∗ z. Because rule Apq → Apr Arq is in G, Apq ⇒∗ x, and the proof is complete.
That completes the proof of Lemma 2.27 and of Theorem 2.20.
We have just proved that pushdown automata recognize the class of context- free languages. This proof allows us to establish a relationship between the reg- ular languages and the context-free languages. Because every regular language is recognized by a finite automaton and every finite automaton is automatically a pushdown automaton that simply ignores its stack, we now know that every regular language is also a context-free language.
COROLLARY 2.32
Every regular language is context free.
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2.3 NON-CONTEXT-FREE LANGUAGES 125
FIGURE 2.33
Relationship of the regular and context-free languages
2.3
NON-CONTEXT-FREE LANGUAGES
In this section we present a technique for proving that certain languages are not context free. Recall that in Section 1.4 we introduced the pumping lemma for showing that certain languages are not regular. Here we present a similar pump- ing lemma for context-free languages. It states that every context-free language has a special value called the pumping length such that all longer strings in the language can be “pumped.” This time the meaning of pumped is a bit more com- plex. It means that the string can be divided into five parts so that the second and the fourth parts may be repeated together any number of times and the resulting string still remains in the language.
THE PUMPING LEMMA FOR CONTEXT-FREE LANGUAGES
THEOREM 2.34
Pumping lemma for context-free languages If A is a context-free language, then there is a number p (the pumping length) where, if s is any string in A of length at least p, then s may be divided into five pieces s = uvxyz satisfying the conditions
1.foreachi≥0, uvixyiz∈A, 2. |vy| > 0, and
3. |vxy| ≤ p.
When s is being divided into uvxyz, condition 2 says that either v or y is not the empty string. Otherwise the theorem would be trivially true. Condition 3
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126 CHAPTER 2 / CONTEXT-FREE LANGUAGES
states that the pieces v, x, and y together have length at most p. This technical condition sometimes is useful in proving that certain languages are not context free.
PROOF IDEA Let A be a CFL and let G be a CFG that generates it. We must show that any sufficiently long string s in A can be pumped and remain in A. The idea behind this approach is simple.
Let s be a very long string in A. (We make clear later what we mean by “very long.”) Because s is in A, it is derivable from G and so has a parse tree. The parse tree for s must be very tall because s is very long. That is, the parse tree must contain some long path from the start variable at the root of the tree to one of the terminal symbols at a leaf. On this long path, some variable symbol R must repeat because of the pigeonhole principle. As the following figure shows, this repetition allows us to replace the subtree under the second occurrence of R with the subtree under the first occurrence of R and still get a legal parse tree. Therefore, we may cut s into five pieces uvxyz as the figure indicates, and we may repeat the second and fourth pieces and obtain a string still in the language. In other words, uvixyiz is in A for any i ≥ 0.
FIGURE 2.35 Surgery on parse trees
Let’s now turn to the details to obtain all three conditions of the pumping lemma. We also show how to calculate the pumping length p.
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2.3 NON-CONTEXT-FREE LANGUAGES 127
PROOF Let G be a CFG for CFL A. Let b be the maximum number of symbols in the right-hand side of a rule (assume at least 2). In any parse tree using this grammar, we know that a node can have no more than b children. In other words, at most b leaves are 1 step from the start variable; at most b2 leaves are within 2 steps of the start variable; and at most bh leaves are within h steps of the start variable. So, if the height of the parse tree is at most h, the length of the string generated is at most bh. Conversely, if a generated string is at least bh + 1 long, each of its parse trees must be at least h + 1 high.
Say |V | is the number of variables in G. We set p, the pumping length, to be b|V |+1. Now if s is a string in A and its length is p or more, its parse tree must beatleast|V|+1high,becauseb|V|+1 ≥b|V| +1.
To see how to pump any such string s, let τ be one of its parse trees. If s has several parse trees, choose τ to be a parse tree that has the smallest number of nodes. We know that τ must be at least |V | + 1 high, so its longest path from the root to a leaf has length at least |V | + 1. That path has at least |V | + 2 nodes; one at a terminal, the others at variables. Hence that path has at least |V | + 1 variables. With G having only |V | variables, some variable R appears more than once on that path. For convenience later, we select R to be a variable that repeats among the lowest |V | + 1 variables on this path.
We divide s into uvxyz according to Figure 2.35. Each occurrence of R has a subtree under it, generating a part of the string s. The upper occurrence of R has a larger subtree and generates vxy, whereas the lower occurrence generates just x with a smaller subtree. Both of these subtrees are generated by the same variable, so we may substitute one for the other and still obtain a valid parse tree. Replacing the smaller by the larger repeatedly gives parse trees for the strings uvixyiz at each i > 1. Replacing the larger by the smaller generates the string uxz. That establishes condition 1 of the lemma. We now turn to conditions 2 and 3.
To get condition 2, we must be sure that v and y are not both ε. If they were, the parse tree obtained by substituting the smaller subtree for the larger would have fewer nodes than τ does and would still generate s. This result isn’t possible because we had already chosen τ to be a parse tree for s with the smallest number of nodes. That is the reason for selecting τ in this way.
In order to get condition 3, we need to be sure that vxy has length at most p. In the parse tree for s the upper occurrence of R generates vxy. We chose R so that both occurrences fall within the bottom |V | + 1 variables on the path, and we chose the longest path in the parse tree, so the subtree where R generates vxy is at most |V | + 1 high. A tree of this height can generate a string of length at most b|V |+1 = p.
For some tips on using the pumping lemma to prove that languages are not context free, review the text preceding Example 1.73 (page 80) where we dis- cuss the related problem of proving nonregularity with the pumping lemma for regular languages.
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128 CHAPTER 2 / CONTEXT-FREE LANGUAGES EXAMPLE 2.36
Use the pumping lemma to show that the language B = {anbncn| n ≥ 0} is not context free.
We assume that B is a CFL and obtain a contradiction. Let p be the pumping length for B that is guaranteed to exist by the pumping lemma. Select the string s = apbpcp. Clearly s is a member of B and of length at least p. The pumping lemma states that s can be pumped, but we show that it cannot. In other words, we show that no matter how we divide s into uvxyz, one of the three conditions of the lemma is violated.
First, condition 2 stipulates that either v or y is nonempty. Then we consider one of two cases, depending on whether substrings v and y contain more than one type of alphabet symbol.
1. When both v and y contain only one type of alphabet symbol, v does not contain both a’s and b’s or both b’s and c’s, and the same holds for y. In this case, the string uv2xy2z cannot contain equal numbers of a’s, b’s, and c’s. Therefore, it cannot be a member of B. That violates condition 1 of the lemma and is thus a contradiction.
2. When either v or y contains more than one type of symbol, uv2xy2z may contain equal numbers of the three alphabet symbols but not in the correct order. Hence it cannot be a member of B and a contradiction occurs.
One of these cases must occur. Because both cases result in a contradiction, a contradiction is unavoidable. So the assumption that B is a CFL must be false. Thus we have proved that B is not a CFL.
EXAMPLE 2.37
Let C = {aibjck|0 ≤ i ≤ j ≤ k}. We use the pumping lemma to show that C is not a CFL. This language is similar to language B in Example 2.36, but proving that it is not context free is a bit more complicated.
Assume that C is a CFL and obtain a contradiction. Let p be the pumping length given by the pumping lemma. We use the string s = apbpcp that we used earlier, but this time we must “pump down” as well as “pump up.” Let s = uvxyz and again consider the two cases that occurred in Example 2.36.
1. When both v and y contain only one type of alphabet symbol, v does not contain both a’s and b’s or both b’s and c’s, and the same holds for y. Note that the reasoning used previously in case 1 no longer applies. The reason is that C contains strings with unequal numbers of a’s, b’s, and c’s as long as the numbers are not decreasing. We must analyze the situation more carefully to show that s cannot be pumped. Observe that because v and y contain only one type of alphabet symbol, one of the symbols a, b, or c doesn’t appear in v or y. We further subdivide this case into three subcases according to which symbol does not appear.
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2.3 NON-CONTEXT-FREE LANGUAGES 129
a. The a’s do not appear. Then we try pumping down to obtain the string uv0xy0z = uxz. That contains the same number of a’s as s does, but it contains fewer b’s or fewer c’s. Therefore, it is not a member of C, and a contradiction occurs.
b. The b’s do not appear. Then either a’s or c’s must appear in v or y be- cause both can’t be the empty string. If a’s appear, the string uv2xy2z contains more a’s than b’s, so it is not in C. If c’s appear, the string uv0xy0z contains more b’s than c’s, so it is not in C. Either way, a contradiction occurs.
c. The c’s do not appear. Then the string uv2xy2z contains more a’s or more b’s than c’s, so it is not in C, and a contradiction occurs.
2. When either v or y contains more than one type of symbol, uv2xy2z will not contain the symbols in the correct order. Hence it cannot be a member of C, and a contradiction occurs.
Thus we have shown that s cannot be pumped in violation of the pumping lemma and that C is not context free.
EXAMPLE 2.38
Let D = {ww| w ∈ {0,1}∗}. Use the pumping lemma to show that D is not a CFL. Assume that D is a CFL and obtain a contradiction. Let p be the pumping length given by the pumping lemma.
This time choosing string s is less obvious. One possibility is the string 0p10p1. It is a member of D and has length greater than p, so it appears to be a good candidate. But this string can be pumped by dividing it as follows, so it is not adequate for our purposes.
0p1 0p1
000···000 0 1 0 000···0001
uvxyz
Let’s try another candidate for s. Intuitively, the string 0p1p0p1p seems to capture more of the “essence” of the language D than the previous candidate did. In fact, we can show that this string does work, as follows.
We show that the string s = 0p1p0p1p cannot be pumped. This time we use condition 3 of the pumping lemma to restrict the way that s can be divided. It says that we can pump s by dividing s = uvxyz, where |vxy| ≤ p.
First, we show that the substring vxy must straddle the midpoint of s. Other- wise, if the substring occurs only in the first half of s, pumping s up to uv2xy2z moves a 1 into the first position of the second half, and so it cannot be of the form ww. Similarly, if vxy occurs in the second half of s, pumping s up to uv2xy2z moves a 0 into the last position of the first half, and so it cannot be of the form ww.
But if the substring vxy straddles the midpoint of s, when we try to pump s down to uxz it has the form 0p1i0j1p, where i and j cannot both be p. This string is not of the form ww. Thus s cannot be pumped, and D is not a CFL.
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130 CHAPTER 2 / CONTEXT-FREE LANGUAGES 2.4
DETERMINISTIC CONTEXT-FREE LANGUAGES
As you recall, deterministic finite automata and nondeterministic finite automata are equivalent in language recognition power. In contrast, nondeterministic pushdown automata are more powerful than their deterministic counterparts. We will show that certain context-free languages cannot be recognized by deter- ministic PDAs—these languages require nondeterministic PDAs. The languages that are recognizable by deterministic pushdown automata (DPDAs) are called deterministic context-free languages (DCFLs). This subclass of the context-free languages is relevant to practical applications, such as the design of parsers in compilers for programming languages, because the parsing problem is gener- ally easier for DCFLs than for CFLs. This section gives a short overview of this important and beautiful subject.
In defining DPDAs, we conform to the basic principle of determinism: at each step of its computation, the DPDA has at most one way to proceed according to its transition function. Defining DPDAs is more complicated than defining DFAs because DPDAs may read an input symbol without popping a stack symbol, and vice versa. Accordingly, we allow ε-moves in the DPDA’s transition function even though ε-moves are prohibited in DFAs. These ε-moves take two forms: ε-input moves corresponding to δ(q, ε, x), and ε-stack moves corresponding to δ(q, a, ε). A move may combine both forms, corresponding to δ(q,ε,ε). If a DPDA can make an ε-move in a certain situation, it is prohibited from making a move in that same situation that involves processing a symbol instead of ε. Otherwise multiple valid computation branches might occur, leading to nondeterministic behavior. The formal definition follows.
DEFINITION 2.39
A deterministic pushdown automaton is a 6-tuple (Q, Σ, Γ, δ, q0, F ),
where Q, Σ, Γ, and F are all finite sets, and
1. Q is the set of states,
2. Σ is the input alphabet,
3. Γ is the stack alphabet,
4 . δ : Q × Σ ε × Γε −→ ( Q × Γε ) ∪ { ∅ } i s t h e t r a n s i t i o n f u n c t i o n , 5. q0 ∈ Q is the start state, and
6. F ⊆ Q is the set of accept states.
The transition function δ must satisfy the following condition. For every q ∈ Q, a ∈ Σ, and x ∈ Γ, exactly one of the values
δ(q,a,x),δ(q,a,ε),δ(q,ε,x), andδ(q,ε,ε) is not ∅.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 131
The transition function may output either a single move of the form (r, y) or it may indicate no action by outputting ∅. To illustrate these possibilities, let’s consider an example. Suppose a DPDA M with transition function δ is in state q, has a as its next input symbol, and has symbol x on the top of its stack. If δ(q, a, x) = (r, y) then M reads a, pops x off the stack, enters state r, and pushes y on the stack. Alternatively, if δ(q, a, x) = ∅ then when M is in state q, it has no move that reads a and pops x. In that case, the condition on δ requires that one of δ(q, ε, x), δ(q, a, ε), or δ(q, ε, ε) is nonempty, and then M moves accordingly. The condition enforces deterministic behavior by preventing the DPDA from taking two different actions in the same situation, such as would be the case if both δ(q, a, x) ̸= ∅ and δ(q, a, ε) ̸= ∅. A DPDA has exactly one legal move in every situation where its stack is nonempty. If the stack is empty, a DPDA can move only if the transition function specifies a move that pops ε. Otherwise the DPDA has no legal move and it rejects without reading the rest of the input.
Acceptance for DPDAs works in the same way it does for PDAs. If a DPDA enters an accept state after it has read the last input symbol of an input string, it accepts that string. In all other cases, it rejects that string. Rejection occurs if the DPDA reads the entire input but doesn’t enter an accept state when it is at the end, or if the DPDA fails to read the entire input string. The latter case may arise if the DPDA tries to pop an empty stack or if the DPDA makes an endless sequence of ε-input moves without reading the input past a certain point.
The language of a DPDA is called a deterministic context-free language.
EXAMPLE 2.40
The language {0n 1n | n ≥ 0} in Example 2.14 is a DCFL. We can easily modify its PDA M1 to be a DPDA by adding transitions for any missing state, input symbol, and stack symbol combinations to a “dead” state from which acceptance isn’t possible.
Examples 2.16 and 2.18 give CFLs {aibjck| i,j,k ≥ 0 and i = j or i = k} and {wwR| w ∈ {0,1}∗}, which are not DCFLs. Problems 2.57 and 2.58 show that nondeterminism is necessary for recognizing these languages.
Arguments involving DPDAs tend to be somewhat technical in nature, and though we strive to emphasize the primary ideas behind the constructions, read- ers may find this section to be more challenging than other sections in the first few chapters. Later material in the book doesn’t depend on this section, so it may be skipped if desired.
We’ll begin with a technical lemma that will simplify the discussion later on. As noted, DPDAs may reject inputs by failing to read the entire input string, but such DPDAs introduce messy cases. Fortunately, the next lemma shows that we can convert a DPDA into one that avoids this inconvenient behavior.
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132 CHAPTER 2 / CONTEXT-FREE LANGUAGES LEMMA 2.41
Every DPDA has an equivalent DPDA that always reads the entire input string.
PROOF IDEA A DPDA may fail to read the entire input if it tries to pop an empty stack or because it makes an endless sequence of ε-input moves. Call the first situation hanging and the second situation looping. We solve the hanging problem by initializing the stack with a special symbol. If that symbol is later popped from the stack before the end of the input, the DPDA reads to the end of the input and rejects. We solve the looping problem by identifying the looping situations, i.e., those from which no further input symbol is ever read, and re- programming the DPDA so that it reads and rejects the input instead of looping. We must adjust these modifications to accommodate the case where hanging or looping occurs on the last symbol of the input. If the DPDA enters an accept state at any point after it has read the last symbol, the modified DPDA accepts instead of rejects.
PROOF Let P = (Q,Σ,Γ,δ,q0,F) be a DPDA. First, add a new start state qstart, an additional accept state qaccept, a new state qreject, as well as other new states as described. Perform the following changes for for every r ∈ Q, a ∈ Σε, a n d x , y ∈ Γε .
First modify P so that, once it enters an accept state, it remains in accepting states until it reads the next input symbol. Add a new accept state qa for every q ∈ Q. For each q ∈ Q, if δ(q,ε,x) = (r,y), set δ(qa,ε,x) = (ra,y), and then if q ∈ F, also change δ so that δ(q,ε,x) = (ra,y). For each q ∈ Q and a ∈ Γ, if δ(q,a,x) = (r,y) set δ(qa,a,x) = (r,y). Let F′ be the set of new and old accept states.
Next, modify P to reject when it tries to pop an empty stack, by initializing the stack with a special new stack symbol $. If P subsequently detects $ while in a non-accepting state, it enters qreject and scans the input to the end. If P detects $ while in an accept state, it enters qaccept. Then, if any input remains unread, it enters qreject and scans the input to the end. Formally, set δ(qstart, ε, ε) = (q0, $). For x ∈ Γ and δ(q,a,x) ̸= ∅, if q ̸∈ F′ then set δ(q,a,$) = (qreject,ε), and if q ∈ F ′ then set δ(q, a, $) = (qaccept, ε). For a ∈ Σ, set δ(qreject, a, ε) = (qreject, ε) and δ(qaccept, a, ε) = (qreject, ε).
Lastly, modify P to reject instead of making an endless sequence of ε-input moves prior to the end of the input. For every q ∈ Q and x ∈ Γ, call (q,x) a looping situation if, when P is started in state q with x ∈ Γ on the top of the stack, it never pops anything below x and it never reads an input symbol. Say the looping situation is accepting if P enters an accept state during its subsequent moves, and otherwise it is rejecting. If (q, x) is an accepting looping situation, set δ(q, ε, x) = (qaccept, ε), whereas if (q, x) is a rejecting looping situation, set δ(q, ε, x) = (qreject, ε).
For simplicity, we’ll assume henceforth that DPDAs read their input to the end.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 133 PROPERTIES OF DCFLS
We’ll explore closure and nonclosure properties of the class of DCFLs, and use these to exhibit a CFL that is not a DCFL.
THEOREM 2.42
The class of DCFLs is closed under complementation.
PROOF IDEA Swapping the accept and non-accept states of a DFA yields a new DFA that recognizes the complementary language, thereby proving that the class of regular languages is closed under complementation. The same approach works for DPDAs, except for one problem. The DPDA may accept its input by entering both accept and non-accept states in a sequence of moves at the end of the input string. Interchanging accept and non-accept states would still accept in this case.
We fix this problem by modifying the DPDA to limit when acceptance can occur. For each symbol of the input, the modified DPDA can enter an accept state only when it is about to read the next symbol. In other words, only reading states—states that always read an input symbol—may be accept states. Then, by swapping acceptance and non-acceptance only among these reading states, we invert the output of the DPDA.
PROOF First modify P as described in the proof of Lemma 2.41 and let (Q, Σ, Γ, δ, q0, F ) be the resulting machine. This machine always reads the en- tire input string. Moreover, once enters an accept state, it remains in accept states until it reads the next input symbol.
In order to carry out the proof idea, we need to identify the reading states. If the DPDA in state q reads an input symbol a ∈ Σ without popping the stack, i.e., δ(q, a, ε) ̸= ∅, designate q to be a reading state. However, if it reads and also pops, the decision to read may depend on the popped symbol, so divide that step into two: a pop and then a read. Thus if δ(q, a, x) = (r, y) for a ∈ Σ and x ∈ Γ, add a new state qx and modify δ so δ(q,ε,x) = (qx,ε) and δ(qx,a,ε) = (r,y). Designate qx to be a reading state. The states qx never pop the stack, so their action is independent of the stack contents. Assign qx to be an accept state if q ∈ F . Finally, remove the accepting state designation from any state which isn’t a reading state. The modified DPDA is equivalent to P , but it enters an accept state at most once per input symbol, when it is about to read the next symbol.
Now, invert which reading states are classified as accepting. The resulting DPDA recognizes the complementary language.
This theorem implies that some CFLs are not DCFLs. Any CFL whose comple- ment isn’t a CFL isn’t a DCFL. Thus A = {ai bj ck | i ̸= j or j ̸= k where i, j, k ≥ 0} is a CFL but not a DCFL. Otherwise A would be a CFL, so the result of Problem 2.18 would incorrectly imply that A ∩ a∗b∗c∗ = {anbncn| n ≥ 0} is context free.
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134 CHAPTER 2 / CONTEXT-FREE LANGUAGES
Problem 2.53 asks you to show that the class of DCFLs isn’t closed under other familiar operations such as union, intersection, star, and reversal.
To simplify arguments, we will occasionally consider endmarked inputs whereby the special endmarker symbol ⊣ is appended to the input string. Here we add ⊣ to the DPDA’s input alphabet. As we show in the next theorem, adding endmarkers doesn’t change the power of DPDAs. However, designing DPDAs on endmarked inputs is often easier because we can take advantage of knowing when the input string ends. For any language A, we write the endmarked language A⊣ to be the collection of strings w⊣ where w ∈ A.
THEOREM 2.43
A is a DCFL if and only if A⊣ is a DCFL.
PROOF IDEA Proving the forward direction of this theorem is routine. Say DPDA P recognizes A. Then DPDA P ′ recognizes A⊣ by simulating P until P ′ reads ⊣. At that point, P ′ accepts if P had entered an accept state during the previous symbol. P ′ doesn’t read any symbols after ⊣.
To prove the reverse direction, let DPDA P recognize A⊣ and construct a DPDA P ′ that recognizes A. As P ′ reads its input, it simulates P . Prior to read- ing each input symbol, P′ determines whether P would accept if that symbol were ⊣. If so, P ′ enters an accept state. Observe that P may operate the stack after it reads ⊣, so determining whether it accepts after reading ⊣ may depend on the stack contents. Of course, P′ cannot afford to pop the entire stack at every input symbol, so it must determine what P would do after reading ⊣, but without popping the stack. Instead, P ′ stores additional information on the stack that allows P ′ to determine immediately whether P would accept. This infor- mation indicates from which states P would eventually accept while (possibly) manipulating the stack, but without reading further input.
PROOF We give proof details of the reverse direction only. As we described in the proof idea, let DPDA P = (Q, Σ∪{⊣}, Γ, δ, q0, F ) recognize A⊣ and construct a DPDA P′ = (Q′,Σ,Γ′,δ′,q0′,F′) that recognizes A. First, modify P so that each of its moves does exactly one of the following operations: read an input symbol; push a symbol onto the stack; or pop a symbol from the stack. Making this modification is straightforward by introducing new states.
P′ simulates P, while maintaining a copy of its stack contents interleaved with additional information on the stack. Every time P ′ pushes one of P ’s stack symbols, P′ follows that by pushing a symbol that represents a subset of P’s states. Thus we set Γ′ = Γ ∪ P (Q). The stack in P ′ interleaves members of Γ with members of P(Q). If R ∈ P(Q) is the top stack symbol, then by starting P in any one of R’s states, P will eventually accept without reading any more input.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 135
Initially, P′ pushes the set R0 on the stack, where R0 contains every state q such that when P is started in q with an empty stack, it eventually accepts without reading any input symbols. Then P ′ begins simulating P . To simulate a pop move, P ′ first pops and discards the set of states that appears as the top stack symbol, then it pops again to obtain the symbol that P would have popped at this point, and uses it to determine the next move of P . Simulating a push move δ(q,ε,ε) = (r,x), where P pushes x as it goes from state q to state r, goes as follows. First P ′ examines the set of states R on the top of its stack, and then it pushes x and after that the set S, where q ∈ S if q ∈ F or if δ(q, ε, x) = (r, ε) and r ∈ R. In other words, S is the set of states that are either accepting immediately, or that would lead to a state in R after popping x. Lastly, P ′ simulates a read move δ(q, a, ε) = (r, ε), by examining the set R on the top of the stack and entering an accept state if r ∈ R. If P ′ is at the end of the input string when it enters this state, it will accept the input. If it is not at the end of the input string, it will continue simulating P, so this accept state must also record P’s state. Thus we create this state as a second copy of P ’s original state, marking it as an accept state in P′.
DETERMINISTIC CONTEXT-FREE GRAMMARS
This section defines deterministic context-free grammars, the counterpart to deterministic pushdown automata. We will show that these two models are equivalent in power, provided that we restrict our attention to endmarked lan- guages, where all strings are terminated with ⊣. Thus the correspondence isn’t quite as strong as we saw in regular expressions and finite automata, or in CFGs and PDAs, where the generating model and the recognizing model describe ex- actly the same class of languages without the need for endmarkers. However, in the case of DPDAs and DCFGs, the endmarkers are necessary because equivalence doesn’t hold otherwise.
In a deterministic automaton, each step in a computation determines the next step. The automaton cannot make choices about how it proceeds because only a single possibility is available at every point. To define determinism in a grammar, observe that computations in automata correspond to derivations in grammars. In a deterministic grammar, derivations are constrained, as you will see.
Derivations in CFGs begin with the start variable and proceed “top down” with a series of substitutions according to the grammar’s rules, until the deriva- tion obtains a string of terminals. For defining DCFGs we take a “bottom up” approach, by starting with a string of terminals and processing the derivation in reverse, employing a series of reduce steps until reaching the start variable. Each reduce step is a reversed substitution, whereby the string of terminals and variables on the right-hand side of a rule is replaced by the variable on the cor- responding left-hand side. The string replaced is called the reducing string. We call the entire reversed derivation a reduction. Deterministic CFGs are defined in terms of reductions that have a certain property.
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136 CHAPTER 2 / CONTEXT-FREE LANGUAGES
More formally, if u and v are strings of variables and terminals, write u v to mean that v can be obtained from u by a reduce step. In other words, u v means the same as v ⇒ u. A reduction from u to v is a sequence
u=u1 u2 …uk =v
and we say that u is reducible to v, written u ∗ v. Thus u ∗ v whenever v ⇒∗ u. A reduction from u is a reduction from u to the start variable. In a leftmost reduction, each reducing string is reduced only after all other reducing strings that lie entirely to its left. With a little thought we can see that a leftmost reduction is a rightmost derivation in reverse.
Here’s the idea behind determinism in CFGs. In a CFG with start variable S and string w in its language, say that a leftmost reduction of w is
w=u1 u2 …uk =S.
First, we stipulate that every ui determines the next reduce step and hence ui+1. Thus w determines its entire leftmost reduction. This requirement implies only that the grammar is unambiguous. To get determinism, we need to go further. In each ui, the next reduce step must be uniquely determined by the prefix of ui up through and including the reducing string h of that reduce step. In other words, the leftmost reduce step in ui doesn’t depend on the symbols in ui to the right of its reducing string.
Introducing terminology will help us make this idea precise. Let w be a string in the language of CFG G, and let ui appear in a leftmost reduction of w. In the reduce step ui ui+1, say that rule T → h was applied in reverse. That means we can write ui = xhy and ui+1 = xT y, where h is the reducing string, x is the part of ui that appears leftward of h, and y is the part of ui that appears rightward of h. Pictorially,
xhyxTy
ui = x1 ···xj h1 ···hk y1 ···yl x1 ···xj T y1 ···yl = ui+1. FIGURE 2.44
Expanded view of xhy xT y
We call h, together with its reducing rule T → h, a handle of ui. In other words, a handle of a string ui that appears in a leftmost reduction of w ∈ L(G) is the occurrence of the reducing string in ui, together with the reducing rule for ui in this reduction. Occasionally we associate a handle with its reducing string only, when we aren’t concerned with the reducing rule. A string that appears in a leftmost reduction of some string in L(G) is called a valid string. We define handles only for valid strings.
A valid string may have several handles, but only if the grammar is ambigu- ous. Unambiguous grammars may generate strings by one parse tree only, and
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 137
therefore the leftmost reductions, and hence the handles, are also unique. In that case, we may refer to the handle of a valid string.
Observe that y, the portion of ui following a handle, is always a string of ter- minals because the reduction is leftmost. Otherwise, y would contain a variable symbol and that could arise only from a previous reduce step whose reducing string was completely to the right of h. But then the leftmost reduction should have reduced the handle at an earlier step.
EXAMPLE 2.45 Consider the grammar G1:
R→S|T
S → aSb | ab
T → aT bb | abb
Its language is B ∪ C where B = {ambm| m ≥ 1} and C = {amb2m| m ≥ 1}. In this leftmost reduction of the string aaabbb ∈ L(G1), we’ve underlined the handle at each step:
b aSb S R. Similarly, this is a leftmost reduction of the string aaabbbbbb:
bbaTbbT R.
In both cases, the leftmost reduction shown happens to be the only reduction possible; but in other grammars where several reductions may occur, we must use a leftmost reduction to define the handles. Notice that the handles of aaabbb and aaabbbbbb are unequal, even though the initial parts of these strings agree. We’ll discuss this point in more detail shortly when we define DCFGs.
A PDA can recognize L(G1) by using its nondeterminism to guess whether its input is in B or in C. Then, after it pushes the a’s on the stack, it pops the a’s and matches each one with b or bb accordingly. Problem 2.55 asks you to show that L(G1) is not a DCFL. If you try to make a DPDA that recognizes this language, you’ll see that the machine cannot know in advance whether the input is in B or in C so it doesn’t know how to match the a’s with the b’s. Contrast this grammar with grammar G2:
R → 1S | 2T
S → aSb | ab
T → aT bb | abb
where the first symbol in the input provides this information. Our definition of DCFGs must include G2 yet exclude G1.
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aaabbb aaSb
aaabbbbbb aaTbb
138 CHAPTER 2 / CONTEXT-FREE LANGUAGES
EXAMPLE 2.46
Let G3 be the following grammar:
S → T⊣
T → T (T ) | ε
This grammar illustrates several points. First, it generates an endmarked lan- guage. We will focus on endmarked languages later on when we prove the equivalence between DPDAs and DCFGs. Second, ε handles may occur in re- ductions, as indicated with short underscores in the leftmost reduction of the string ()()⊣:
()()⊣ T (
)()⊣ T(T)()⊣ T()⊣ T(T)⊣ T⊣ S.
Handles play an important role in defining DCFGs because handles determine reductions. Once we know the handle of a string, we know the next reduce step. To make sense of the coming definition, keep our goal in mind: we aim to define DCFGs so that they correspond to DPDAs. We’ll establish that correspondence by showing how to convert DCFGs to equivalent DPDAs, and vice versa. For this conversion to work, the DPDA needs to find handles so that it can find reductions. But finding a handle may be tricky. It seems that we need to know a string’s next reduce step to identify its handle, but a DPDA doesn’t know the reduction in advance. We’ll solve this by restricting handles in a DCFG so that the DPDA can find them more easily.
To motivate the definition, consider ambiguous grammars, where some strings have several handles. Selecting a specific handle may require advance knowledge of which parse tree derives the string, information that is certainly unavailable to the DPDA. We’ll see that DCFGs are unambiguous so handles are unique. However, uniqueness alone is unsatisfactory for defining DCFGs as grammar G1 in Example 2.45 shows.
Why don’t unique handles imply that we have a DCFG? The answer is evident by examining the handles in G1. If w ∈ B, the handle is ab, whereas if w ∈ C, the handle is abb. Though w determines which of these cases applies, discovering which of ab or abb is the handle may require examining all of w, and a DPDA hasn’t read the entire input when it needs to select the handle.
In order to define DCFGs that correspond to DPDAs, we impose a stronger requirement on the handles. The initial part of a valid string, up to and including its handle, must be sufficient to determine the handle. Thus, if we are reading a valid string from left to right, as soon as we read the handle we know we have it. We don’t need to read beyond the handle in order to identify the handle. Recall that the unread part of the valid string contains only terminals because the valid string has been obtained by a leftmost reduction of an initial string of terminals, and the unread part hasn’t been processed yet. Accordingly, we say that a handle h of a valid string v = xhy is a forced handle if h is the unique handle in every valid string xhyˆ where yˆ ∈ Σ∗.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 139
DEFINITION 2.47
A deterministic context-free grammar is a context-free grammar such that every valid string has a forced handle.
For simplicity, we’ll assume throughout this section on deterministic context- free languages that the start variable of a CFG doesn’t appear on the right-hand side of any rule and that every variable in a grammar appears in a reduction of some string in the grammar’s language, i.e., grammars contain no useless vari- ables.
Though our definition of DCFGs is mathematically precise, it doesn’t give any obvious way to determine whether a CFG is deterministic. Next we’ll present a procedure to do exactly that, called the DK-test. We’ll also use the construction underlying the DK-test to enable a DPDA to find handles, when we show how to convert a DCFG to a DPDA.
The DK-test relies on one simple but surprising fact. For any CFG G we can construct an associated DFA DK that can identify handles. Specifically, DK accepts its input z if
1. z is the prefix of some valid string v = zy, and 2. z ends with a handle of v.
Moreover, each accept state of DK indicates the associated reducing rule(s). In a general CFG, multiple reducing rules may apply, depending on which valid v extends z. But in a DCFG, as we’ll see, each accept state corresponds to exactly one reducing rule.
We will describe the DK-test after we’ve presented DK formally and estab- lished its properties, but here’s the plan. In a DCFG, all handles are forced. Thus if zy is a valid string with a prefix z that ends in a handle of zy, that handle is unique, and it is also the handle for all valid strings zyˆ. For these properties to hold, each of DK’s accept states must be associated with a single handle and hence with a single applicable reducing rule. Moreover, the accept state must not have an outgoing path that leads to an accept state by reading a string in Σ∗. Otherwise, the handle of zy would not be unique or it would depend on y. In the DK-test, we construct DK and then conclude that G is deterministic if all of its accept states have these properties.
To construct DFA DK, we’ll construct an equivalent NFA K and convert K to DK1 via the subset construction introduced in Theorem 1.39. To understand K, first consider an NFA J that performs a simpler task. It accepts every input string that ends with the right-hand side of any rule. Constructing J is easy. It guesses which rule to use and it also guesses the point at which to start matching the input with that rule’s right-hand side. As it matches the input, J keeps track
1The name DK is a mnemonic for “deterministic K” but it also stands for Donald Knuth, who first proposed this idea.
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140 CHAPTER 2 / CONTEXT-FREE LANGUAGES
of its progress through the chosen right-hand side. We represent this progress by placing a dot in the corresponding point in the rule, yielding a dotted rule, also called an item in some other treatments of this material. Thus for each rule B → u1u2 ···uk with k symbols on the right-hand side, we get k + 1 dotted rules:
B → .u1u2 ···uk B → u1.u2 ···uk
.
B → u1u2 ···.uk
B → u1u2 ···uk.
Each of these dotted rules corresponds to one state of J. We indicate the state
associated with the dotted rule B → u.v with a box around it, . The ✁
accept states ✁ correspond to the completed rules that have the dot at ✁
the end. We add a separate start state with a self-loop on all symbols and an
ε-move to for each rule B → u. Thus J accepts if the match completes ✁
successfully at the end of the input. If a mismatch occurs or if the end of the match doesn’t coincide with the end of the input, this branch of J ’s computation rejects.
NFA K operates similarly, but it is more judicious about choosing a rule for
matching. Only potential reducing rules are allowed. Like J, its states cor-
respond to all dotted rules. It has a special start state that has an ε-move to
✁for every rule involving the start variable S1. On each branch of its computation, K matches a potential reducing rule with a substring of the input. If that rule’s right-hand side contains a variable, K may nondeterministically switch to some rule that expands that variable. Lemma 2.48 formalizes this idea. First we describe K in detail.
The transitions come in two varieties: shift-moves and ε-moves. The shift- moves appear for every a that is a terminal or variable, and every rule B → uav:
B u•av a B ua•v The ε-moves appear for all rules B → uCv and C → r:
B u•Cv ε C •r
The accept states are all ✁corresponding to a completed rule. Accept ✁
states have no outgoing transitions and are written with a double box.
The next lemma and its corollary prove that K accepts all strings z that end with handles for some valid extension of z. Because K is nondeterministic, we say that it “may” enter a state to mean that K does enter that state on some branch of its nondeterminism.
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✄ ✄
✂ ✂
B → u.
B → u.v ✂
✄
B → .u ✂
✄
S1 → .u ✂
✄
✄ ✄
✂ ✂
B → u.
2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 141 LEMMA 2.48
K may enter state ✁on reading input z iff z = xu and xuvy is a valid
string with handle uv and reducing rule T → uv, for some y ∈ Σ∗.
PROOF IDEA K operates by matching a selected rule’s right-hand side with a portion of the input. If that match completes successfully, it accepts. If that right-hand side contains a variable C, either of two situations may arise. If C is the next input symbol, then matching the selected rule simply continues. If C has been expanded, the input will contain symbols derived from C, so K nondeterministically selects a substitution rule for C and starts matching from the beginning of the right-hand side of that rule. It accepts when the right-hand side of the currently selected rule has been matched completely.
PROOF First we prove the forward direction. Assume that K on w enters
. Examine K’s path from its start state to . Think of the path ✁✁
as runs of shift-moves separated by ε-moves. The shift-moves are transitions between states sharing the same rule, shifting the dot rightward over symbols read from the input. In the ith run, say that the rule is Si → uiSi+1vi, where Si+1 is the variable expanded in the next run. The penultimate run is for rule Sl → ulTvl, and the final run has rule T → uv.
Input z must then equal u1u2 . . . ulu = xu because the strings ui and u were the shift-move symbols read from the input. Letting y′ = vl . . . v2v1, we see that xuvy′ is derivable in G because the rules above give the derivation as shown in the parse tree illustrated in Figure 2.49.
S1
S2
S3 T
……
x uv y’
FIGURE 2.49
Parse tree leading to xuvy′
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T → u.v ✂
✄
T → u.v ✂
✄
T → u.v ✂
✄
…
142 CHAPTER 2 / CONTEXT-FREE LANGUAGES
To obtain a valid string, fully expand all variables that appear in y′ until each variable derives some string of terminals, and call the resulting string y. The string xuvy is valid because it occurs in a leftmost reduction of w ∈ L(G), a string of terminals obtained by fully expanding all variables in xuvy.
As is evident from the figure below, uv is the handle in the reduction and its reducing rule is T → uv.
S1
S2
S3 T
…
x uv y
FIGURE 2.50
Parse tree leading to valid string xuvy with handle uv
Now we prove the reverse direction of the lemma. Assume that string xuvy is
a valid string with handle uvand reducing rule T → uv. Show that K on input
xu may enter state . ✁
The parse tree for xuvy appears in the preceding figure. It is rooted at the start variable S1 and it must contain the variable T because T → uv is the first reduce step in the reduction of xuvy. Let S2, . . . , Sl be the variables on the path from S1 to T as shown. Note that all variables in the parse tree that appear leftward of this path must be unexpanded, or else uv wouldn’t be the handle.
In this parse tree, each Si leads to Si+1 by some rule Si → uiSi+1vi. Thus the grammar must contain the following rules for some strings ui and vi.
S1 → u1S2v1 S2 → u2S3v2
.
Sl →ulTvl T → uv
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T → u.v ✂
✄
…
…
2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 143
K contains the following path from its start state to state on reading
input z = xu. First, K makes an ε-move to
ing the symbols of u1, it performs the corresponding shift-moves until it enters
✁
at the end of u . Then it makes an ε-move to and ✁1✁
continues with shift-moves on reading u2 until it reaches and so
on. After reading ul it enters which leads by an ε-move to ✁
and finally after reading u it is in . ✁
The following corollary shows that K accepts all strings ending with a handle of some valid extension. It follows from Lemma 2.48 by taking u = h and v = ε.
COROLLARY 2.51
K may enter accept state ✁on input z iff z = xh and h is a handle of
✁
some valid string xhy with reducing rule T → h.
Finally, we convert NFA K to DFA DK by using the subset construction in the proof of Theorem 1.39 on page 55 and then removing all states that are un- reachable from the start state. Each of DK’s states thus contains one or more dotted rules. Each accept state contains at least one completed rule. We can ap- ply Lemma 2.48 and Corollary 2.51 to DK by referring to the states that contain the indicated dotted rules.
Now we are ready to describe the DK-test.
Starting with a CFG G, construct the associated DFA DK. Determine whether G is deterministic by examining DK’s accept states. The DK-test stipulates that every accept state contains
1. exactly one completed rule, and
2. no dotted rule in which a terminal symbol immediately follows the dot,
i.e., no dotted rule of the form B → u.av for a ∈ Σ. THEOREM 2.52
G passes the DK-test iff G is a DCFG.
PROOF IDEA We’ll show that the DK-test passes if and only if all handles are forced. Equivalently, the test fails iff some handle isn’t forced. First, suppose that some valid string has an unforced handle. If we run DK on this string, Corollary 2.51 says that DK enters an accept state at the end of the handle. The DK-test fails because that accept state has either a second completed rule or an outgoing path leading to an accept state, where the outgoing path begins with a terminal symbol. In the latter case, the accept state would contain a dotted rule with a terminal symbol following the dot.
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T → u.v ✂
✄
✄
Sl → ul ✂
.T vl
S1 → .u1S2v1 ✂
✄
✁
. Then, while read-
✄
S1 → u1 ✂
.S2 v1
✄
✂
S2 → .u2S3v2
✄
S2 → u2 ✂
.S3 v2
✁
T → .uv ✂
✄
✁
T → u.v ✂
✄
✄ ✄
T → h.
✂ ✂
144 CHAPTER 2 / CONTEXT-FREE LANGUAGES
Conversely, if the DK-test fails because an accept state has two completed rules, extend the associated string to two valid strings with differing handles at that point. Similarly, if it has a completed rule and a dotted rule with a terminal following the dot, employ Lemma 2.48 to get two valid extensions with differing handles at that point. Constructing the valid extension corresponding to the second rule is a bit delicate.
PROOF Start with the forward direction. Assume that G isn’t deterministic and show that it fails the DK-test. Take a valid string xhy that has an unforced handle h. Hence some valid string xhy′ has a different handle hˆ ̸= h, where y′ is a string of terminals. We can thus write xhy′ as xhy′ = xˆhˆyˆ.
If xh = xˆhˆ, the reducing rules differ because h and hˆ aren’t the same handle. Therefore, input xh sends DK to a state that contains two completed rules, a violation of the DK-test.
If xh ̸= xˆhˆ, one of these extends the other. Assume that xh is the proper prefix of xˆhˆ. The argument is the same with the strings interchanged and y in place of y′, if xˆhˆ is the shorter string. Let q be the state that DK enters on input xh. State q must be accepting because h is a handle of xhy. A transition arrow must exit q because xˆhˆ sends DK to an accept state via q. Furthermore, that transition arrow is labeled with a terminal symbol, because y′ ∈ Σ+. Here y′ ̸= ε because xˆhˆ extends xh. Hence q contains a dotted rule with a terminal symbol immediately following the dot, violating the DK-test.
To prove the reverse direction, assume G fails the DK-test at some accept state q, and show that G isn’t deterministic by exhibiting an unforced handle. Because q is accepting, it has a completed rule T → h.. Let z be a string that leads DK to q. Then z = xh where some valid string xhy has handle h with reducing rule T → h, for y ∈ Σ∗. Now we consider two cases, depending on how the DK-test fails.
First, say q has another completed rule B → hˆ.. Then some valid string xhy′ must have a different handle hˆ with reducing rule B → hˆ. Therefore, h isn’t a forced handle.
Second, say q contains a rule B → u.av where a ∈ Σ. Because xh takes DK to q, we have xh = xˆu, where xˆuavyˆ is valid and has a handle uav with reducing rule B → uav, for some yˆ ∈ Σ∗. To show that h is unforced, fully expand all variables in v to get the result v′ ∈ Σ∗, then let y′ = av′yˆ and notice that y′ ∈ Σ∗. The following leftmost reduction shows that xhy′ is a valid string and h is not the handle.
xhy′ =xhav′yˆ=xˆuav′yˆ∗ xˆuavyˆxˆByˆ∗ S
where S is the start variable. We know that xˆ uav yˆ is valid and we can obtain xˆ uav′yˆ from it by using a rightmost derivation so xˆ uav′ yˆ is also valid. More- over, the handle of xˆ uav′yˆ either lies inside v′ (if v ̸= v′) or is uav (if v = v′). In either case, the handle includes a or follows a and thus cannot be h because h fully precedes a. Hence h isn’t a forced handle.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 145
When building the DFA DK in practice, a direct construction may be faster than first constructing the NFA K. Begin by adding a dot at the initial point in all rules involving the start variable and place these now-dotted rules into DK’s start state. If a dot precedes a variable C in any of these rules, place dots at the initial position in all rules that have C on the left-hand side and add these rules to the state, continuing this process until no new dotted rules are obtained. For any symbol c that follows a dot, add an outgoing edge labeled c to a new state containing the dotted rules obtained by shifting the dot across the c in any of the dotted rules where the dot precedes the c, and add rules corresponding to the rules where a dot precedes a variable as before.
EXAMPLE 2.53
Here we illustrate how the DK-test fails for the following grammar.
S → E⊣ E→E+T|T T→Txa|a
S •E⊣ E •E+T E •T
T •T×a T•a
T
E S E•⊣ E E•+T
+
a
×
⊣
T
S E⊣•
E E+•T
T •T×a aT•a
E E+T• T T•×a
×
T Tוa a
T T×a• T T×a•
Ta•
ET•
T T•×a
FIGURE 2.54
Example of a failed DK-test
Notice the two problematic states at the lower left and the second from the top right, where an accept state contains a dotted rule where a terminal symbol follows the dot.
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146 CHAPTER 2 / CONTEXT-FREE LANGUAGES EXAMPLE 2.55
Here is the DFA DK showing that the grammar below is a DCFG. S → T⊣
T → T (T ) | ε
T S T•⊣ ⊣
T T•(T) (
T T(T•) ) T T•(T)
FIGURE 2.56
Example of a DK-test that passes
Observe that all accept states satisfy the DK-test conditions. RELATIONSHIP OF DPDAS AND DCFGS
In this section we will show that DPDAs and DCFGs describe the same class of endmarked languages. First, we will demonstrate how to convert DCFGs to equivalent DPDAs. This conversion works in all cases. Second, we will show how to do the reverse conversion, from DPDAs to equivalent DCFGs. The latter conversion works only for endmarked languages. We restrict the equivalence to endmarked languages, because the models are not equivalent without this re- striction. We showed earlier that endmarkers don’t affect the class of languages that DPDAs recognize, but they do affect the class of languages that DCFGs gen- erate. Without endmarkers, DCFGs generate only a subclass of the DCFLs—those that are prefix-free (see Problem 2.52). Note that every endmarked language is prefix-free.
THEOREM 2.57
An endmarked language is generated by a deterministic context-free grammar if
and only if it is deterministic context free.
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S •T⊣ T •T(T) T•
S T⊣•
T (T)•
T T(•T) T T •T(T) T•(
2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 147
We have two directions to prove. First we will show that every DCFG has an equivalent DPDA. Then we will show that every DPDA that recognizes an end- marked language has an equivalent DCFG. We handle these two directions in separate lemmas.
LEMMA 2.58
Every DCFG has an equivalent DPDA.
PROOF IDEA We show how to convert a DCFG G to an equivalent DPDA P . P uses the DFA DK to operate as follows. It simulates DK on the symbols it reads from the input until DK accepts. As shown in the proof of Theorem 2.52, DK’s accept state indicates a specific dotted rule because G is deterministic, and that rule identifies a handle for some valid string extending the input it has seen so far. Moreover, this handle applies to every valid extension because G is deterministic, and in particular it will apply to the full input to P , if that input is in L(G). So P can use this handle to identify the first reduce step for its input string, even though it has read only a part of its input at this point.
How does P identify the second and subsequent reduce steps? One idea is to perform the reduce step directly on the input string, and then run the modified input through DK as we did above. But the input can be neither modified nor reread so this idea doesn’t work. Another approach would be to copy the input to the stack and carry out the reduce step there, but then P would need to pop the entire stack to run the modified input through DK and so the modified input would not remain available for later steps.
The trick here is to store the states of DK on the stack, instead of storing the input string there. Every time P reads an input symbol and simulates a move in DK, it records DK’s state by pushing it on the stack. When it performs a reduce step using reducing rule T → u, it pops |u| states off the stack, revealing the state DK was in prior to reading u. It resets DK to that state, then simulates it on input T and pushes the resulting state on the stack. Then P proceeds by reading and processing input symbols as before.
When P pushes the start variable on the stack, it has found a reduction of its input to the start variable, so it enters an accept state.
Next we prove the other direction of Theorem 2.57.
LEMMA 2.59
Every DPDA that recognizes an endmarked language has an equivalent DCFG.
PROOF IDEA This proof is a modification of the construction in Lemma 2.27 on page 121 that describes the conversion of a PDA P to an equivalent CFG G.
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148 CHAPTER 2 / CONTEXT-FREE LANGUAGES
Here P and G are deterministic. In the proof idea for Lemma 2.27, we altered P to empty its stack and enter a specific accept state qaccept when it accepts. A PDA cannot directly determine that it is at the end of its input, so P uses its nondeterminism to guess that it is in that situation. We don’t want to introduce nondeterminism in constructing DPDA P . Instead we use the assumption that L(P ) is endmarked. We modify P to empty its stack and enter qaccept when it enters one of its original accept states after it has read the endmarker ⊣.
Next we apply the grammar construction to obtain G. Simply applying the original construction to a DPDA produces a nearly deterministic grammar be- cause the CFG’s derivations closely correspond to the DPDA’s computations. That grammar fails to be deterministic in one minor, fixable way.
The original construction introduces rules of the form Apq → AprArq and these may cause ambiguity. These rules cover the case where Apq generates a string that takes P from state p to state q with its stack empty at both ends, and the stack empties midway. The substitution corresponds to dividing the computation at that point. But if the stack empties several times, several divisions are possible. Each of these divisions yields different parse trees, so the resulting grammar is ambiguous. We fix this problem by modifying the grammar to divide the computation only at the very last point where the stack empties midway, thereby removing this ambiguity. For illustration, a similar but simpler situation occurs in the ambiguous grammar
S → T⊣
T → T T | (T ) | ε
which is equivalent to the unambiguous, and deterministic, grammar
S → T⊣
T → T (T ) | ε.
Next we show the modified grammar is deterministic by using the DK-test. The grammar is designed to simulate the DPDA. As we proved in Lemma 2.27, Apq generates exactly those strings on which P goes from state p on empty stack to state q on empty stack. We’ll prove G’s determinism using P ’s determinism so we will find it useful to define P ’s computation on valid strings to observe its action on handles. Then we can use P ’s deterministic behavior to show that handles are forced.
PROOF Say that P = (Q, Σ, Γ, δ, q0, {qaccept}) and construct G. The start variable is Aq0 ,qaccept . The construction on page 121 contains parts 1, 2, and 3, repeated here for convenience.
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 149
1. For each p,q,r,s ∈ Q, u ∈ Γ, and a,b ∈ Σε, if δ(p,a,ε) contains (r,u) and δ(s, b, u) contains (q, ε), put the rule Apq → aArsb in G.
2. For each p,q,r ∈ Q, put the rule Apq → AprArq in G.
3. Foreachp∈Q,puttheruleApp →εinG.
We modify the construction to avoid introducing ambiguity, by combining rules of types 1 and 2 into a single type 1-2 rule that achieves the same effect.
1-2. For each p,q,r,s,t ∈ Q, u ∈ Γ, and a,b ∈ Σε, if δ(r,a,ε) = (s,u) and δ(t, b, u) = (q, ε), put the rule Apq → ApraAstb in G.
To see that the modified grammar generates the same language, consider any derivation in the original grammar. For each substitution due to a type 2 rule Apq →AprArq,wecanassumethatrisP’sstatewhenitisattherightmostpoint where the stack becomes empty midway by modifying the proof of Claim 2.31 on page 123 to select r in this way. Then the subsequent substitution of Arq must expand it using a type 1 rule Arq → aAstb. We can combine these two substitutions into a single type 1-2 rule Apq → ApraAstb.
Conversely, in a derivation using the modified grammar, if we replace each type 1-2 rule Apq → ApraAstb by the type 2 rule Apq → AprArq followed by the type 1 rule Arq → aAstb, we get the same result.
Now we use the DK-test to show that G is deterministic. To do that, we’ll analyze how P operates on valid strings by extending its input alphabet and tran- sition function to process variable symbols in addition to terminal symbols. We add all symbols Apq to P ’s input alphabet and we extend its transition function δ by defining δ(p, Apq , ε) = (q, ε). Set all other transitions involving Apq to ∅. To preserve P ’s deterministic behavior, if P reads Apq from the input then disallow an ε-input move.
The following claim applies to a derivation of any string w in L(G) such as Aq0,qaccept =v0 ⇒v1 ⇒···⇒vi ⇒···⇒vk =w.
CLAIM 2.60
If P reads vi containing a variable Apq , it enters state p just prior to reading Apq .
The proof uses induction on i, the number of steps to derive vi from Aq0 ,qaccept . Basis: i = 0.
In this case, vi = Aq0 ,qaccept and P starts in state q0 so the basis is true.
Induction step: Assume the claim for i and prove it for i + 1.
First consider the case where vi = xApqy and Apq is the variable substituted in the step vi ⇒ vi+1. The induction hypothesis implies that P enters state p after it reads x, prior to reading symbol Apq . According to G’s construction the substitution rules may be of two types:
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150 CHAPTER 2 / CONTEXT-FREE LANGUAGES 1. Apq → ApraAstb or
2. App → ε.
Thus either vi+1 = xApraAstby or vi+1 = xy, depending on which type of rule was used. In the first case, when P reads ApraAstb in vi+1, we know it starts in state p, because it has just finished reading x. As P reads ApraAstb in vi+1, it enters the sequence of states r, s, t, and q, due to the substitution rule’s construction. Therefore, it enters state p just prior to reading Apr and it enters state s just prior to reading Ast, thereby establishing the claim for these two occurrences of variables. The claim holds on occurrences of variables in the y part because, after P reads b it enters state q and then it reads string y. On input vi, it also enters q just before reading y, so the computations agree on the y parts of vi and vi+1. Obviously, the computations agree on the x parts. Therefore, the claim holds for vi+1. In the second case, no new variables are introduced, so we only need to observe that the computations agree on the x and y parts of vi and vi+1. This proves the claim.
CLAIM 2.61
G passes the DK-test.
We show that each of DK’s accept states satisfies the DK-test requirements. Select one of these accept states. It contains a completed rule R. This com- pleted rule may have one of two forms:
1. Apq → ApraAstb. 2. App → .
In both situations, we need to show that the accept state cannot contain a. another completed rule, and
b. a dotted rule that has a terminal symbol immediately after the dot.
We consider each of these four cases separately. In each case, we start by
considering a string z on which DK goes to the accept state we selected above.
Case 1a. Here R is a completed type 1-2 rule. For any rule in this accept state, z must end with the symbols preceding the dot in that rule because DK goes to that state on z. Hence the symbols preceding the dot must be consistent in all such rules. These symbols are ApraAstb in R so any other type 1-2 completed rule must have exactly the same symbols on the right-hand side. It follows that the variables on the left-hand side must also agree, so the rules must be the same.
Suppose the accept state contains R and some type 3 completed ε-rule T . From R we know that z ends with ApraAstb. Moreover, we know that P pops its stack at the very end of z because a pop occurs at that point in R, due to G’s construction. According to the way we build DK, a completed ε-rule in a state must derive from a dotted rule that resides in the same state, where the dot isn’t at the very beginning and the dot immediately precedes some variable. (An
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2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 151
exception occurs at DK’s start state, where this dot may occur at the beginning of the rule, but this accept state cannot be the start state because it contains a completed type 1-2 rule.) In G, that means T derives from a type 1-2 dotted rule where the dot precedes the second variable. From G’s construction a push occurs just before the dot. This implies that P does a push move at the very end of z, contradicting our previous statement. Thus the completed ε-rule T cannot exist. Either way, a second completed rule of either type cannot occur in this accept state.
Case 2a. Here R is a completed ε-rule App → .. We show that no other com- pleted ε-rule Aqq → . can coexist with R. If it does, the preceding claim shows that P must be in p after reading z and it must also be in q after reading z. Hence p = q and therefore the two completed ε-rules are the same.
Case 1b. Here R is a completed type 1-2 rule. From Case 1a, we know that P pops its stack at the end of z. Suppose the accept state also contains a dotted rule T where a terminal symbol immediately follows the dot. From T we know that P doesn’t pop its stack at the end of z. This contradiction shows that this situation cannot arise.
Case 2b. Here R is a completed ε-rule. Assume that the accept state also contains a dotted rule T where a terminal symbol immediately follows the dot. Because T is of type 1-2, a variable symbol immediately precedes the dot, and thus z ends with that variable symbol. Moreover, after P reads z it is prepared to read a non-ε input symbol because a terminal follows the dot. As in Case 1a, the completed ε-rule R derives from a type 1-2 dotted rule S where the dot immediately precedes the second variable. (Again this accept state cannot be DK ’s start state because the dot doesn’t occur at the beginning of T .) Thus some symbol aˆ ∈ Σε immediately precedes the dot in S and so z ends with aˆ. Either aˆ ∈ Σ or aˆ = ε, but because z ends with a variable symbol, aˆ ̸∈ Σ so aˆ = ε. Therefore, after P reads z but before it makes the ε-input move to process aˆ, it is prepared to read an ε input. We also showed above that P is prepared to read a non-ε input symbol at this point. But a DPDA isn’t allowed to make both an ε-input move and a move that reads a non-ε input symbol at a given state and stack, so the above situation is impossible. Thus this situation cannot occur.
PARSING AND LR(K) GRAMMARS
Deterministic context-free languages are of major practical importance. Their algorithms for membership and parsing are based on DPDAs and are therefore ef- ficient, and they encompass a rich class of CFLs that include most programming languages. However, DCFGs are sometimes inconvenient for expressing partic- ular DCFLs. The requirement that all handles are forced is often an obstacle to designing intuitive DCFGs.
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152 CHAPTER 2 / CONTEXT-FREE LANGUAGES
Fortunately, a broader class of grammars called the LR(k) grammars gives us the best of both worlds. They are close enough to DCFGs to allow direct conver- sion into DPDAs. Yet they are expressive enough for many applications.
Algorithms for LR(k) grammars introduce lookahead. In a DCFG, all handles are forced. A handle depends only on the symbols in a valid string up through and including the handle, but not on terminal symbols that follow the handle. In an LR(k) grammar, a handle may also depend on symbols that follow the handle, but only on the first k of these. The acronym LR(k) stands for: Left to right input
To make this precise, let h be a handle of a valid string v = xhy. Say that h is forced by lookahead k if h is the unique handle of every valid string xhyˆ where yˆ ∈ Σ∗ and where y and yˆ agree on their first k symbols. (If either string is shorter than k, the strings must agree up to the length of the shorter one.)
ightmost derivations (or equivalently, leftmost reductions), and k symbols of lookahead.
processing, R
DEFINITION 2.62
An LR(k) grammar is a context-free grammar such that the handle of every valid string is forced by lookahead k.
Thus a DCFG is the same as an LR(0) grammar. We can show that for every k we can convert LR(k) grammars to DPDAs. We’ve already shown that DPDAs are equivalent to LR(0) grammars. Hence LR(k) grammars are equivalent in power for all k and all describe exactly the DCFLs. The following example shows that LR(1) grammars are more convenient than DCFGs for specifying certain languages.
To avoid cumbersome notation and technical details, we will show how to convert LR(k) grammars to DPDAs only for the special case where k = 1. The conversion in the general case works in essentially the same way.
To begin, we’ll present a variant of the DK-test, modified for LR(1) grammars.
We call it the DK-test with lookahead 1, or simply the DK1-test. As before,
we’ll construct an NFA, called K1 here, and convert it to a DFA DK1. Each of
K1’s states has a dotted rule T → u.v and now also a terminal symbol a, called
the lookahead symbol, shown as . This state indicates that K has ✁1
recently read the string u, which would be a part of a handle uv provided that v follows after u and a follows after v.
The formal construction works much as before. The start state has an ε-move
to ✁for every rule involving the start variable S1 and every a ∈ Σ.
The shift transitions take to on input x where x is
✁ ✁for each rule C → r, where b is the first symbol of any string of
✁
✄
C→.r b ✂
✄
T→u.v a ✂
✄
S1→.u a ✂
T→u.xv a ✂
✄
T→ux.v a ✂
✄
a variable symbol or terminal symbol. The ε-transitions take to
terminals that can be derived from v. If v derives ε, add b = a. The accept states
are all for completed rules B → u. and a ∈ Σ. ✁
✁✁
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T→u.Cv a ✂
✄
✄ ✄
✂ ✂
B→u. a
2.4 DETERMINISTIC CONTEXT-FREE LANGUAGES 153 Let R1 be a completed rule with lookahead symbol a1, and let R2 be a dotted
rule with lookahead symbol a2. Say that R1 and R2 are consistent if 1. R2 is completed and a1 = a2, or
2. R2 is not completed and a1 immediately follows its dot.
Now we are ready to describe the DK1-test. Construct the DFA DK1. The test stipulates that every accept state must not contain any two consistent dotted rules.
THEOREM 2.63
G passes the DK1-test iff G is an LR(1) grammar.
PROOF IDEA Corollary 2.51 still applies to DK1 because we can ignore the lookahead symbols.
EXAMPLE 2.64
This example shows that the following grammar passes the DK1-test. Recall that in Example 2.53 this grammar was shown to fail the DK-test. Hence it is an example of a grammar that is LR(1) but not a DCFG.
S •E⊣ a+×⊣ E•E+T+⊣ E•T +⊣ T •T×a ×+⊣ T •a ×+⊣
T
S → E⊣ E→E+T|T T→Txa|a
E S E•⊣ a+×⊣ ⊣ EE•+T+⊣
S E⊣• a+×⊣
E E+T• +⊣ T T•×a ×+⊣
×
T Tוa ×+⊣ a
T T×a• ×+⊣
+
a
×
a
E E+•T T •T×a T •a
T a•
+⊣ T ×+⊣
×+⊣
×+⊣
ET• +⊣ T T•×a ×+⊣
FIGURE 2.65 Passing the DK1-test
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154 CHAPTER 2 / CONTEXT-FREE LANGUAGES THEOREM 2.66
An endmarked language is generated by an LR(1) grammar iff it is a DCFL.
We’ve already shown that every DCFL has an LR(0) grammar, because an LR(0) grammar is the same as a DCFG. That proves the reverse direction of the the- orem. What remains is the following lemma, which shows how to convert an LR(1) grammar to a DPDA.
LEMMA 2.67
Every LR(1) grammar has an equivalent DPDA.
PROOF IDEA We construct P1, a modified version of the DPDA P that we presented in Lemma 2.67. P1 reads its input and simulates DK1, while using the stack to keep track of the state DK1 would be in if all reduce steps were applied to this input up to this point. Moreover, P1 reads 1 symbol ahead and stores this lookahead information in its finite state memory. Whenever DK1 reaches an accept state, P1 consults its lookahead to see whether to perform a reduce step, and which step to do if several possibilities appear in this state. Only one option can apply because the grammar is LR(1).
EXERCISES
2.1
Recall the CFG G4 that we gave in Example 2.4. For convenience, let’s rename its variables with single letters as follows.
E→E+T|T T→TxF|F F → (E) | a
Give parse trees and derivations for each string.
a. a c. a+a+a b. a+a d. ((a))
a. UsethelanguagesA={ambncn|m,n≥0}andB={anbncm|m,n≥0} together with Example 2.36 to show that the class of context-free languages is not closed under intersection.
b. Use part (a) and DeMorgan’s law (Theorem 0.20) to show that the class of context-free languages is not closed under complementation.
2.2
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A 2.3
EXERCISES 155 Answer each part for the following context-free grammar G.
R → XRX | S
S → aT b | bT a
T →XTX|X|ε
2.4
Give context-free grammars that generate the following languages. In all parts, the alphabet Σ is {0,1}.
Aa. {w| w contains at least three 1s}
b. {w| w starts and ends with the same symbol}
c. {w| the length of w is odd}
Ad. {w| the length of w is odd and its middle symbol is a 0}
e. {w| w = wR, that is, w is a palindrome}
f. The empty set
Give informal descriptions and state diagrams of pushdown automata for the lan-
guages in Exercise 2.4.
Give context-free grammars generating the following languages.
Aa. The set of strings over the alphabet {a,b} with more a’s than b’s b. The complement of the language {an bn | n ≥ 0}
Ac. {w#x| wR is a substring of x for w, x ∈ {0,1}∗}
d. {x1#x2#···#xk|k≥1,eachxi ∈{a,b}∗,andforsomeiandj, xi =xRj }
Give informal English descriptions of PDAs for the languages in Exercise 2.6.
Show that the string the girl touches the boy with the flower has two different leftmost derivations in grammar G2 on page 103. Describe in English the two different meanings of this sentence.
Give a context-free grammar that generates the language
A = {ai bj ck | i = j or j = k where i, j, k ≥ 0}.
Is your grammar ambiguous? Why or why not?
Give an informal description of a pushdown automaton that recognizes the lan-
guage A in Exercise 2.9.
Convert the CFG G4 given in Exercise 2.1 to an equivalent PDA, using the proce-
dure given in Theorem 2.20.
2.5 2.6
A 2.7 A2.8
2.9
2.10 2.11
X→a|b
a. What are the variables of G?
b. What are the terminals of G?
c. Which is the start variable of G?
d. Give three strings in L(G).
e. Give three strings not in L(G).
f. True or False: T ⇒ aba.
g. True or False: T ⇒∗ aba.
h. TrueorFalse:T⇒T.
i. TrueorFalse:T⇒∗ T.
j. True or False: XXX ⇒∗ aba. k. True or False: X ⇒∗ aba.
l. TrueorFalse:T⇒∗ XX.
m. TrueorFalse:T⇒∗ XXX.
n. True or False: S ⇒∗ ε.
o. Give a description in English of L(G).
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156 CHAPTER 2 / CONTEXT-FREE LANGUAGES
2.12 Convert the CFG G given in Exercise 2.3 to an equivalent PDA, using the procedure
given in Theorem 2.20.
2.13 Let G = (V,Σ,R,S) be the following grammar. V = {S,T,U}; Σ = {0,#}; and
R is the set of rules:
S→TT|U
T → 0T | T 0 | # U → 0U00 | #
a. Describe L(G) in English.
b. Prove that L(G) is not regular.
2.14 Convert the following CFG into an equivalent CFG in Chomsky normal form, using the procedure given in Theorem 2.9.
A → BAB | B | ε B → 00 | ε
2.15 Give a counterexample to show that the following construction fails to prove that the class of context-free languages is closed under star. Let A be a CFL that is generated by the CFG G = (V,Σ,R,S). Add the new rule S → SS and call the resulting grammar G′. This grammar is supposed to generate A∗.
2.16 Show that the class of context-free languages is closed under the regular operations, union, concatenation, and star.
2.17 Use the results of Exercise 2.16 to give another proof that every regular language is context free, by showing how to convert a regular expression directly to an equiv- alent context-free grammar.
PROBLEMS
A 2.18
⋆ 2.19
2.20 ⋆ 2.21 ⋆ 2.22
a. Let C be a context-free language and R be a regular language. Prove that the language C ∩ R is context free.
b. Let A = {w| w ∈ {a, b, c}∗ and w contains equal numbers of a’s, b’s, and c’s}. Use part (a) to show that A is not a CFL.
Let CFG G be the following grammar.
S → aSb | bY | Y a
Y → bY | aY | ε
Give a simple description of L(G) in English. Use that description to give a CFG
for L(G), the complement of L(G). LetA/B={w|wx∈Aforsomex∈B}.ShowthatifAiscontextfreeandBis
regular, then A/B is context free.
Let Σ = {a,b}. Give a CFG generating the language of strings with twice as many
a’s as b’s. Prove that your grammar is correct.
Let C = {x#y| x, y ∈ {0,1}∗ and x ̸= y}. Show that C is a context-free language.
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⋆ 2.23 ⋆ 2.24
2.25
2.26 ⋆ 2.27
PROBLEMS 157 Let D = {xy|x, y ∈ {0,1}∗ and |x| = |y| but x ̸= y}. Show that D is a context-free
language.
Let E = {aibj | i ̸= j and 2i ̸= j}. Show that E is a context-free language.
For any language A, let SUFFIX(A) = {v| uv ∈ A for some string u}. Show that the class of context-free languages is closed under the SUFFIX operation.
Show that if G is a CFG in Chomsky normal form, then for any string w ∈ L(G) of length n ≥ 1, exactly 2n − 1 steps are required for any derivation of w.
Let G = (V, Σ, R, ⟨STMT⟩) be the following grammar.
⟨STMT⟩ → ⟨ASSIGN⟩ | ⟨IF-THEN⟩ | ⟨IF-THEN-ELSE⟩ ⟨IF-THEN⟩ → if condition then ⟨STMT⟩
⟨IF-THEN-ELSE⟩ → if condition then ⟨STMT⟩ else ⟨STMT⟩ ⟨ASSIGN⟩ → a:=1
Σ = {if, condition, then, else, a:=1}
V ={⟨STMT⟩,⟨IF-THEN⟩,⟨IF-THEN-ELSE⟩,⟨ASSIGN⟩}
G is a natural-looking grammar for a fragment of a programming language, but G is ambiguous.
a. Show that G is ambiguous.
b. Give a new unambiguous grammar for the same language.
Give unambiguous CFGs for the following languages.
a. {w| in every prefix of w the number of a’s is at least the number of b’s}
b. {w| the number of a’s and the number of b’s in w are equal}
c. {w| the number of a’s is at least the number of b’s in w}
Show that the language A in Exercise 2.9 is inherently ambiguous.
Use the pumping lemma to show that the following languages are not context free.
a. {0n1n0n1n| n ≥ 0}
Ab. {0n#02n#03n| n ≥ 0}
Ac. {w#t| w is a substring of t, where w, t ∈ {a, b}∗}
d. {t1#t2#···#tk|k≥2,eachti ∈{a,b}∗,andti =tj forsomei̸=j}
Let B be the language of all palindromes over {0,1} containing equal numbers of
0s and 1s. Show that B is not context free.
Let Σ = {1,2,3,4} and C = {w ∈ Σ∗| in w, the number of 1s equals the number of 2s, and the number of 3s equals the number of 4s}. Show that C is not context free.
Show that F = {ai bj | i = kj for some positive integer k} is not context free.
Consider the language B = L(G), where G is the grammar given in Exercise 2.13. The pumping lemma for context-free languages, Theorem 2.34, states the exis- tence of a pumping length p for B. What is the minimum value of p that works in the pumping lemma? Justify your answer.
Let G be a CFG in Chomsky normal form that contains b variables. Show that if G generates some string with a derivation having at least 2b steps, L(G) is infinite.
⋆ 2.28
⋆ 2.29 2.30
2.31 2.32
⋆ 2.33 2.34
2.35
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158
CHAPTER 2 / CONTEXT-FREE LANGUAGES
2.36 ⋆ 2.37
A 2.38 2.39 ⋆2.40
⋆ 2.41
⋆2.42 2.43
Give an example of a language that is not context free but that acts like a CFL in the pumping lemma. Prove that your example works. (See the analogous example for regular languages in Problem 1.54.)
Prove the following stronger form of the pumping lemma, wherein both pieces v and y must be nonempty when the string s is broken up.
If A is a context-free language, then there is a number k where, if s is any string in A of length at least k, then s may be divided into five pieces, s = uvxyz, satisfying the conditions:
a. foreachi≥0, uvixyiz∈A, b. v̸=εandy̸=ε,and
c. |vxy| ≤ k.
Refer to Problem 1.41 for the definition of the perfect shuffle operation. Show that the class of context-free languages is not closed under perfect shuffle.
Refer to Problem 1.42 for the definition of the shuffle operation. Show that the class of context-free languages is not closed under shuffle.
Say that a language is prefix-closed if all prefixes of every string in the language are also in the language. Let C be an infinite, prefix-closed, context-free language. Show that C contains an infinite regular subset.
Read the definitions of NOPREFIX (A) and NOEXTEND (A) in Problem 1.40.
a. Show that the class of CFLs is not closed under NOPREFIX .
b. Show that the class of CFLs is not closed under NOEXTEND.
LetY={w|w=t1#t2#···#tk fork≥0,eachti∈1∗,andti̸=tj wheneveri̸=j}.
Here Σ = {1, #}. Prove that Y is not context free.
For strings w and t, write w t if the symbols of w are a permutation of the symbols of t. In other words, w t if t and w have the same symbols in the same quantities, but possibly in a different order.
For any string w, define SCRAMBLE(w) = {t| t w}. For any language A, let SCRAMBLE(A) = {t| t ∈ SCRAMBLE(w) for some w ∈ A}.
a. Show that if Σ = {0,1}, then the SCRAMBLE of a regular language is con- text free.
b. What happens in part (a) if Σ contains three or more symbols? Prove your answer.
IfAandBarelanguages,defineA⋄B={xy|x∈Aandy∈Band|x|=|y|}. Show that if A and B are regular languages, then A ⋄ B is a CFL.
Let A = {wtwR| w,t ∈ {0,1}∗ and |w| = |t|}. Prove that A is not a CFL. Consider the following CFG G:
S → SS | T T → aTb|ab
Describe L(G) and show that G is ambiguous. Give an unambiguous grammar H where L(H) = L(G) and sketch a proof that H is unambiguous.
2.44
⋆ 2.45 2.46
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2.47
2.48
⋆2.49 ⋆ 2.50
2.51 A⋆ 2.52 ⋆ 2.53
2.54
PROBLEMS 159 Let Σ = {0,1} and let B be the collection of strings that contain at least one 1 in
their second half. In other words, B = {uv| u ∈ Σ∗, v ∈ Σ∗1Σ∗ and |u| ≥ |v|}. a. Give a PDA that recognizes B.
b. Give a CFG that generates B.
Let Σ = {0,1}. Let C1 be the language of all strings that contain a 1 in their middle third. Let C2 be the language of all strings that contain two 1s in their middle third. So C1 = {xyz| x,z ∈ Σ∗ and y ∈ Σ∗1Σ∗, where |x| = |z| ≥ |y|} and C2 = {xyz| x,z ∈ Σ∗ and y ∈ Σ∗1Σ∗1Σ∗, where |x| = |z| ≥ |y|}.
a. Show that C1 is a CFL.
b. Show that C2 is not a CFL.
We defined the rotational closure of language A to be RC(A) = {yx| xy ∈ A}. Show that the class of CFLs is closed under rotational closure.
We defined the CUT of language A to be CUT(A) = {yxz| xyz ∈ A}. Show that the class of CFLs is not closed under CUT.
Show that every DCFG is an unambiguous CFG.
Show that every DCFG generates a prefix-free language.
Show that the class of DCFLs is not closed under the following operations:
a. Union
b. Intersection
c. Concatenation
d. Star
e. Reversal
Let G be the following grammar:
S → T⊣
T →TaTb|TbTa|ε
a. Show that L(G) = {w⊣| w contains equal numbers of a’s and b’s}. Use a
proof by induction on the length of w.
b. Use the DK -test to show that G is a DCFG.
c. Describe a DPDA that recognizes L(G).
Let G1 be the following grammar that we introduced in Example 2.45. Use the DK-test to show that G1 is not a DCFG.
R→S|T
S → aSb | ab
T → aT bb | abb
2.55
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160
CHAPTER 2 / CONTEXT-FREE LANGUAGES
⋆ 2.56
⋆2.57 ⋆2.58
⋆ 2.59
Let A = L(G1) where G1 is defined in Problem 2.55. Show that A is not a DCFL. (Hint: Assume that A is a DCFL and consider its DPDA P . Modify P so that its input alphabet is {a, b, c}. When it first enters an accept state, it pretends that c’s are b’s in the input from that point on. What language would the modified P accept?)
LetB={aibjck|i,j,k≥0andi=jori=k}.ProvethatBisnotaDCFL.
Let C = {wwR| w ∈ {0,1}∗}. Prove that C is not a DCFL. (Hint: Suppose that when some DPDA P is started in state q with symbol x on the top of its stack, P never pops its stack below x, no matter what input string P reads from that point on. In that case, the contents of P ’s stack at that point cannot affect its subsequent behavior, so P ’s subsequent behavior can depend only on q and x.)
If we disallow ε-rules in CFGs, we can simplify the DK-test. In the simplified test, we only need to check that each of DK’s accept states has a single rule. Prove that a CFG without ε-rules passes the simplified DK-test iff it is a DCFG.
SELECTED SOLUTIONS
2.3 (a) R,X,S,T; (b) a,b; (c) R; (d) Three strings in L(G) are ab,ba, and aab; (e)ThreestringsnotinL(G)area,b,andε; (f)False; (g)True; (h)False; (i) True; (j) True; (k) False; (l) True; (m) True; (n) False; (o) L(G) consists of all strings over a and b that are not palindromes.
2.4 (a) S → R1R1R1R R → 0R|1R|ε
2.6 (a) S → TaT
T → TT |aTb|bTa|a|ε
T generates all strings with at least as many a’s as b’s, and S forces an extra a.
(d) S → 0|0S0|0S1|1S0|1S1
(c) S → TX
T → 0T0|1T1|#X
X → 0X |1X |ε
2.7 (a) The PDA uses its stack to count the number of a’s minus the number of b’s. It enters an accepting state whenever this count is positive. In more detail, it operates as follows. The PDA scans across the input. If it sees a b and its top stack symbol is an a, it pops the stack. Similarly, if it scans an a and its top stack symbol is a b, it pops the stack. In all other cases, it pushes the input symbol onto the stack. After the PDA finishes the input, if a is on top of the stack, it accepts. Otherwise it rejects.
(c) The PDA scans across the input string and pushes every symbol it reads until it reads a #. If a # is never encountered, it rejects. Then, the PDA skips over part of the input, nondeterministically deciding when to stop skipping. At that point, it compares the next input symbols with the symbols it pops off the stack. At any disagreement, or if the input finishes while the stack is nonempty, this branch of the computation rejects. If the stack becomes empty, the machine reads the rest of the input and accepts.
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2.8 Here is one derivation:
⟨SENTENCE⟩ → ⟨NOUN-PHRASE⟩⟨VERB-PHRASE⟩ → ⟨CMPLX-NOUN⟩⟨VERB-PHRASE⟩ → ⟨ARTICLE⟩⟨NOUN⟩⟨VERB-PHRASE⟩ →
The ⟨NOUN⟩⟨VERB-PHRASE⟩ →
The girl ⟨VERB-PHRASE⟩ →
The girl ⟨CMPLX-VERB⟩⟨PREP-PHRASE⟩ →
The girl ⟨VERB⟩⟨NOUN-PHRASE⟩⟨PREP-PHRASE⟩ → The girl touches ⟨NOUN-PHRASE⟩⟨PREP-PHRASE⟩ → The girl touches ⟨CMPLX-NOUN⟩⟨PREP-PHRASE⟩ → The girl touches ⟨ARTICLE⟩⟨NOUN⟩⟨PREP-PHRASE⟩ → The girl touches the ⟨NOUN⟩⟨PREP-PHRASE⟩ →
The girl touches the boy ⟨PREP-PHRASE⟩ →
The girl touches the boy ⟨PREP⟩⟨CMPLX-NOUN⟩ → The girl touches the boy with ⟨CMPLX-NOUN⟩ → The girl touches the boy with ⟨ARTICLE⟩⟨NOUN⟩ → The girl touches the boy with the ⟨NOUN⟩ →
The girl touches the boy with the flower
Here is another leftmost derivation:
⟨SENTENCE⟩ → ⟨NOUN-PHRASE⟩⟨VERB-PHRASE⟩ → ⟨CMPLX-NOUN⟩⟨VERB-PHRASE⟩ → ⟨ARTICLE⟩⟨NOUN⟩⟨VERB-PHRASE⟩ →
The ⟨NOUN⟩⟨VERB-PHRASE⟩ →
The girl ⟨VERB-PHRASE⟩ →
The girl ⟨CMPLX-VERB⟩ →
The girl ⟨VERB⟩⟨NOUN-PHRASE⟩ →
The girl touches ⟨NOUN-PHRASE⟩ →
The girl touches ⟨CMPLX-NOUN⟩⟨PREP-PHRASE⟩ → The girl touches ⟨ARTICLE⟩⟨NOUN⟩⟨PREP-PHRASE⟩ → The girl touches the ⟨NOUN⟩⟨PREP-PHRASE⟩ →
The girl touches the boy ⟨PREP-PHRASE⟩ →
The girl touches the boy ⟨PREP⟩⟨CMPLX-NOUN⟩ → The girl touches the boy with ⟨CMPLX-NOUN⟩ → The girl touches the boy with ⟨ARTICLE⟩⟨NOUN⟩ → The girl touches the boy with the ⟨NOUN⟩ →
The girl touches the boy with the flower
Each of these derivations corresponds to a different English meaning. In the first derivation, the sentence means that the girl used the flower to touch the boy. In the second derivation, the boy is holding the flower when the girl touches her.
2.18 (a) Let C be a context-free language and R be a regular language. Let P be the PDA that recognizes C, and D be the DFA that recognizes R. If Q is the set of states of P and Q′ is the set of states of D, we construct a PDA P ′ that recognizes C∩RwiththesetofstatesQ×Q′.P′ willdowhatP doesandalsokeeptrackof thestatesofD.Itacceptsastringwifandonlyifitstopsatastateq∈FP ×FD, where FP is the set of accept states of P and FD is the set of accept states of D. Since C ∩ R is recognized by P ′ , it is context free.
(b) Let R be the regular language a∗b∗c∗. If A were a CFL then A ∩ R would be a CFL by part (a). However, A ∩ R = {anbncn| n ≥ 0}, and Example 2.36 proves that A ∩ R is not context free. Thus A is not a CFL.
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SELECTED SOLUTIONS 161
162 CHAPTER 2 / CONTEXT-FREE LANGUAGES
2.30 (b) Let B = {0n#02n#03n| n ≥ 0}. Let p be the pumping length given by the pumping lemma. Let s = 0p#02p#03p. We show that s = uvxyz cannot be pumped.
Neither v nor y can contain #, otherwise uv2xy2z contains more than two #s. Therefore, if we divide s into three segments by #’s: 0p,02p,and 03p, at least one of the segments is not contained within either v or y. Hence uv2xy2z is not in B because the 1 : 2 : 3 length ratio of the segments is not maintained.
(c) Let C = {w#t| w is a substring of t, where w,t ∈ {a,b}∗}. Let p be the pumping length given by the pumping lemma. Let s = apbp#apbp. We show that the string s = uvxyz cannot be pumped.
Neither v nor y can contain #, otherwise uv0xy0z does not contain # and therefore is not in C. If both v and y occur on the left-hand side of the #, the string uv2xy2z cannot be in C because it is longer on the left-hand side of the #. Similarly, if both strings occur on the right-hand side of the #, the string uv0xy0z cannot be in C because it is again longer on the left-hand side of the #. If one of v and y is empty (both cannot be empty), treat them as if both occurred on the same side of the # as above.
The only remaining case is where both v and y are nonempty and straddle the #. But then v consists of b’s and y consists of a’s because of the third pumping lemma condition |vxy| ≤ p. Hence, uv2xy2z contains more b’s on the left-hand side of the #, so it cannot be a member of C.
2.38 Let A be the language {0k1k| k ≥ 0} and let B be the language {akb3k| k ≥ 0}. The perfect shuffle of A and B is the language C = {(0a)k(0b)k(1b)2k| k ≥ 0}. Languages A and B are easily seen to be CFLs, but C is not a CFL, as follows. If C were a CFL, let p be the pumping length given by the pumping lemma, and let s be the string (0a)p(0b)p(1b)2p. Because s is longer than p and s ∈ C, we can divide s = uvxyz satisfying the pumping lemma’s three conditions. Strings in C are exactly one-fourth 1s and one-eighth a’s. In order for uv2xy2z to have that property, the string vxy must contain both 1s and a’s. But that is impossible, because the 1s and a’s are separated by 2p symbols in s yet the third condition says that |vxy| ≤ p. Hence C is not context free.
2.52 We use a proof by contradiction. Assume that w and wz are two unequal strings in L(G), where G is a DCFG. Both are valid strings so both have handles, and these handles must agree because we can write w = xhy and wz = xhyz = xhyˆ where h is the handle of w. Hence, the first reduce steps of w and wz produce valid strings u and uz, respectively. We can continue this process until we obtain S1 and S1z where S1 is the start variable. However, S1 does not appear on the right-hand side of any rule so we cannot reduce S1z. That gives a contradiction.
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PART TWO
COMPUTABILITY THEORY
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3
THE CHURCH-TURING THESIS
So far in our development of the theory of computation, we have presented sev- eral models of computing devices. Finite automata are good models for devices that have a small amount of memory. Pushdown automata are good models for devices that have an unlimited memory that is usable only in the last in, first out manner of a stack. We have shown that some very simple tasks are beyond the capabilities of these models. Hence they are too restricted to serve as models of general purpose computers.
3.1
TURING MACHINES
We turn now to a much more powerful model, first proposed by Alan Turing in 1936, called the Turing machine. Similar to a finite automaton but with an unlimited and unrestricted memory, a Turing machine is a much more accurate model of a general purpose computer. A Turing machine can do everything that a real computer can do. Nonetheless, even a Turing machine cannot solve certain problems. In a very real sense, these problems are beyond the theoretical limits of computation.
The Turing machine model uses an infinite tape as its unlimited memory. It has a tape head that can read and write symbols and move around on the tape.
165
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166 CHAPTER 3 / THE CHURCH—TURING THESIS
Initially the tape contains only the input string and is blank everywhere else. If the machine needs to store information, it may write this information on the tape. To read the information that it has written, the machine can move its head back over it. The machine continues computing until it decides to produce an output. The outputs accept and reject are obtained by entering designated accepting and rejecting states. If it doesn’t enter an accepting or a rejecting state, it will go on forever, never halting.
FIGURE 3.1
Schematic of a Turing machine
The following list summarizes the differences between finite automata and Turing machines.
1. A Turing machine can both write on the tape and read from it.
2. The read–write head can move both to the left and to the right.
3. The tape is infinite.
4. The special states for rejecting and accepting take effect immediately.
Let’s introduce a Turing machine M1 for testing membership in the language B = {w#w| w ∈ {0,1}∗}. We want M1 to accept if its input is a member of B and to reject otherwise. To understand M1 better, put yourself in its place by imagining that you are standing on a mile-long input consisting of millions of characters. Your goal is to determine whether the input is a member of B—that is, whether the input comprises two identical strings separated by a # symbol. The input is too long for you to remember it all, but you are allowed to move back and forth over the input and make marks on it. The obvious strategy is to zig-zag to the corresponding places on the two sides of the # and determine whether they match. Place marks on the tape to keep track of which places correspond.
We design M1 to work in that way. It makes multiple passes over the input string with the read–write head. On each pass it matches one of the characters on each side of the # symbol. To keep track of which symbols have been checked already, M1 crosses off each symbol as it is examined. If it crosses off all the symbols, that means that everything matched successfully, and M1 goes into an accept state. If it discovers a mismatch, it enters a reject state. In summary, M1’s algorithm is as follows.
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M1 = “On input string w:
1. Zig-zag across the tape to corresponding positions on either
side of the # symbol to check whether these positions contain the same symbol. If they do not, or if no # is found, reject. Cross off symbols as they are checked to keep track of which symbols correspond.
2. When all symbols to the left of the # have been crossed off, check for any remaining symbols to the right of the #. If any symbols remain, reject; otherwise, accept.”
The following figure contains several nonconsecutive snapshots of M1’s tape after it is started on input 011000#011000.
3.1 TURING MACHINES 167
FIGURE 3.2
Snapshots of Turing machine M1 computing on input 011000#011000
This description of Turing machine M1 sketches the way it functions but does not give all its details. We can describe Turing machines in complete detail by giving formal descriptions analogous to those introduced for finite and push- down automata. The formal descriptions specify each of the parts of the formal definition of the Turing machine model to be presented shortly. In actuality, we almost never give formal descriptions of Turing machines because they tend to be very big.
FORMAL DEFINITION OF A TURING MACHINE
The heart of the definition of a Turing machine is the transition function δ be- cause it tells us how the machine gets from one step to the next. For a Turing machine, δ takes the form: Q × Γ −→ Q × Γ × {L, R}. That is, when the machine
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168 CHAPTER 3 / THE CHURCH—TURING THESIS
is in a certain state q and the head is over a tape square containing a symbol a, and if δ(q, a) = (r, b, L), the machine writes the symbol b replacing the a, and goes to state r. The third component is either L or R and indicates whether the head moves to the left or right after writing. In this case, the L indicates a move to the left.
DEFINITION 3.3
A Turing machine is a 7-tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where
Q, Σ, Γ are all finite sets and
1. Q is the set of states,
2. Σ is the input alphabet not containing the blank symbol ␣, 3. Γisthetapealphabet,where␣∈ΓandΣ⊆Γ,
4. δ: Q×Γ−→Q×Γ×{L,R}isthetransitionfunction,
5. q0 ∈ Q is the start state,
6. qaccept ∈ Q is the accept state, and
7. qreject ∈ Q is the reject state, where qreject ̸= qaccept.
A Turing machine M = (Q, Σ, Γ, δ, q0, qaccept, qreject) computes as follows. Ini- tially, M receives its input w = w1w2 . . . wn ∈ Σ∗ on the leftmost n squares of the tape, and the rest of the tape is blank (i.e., filled with blank symbols). The head starts on the leftmost square of the tape. Note that Σ does not contain the blank symbol, so the first blank appearing on the tape marks the end of the input. Once M has started, the computation proceeds according to the rules described by the transition function. If M ever tries to move its head to the left off the left-hand end of the tape, the head stays in the same place for that move, even though the transition function indicates L. The computation continues until it enters either the accept or reject states, at which point it halts. If neither occurs, M goes on forever.
As a Turing machine computes, changes occur in the current state, the cur- rent tape contents, and the current head location. A setting of these three items is called a configuration of the Turing machine. Configurations often are rep- resented in a special way. For a state q and two strings u and v over the tape alphabet Γ, we write u q v for the configuration where the current state is q, the current tape contents is uv, and the current head location is the first symbol of v. The tape contains only blanks following the last symbol of v. For example, 1011q701111 represents the configuration when the tape is 101101111, the cur- rent state is q7, and the head is currently on the second 0. Figure 3.4 depicts a Turing machine with that configuration.
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3.1 TURING MACHINES 169
FIGURE 3.4
A Turing machine with configuration 1011q701111
Here we formalize our intuitive understanding of the way that a Turing ma- chine computes. Say that configuration C1 yields configuration C2 if the Turing machine can legally go from C1 to C2 in a single step. We define this notion formally as follows.
Supposethatwehavea,b,andcinΓ,aswellasuandvinΓ∗ andstatesqi and qj . In that case, ua qi bv and u qj acv are two configurations. Say that
ua qi bv yields u qj acv
if in the transition function δ(qi,b) = (qj,c,L). That handles the case where the
Turing machine moves leftward. For a rightward move, say that ua qi bv yields uac qj v
if δ(qi,b) = (qj,c,R).
Special cases occur when the head is at one of the ends of the configuration.
For the left-hand end, the configuration qi bv yields qj cv if the transition is left- moving (because we prevent the machine from going off the left-hand end of the tape), and it yields c qj v for the right-moving transition. For the right-hand end, the configuration ua qi is equivalent to ua qi ␣ because we assume that blanks follow the part of the tape represented in the configuration. Thus we can handle this case as before, with the head no longer at the right-hand end.
The start configuration of M on input w is the configuration q0 w, which indicates that the machine is in the start state q0 with its head at the leftmost position on the tape. In an accepting configuration, the state of the configuration is qaccept. In a rejecting configuration, the state of the configuration is qreject. Accepting and rejecting configurations are halting configurations and do not yield further configurations. Because the machine is defined to halt when in the states qaccept and qreject, we equivalently could have defined the transition function tohavethemorecomplicatedformδ:Q′×Γ−→Q×Γ×{L,R},whereQ′ isQ without qaccept and qreject. A Turing machine M accepts input w if a sequence of configurations C1, C2, . . . , Ck exists, where
1. C1 is the start configuration of M on input w, 2. each Ci yields Ci+1, and
3. Ck is an accepting configuration.
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170 CHAPTER 3 / THE CHURCH—TURING THESIS
The collection of strings that M accepts is the language of M, or the lan-
guage recognized by M, denoted L(M).
DEFINITION 3.5
Call a language Turing-recognizable if some Turing machine recognizes it.1
When we start a Turing machine on an input, three outcomes are possible. The machine may accept, reject, or loop. By loop we mean that the machine simply does not halt. Looping may entail any simple or complex behavior that never leads to a halting state.
A Turing machine M can fail to accept an input by entering the qreject state and rejecting, or by looping. Sometimes distinguishing a machine that is looping from one that is merely taking a long time is difficult. For this reason, we prefer Turing machines that halt on all inputs; such machines never loop. These ma- chines are called deciders because they always make a decision to accept or reject. A decider that recognizes some language also is said to decide that language.
DEFINITION 3.6
Call a language Turing-decidable or simply decidable if some Turing machine decides it.2
Next, we give examples of decidable languages. Every decidable language is Turing-recognizable. We present examples of languages that are Turing- recognizable but not decidable after we develop a technique for proving un- decidability in Chapter 4.
EXAMPLES OF TURING MACHINES
As we did for finite and pushdown automata, we can formally describe a partic- ular Turing machine by specifying each of its seven parts. However, going to that level of detail can be cumbersome for all but the tiniest Turing machines. Accordingly, we won’t spend much time giving such descriptions. Mostly we
1It is called a recursively enumerable language in some other textbooks. 2It is called a recursive language in some other textbooks.
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will give only higher level descriptions because they are precise enough for our purposes and are much easier to understand. Nevertheless, it is important to remember that every higher level description is actually just shorthand for its formal counterpart. With patience and care we could describe any of the Turing machines in this book in complete formal detail.
To help you make the connection between the formal descriptions and the higher level descriptions, we give state diagrams in the next two examples. You may skip over them if you already feel comfortable with this connection.
EXAMPLE 3.7
Here we describe a Turing machine (TM) M2 that decides A = {02n | n ≥ 0}, the
language consisting of all strings of 0s whose length is a power of 2.
M2 = “On input string w:
1. Sweep left to right across the tape, crossing off every other 0.
2. If in stage 1 the tape contained a single 0, accept.
3. If in stage 1 the tape contained more than a single 0 and the
number of 0s was odd, reject.
4. Return the head to the left-hand end of the tape.
5. Go to stage 1.”
Each iteration of stage 1 cuts the number of 0s in half. As the machine sweeps across the tape in stage 1, it keeps track of whether the number of 0s seen is even or odd. If that number is odd and greater than 1, the original number of 0s in the input could not have been a power of 2. Therefore, the machine rejects in this instance. However, if the number of 0s seen is 1, the original number must have been a power of 2. So in this case, the machine accepts.
Now we give the formal description of M2 = (Q, Σ, Γ, δ, q1, qaccept, qreject):
• Q={q1,q2,q3,q4,q5,qaccept,qreject},
• Σ={0},and
• Γ={0,x,␣}.
• Wedescribeδwithastatediagram(seeFigure3.8).
• Thestart,accept,andrejectstatesareq1,qaccept,andqreject,respectively.
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3.1 TURING MACHINES 171
172 CHAPTER 3 / THE CHURCH—TURING THESIS
FIGURE 3.8
State diagram for Turing machine M2
In this state diagram, the label 0→␣,R appears on the transition from q1 to q2. This label signifies that when in state q1 with the head reading 0, the machine goes to state q2, writes ␣, and moves the head to the right. In other words, δ(q1,0) = (q2,␣,R). For clarity we use the shorthand 0→R in the transition from q3 to q4, to mean that the machine moves to the right when reading 0 in state q3 but doesn’t alter the tape, so δ(q3,0) = (q4,0,R).
This machine begins by writing a blank symbol over the leftmost 0 on the tape so that it can find the left-hand end of the tape in stage 4. Whereas we would normally use a more suggestive symbol such as # for the left-hand end delimiter, we use a blank here to keep the tape alphabet, and hence the state diagram, small. Example 3.11 gives another method of finding the left-hand end of the tape.
Next we give a sample run of this machine on input 0000. The starting con- figuration is q10000. The sequence of configurations the machine enters appears as follows; read down the columns and left to right.
q1 0000 ␣q2 000 ␣xq3 00 ␣x0q4 0 ␣x0xq3 ␣ ␣x0q5 x␣ ␣xq5 0x␣
␣q5 x0x␣ q5 ␣x0x␣ ␣q2 x0x␣ ␣xq2 0x␣ ␣xxq3 x␣ ␣xxxq3 ␣ ␣xxq5 x␣
␣xq5 xx␣ ␣q5 xxx␣
q5 ␣xxx␣ ␣q2 xxx␣ ␣xq2 xx␣ ␣xxq2 x␣ ␣xxxq2 ␣ ␣xxx␣qaccept
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EXAMPLE 3.9
ThefollowingisaformaldescriptionofM1 =(Q,Σ,Γ,δ,q1,qaccept,qreject),the Turing machine that we informally described (page 167) for deciding the lan- guage B = {w#w| w ∈ {0,1}∗}.
• Q={q1,…,q8,qaccept,qreject},
• Σ={0,1,#},andΓ={0,1,#,x,␣}.
• Wedescribeδwithastatediagram(seethefollowingfigure).
• Thestart,accept,andrejectstatesareq1,qaccept,andqreject,respectively.
3.1 TURING MACHINES 173
FIGURE 3.10
State diagram for Turing machine M1
In Figure 3.10, which depicts the state diagram of TM M1, you will find the label 0,1→R on the transition going from q3 to itself. That label means that the machine stays in q3 and moves to the right when it reads a 0 or a 1 in state q3. It doesn’t change the symbol on the tape.
Stage 1 is implemented by states q1 through q7, and stage 2 by the remaining states. To simplify the figure, we don’t show the reject state or the transitions going to the reject state. Those transitions occur implicitly whenever a state lacks an outgoing transition for a particular symbol. Thus because in state q5 no outgoing arrow with a # is present, if a # occurs under the head when the machine is in state q5, it goes to state qreject. For completeness, we say that the head moves right in each of these transitions to the reject state.
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174 CHAPTER 3 / THE CHURCH—TURING THESIS EXAMPLE 3.11
Here, a TM M3 is doing some elementary arithmetic. It decides the language C = {ai bj ck | i × j = k and i, j, k ≥ 1}.
M3 = “On input string w:
1. Scan the input from left to right to determine whether it is a
member of a+b+c+ and reject if it isn’t.
2. Return the head to the left-hand end of the tape.
3. Cross off an a and scan to the right until a b occurs. Shuttle
between the b’s and the c’s, crossing off one of each until all b’s are gone. If all c’s have been crossed off and some b’s remain, reject .
4. Restore the crossed off b’s and repeat stage 3 if there is another a to cross off. If all a’s have been crossed off, determine whether all c’s also have been crossed off. If yes, accept; otherwise, reject .”
Let’s examine the four stages of M3 more closely. In stage 1, the machine operates like a finite automaton. No writing is necessary as the head moves from left to right, keeping track by using its states to determine whether the input is in the proper form.
Stage 2 looks equally simple but contains a subtlety. How can the TM find the left-hand end of the input tape? Finding the right-hand end of the input is easy because it is terminated with a blank symbol. But the left-hand end has no terminator initially. One technique that allows the machine to find the left- hand end of the tape is for it to mark the leftmost symbol in some way when the machine starts with its head on that symbol. Then the machine may scan left until it finds the mark when it wants to reset its head to the left-hand end. Example 3.7 illustrated this technique; a blank symbol marks the left-hand end.
A trickier method of finding the left-hand end of the tape takes advantage of the way that we defined the Turing machine model. Recall that if the machine tries to move its head beyond the left-hand end of the tape, it stays in the same place. We can use this feature to make a left-hand end detector. To detect whether the head is sitting on the left-hand end, the machine can write a special symbol over the current position while recording the symbol that it replaced in the control. Then it can attempt to move the head to the left. If it is still over the special symbol, the leftward move didn’t succeed, and thus the head must have been at the left-hand end. If instead it is over a different symbol, some symbols remained to the left of that position on the tape. Before going farther, the machine must be sure to restore the changed symbol to the original.
Stages 3 and 4 have straightforward implementations and use several states each.
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EXAMPLE 3.12
Here, a TM M4 is solving what is called the element distinctness problem. It is given a list of strings over {0,1} separated by #s and its job is to accept if all the strings are different. The language is
E={#x1#x2#···#xl|eachxi ∈{0,1}∗ andxi ̸=xj foreachi̸=j}.
Machine M4 works by comparing x1 with x2 through xl, then by comparing x2 with x3 through xl, and so on. An informal description of the TM M4 deciding this language follows.
M4 = “On input w:
1. Place a mark on top of the leftmost tape symbol. If that symbol
was a blank, accept. If that symbol was a #, continue with the
next stage. Otherwise, reject.
2. Scan right to the next # and place a second mark on top of it. If
no # is encountered before a blank symbol, only x1 was present,
so accept.
3. By zig-zagging, compare the two strings to the right of the
marked #s. If they are equal, reject.
4. Move the rightmost of the two marks to the next # symbol to
the right. If no # symbol is encountered before a blank sym- bol, move the leftmost mark to the next # to its right and the rightmost mark to the # after that. This time, if no # is available for the rightmost mark, all the strings have been compared, so accept .
5. Go to stage 3.”
This machine illustrates the technique of marking tape symbols. In stage 2, the machine places a mark above a symbol, # in this case. In the actual imple- mentation, the machine has two different symbols, # and #•, in its tape alphabet. Saying that the machine places a mark above a # means that the machine writes the symbol #• at that location. Removing the mark means that the machine writes the symbol without the dot. In general, we may want to place marks over vari- ous symbols on the tape. To do so, we merely include versions of all these tape symbols with dots in the tape alphabet.
We conclude from the preceding examples that the described languages A, B, C, and E are decidable. All decidable languages are Turing-recognizable, so these languages are also Turing-recognizable. Demonstrating a language that is Turing-recognizable but undecidable is more difficult. We do so in Chapter 4.
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3.1 TURING MACHINES 175
176 CHAPTER 3 / THE CHURCH—TURING THESIS 3.2
VARIANTS OF TURING MACHINES
Alternative definitions of Turing machines abound, including versions with mul- tiple tapes or with nondeterminism. They are called variants of the Turing machine model. The original model and its reasonable variants all have the same power—they recognize the same class of languages. In this section, we de- scribe some of these variants and the proofs of equivalence in power. We call this invariance to certain changes in the definition robustness. Both finite automata and pushdown automata are somewhat robust models, but Turing machines have an astonishing degree of robustness.
To illustrate the robustness of the Turing machine model, let’s vary the type of transition function permitted. In our definition, the transition function forces the head to move to the left or right after each step; the head may not simply stay put. Suppose that we had allowed the Turing machine the ability to stay put. The transition function would then have the form δ : Q × Γ−→ Q × Γ × {L, R, S}. Might this feature allow Turing machines to recognize additional languages, thus adding to the power of the model? Of course not, because we can convert any TM with the “stay put” feature to one that does not have it. We do so by replacing each stay put transition with two transitions: one that moves to the right and the second back to the left.
This small example contains the key to showing the equivalence of TM vari- ants. To show that two models are equivalent, we simply need to show that one can simulate the other.
MULTITAPE TURING MACHINES
A multitape Turing machine is like an ordinary Turing machine with several tapes. Each tape has its own head for reading and writing. Initially the input appears on tape 1, and the others start out blank. The transition function is changed to allow for reading, writing, and moving the heads on some or all of the tapes simultaneously. Formally, it is
δ : Q × Γ k −→ Q × Γ k × { L , R , S } k , where k is the number of tapes. The expression
δ(qi,a1,…,ak) = (qj,b1,…,bk,L,R,…,L)
means that if the machine is in state qi and heads 1 through k are reading symbols a1 through ak, the machine goes to state qj, writes symbols b1 through bk, and directs each head to move left or right, or to stay put, as specified.
Multitape Turing machines appear to be more powerful than ordinary Turing machines, but we can show that they are equivalent in power. Recall that two machines are equivalent if they recognize the same language.
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3.2 VARIANTS OF TURING MACHINES 177
THEOREM 3.13
Every multitape Turing machine has an equivalent single-tape Turing machine.
PROOF We show how to convert a multitape TM M to an equivalent single- tape TM S. The key idea is to show how to simulate M with S.
Say that M has k tapes. Then S simulates the effect of k tapes by storing their information on its single tape. It uses the new symbol # as a delimiter to separate the contents of the different tapes. In addition to the contents of these tapes, S must keep track of the locations of the heads. It does so by writing a tape symbol with a dot above it to mark the place where the head on that tape would be. Think of these as “virtual” tapes and heads. As before, the “dotted” tape symbols are simply new symbols that have been added to the tape alphabet. The following figure illustrates how one tape can be used to represent three tapes.
FIGURE 3.14
Representing three tapes with one
S=“Oninputw=w1 ···wn:
1. First S puts its tape into the format that represents all k tapes
of M . The formatted tape contains
# w• w · · · w # ␣• # ␣• # · · · # .
2. To simulate a single move, S scans its tape from the first #, which marks the left-hand end, to the (k + 1)st #, which marks the right-hand end, in order to determine the symbols under the virtual heads. Then S makes a second pass to update the tapesaccordingtothewaythatM’stransitionfunctiondictates.
3. If at any point S moves one of the virtual heads to the right onto a #, this action signifies that M has moved the corresponding head onto the previously unread blank portion of that tape. So S writes a blank symbol on this tape cell and shifts the tape contents, from this cell until the rightmost #, one unit to the right. Then it continues the simulation as before.”
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12n
178 CHAPTER 3 / THE CHURCH—TURING THESIS COROLLARY 3.15
A language is Turing-recognizable if and only if some multitape Turing machine recognizes it.
PROOF A Turing-recognizable language is recognized by an ordinary (single- tape) Turing machine, which is a special case of a multitape Turing machine. That proves one direction of this corollary. The other direction follows from Theorem 3.13.
NONDETERMINISTIC TURING MACHINES
A nondeterministic Turing machine is defined in the expected way. At any point in a computation, the machine may proceed according to several possibilities. The transition function for a nondeterministic Turing machine has the form
δ : Q × Γ −→ P ( Q × Γ × { L , R } ) .
The computation of a nondeterministic Turing machine is a tree whose branches correspond to different possibilities for the machine. If some branch of the com- putation leads to the accept state, the machine accepts its input. If you feel the need to review nondeterminism, turn to Section 1.2 (page 47). Now we show that nondeterminism does not affect the power of the Turing machine model.
THEOREM 3.16
Every nondeterministic Turing machine has an equivalent deterministic Turing
machine.
PROOF IDEA We can simulate any nondeterministic TM N with a determin- istic TM D. The idea behind the simulation is to have D try all possible branches of N’s nondeterministic computation. If D ever finds the accept state on one of these branches, D accepts. Otherwise, D’s simulation will not terminate.
We view N’s computation on an input w as a tree. Each branch of the tree represents one of the branches of the nondeterminism. Each node of the tree is a configuration of N. The root of the tree is the start configuration. The TM D searches this tree for an accepting configuration. Conducting this search carefully is crucial lest D fail to visit the entire tree. A tempting, though bad, idea is to have D explore the tree by using depth-first search. The depth-first search strategy goes all the way down one branch before backing up to explore other branches. If D were to explore the tree in this manner, D could go forever down one infinite branch and miss an accepting configuration on some other branch. Hence we design D to explore the tree by using breadth-first search instead. This strategy explores all branches to the same depth before going on to explore any branch to the next depth. This method guarantees that D will visit every node in the tree until it encounters an accepting configuration.
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3.2 VARIANTS OF TURING MACHINES 179
PROOF The simulating deterministic TM D has three tapes. By Theo- rem 3.13, this arrangement is equivalent to having a single tape. The machine D uses its three tapes in a particular way, as illustrated in the following figure. Tape 1 always contains the input string and is never altered. Tape 2 maintains a copy of N’s tape on some branch of its nondeterministic computation. Tape 3 keeps track of D’s location in N’s nondeterministic computation tree.
FIGURE 3.17
Deterministic TM D simulating nondeterministic TM N
Let’s first consider the data representation on tape 3. Every node in the tree can have at most b children, where b is the size of the largest set of possible choices given by N’s transition function. To every node in the tree we assign an address that is a string over the alphabet Γb = {1, 2, . . . , b}. We assign the address 231 to the node we arrive at by starting at the root, going to its 2nd child, going to that node’s 3rd child, and finally going to that node’s 1st child. Each symbol in the string tells us which choice to make next when simulating a step in one branch in N’s nondeterministic computation. Sometimes a symbol may not correspond to any choice if too few choices are available for a configuration. In that case, the address is invalid and doesn’t correspond to any node. Tape 3 contains a string over Γb. It represents the branch of N’s computation from the root to the node addressed by that string unless the address is invalid. The empty string is the address of the root of the tree. Now we are ready to describe D.
1. Initially, tape 1 contains the input w, and tapes 2 and 3 are empty.
2. Copy tape 1 to tape 2 and initialize the string on tape 3 to be ε.
3. Use tape 2 to simulate N with input w on one branch of its nondeterminis- tic computation. Before each step of N , consult the next symbol on tape 3 to determine which choice to make among those allowed by N’s transition function. If no more symbols remain on tape 3 or if this nondeterministic choice is invalid, abort this branch by going to stage 4. Also go to stage 4 if a rejecting configuration is encountered. If an accepting configuration is encountered, accept the input.
4. Replace the string on tape 3 with the next string in the string ordering. Simulate the next branch of N’s computation by going to stage 2.
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180 CHAPTER 3 / THE CHURCH—TURING THESIS COROLLARY 3.18
A language is Turing-recognizable if and only if some nondeterministic Turing machine recognizes it.
PROOF Any deterministic TM is automatically a nondeterministic TM, and so one direction of this corollary follows immediately. The other direction follows from Theorem 3.16.
We can modify the proof of Theorem 3.16 so that if N always halts on all branches of its computation, D will always halt. We call a nondeterministic Tur- ing machine a decider if all branches halt on all inputs. Exercise 3.3 asks you to modify the proof in this way to obtain the following corollary to Theorem 3.16.
COROLLARY 3.19
A language is decidable if and only if some nondeterministic Turing machine
decides it.
ENUMERATORS
As we mentioned earlier, some people use the term recursively enumerable lan- guage for Turing-recognizable language. That term originates from a type of Turing machine variant called an enumerator. Loosely defined, an enumera- tor is a Turing machine with an attached printer. The Turing machine can use that printer as an output device to print strings. Every time the Turing machine wants to add a string to the list, it sends the string to the printer. Exercise 3.4 asks you to give a formal definition of an enumerator. The following figure depicts a schematic of this model.
FIGURE 3.20
Schematic of an enumerator
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3.2 VARIANTS OF TURING MACHINES 181
An enumerator E starts with a blank input on its work tape. If the enumerator doesn’t halt, it may print an infinite list of strings. The language enumerated by E is the collection of all the strings that it eventually prints out. Moreover, E may generate the strings of the language in any order, possibly with repetitions. Now we are ready to develop the connection between enumerators and Turing- recognizable languages.
THEOREM 3.21
A language is Turing-recognizable if and only if some enumerator enumerates it.
PROOF First we show that if we have an enumerator E that enumerates a language A, a TM M recognizes A. The TM M works in the following way.
M = “On input w:
1. Run E. Every time that E outputs a string, compare it with w.
2. If w ever appears in the output of E, accept.”
Clearly, M accepts those strings that appear on E’s list.
Now we do the other direction. If TM M recognizes a language A, we can
constructthefollowingenumeratorEforA.Saythats1,s2,s3,… isalistofall possible strings in Σ∗.
E = “Ignore the input.
1. 2. 3.
Repeat the following for i = 1,2,3,… .
Run M for i steps on each input, s1,s2,…,si.
If any computations accept, print out the corresponding sj .”
If M accepts a particular string s, eventually it will appear on the list generated by E. In fact, it will appear on the list infinitely many times because M runs from the beginning on each string for each repetition of step 1. This procedure gives the effect of running M in parallel on all possible input strings.
EQUIVALENCE WITH OTHER MODELS
So far we have presented several variants of the Turing machine model and have shown them to be equivalent in power. Many other models of general pur- pose computation have been proposed. Some of these models are very much like Turing machines, but others are quite different. All share the essential fea- ture of Turing machines—namely, unrestricted access to unlimited memory— distinguishing them from weaker models such as finite automata and pushdown automata. Remarkably, all models with that feature turn out to be equivalent in power, so long as they satisfy reasonable requirements.3
3For example, one requirement is the ability to perform only a finite amount of work in a single step.
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182 CHAPTER 3 / THE CHURCH—TURING THESIS
To understand this phenomenon, consider the analogous situation for pro- gramming languages. Many, such as Pascal and LISP, look quite different from one another in style and structure. Can some algorithm be programmed in one of them and not the others? Of course not—we can compile LISP into Pascal and Pascal into LISP, which means that the two languages describe exactly the same class of algorithms. So do all other reasonable programming languages. The widespread equivalence of computational models holds for precisely the same reason. Any two computational models that satisfy certain reasonable re- quirements can simulate one another and hence are equivalent in power.
This equivalence phenomenon has an important philosophical corollary. Even though we can imagine many different computational models, the class of algorithms that they describe remains the same. Whereas each individual computational model has a certain arbitrariness to its definition, the underlying class of algorithms that it describes is natural because the other models arrive at the same, unique class. This phenomenon has had profound implications for mathematics, as we show in the next section.
3.3
THE DEFINITION OF ALGORITHM
Informally speaking, an algorithm is a collection of simple instructions for car- rying out some task. Commonplace in everyday life, algorithms sometimes are called procedures or recipes. Algorithms also play an important role in mathemat- ics. Ancient mathematical literature contains descriptions of algorithms for a variety of tasks, such as finding prime numbers and greatest common divisors. In contemporary mathematics, algorithms abound.
Even though algorithms have had a long history in mathematics, the notion of algorithm itself was not defined precisely until the twentieth century. Before that, mathematicians had an intuitive notion of what algorithms were, and relied upon that notion when using and describing them. But that intuitive notion was insufficient for gaining a deeper understanding of algorithms. The following story relates how the precise definition of algorithm was crucial to one important mathematical problem.
HILBERT’S PROBLEMS
In 1900, mathematician David Hilbert delivered a now-famous address at the International Congress of Mathematicians in Paris. In his lecture, he identified 23 mathematical problems and posed them as a challenge for the coming century. The tenth problem on his list concerned algorithms.
Before describing that problem, let’s briefly discuss polynomials. A polyno- mial is a sum of terms, where each term is a product of certain variables and a
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3.3 THE DEFINITION OF ALGORITHM 183
constant, called a coefficient. For example,
6 · x · x · x · y · z · z = 6x3yz2
is a term with coefficient 6, and
6x3yz2 + 3xy2 − x3 − 10
is a polynomial with four terms, over the variables x, y, and z. For this discus- sion, we consider only coefficients that are integers. A root of a polynomial is an assignment of values to its variables so that the value of the polynomial is 0. This polynomial has a root at x = 5, y = 3, and z = 0. This root is an integral root because all the variables are assigned integer values. Some polynomials have an integral root and some do not.
Hilbert’s tenth problem was to devise an algorithm that tests whether a poly- nomial has an integral root. He did not use the term algorithm but rather “a process according to which it can be determined by a finite number of oper- ations.”4 Interestingly, in the way he phrased this problem, Hilbert explicitly asked that an algorithm be “devised.” Thus he apparently assumed that such an algorithm must exist—someone need only find it.
As we now know, no algorithm exists for this task; it is algorithmically unsolv- able. For mathematicians of that period to come to this conclusion with their intuitive concept of algorithm would have been virtually impossible. The intu- itive concept may have been adequate for giving algorithms for certain tasks, but it was useless for showing that no algorithm exists for a particular task. Proving that an algorithm does not exist requires having a clear definition of algorithm. Progress on the tenth problem had to wait for that definition.
The definition came in the 1936 papers of Alonzo Church and Alan Tur- ing. Church used a notational system called the λ-calculus to define algorithms. Turing did it with his “machines.” These two definitions were shown to be equivalent. This connection between the informal notion of algorithm and the precise definition has come to be called the Church–Turing thesis.
The Church–Turing thesis provides the definition of algorithm necessary to resolve Hilbert’s tenth problem. In 1970, Yuri Matijasevic ̆, building on the work of Martin Davis, Hilary Putnam, and Julia Robinson, showed that no algorithm exists for testing whether a polynomial has integral roots. In Chapter 4 we de- velop the techniques that form the basis for proving that this and other problems are algorithmically unsolvable.
FIGURE 3.22
The Church–Turing thesis
4Translated from the original German.
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Intuitive notion equals Turing machine of algorithms algorithms
184 CHAPTER 3 / THE CHURCH—TURING THESIS
Let’s phrase Hilbert’s tenth problem in our terminology. Doing so helps to
introduce some themes that we explore in Chapters 4 and 5. Let D = {p| p is a polynomial with an integral root}.
Hilbert’s tenth problem asks in essence whether the set D is decidable. The answer is negative. In contrast, we can show that D is Turing-recognizable. Before doing so, let’s consider a simpler problem. It is an analog of Hilbert’s tenth problem for polynomials that have only a single variable, such as 4×3 − 2×2 + x − 7. Let
D1 = {p| p is a polynomial over x with an integral root}. Here is a TM M1 that recognizes D1:
M1 = “On input ⟨p⟩: where p is a polynomial over the variable x.
1. Evaluate p with x set successively to the values 0, 1, −1, 2, −2, 3,
−3, . . . . If at any point the polynomial evaluates to 0, accept .”
If p has an integral root, M1 eventually will find it and accept. If p does not have an integral root, M1 will run forever. For the multivariable case, we can present a similar TM M that recognizes D. Here, M goes through all possible settings of its variables to integral values.
Both M1 and M are recognizers but not deciders. We can convert M1 to be a decider for D1 because we can calculate bounds within which the roots of a single variable polynomial must lie and restrict the search to these bounds. In Problem 3.21 you are asked to show that the roots of such a polynomial must lie between the values
± k cmax , c1
where k is the number of terms in the polynomial, cmax is the coefficient with the largest absolute value, and c1 is the coefficient of the highest order term. If a root is not found within these bounds, the machine rejects. Matijasevic ̆’s theorem shows that calculating such bounds for multivariable polynomials is impossible.
TERMINOLOGY FOR DESCRIBING TURING MACHINES
We have come to a turning point in the study of the theory of computation. We continue to speak of Turing machines, but our real focus from now on is on al- gorithms. That is, the Turing machine merely serves as a precise model for the definition of algorithm. We skip over the extensive theory of Turing machines themselves and do not spend much time on the low-level programming of Tur- ing machines. We need only to be comfortable enough with Turing machines to believe that they capture all algorithms.
With that in mind, let’s standardize the way we describe Turing machine algo- rithms. Initially, we ask: What is the right level of detail to give when describing
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3.3 THE DEFINITION OF ALGORITHM 185
such algorithms? Students commonly ask this question, especially when prepar- ing solutions to exercises and problems. Let’s entertain three possibilities. The first is the formal description that spells out in full the Turing machine’s states, transition function, and so on. It is the lowest, most detailed level of description. The second is a higher level of description, called the implementation descrip- tion, in which we use English prose to describe the way that the Turing machine moves its head and the way that it stores data on its tape. At this level we do not give details of states or transition function. The third is the high-level description, wherein we use English prose to describe an algorithm, ignoring the implemen- tation details. At this level we do not need to mention how the machine manages its tape or head.
In this chapter, we have given formal and implementation-level descriptions of various examples of Turing machines. Practicing with lower level Turing ma- chine descriptions helps you understand Turing machines and gain confidence in using them. Once you feel confident, high-level descriptions are sufficient.
We now set up a format and notation for describing Turing machines. The in- put to a Turing machine is always a string. If we want to provide an object other than a string as input, we must first represent that object as a string. Strings can easily represent polynomials, graphs, grammars, automata, and any combi- nation of those objects. A Turing machine may be programmed to decode the representation so that it can be interpreted in the way we intend. Our nota- tion for the encoding of an object O into its representation as a string is ⟨O⟩. If we have several objects O1 , O2 , . . . , Ok , we denote their encoding into a single string ⟨O1,O2,…,Ok⟩. The encoding itself can be done in many reasonable ways. It doesn’t matter which one we pick because a Turing machine can always translate one such encoding into another.
In our format, we describe Turing machine algorithms with an indented seg- ment of text within quotes. We break the algorithm into stages, each usually involving many individual steps of the Turing machine’s computation. We indi- cate the block structure of the algorithm with further indentation. The first line of the algorithm describes the input to the machine. If the input description is simply w, the input is taken to be a string. If the input description is the encod- ing of an object as in ⟨A⟩, the Turing machine first implicitly tests whether the input properly encodes an object of the desired form and rejects it if it doesn’t.
EXAMPLE 3.23
Let A be the language consisting of all strings representing undirected graphs that are connected. Recall that a graph is connected if every node can be reached from every other node by traveling along the edges of the graph. We write
A = {⟨G⟩| G is a connected undirected graph}.
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186 CHAPTER 3 / THE CHURCH—TURING THESIS
The following is a high-level description of a TM M that decides A. M = “On input ⟨G⟩, the encoding of a graph G:
1. 2. 3.
Select the first node of G and mark it.
Repeat the following stage until no new nodes are marked:
For each node in G, mark it if it is attached by an edge to a
node that is already marked.
4. Scan all the nodes of G to determine whether they all are
marked. If they are, accept; otherwise, reject.”
For additional practice, let’s examine some implementation-level details of Turing machine M. Usually we won’t give this level of detail in the future and you won’t need to either, unless specifically requested to do so in an exercise. First, we must understand how ⟨G⟩ encodes the graph G as a string. Consider an encoding that is a list of the nodes of G followed by a list of the edges of G. Each node is a decimal number, and each edge is the pair of decimal numbers that represent the nodes at the two endpoints of the edge. The following figure depicts such a graph and its encoding.
FIGURE 3.24
A graph G and its encoding ⟨G⟩
When M receives the input ⟨G⟩, it first checks to determine whether the input is the proper encoding of some graph. To do so, M scans the tape to be sure that there are two lists and that they are in the proper form. The first list should be a list of distinct decimal numbers, and the second should be a list of pairs of decimal numbers. Then M checks several things. First, the node list should contain no repetitions; and second, every node appearing on the edge list should also appear on the node list. For the first, we can use the procedure given in Example 3.12 for TM M4 that checks element distinctness. A similar method works for the second check. If the input passes these checks, it is the encoding of some graph G. This verification completes the input check, and M goes on to stage 1.
For stage 1, M marks the first node with a dot on the leftmost digit.
For stage 2, M scans the list of nodes to find an undotted node n1 and flags it by marking it differently—say, by underlining the first symbol. Then M scans the list again to find a dotted node n2 and underlines it, too.
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Now M scans the list of edges. For each edge, M tests whether the two underlined nodes n1 and n2 are the ones appearing in that edge. If they are, M dots n1, removes the underlines, and goes on from the beginning of stage 2. If they aren’t, M checks the next edge on the list. If there are no more edges, {n1,n2} is not an edge of G. Then M moves the underline on n2 to the next dotted node and now calls this node n2. It repeats the steps in this paragraph to check, as before, whether the new pair {n1,n2} is an edge. If there are no more dotted nodes, n1 is not attached to any dotted nodes. Then M sets the underlines so that n1 is the next undotted node and n2 is the first dotted node and repeats the steps in this paragraph. If there are no more undotted nodes, M has not been able to find any new nodes to dot, so it moves on to stage 4.
For stage 4, M scans the list of nodes to determine whether all are dotted. If they are, it enters the accept state; otherwise, it enters the reject state. This completes the description of TM M .
EXERCISES
EXERCISES 187
3.1
3.2
This exercise concerns TM M2, whose description and state diagram appear in Ex- ample 3.7. In each of the parts, give the sequence of configurations that M2 enters when started on the indicated input string.
a. 0. Ab. 00.
c. 000.
d. 000000.
This exercise concerns TM M1, whose description and state diagram appear in Ex- ample 3.9. In each of the parts, give the sequence of configurations that M1 enters when started on the indicated input string.
Aa. 11.
b. 1#1.
c. 1##1. d. 10#11. e. 10#10.
Modify the proof of Theorem 3.16 to obtain Corollary 3.19, showing that a lan- guage is decidable iff some nondeterministic Turing machine decides it. (You may assume the following theorem about trees. If every node in a tree has finitely many children and every branch of the tree has finitely many nodes, the tree itself has finitely many nodes.)
Give a formal definition of an enumerator. Consider it to be a type of two-tape Turing machine that uses its second tape as the printer. Include a definition of the enumerated language.
A 3.3
3.4
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188
CHAPTER 3 / THE CHURCH—TURING THESIS
A 3.5
3.6
3.7
3.8
Examine the formal definition of a Turing machine to answer the following ques- tions, and explain your reasoning.
a. Can a Turing machine ever write the blank symbol ␣ on its tape?
b. Can the tape alphabet Γ be the same as the input alphabet Σ?
c. Can a Turing machine’s head ever be in the same location in two successive steps?
d. Can a Turing machine contain just a single state?
In Theorem 3.21, we showed that a language is Turing-recognizable iff some enu- merator enumerates it. Why didn’t we use the following simpler algorithm for the forward direction of the proof? As before, s1 , s2 , . . . is a list of all strings in Σ∗ .
E = “Ignore the input.
1. Repeatthefollowingfori=1,2,3,….
2. Run M on si.
3. If it accepts, print out si.”
Explain why the following is not a description of a legitimate Turing machine.
Mbad =“Oninput⟨p⟩,apolynomialovervariablesx1,…,xk:
1. Try all possible settings of x1, . . . , xk to integer values.
2. Evaluate p on all of these settings.
3. If any of these settings evaluates to 0, accept ; otherwise, reject .”
Give implementation-level descriptions of Turing machines that decide the follow- ing languages over the alphabet {0,1}.
Aa. {w| w contains an equal number of 0s and 1s}
b. {w| w contains twice as many 0s as 1s}
c. {w| w does not contain twice as many 0s as 1s}
PROBLEMS
3.9
A 3.10
Let a k-PDA be a pushdown automaton that has k stacks. Thus a 0-PDA is an NFA and a 1-PDA is a conventional PDA. You already know that 1-PDAs are more powerful (recognize a larger class of languages) than 0-PDAs.
a. Show that 2-PDAs are more powerful than 1-PDAs.
b. Show that 3-PDAs are not more powerful than 2-PDAs.
(Hint: Simulate a Turing machine tape with two stacks.)
Say that a write-once Turing machine is a single-tape TM that can alter each tape square at most once (including the input portion of the tape). Show that this variant Turing machine model is equivalent to the ordinary Turing machine model. (Hint: As a first step, consider the case whereby the Turing machine may alter each tape square at most twice. Use lots of tape.)
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3.11 A Turing machine with doubly infinite tape is similar to an ordinary Turing ma- chine, but its tape is infinite to the left as well as to the right. The tape is initially filled with blanks except for the portion that contains the input. Computation is defined as usual except that the head never encounters an end to the tape as it moves leftward. Show that this type of Turing machine recognizes the class of Turing-recognizable languages.
3.12 A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form
δ: Q×Γ−→Q×Γ×{R,RESET}.
If δ(q, a) = (r, b, RESET), when the machine is in state q reading an a, the ma- chine’s head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.
3.13 A Turing machine with stay put instead of left is similar to an ordinary Turing machine, but the transition function has the form
δ : Q × Γ −→ Q × Γ × { R , S } .
At each point, the machine can move its head right or let it stay in the same posi- tion. Show that this Turing machine variant is not equivalent to the usual version. What class of languages do these machines recognize?
3.14 A queue automaton is like a push-down automaton except that the stack is replaced by a queue. A queue is a tape allowing symbols to be written only on the left-hand end and read only at the right-hand end. Each write operation (we’ll call it a push) adds a symbol to the left-hand end of the queue and each read operation (we’ll call it a pull) reads and removes a symbol at the right-hand end. As with a PDA, the input is placed on a separate read-only input tape, and the head on the input tape can move only from left to right. The input tape contains a cell with a blank symbol following the input, so that the end of the input can be detected. A queue automaton accepts its input by entering a special accept state at any time. Show that a language can be recognized by a deterministic queue automaton iff the language is Turing-recognizable.
3.15 Show that the collection of decidable languages is closed under the operation of
Aa. union. d. complementation.
b. concatenation. e. intersection.
c. star.
3.16 Show that the collection of Turing-recognizable languages is closed under the op- eration of
⋆ 3.17
Aa. union. d. intersection.
b. concatenation. e. homomorphism.
c. star.
Let B = {⟨M1 ⟩, ⟨M2 ⟩, . . .} be a Turing-recognizable language consisting of TM descriptions. Show that there is a decidable language C consisting of TM descrip- tions such that every machine described in B has an equivalent machine in C and vice versa.
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PROBLEMS 189
190
CHAPTER 3 / THE CHURCH—TURING THESIS
⋆ 3.18 ⋆3.19 ⋆ 3.20
3.21
A 3.22
Show that a language is decidable iff some enumerator enumerates the language in the standard string order.
Show that every infinite Turing-recognizable language has an infinite decidable subset.
Show that single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.
Letc1xn+c2xn−1+···+cnx+cn+1 beapolynomialwitharootatx=x0.Let cmax be the largest absolute value of a ci. Show that
|x0 | < (n + 1) cmax . |c1 |
Let A be the language containing only the single string s, where s = 0 if life never will be found on Mars.
1 if life will be found on Mars someday.
Is A decidable? Why or why not? For the purposes of this problem, assume that the question of whether life will be found on Mars has an unambiguous YES or NO answer.
SELECTED SOLUTIONS
3.1 (b) q1 00, ␣q2 0, ␣xq3 ␣, ␣q5 x␣, q5 ␣x␣, ␣q2 x␣, ␣xq2 ␣, ␣x␣qaccept .
3.2 (a) q1 11, xq3 1, x1q3 ␣, x1␣qreject .
3.3 We prove both directions of the iff. First, if a language L is decidable, it can be decided by a deterministic Turing machine, and that is automatically a nondeter- ministic Turing machine.
Second, if a language L is decided by a nondeterministic TM N, we modify the deterministic TM D that was given in the proof of Theorem 3.16 as follows.
Move stage 4 to be stage 5.
Add new stage 4: Reject if all branches of N ’s nondeterminism have rejected.
We argue that this new TM D′ is a decider for L. If N accepts its input, D′ will eventually find an accepting branch and accept, too. If N rejects its input, all of its branches halt and reject because it is a decider. Hence each of the branches has finitely many nodes, where each node represents one step of N ’s computation along that branch. Therefore, N ’s entire computation tree on this input is finite, by virtue of the theorem about trees given in the statement of the exercise. Consequently, D′ will halt and reject when this entire tree has been explored.
3.5 (a) Yes. The tape alphabet Γ contains ␣. A Turing machine can write any characters in Γ on its tape.
(b) No. Σ never contains ␣, but Γ always contains ␣. So they cannot be equal.
(c) Yes. If the Turing machine attempts to move its head off the left-hand end of
the tape, it remains on the same tape cell.
(d) No. Any Turing machine must contain two distinct states: qaccept and qreject. So, a Turing machine contains at least two states.
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3.8 (a) “On input string w:
1. Scan the tape and mark the first 0 that has not been marked. If
no unmarked 0 is found, go to stage 4. Otherwise, move the
head back to the front of the tape.
2. Scan the tape and mark the first 1 that has not been marked. If
no unmarked 1 is found, reject .
3. Move the head back to the front of the tape and go to stage 1.
4. Move the head back to the front of the tape. Scan the tape to see
if any unmarked 1s remain. If none are found, accept ; otherwise, reject .”
3.10 We first simulate an ordinary Turing machine by a write-twice Turing machine. The write-twice machine simulates a single step of the original machine by copying the entire tape over to a fresh portion of the tape to the right-hand side of the currently used portion. The copying procedure operates character by character, marking a character as it is copied. This procedure alters each tape square twice: once to write the character for the first time, and again to mark that it has been copied. The position of the original Turing machine’s tape head is marked on the tape. When copying the cells at or adjacent to the marked position, the tape content is updated according to the rules of the original Turing machine.
To carry out the simulation with a write-once machine, operate as before, except that each cell of the previous tape is now represented by two cells. The first of these contains the original machine’s tape symbol and the second is for the mark used in the copying procedure. The input is not presented to the machine in the format with two cells per symbol, so the very first time the tape is copied, the copying marks are put directly over the input symbols.
3.15 (a) For any two decidable languages L1 and L2, let M1 and M2 be the TMs that decide them. We construct a TM M′ that decides the union of L1 and L2:
“On input w:
1. Run M1 on w. If it accepts, accept .
2. Run M2 on w. If it accepts, accept . Otherwise, reject .”
M ′ accepts w if either M1 or M2 accepts it. If both reject, M ′ rejects.
3.16 (a) For any two Turing-recognizable languages L1 and L2, let M1 and M2 be the TMs that recognize them. We construct a TM M′ that recognizes the union of L1 and L2:
“On input w:
1. Run M1 and M2 alternately on w step by step. If either accepts,
accept. If both halt and reject, reject.”
If either M1 or M2 accepts w, M′ accepts w because the accepting TM arrives to its accepting state after a finite number of steps. Note that if both M1 and M2 reject and either of them does so by looping, then M ′ will loop.
3.22 The language A is one of the two languages {0} or {1}. In either case, the language is finite and hence decidable. If you aren’t able to determine which of these two languages is A, you won’t be able to describe the decider for A. However, you can give two Turing machines, one of which is A’s decider.
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SELECTED SOLUTIONS 191
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4
DECIDABILITY
In Chapter 3 we introduced the Turing machine as a model of a general purpose computer and defined the notion of algorithm in terms of Turing machines by means of the Church–Turing thesis.
In this chapter we begin to investigate the power of algorithms to solve prob- lems. We demonstrate certain problems that can be solved algorithmically and others that cannot. Our objective is to explore the limits of algorithmic solv- ability. You are probably familiar with solvability by algorithms because much of computer science is devoted to solving problems. The unsolvability of certain problems may come as a surprise.
Why should you study unsolvability? After all, showing that a problem is unsolvable doesn’t appear to be of any use if you have to solve it. You need to study this phenomenon for two reasons. First, knowing when a problem is algorithmically unsolvable is useful because then you realize that the problem must be simplified or altered before you can find an algorithmic solution. Like any tool, computers have capabilities and limitations that must be appreciated if they are to be used well. The second reason is cultural. Even if you deal with problems that clearly are solvable, a glimpse of the unsolvable can stimulate your imagination and help you gain an important perspective on computation.
193
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194 CHAPTER 4 / DECIDABILITY 4.1
DECIDABLE LANGUAGES
In this section we give some examples of languages that are decidable by al- gorithms. We focus on languages concerning automata and grammars. For example, we present an algorithm that tests whether a string is a member of a context-free language (CFL). These languages are interesting for several reasons. First, certain problems of this kind are related to applications. This problem of testing whether a CFG generates a string is related to the problem of recogniz- ing and compiling programs in a programming language. Second, certain other problems concerning automata and grammars are not decidable by algorithms. Starting with examples where decidability is possible helps you to appreciate the undecidable examples.
DECIDABLE PROBLEMS CONCERNING REGULAR LANGUAGES
We begin with certain computational problems concerning finite automata. We give algorithms for testing whether a finite automaton accepts a string, whether the language of a finite automaton is empty, and whether two finite automata are equivalent.
Note that we chose to represent various computational problems by lan- guages. Doing so is convenient because we have already set up terminology for dealing with languages. For example, the acceptance problem for DFAs of testing whether a particular deterministic finite automaton accepts a given string can be expressed as a language, ADFA. This language contains the encodings of all DFAs together with strings that the DFAs accept. Let
ADFA = {⟨B, w⟩| B is a DFA that accepts input string w}.
The problem of testing whether a DFA B accepts an input w is the same as the problem of testing whether ⟨B, w⟩ is a member of the language ADFA. Similarly, we can formulate other computational problems in terms of testing membership in a language. Showing that the language is decidable is the same as showing that the computational problem is decidable.
In the following theorem we show that ADFA is decidable. Hence this theorem shows that the problem of testing whether a given finite automaton accepts a given string is decidable.
THEOREM 4.1
ADFA is a decidable language.
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PROOF IDEA We simply need to present a TM M that decides ADFA. M = “On input ⟨B, w⟩, where B is a DFA and w is a string:
PROOF
1. 2.
4.1 DECIDABLE LANGUAGES 195
Simulate B on input w.
If the simulation ends in an accept state, accept. If it ends in a nonaccepting state, reject.”
We mention just a few implementation details of this proof. For those of you familiar with writing programs in any standard programming lan- guage, imagine how you would write a program to carry out the simulation.
First, let’s examine the input ⟨B, w⟩. It is a representation of a DFA B together with a string w. One reasonable representation of B is simply a list of its five components: Q, Σ, δ, q0, and F. When M receives its input, M first determines whether it properly represents a DFA B and a string w. If not, M rejects.
Then M carries out the simulation directly. It keeps track of B’s current state and B’s current position in the input w by writing this information down on its tape. Initially, B’s current state is q0 and B’s current input position is the leftmost symbol of w. The states and position are updated according to the specified transition function δ. When M finishes processing the last symbol of w, M accepts the input if B is in an accepting state; M rejects the input if B is in a nonaccepting state.
We can prove a similar theorem for nondeterministic finite automata. Let ANFA = {⟨B, w⟩| B is an NFA that accepts input string w}.
THEOREM 4.2
ANFA is a decidable language.
PROOF We present a TM N that decides ANFA. We could design N to operate like M, simulating an NFA instead of a DFA. Instead, we’ll do it differently to illustrate a new idea: Have N use M as a subroutine. Because M is designed to work with DFAs, N first converts the NFA it receives as input to a DFA before passing it to M .
N = “On input ⟨B, w⟩, where B is an NFA and w is a string:
1. Convert NFA B to an equivalent DFA C, using the procedure for
this conversion given in Theorem 1.39.
2. Run TM M from Theorem 4.1 on input ⟨C, w⟩.
3. If M accepts, accept; otherwise, reject.”
Running TM M in stage 2 means incorporating M into the design of N as a subprocedure.
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196 CHAPTER 4 / DECIDABILITY
Similarly, we can determine whether a regular expression generates a given
string. Let AREX = {⟨R, w⟩| R is a regular expression that generates string w}. THEOREM 4.3
AREX is a decidable language.
PROOF The following TM P decides AREX.
P = “On input ⟨R, w⟩, where R is a regular expression and w is a string:
1. Convert regular expression R to an equivalent NFA A by using
the procedure for this conversion given in Theorem 1.54.
2. Run TM N on input ⟨A, w⟩.
3. If N accepts, accept; if N rejects, reject.”
Theorems 4.1, 4.2, and 4.3 illustrate that, for decidability purposes, it is equivalent to present the Turing machine with a DFA, an NFA, or a regular ex- pression because the machine can convert one form of encoding to another.
Now we turn to a different kind of problem concerning finite automata: emptiness testing for the language of a finite automaton. In the preceding three theorems we had to determine whether a finite automaton accepts a particular string. In the next proof we must determine whether or not a finite automaton accepts any strings at all. Let
EDFA ={⟨A⟩|AisaDFAandL(A)=∅}. EDFA is a decidable language.
PROOF A DFA accepts some string iff reaching an accept state from the start state by traveling along the arrows of the DFA is possible. To test this condition, we can design a TM T that uses a marking algorithm similar to that used in Example 3.23.
T = “On input ⟨A⟩, where A is a DFA:
1. Mark the start state of A.
2. Repeat until no new states get marked:
3. Mark any state that has a transition coming into it from any
state that is already marked.
4. If no accept state is marked, accept; otherwise, reject.”
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THEOREM 4.4
4.1 DECIDABLE LANGUAGES 197 The next theorem states that determining whether two DFAs recognize the
same language is decidable. Let
EQDFA ={⟨A,B⟩|AandBareDFAsandL(A)=L(B)}.
THEOREM 4.5
EQDFA is a decidable language.
PROOF To prove this theorem, we use Theorem 4.4. We construct a new DFA C from A and B, where C accepts only those strings that are accepted by either A or B but not by both. Thus, if A and B recognize the same language, C will accept nothing. The language of C is
L(C) = L(A) ∩ L(B) ∪ L(A) ∩ L(B).
This expression is sometimes called the symmetric difference of L(A) and L(B) and is illustrated in the following figure. Here, L(A) is the complement of L(A). The symmetric difference is useful here because L(C) = ∅ iff L(A) = L(B). We can construct C from A and B with the constructions for proving the class of regular languages closed under complementation, union, and intersection. These constructions are algorithms that can be carried out by Turing machines. Once we have constructed C, we can use Theorem 4.4 to test whether L(C) is empty. If it is empty, L(A) and L(B) must be equal.
F = “On input ⟨A, B⟩, where A and B are DFAs:
1. Construct DFA C as described.
2. Run TM T from Theorem 4.4 on input ⟨C⟩.
3. If T accepts, accept. If T rejects, reject.”
FIGURE 4.6
The symmetric difference of L(A) and L(B)
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198 CHAPTER 4 / DECIDABILITY DECIDABLE PROBLEMS CONCERNING
CONTEXT-FREE LANGUAGES
Here, we describe algorithms to determine whether a CFG generates a particular string and to determine whether the language of a CFG is empty. Let
ACFG = {⟨G, w⟩| G is a CFG that generates string w}. THEOREM 4.7
ACFG is a decidable language.
PROOF IDEA For CFG G and string w, we want to determine whether G generates w. One idea is to use G to go through all derivations to determine whether any is a derivation of w. This idea doesn’t work, as infinitely many derivations may have to be tried. If G does not generate w, this algorithm would never halt. This idea gives a Turing machine that is a recognizer, but not a decider, for ACFG.
To make this Turing machine into a decider, we need to ensure that the al- gorithm tries only finitely many derivations. In Problem 2.26 (page 157) we showed that if G were in Chomsky normal form, any derivation of w has 2n − 1 steps, where n is the length of w. In that case, checking only derivations with 2n − 1 steps to determine whether G generates w would be sufficient. Only finitely many such derivations exist. We can convert G to Chomsky normal form by using the procedure given in Section 2.1.
PROOF The TM S for ACFG follows.
S = “On input ⟨G, w⟩, where G is a CFG and w is a string:
1. Convert G to an equivalent grammar in Chomsky normal form.
2. List all derivations with 2n − 1 steps, where n is the length of w;
except if n = 0, then instead list all derivations with one step.
3. If any of these derivations generate w, accept; if not, reject.”
The problem of determining whether a CFG generates a particular string is related to the problem of compiling programming languages. The algorithm in TM S is very inefficient and would never be used in practice, but it is easy to de- scribe and we aren’t concerned with efficiency here. In Part Three of this book, we address issues concerning the running time and memory use of algorithms. In the proof of Theorem 7.16, we describe a more efficient algorithm for rec- ognizing general context-free languages. Even greater efficiency is possible for recognizing deterministic context-free languages.
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4.1 DECIDABLE LANGUAGES 199
Recall that we have given procedures for converting back and forth between CFGs and PDAs in Theorem 2.20. Hence everything we say about the decidability of problems concerning CFGs applies equally well to PDAs.
Let’s turn now to the emptiness testing problem for the language of a CFG. As we did for DFAs, we can show that the problem of determining whether a CFG generates any strings at all is decidable. Let
ECFG ={⟨G⟩|GisaCFGandL(G)=∅}. ECFG is a decidable language.
PROOF IDEA To find an algorithm for this problem, we might attempt to use TM S from Theorem 4.7. It states that we can test whether a CFG generates some particular string w. To determine whether L(G) = ∅, the algorithm might try going through all possible w’s, one by one. But there are infinitely many w’s to try, so this method could end up running forever. We need to take a different approach.
In order to determine whether the language of a grammar is empty, we need to test whether the start variable can generate a string of terminals. The algo- rithm does so by solving a more general problem. It determines for each variable whether that variable is capable of generating a string of terminals. When the algorithm has determined that a variable can generate some string of terminals, the algorithm keeps track of this information by placing a mark on that variable.
First, the algorithm marks all the terminal symbols in the grammar. Then, it scans all the rules of the grammar. If it ever finds a rule that permits some vari- able to be replaced by some string of symbols, all of which are already marked, the algorithm knows that this variable can be marked, too. The algorithm con- tinues in this way until it cannot mark any additional variables. The TM R implements this algorithm.
PROOF
R = “On input ⟨G⟩, where G is a CFG:
1. Mark all terminal symbols in G.
2. Repeat until no new variables get marked:
3. MarkanyvariableAwhereGhasaruleA→U1U2···Uk and
each symbol U1, . . . , Uk has already been marked.
4. If the start variable is not marked, accept; otherwise, reject.”
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THEOREM 4.8
200 CHAPTER 4 / DECIDABILITY
Next, we consider the problem of determining whether two context-free
grammars generate the same language. Let
EQCFG = {⟨G, H⟩| G and H are CFGs and L(G) = L(H)}.
Theorem 4.5 gave an algorithm that decides the analogous language EQDFA for finite automata. We used the decision procedure for EDFA to prove that EQDFA is decidable. Because ECFG also is decidable, you might think that we can use a similar strategy to prove that EQCFG is decidable. But something is wrong with this idea! The class of context-free languages is not closed under comple- mentation or intersection, as you proved in Exercise 2.2. In fact, EQCFG is not decidable. The technique for proving so is presented in Chapter 5.
Now we show that context-free languages are decidable by Turing machines.
THEOREM 4.9
Every context-free language is decidable.
PROOF IDEA Let A be a CFL. Our objective is to show that A is decidable. One (bad) idea is to convert a PDA for A directly into a TM. That isn’t hard to do because simulating a stack with the TM’s more versatile tape is easy. The PDA for A may be nondeterministic, but that seems okay because we can convert it into a nondeterministic TM and we know that any nondeterministic TM can be converted into an equivalent deterministic TM. Yet, there is a difficulty. Some branches of the PDA’s computation may go on forever, reading and writing the stack without ever halting. The simulating TM then would also have some non- halting branches in its computation, and so the TM would not be a decider. A different idea is necessary. Instead, we prove this theorem with the TM S that we designed in Theorem 4.7 to decide ACFG.
PROOF Let G be a CFG for A and design a TM MG that decides A. We build a copy of G into MG. It works as follows.
MG = “On input w:
1. Run TM S on input ⟨G, w⟩.
2. If this machine accepts, accept; if it rejects, reject.”
Theorem 4.9 provides the final link in the relationship among the four main classes of languages that we have described so far: regular, context-free, decid- able, and Turing-recognizable. Figure 4.10 depicts this relationship.
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4.2 UNDECIDABILITY 201
FIGURE 4.10
The relationship among classes of languages
4.2
UNDECIDABILITY
In this section, we prove one of the most philosophically important theorems of the theory of computation: There is a specific problem that is algorithmically unsolvable. Computers appear to be so powerful that you may believe that all problems will eventually yield to them. The theorem presented here demon- strates that computers are limited in a fundamental way.
What sorts of problems are unsolvable by computer? Are they esoteric, dwelling only in the minds of theoreticians? No! Even some ordinary prob- lems that people want to solve turn out to be computationally unsolvable.
In one type of unsolvable problem, you are given a computer program and a precise specification of what that program is supposed to do (e.g., sort a list of numbers). You need to verify that the program performs as specified (i.e., that it is correct). Because both the program and the specification are mathe- matically precise objects, you hope to automate the process of verification by feeding these objects into a suitably programmed computer. However, you will be disappointed. The general problem of software verification is not solvable by computer.
In this section and in Chapter 5, you will encounter several computationally unsolvable problems. We aim to help you develop a feeling for the types of problems that are unsolvable and to learn techniques for proving unsolvability.
Now we turn to our first theorem that establishes the undecidability of a spe- cific language: the problem of determining whether a Turing machine accepts a
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202 CHAPTER 4 / DECIDABILITY
given input string. We call it ATM by analogy with ADFA and ACFG. But, whereas
ADFA and ACFG were decidable, ATM is not. Let
ATM = {⟨M,w⟩| M is a TM and M accepts w}.
THEOREM 4.11 ATM is undecidable.
Before we get to the proof, let’s first observe that ATM is Turing-recognizable. Thus, this theorem shows that recognizers are more powerful than deciders. Requiring a TM to halt on all inputs restricts the kinds of languages that it can recognize. The following Turing machine U recognizes ATM.
U =“Oninput⟨M,w⟩,whereM isaTMandwisastring:
1. Simulate M on input w.
2. If M ever enters its accept state, accept; if M ever enters its
reject state, reject.”
Note that this machine loops on input ⟨M, w⟩ if M loops on w, which is why this machine does not decide ATM. If the algorithm had some way to determine that M was not halting on w, it could reject in this case. However, an algorithm has no way to make this determination, as we shall see.
The Turing machine U is interesting in its own right. It is an example of the universal Turing machine first proposed by Alan Turing in 1936. This machine is called universal because it is capable of simulating any other Turing machine from the description of that machine. The universal Turing machine played an important early role in the development of stored-program computers.
THE DIAGONALIZATION METHOD
The proof of the undecidability of ATM uses a technique called diagonalization, discovered by mathematician Georg Cantor in 1873. Cantor was concerned with the problem of measuring the sizes of infinite sets. If we have two infinite sets, how can we tell whether one is larger than the other or whether they are of the same size? For finite sets, of course, answering these questions is easy. We simply count the elements in a finite set, and the resulting number is its size. But if we try to count the elements of an infinite set, we will never finish! So we can’t use the counting method to determine the relative sizes of infinite sets.
For example, take the set of even integers and the set of all strings over {0,1}. Both sets are infinite and thus larger than any finite set, but is one of the two larger than the other? How can we compare their relative size?
Cantor proposed a rather nice solution to this problem. He observed that two finite sets have the same size if the elements of one set can be paired with the elements of the other set. This method compares the sizes without resorting to counting. We can extend this idea to infinite sets. Here it is more precisely.
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Alternative common terminology for these types of functions is injective for one-to-one, surjective for onto, and bijective for one-to-one and onto.
EXAMPLE 4.13
Let N be the set of natural numbers {1,2,3,...} and let E be the set of even natural numbers {2, 4, 6, . . .}. Using Cantor’s definition of size, we can see that N and E have the same size. The correspondence f mapping N to E is simply f (n) = 2n. We can visualize f more easily with the help of a table.
n f(n) 12 24 36
. .
Of course, this example seems bizarre. Intuitively, E seems smaller than N be- cause E is a proper subset of N . But pairing each member of N with its own member of E is possible, so we declare these two sets to be the same size.
EXAMPLE 4.15
Now we turn to an even stranger example. If we let Q = { m | m, n ∈ N } be the
n
set of positive rational numbers, Q seems to be much larger than N. Yet these two sets are the same size according to our definition. We give a correspondence with N to show that Q is countable. One easy way to do so is to list all the
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4.2 UNDECIDABILITY 203
DEFINITION 4.12
Assume that we have sets A and B and a function f from A to B. Say that f is one-to-one if it never maps two different elements to the same place—that is, if f(a) ̸= f(b) whenever a ̸= b. Say that f is onto if it hits every element of B—that is, if for every b ∈ B there is an a ∈ A such that f (a) = b. Say that A and B are the same size if there is a one-to-one, onto function f : A−→B. A function that is both one-to-one and onto is called a correspondence. In a correspondence, every element of A maps to a unique element of B and each element of B has a unique element of A mapping to it. A correspondence is simply a way of pairing the elements of A with the elements of B.
DEFINITION 4.14
A set A is countable if either it is finite or it has the same size as N .
204 CHAPTER 4 / DECIDABILITY
elements of Q. Then we pair the first element on the list with the number 1 from N , the second element on the list with the number 2 from N , and so on. We must ensure that every member of Q appears only once on the list.
To get this list, we make an infinite matrix containing all the positive ratio-
nal numbers, as shown in Figure 4.16. The ith row contains all numbers with
numerator i and the jth column has all numbers with denominator j. So the
number i occurs in the ith row and jth column. j
Now we turn this matrix into a list. One (bad) way to attempt it would be to begin the list with all the elements in the first row. That isn’t a good approach because the first row is infinite, so the list would never get to the second row. Instead we list the elements on the diagonals, which are superimposed on the diagram, starting from the corner. The first diagonal contains the single element
1 , and the second diagonal contains the two elements 2 and 1 . So the first 112112
three elements on the list are 1 , 1 , and 2 . In the third diagonal, a complication arises. It contains 3 , 2 , and 1 . If we simply added these to the list, we would
12123
repeat 1 = 2 . We avoid doing so by skipping an element when it would cause
a repetition. So we add only the two new elements 3 and 1 . Continuing in this
way, we obtain a list of all the elements of Q.
13
FIGURE 4.16
A correspondence of N and Q
After seeing the correspondence of N and Q, you might think that any two infinite sets can be shown to have the same size. After all, you need only demon- strate a correspondence, and this example shows that surprising correspondences do exist. However, for some infinite sets, no correspondence with N exists. These sets are simply too big. Such sets are called uncountable.
The set of real numbers is an example of an uncountable set. A real number is one that has a decimal representation. The numbers π = 3.1415926 . . . and
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√2 = 1.4142135... are examples of real numbers. Let R be the set of real numbers. Cantor proved that R is uncountable. In doing so, he introduced the diagonalization method.
THEOREM 4.17 R is uncountable.
PROOF In order to show that R is uncountable, we show that no correspon- dence exists between N and R. The proof is by contradiction. Suppose that a correspondence f existed between N and R. Our job is to show that f fails to work as it should. For it to be a correspondence, f must pair all the members of N with all the members of R. But we will find an x in R that is not paired with anything in N , which will be our contradiction.
The way we find this x is by actually constructing it. We choose each digit of x to make x different from one of the real numbers that is paired with an element of N . In the end, we are sure that x is different from any real number that is paired.
We can illustrate this idea by giving an example. Suppose that the correspon- dence f exists. Let f(1) = 3.14159..., f(2) = 55.55555..., f(3) = ..., and so on, just to make up some values for f. Then f pairs the number 1 with 3.14159 . . . , the number 2 with 55.55555 . . . , and so on. The following table shows a few values of a hypothetical correspondence f between N and R.
4.2 UNDECIDABILITY 205
n
1 2 3 4
.
f(n) 3.14159 . . .
55.55555 . . . 0.12345 . . . 0.50000 . . .
.
We construct the desired x by giving its decimal representation. It is a num- ber between 0 and 1, so all its significant digits are fractional digits following the decimal point. Our objective is to ensure that x ̸= f(n) for any n. To ensure that x ̸= f(1), we let the first digit of x be anything different from the first fractional digit 1 of f(1) = 3.14159... . Arbitrarily, we let it be 4. To ensure that x ̸= f (2), we let the second digit of x be anything different from the second fractional digit 5 of f (2) = 55.555555 . . . . Arbitrarily, we let it be 6. The third fractional digit of f (3) = 0.12345 . . . is 3, so we let x be anything different— say, 4. Continuing in this way down the diagonal of the table for f, we obtain all the digits of x, as shown in the following table. We know that x is not f(n) for any n because it differs from f (n) in the nth fractional digit. (A slight prob- lem arises because certain numbers, such as 0.1999 . . . and 0.2000 . . . , are equal even though their decimal representations are different. We avoid this problem by never selecting the digits 0 or 9 when we construct x.)
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206 CHAPTER 4 / DECIDABILITY
n
1 2 3 4
.
f(n)
4159...
3.1 55.55555 . . .
0.123
45... x = 0.4641 ...
0.50000 . . . .
The preceding theorem has an important application to the theory of com- putation. It shows that some languages are not decidable or even Turing- recognizable, for the reason that there are uncountably many languages yet only countably many Turing machines. Because each Turing machine can recognize a single language and there are more languages than Turing machines, some languages are not recognized by any Turing machine. Such languages are not Turing-recognizable, as we state in the following corollary.
COROLLARY 4.18
Some languages are not Turing-recognizable.
PROOF To show that the set of all Turing machines is countable, we first observe that the set of all strings Σ∗ is countable for any alphabet Σ. With only finitely many strings of each length, we may form a list of Σ∗ by writing down all strings of length 0, length 1, length 2, and so on.
The set of all Turing machines is countable because each Turing machine M has an encoding into a string ⟨M⟩. If we simply omit those strings that are not legal encodings of Turing machines, we can obtain a list of all Turing machines.
To show that the set of all languages is uncountable, we first observe that the set of all infinite binary sequences is uncountable. An infinite binary sequence is an unending sequence of 0s and 1s. Let B be the set of all infinite binary sequences. We can show that B is uncountable by using a proof by diagonalization similar to the one we used in Theorem 4.17 to show that R is uncountable.
Let L be the set of all languages over alphabet Σ. We show that L is un- countable by giving a correspondence with B, thus showing that the two sets are the same size. Let Σ∗ = {s1, s2, s3, . . .}. Each language A ∈ L has a unique sequenceinB.Theithbitofthatsequenceisa1ifsi ∈Aandisa0ifsi ̸∈A, which is called the characteristic sequence of A. For example, if A were the lan- guage of all strings starting with a 0 over the alphabet {0,1}, its characteristic sequence χA would be
Σ∗={ ε, 0, 1, 00, 01, 10, 11, 000,001, ··· } ; A={ 0, 00, 01, 000,001,···};
χA= 0 1 0 1 1 0 0 1 1 ··· .
The function f : L−→B, where f(A) equals the characteristic sequence of A, is one-to-one and onto, and hence is a correspondence. Therefore, as B is uncountable, L is uncountable as well.
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Thus we have shown that the set of all languages cannot be put into a corre- spondence with the set of all Turing machines. We conclude that some languages are not recognized by any Turing machine.
AN UNDECIDABLE LANGUAGE
Now we are ready to prove Theorem 4.11, the undecidability of the language ATM = {⟨M,w⟩| M is a TM and M accepts w}.
PROOF We assume that ATM is decidable and obtain a contradiction. Sup- pose that H is a decider for ATM. On input ⟨M, w⟩, where M is a TM and w is a string, H halts and accepts if M accepts w. Furthermore, H halts and rejects if M fails to accept w. In other words, we assume that H is a TM, where
H⟨M, w⟩ = accept if M accepts w
reject if M does not accept w.
Now we construct a new Turing machine D with H as a subroutine. This new TM calls H to determine what M does when the input to M is its own description ⟨M ⟩. Once D has determined this information, it does the opposite. That is, it rejects if M accepts and accepts if M does not accept. The following is a description of D.
D=“Oninput⟨M⟩,whereM isaTM:
1. RunHoninput⟨M,⟨M⟩⟩.
2. Output the opposite of what H outputs. That is, if H accepts,
reject; and if H rejects, accept.”
Don’t be confused by the notion of running a machine on its own description! That is similar to running a program with itself as input, something that does occasionally occur in practice. For example, a compiler is a program that trans- lates other programs. A compiler for the language Python may itself be written in Python, so running that program on itself would make sense. In summary,
D⟨M⟩ = accept if M does not accept ⟨M⟩ reject if M accepts ⟨M⟩.
What happens when we run D with its own description ⟨D⟩ as input? In that
case, we get
D⟨D⟩ = accept if D does not accept ⟨D⟩
reject if D accepts ⟨D⟩.
No matter what D does, it is forced to do the opposite, which is obviously a
contradiction. Thus, neither TM D nor TM H can exist.
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4.2 UNDECIDABILITY 207
208 CHAPTER 4 / DECIDABILITY
Let’s review the steps of this proof. Assume that a TM H decides ATM. Use H to build a TM D that takes an input ⟨M ⟩, where D accepts its input ⟨M ⟩ exactly when M does not accept its input ⟨M⟩. Finally, run D on itself. Thus, the machines take the following actions, with the last line being the contradiction.
• Haccepts⟨M,w⟩exactlywhenMacceptsw. • Drejects⟨M⟩exactlywhenMaccepts⟨M⟩. • Drejects⟨D⟩exactlywhenDaccepts⟨D⟩.
Where is the diagonalization in the proof of Theorem 4.11? It becomes ap- parent when you examine tables of behavior for TMs H and D. In these tables we list all TM s down the rows, M1 , M2 , . . . , and all their descriptions across the columns, ⟨M1⟩, ⟨M2⟩, . . . . The entries tell whether the machine in a given row accepts the input in a given column. The entry is accept if the machine accepts the input but is blank if it rejects or loops on that input. We made up the entries in the following figure to illustrate the idea.
⟨M1⟩ M1 accept M2 accept
M3
M4 accept
⟨M2⟩ ⟨M3⟩ ⟨M4⟩ · · · accept
accept accept accept accept
···
.
Entry i, j is accept if Mi accepts ⟨Mj ⟩
.
FIGURE 4.19
In the following figure, the entries are the results of running H on inputs corresponding to Figure 4.19. So if M3 does not accept input ⟨M2⟩, the entry for row M3 and column ⟨M2⟩ is reject because H rejects input ⟨M3, ⟨M2⟩⟩.
⟨M1⟩ ⟨M2⟩ ⟨M3⟩ ⟨M4⟩ · · · M1 accept reject accept reject
M2 accept accept accept accept
M3
. .
···
reject reject reject reject M4 accept accept reject reject
FIGURE 4.20
Entry i, j is the value of H on input ⟨Mi, ⟨Mj ⟩⟩
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In the following figure, we added D to Figure 4.20. By our assumption, H is a TM and so is D. Therefore, it must occur on the list M1, M2, ... of all TMs. Note that D computes the opposite of the diagonal entries. The contradiction occurs at the point of the question mark where the entry must be the opposite of itself.
⟨M1⟩ ⟨M2⟩ ⟨M3⟩ ⟨M4⟩ ··· ⟨D⟩ ···
4.2 UNDECIDABILITY 209
M1 accept reject M2 accept accept M3 reject reject M4 accept accept
accept accept reject reject
accept
reject accept reject reject
accept
accept
··· accept ··· reject
accept
...
...
.
D
.
reject
reject
. .
?
FIGURE 4.21
If D is in the figure, a contradiction occurs at “?”
A TURING-UNRECOGNIZABLE LANGUAGE
In the preceding section, we exhibited a language—namely, ATM—that is un- decidable. Now we exhibit a language that isn’t even Turing-recognizable. Note that ATM will not suffice for this purpose because we showed that ATM is Turing-recognizable (page 202). The following theorem shows that if both a language and its complement are Turing-recognizable, the language is decid- able. Hence for any undecidable language, either it or its complement is not Turing-recognizable. Recall that the complement of a language is the language consisting of all strings that are not in the language. We say that a language is co- Turing-recognizable if it is the complement of a Turing-recognizable language.
THEOREM 4.22
A language is decidable iff it is Turing-recognizable and co-Turing-recognizable.
In other words, a language is decidable exactly when both it and its complement are Turing-recognizable.
PROOF We have two directions to prove. First, if A is decidable, we can easily see that both A and its complement A are Turing-recognizable. Any decidable language is Turing-recognizable, and the complement of a decidable language also is decidable.
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210 CHAPTER 4 / DECIDABILITY
For the other direction, if both A and A are Turing-recognizable, we let M1 be the recognizer for A and M2 be the recognizer for A. The following Turing machine M is a decider for A.
M = “On input w:
1. Run both M1 and M2 on input w in parallel.
2. If M1 accepts, accept; if M2 accepts, reject.”
Running the two machines in parallel means that M has two tapes, one for simu- lating M1 and the other for simulating M2. In this case, M takes turns simulating one step of each machine, which continues until one of them accepts.
Now we show that M decides A. Every string w is either in A or A. There- fore, either M1 or M2 must accept w. Because M halts whenever M1 or M2 accepts, M always halts and so it is a decider. Furthermore, it accepts all strings in A and rejects all strings not in A. So M is a decider for A, and thus A is decidable.
COROLLARY 4.23
ATM is not Turing-recognizable.
PROOF We know that ATM is Turing-recognizable. If ATM also were Turing- recognizable, ATM would be decidable. Theorem 4.11 tells us that ATM is not decidable, so ATM must not be Turing-recognizable.
EXERCISES
A4.1 Answer all parts for the following DFA M and give reasons for your answers.
a. Is ⟨M, 0100⟩ ∈ ADFA? b. Is ⟨M, 011⟩ ∈ ADFA? c. Is ⟨M⟩ ∈ ADFA?
d. Is ⟨M, 0100⟩ ∈ AREX? e. Is ⟨M⟩ ∈ EDFA?
f. Is ⟨M, M⟩ ∈ EQDFA?
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PROBLEMS 211
4.2 Consider the problem of determining whether a DFA and a regular expression are
equivalent. Express this problem as a language and show that it is decidable.
4.3 Let ALLDFA = {⟨A⟩| A is a DFA and L(A) = Σ∗ }. Show that ALLDFA is decidable.
4.4 Let AεCFG = {⟨G⟩| G is a CFG that generates ε}. Show that AεCFG is decidable.
A4.5 Let ETM = {⟨M⟩| M is a TM and L(M) = ∅}. Show that ETM, the complement of ETM, is Turing-recognizable.
4.6 Let X be the set {1,2,3,4,5} and Y be the set {6,7,8,9,10}. We describe the functions f: X−→Y and g: X−→Y in the following tables. Answer each part and give a reason for each negative answer.
n f(n) n g(n) 16 110 27 29 36 38 47 47 56 56
Aa. Is f one-to-one?
b. Is f onto?
c. Is f a correspondence?
Ad. Is g one-to-one? e. Is g onto?
f. Is g a correspondence?
4.7 Let B be the set of all infinite sequences over {0,1}. Show that B is uncountable
using a proof by diagonalization.
4.8 LetT ={(i,j,k)|i,j,k∈N}.ShowthatT iscountable.
4.9 Review the way that we define sets to be the same size in Definition 4.12 (page 203). Show that “is the same size” is an equivalence relation.
PROBLEMS
A4.10 4.11 A 4.12 4.13 A 4.14
Let INFINITEDFA = {⟨A⟩| A is a DFA and L(A) is an infinite language}. Show that INFINITEDFA is decidable.
Let INFINITE PDA = {⟨M ⟩| M is a PDA and L(M ) is an infinite language}. Show that INFINITEPDA is decidable.
Let A = {⟨M ⟩| M is a DFA that doesn’t accept any string containing an odd num- ber of 1s}. Show that A is decidable.
Let A = {⟨R, S⟩| R and S are regular expressions and L(R) ⊆ L(S)}. Show that A is decidable.
Let Σ = {0,1}. Show that the problem of determining whether a CFG generates some string in 1∗ is decidable. In other words, show that
{⟨G⟩|GisaCFGover{0,1}and1∗ ∩L(G)̸=∅} is a decidable language.
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212
CHAPTER 4 / DECIDABILITY
⋆ 4.15
4.16
4.17
⋆ 4.18
⋆ 4.19 4.20
4.21 4.22
A⋆ 4.23
4.24
A⋆4.25
⋆4.26 ⋆ 4.27 4.28 4.29 4.30
4.31
Show that the problem of determining whether a CFG generates all strings in 1∗ is decidable. In other words, show that {⟨G⟩| G is a CFG over {0,1} and 1∗ ⊆ L(G)} is a decidable language.
Let A = {⟨R⟩| R is a regular expression describing a language containing at least one string w that has 111 as a substring (i.e., w = x111y for some x and y)}. Show that A is decidable.
Prove that EQ DFA is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.
Let C be a language. Prove that C is Turing-recognizable iff a decidable language D exists such that C = {x| ∃y (⟨x, y⟩ ∈ D)}.
Prove that the class of decidable languages is not closed under homomorphism.
Let A and B be two disjoint languages. Say that language C separates A and B if A ⊆ C and B ⊆ C. Show that any two disjoint co-Turing-recognizable languages are separable by some decidable language.
Let S = {⟨M⟩| M is a DFA that accepts wR whenever it accepts w}. Show that S is decidable.
Let PREFIX-FREEREX = {⟨R⟩| R is a regular expression and L(R) is prefix-free}. Show that PREFIX-FREEREX is decidable. Why does a similar approach fail to show that PREFIX-FREECFG is decidable?
Say that an NFA is ambiguous if it accepts some string along two different com- putation branches. Let AMBIGNFA = {⟨N⟩| N is an ambiguous NFA}. Show that AMBIGNFA is decidable. (Suggestion: One elegant way to solve this problem is to construct a suitable DFA and then run EDFA on it.)
A useless state in a pushdown automaton is never entered on any input string. Con- sider the problem of determining whether a pushdown automaton has any useless states. Formulate this problem as a language and show that it is decidable.
Let BALDFA = {⟨M⟩| M is a DFA that accepts some string containing an equal number of 0s and 1s}. Show that BALDFA is decidable. (Hint: Theorems about CFLs are helpful here.)
Let PALDFA = {⟨M⟩| M is a DFA that accepts some palindrome}. Show that PALDFA is decidable. (Hint: Theorems about CFLs are helpful here.)
Let E = {⟨M ⟩| M is a DFA that accepts some string with more 1s than 0s}. Show that E is decidable. (Hint: Theorems about CFLs are helpful here.)
LetC={⟨G,x⟩|GisaCFGxisasubstringofsomey∈L(G)}.ShowthatCis decidable. (Hint: An elegant solution to this problem uses the decider for ECFG.)
Let CCFG = {⟨G, k⟩| G is a CFG and L(G) contains exactly k strings where k ≥ 0 or k = ∞}. Show that CCFG is decidable.
Let A be a Turing-recognizable language consisting of descriptions of Turing ma- chines, {⟨M1 ⟩, ⟨M2 ⟩, . . .}, where every Mi is a decider. Prove that some decidable language D is not decided by any decider Mi whose description appears in A. (Hint: You may find it helpful to consider an enumerator for A.)
Say that a variable A in CFL G is usable if it appears in some derivation of some string w ∈ G. Given a CFG G and a variable A, consider the problem of testing whether A is usable. Formulate this problem as a language and show that it is decidable.
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4.32 The proof of Lemma 2.41 says that (q, x) is a looping situation for a DPDA P if when P is started in state q with x ∈ Γ on the top of the stack, it never pops anything below x and it never reads an input symbol. Show that F is decidable, where F = {⟨P, q, x⟩| (q, x) is a looping situation for P }.
SELECTED SOLUTIONS
4.1 (a) Yes. The DFA M accepts 0100.
(b) No. M doesn’t accept 011.
(c) No. This input has only a single component and thus is not of the correct form.
(d) No. The first component is not a regular expression and so the input is not of the correct form.
(e) No. M’s language isn’t empty.
(f) Yes. M accepts the same language as itself.
4.5 Lets1,s2,...bealistofallstringsinΣ∗.ThefollowingTMrecognizesETM.
“On input ⟨M⟩, where M is a TM:
1. Repeatthefollowingfori=1,2,3,....
2. Run M for i steps on each input, s1,s2,...,si.
3. If M has accepted any of these, accept . Otherwise, continue.”
4.6 (a) No, f is not one-to-one because f(1) = f(3). (d) Yes, g is one-to-one.
4.10 The following TM I decides INFINITE DFA .
I = “On input ⟨A⟩, where A is a DFA:
1. Let k be the number of states of A.
2. Construct a DFA D that accepts all strings of length k or more.
3. Construct a DFA M such that L(M) = L(A) ∩ L(D).
4. Test L(M ) = ∅ using the EDFA decider T from Theorem 4.4.
5. If T accepts, reject ; if T rejects, accept .”
This algorithm works because a DFA that accepts infinitely many strings must ac- cept arbitrarily long strings. Therefore, this algorithm accepts such DFAs. Con- versely, if the algorithm accepts a DFA, the DFA accepts some string of length k or more, where k is the number of states of the DFA. This string may be pumped in the manner of the pumping lemma for regular languages to obtain infinitely many accepted strings.
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SELECTED SOLUTIONS 213
214 CHAPTER 4 / DECIDABILITY 4.12 The following TM decides A.
“On input ⟨M⟩:
1. Construct a DFA O that accepts every string containing an odd
number of 1s.
2. Construct a DFA B such that L(B) = L(M) ∩ L(O).
3. Test whether L(B) = ∅ using the EDFA decider T from Theo-
rem 4.4.
4. If T accepts, accept ; if T rejects, reject .”
4.14 You showed in Problem 2.18 that if C is a context-free language and R is a regular language, then C ∩ R is context free. Therefore, 1∗ ∩ L(G) is context free. The following TM decides the language of this problem.
“On input ⟨G⟩:
1. Construct CFG H such that L(H) = 1∗ ∩ L(G).
2. Test whether L(H) = ∅ using the ECFG decider R from Theo-
rem 4.8.
3. If R accepts, reject ; if R rejects, accept .”
4.23 The following procedure decides AMBIGNFA. Given an NFA N, we design a DFA D that simulates N and accepts a string iff it is accepted by N along two different computational branches. Then we use a decider for EDFA to determine whether D accepts any strings.
Our strategy for constructing D is similar to the NFA-to-DFA conversion in the proof of Theorem 1.39. We simulate N by keeping a pebble on each active state. We begin by putting a red pebble on the start state and on each state reachable from the start state along ε transitions. We move, add, and remove pebbles in accordance with N’s transitions, preserving the color of the pebbles. Whenever two or more pebbles are moved to the same state, we replace its pebbles with a blue pebble. After reading the input, we accept if a blue pebble is on an accept state of N or if two different accept states of N have red pebbles on them.
The DFA D has a state corresponding to each possible position of pebbles. For each state of N , three possibilities occur: It can contain a red pebble, a blue pebble, or no pebble. Thus, if N has n states, D will have 3n states. Its start state, accept states, and transition function are defined to carry out the simulation.
4.25 The language of all strings with an equal number of 0s and 1s is a context-free language, generated by the grammar S → 1S0S | 0S1S | ε. Let P be the PDA that recognizes this language. Build a TM M for BALDFA, which operates as follows. On input ⟨B⟩, where B is a DFA, use B and P to construct a new PDA R that recognizes the intersection of the languages of B and P . Then test whether R’s language is empty. If its language is empty, reject ; otherwise, accept .
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5
REDUCIBILITY
In Chapter 4 we established the Turing machine as our model of a general pur- pose computer. We presented several examples of problems that are solvable on a Turing machine and gave one example of a problem, ATM, that is compu- tationally unsolvable. In this chapter we examine several additional unsolvable problems. In doing so, we introduce the primary method for proving that prob- lems are computationally unsolvable. It is called reducibility.
A reduction is a way of converting one problem to another problem in such a way that a solution to the second problem can be used to solve the first problem. Such reducibilities come up often in everyday life, even if we don’t usually refer to them in this way.
For example, suppose that you want to find your way around a new city. You know that doing so would be easy if you had a map. Thus, you can reduce the problem of finding your way around the city to the problem of obtaining a map of the city.
Reducibility always involves two problems, which we call A and B. If A re- duces to B, we can use a solution to B to solve A. So in our example, A is the problem of finding your way around the city and B is the problem of obtaining a map. Note that reducibility says nothing about solving A or B alone, but only about the solvability of A in the presence of a solution to B.
The following are further examples of reducibilities. The problem of travel- ing from Boston to Paris reduces to the problem of buying a plane ticket between the two cities. That problem in turn reduces to the problem of earning the money for the ticket. And that problem reduces to the problem of finding a job.
215
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216 CHAPTER 5 / REDUCIBILITY
Reducibility also occurs in mathematical problems. For example, the problem of measuring the area of a rectangle reduces to the problem of measuring its length and width. The problem of solving a system of linear equations reduces to the problem of inverting a matrix.
Reducibility plays an important role in classifying problems by decidability, and later in complexity theory as well. When A is reducible to B, solving A cannot be harder than solving B because a solution to B gives a solution to A. In terms of computability theory, if A is reducible to B and B is decidable, A also is decidable. Equivalently, if A is undecidable and reducible to B, B is undecidable. This last version is key to proving that various problems are undecidable.
In short, our method for proving that a problem is undecidable will be to show that some other problem already known to be undecidable reduces to it.
5.1
UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY
We have already established the undecidability of ATM, the problem of deter- mining whether a Turing machine accepts a given input. Let’s consider a related problem, HALT TM , the problem of determining whether a Turing machine halts (by accepting or rejecting) on a given input. This problem is widely known as the halting problem. We use the undecidability of ATM to prove the undecidability of the halting problem by reducing ATM to HALTTM. Let
HALTTM ={⟨M,w⟩|MisaTMandMhaltsoninputw}. THEOREM 5.1
HALT TM is undecidable.
PROOF IDEA This proof is by contradiction. We assume that HALTTM is decidable and use that assumption to show that ATM is decidable, contradicting Theorem 4.11. The key idea is to show that ATM is reducible to HALTTM.
Let’s assume that we have a TM R that decides HALTTM. Then we use R to construct S, a TM that decides ATM. To get a feel for the way to construct S, pretend that you are S. Your task is to decide ATM. You are given an input of the form ⟨M, w⟩. You must output accept if M accepts w, and you must output reject if M loops or rejects on w. Try simulating M on w. If it accepts or rejects, do the same. But you may not be able to determine whether M is looping, and in that case your simulation will not terminate. That’s bad because you are a decider and thus never permitted to loop. So this idea by itself does not work.
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5.1 UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 217
Instead, use the assumption that you have TM R that decides HALTTM. With R, you can test whether M halts on w. If R indicates that M doesn’t halt on w, reject because ⟨M, w⟩ isn’t in ATM. However, if R indicates that M does halt on w, you can do the simulation without any danger of looping.
Thus, if TM R exists, we can decide ATM, but we know that ATM is unde- cidable. By virtue of this contradiction, we can conclude that R does not exist. Therefore, HALTTM is undecidable.
PROOF Let’s assume for the purpose of obtaining a contradiction that TM R decides HALTTM. We construct TM S to decide ATM, with S operating as follows.
S = “On input ⟨M,w⟩, an encoding of a TM M and a string w:
1. Run TM R on input ⟨M, w⟩.
2. If R rejects, reject.
3. If R accepts, simulate M on w until it halts.
4. If M has accepted, accept; if M has rejected, reject.”
Clearly, if R decides HALTTM, then S decides ATM. Because ATM is unde-
cidable, HALT TM also must be undecidable.
Theorem 5.1 illustrates our strategy for proving that a problem is undecid- able. This strategy is common to most proofs of undecidability, except for the undecidability of ATM itself, which is proved directly via the diagonalization method.
We now present several other theorems and their proofs as further examples of the reducibility method for proving undecidability. Let
ETM ={⟨M⟩|M isaTMandL(M)=∅}.
THEOREM 5.2 ETM is undecidable.
PROOF IDEA We follow the pattern adopted in Theorem 5.1. We assume that ETM is decidable and then show that ATM is decidable—a contradiction. Let R be a TM that decides ETM. We use R to construct TM S that decides ATM. How will S work when it receives input ⟨M, w⟩?
One idea is for S to run R on input ⟨M ⟩ and see whether it accepts. If it does, we know that L(M) is empty and therefore that M does not accept w. But if R rejects ⟨M ⟩, all we know is that L(M ) is not empty and therefore that M accepts some string—but we still do not know whether M accepts the particular string w. So we need to use a different idea.
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218 CHAPTER 5 / REDUCIBILITY
Instead of running R on ⟨M ⟩, we run R on a modification of ⟨M ⟩. We modify ⟨M⟩ to guarantee that M rejects all strings except w, but on input w it works as usual. Then we use R to determine whether the modified machine recognizes the empty language. The only string the machine can now accept is w, so its language will be nonempty iff it accepts w. If R accepts when it is fed a descrip- tion of the modified machine, we know that the modified machine doesn’t accept anything and that M doesn’t accept w.
PROOF Let’s write the modified machine described in the proof idea using our standard notation. We call it M1.
M1 = “On input x:
1. If x ̸= w, reject.
2. Ifx=w,runM oninputwandaccept ifM does.”
This machine has the string w as part of its description. It conducts the test of whether x = w in the obvious way, by scanning the input and comparing it character by character with w to determine whether they are the same.
Putting all this together, we assume that TM R decides ETM and construct TM S that decides ATM as follows.
S = “On input ⟨M,w⟩, an encoding of a TM M and a string w:
1. Use the description of M and w to construct the TM M1 just
described.
2. Run R on input ⟨M1 ⟩.
3. If R accepts, reject ; if R rejects, accept .”
Note that S must actually be able to compute a description of M1 from a description of M and w. It is able to do so because it only needs to add extra states to M that perform the x = w test.
If R were a decider for ETM, S would be a decider for ATM. A decider for ATM cannot exist, so we know that ETM must be undecidable.
Another interesting computational problem regarding Turing machines con- cerns determining whether a given Turing machine recognizes a language that also can be recognized by a simpler computational model. For example, we let REGULARTM be the problem of determining whether a given Turing machine has an equivalent finite automaton. This problem is the same as determining whether the Turing machine recognizes a regular language. Let
REGULARTM = {⟨M ⟩| M is a TM and L(M ) is a regular language}.
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5.1 UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 219 THEOREM 5.3
REGULARTM is undecidable.
PROOF IDEA As usual for undecidability theorems, this proof is by reduction from ATM. We assume that REGULARTM is decidable by a TM R and use this assumption to construct a TM S that decides ATM. Less obvious now is how to use R’s ability to assist S in its task. Nonetheless, we can do so.
The idea is for S to take its input ⟨M, w⟩ and modify M so that the result- ing TM recognizes a regular language if and only if M accepts w. We call the modified machine M2. We design M2 to recognize the nonregular language {0n1n| n ≥ 0} if M does not accept w, and to recognize the regular language Σ∗ if M accepts w. We must specify how S can construct such an M2 from M and w. Here, M2 works by automatically accepting all strings in {0n1n| n ≥ 0}. In addition, if M accepts w, M2 accepts all other strings.
Note that the TM M2 is not constructed for the purposes of actually running it on some input—a common confusion. We construct M2 only for the purpose of feeding its description into the decider for REGULARTM that we have assumed to exist. Once this decider returns its answer, we can use it to obtain the answer to whether M accepts w. Thus, we can decide ATM, a contradiction.
PROOF We let R be a TM that decides REGULARTM and construct TM S to decide ATM. Then S works in the following manner.
S = “On input ⟨M, w⟩, where M is a TM and w is a string:
1. Construct the following TM M2.
M2 = “On input x:
1. Ifxhastheform0n1n,accept.
2. If x does not have this form, run M on input w and
accept if M accepts w.”
2. Run R on input ⟨M2⟩.
3. If R accepts, accept; if R rejects, reject.”
Similarly, the problems of testing whether the language of a Turing machine is a context-free language, a decidable language, or even a finite language can be shown to be undecidable with similar proofs. In fact, a general result, called Rice’s theorem, states that determining any property of the languages recognized by Turing machines is undecidable. We give Rice’s theorem in Problem 5.28.
So far, our strategy for proving languages undecidable involves a reduction from ATM. Sometimes reducing from some other undecidable language, such as ETM, is more convenient when we are showing that certain languages are undecidable. Theorem 5.4 shows that testing the equivalence of two Turing
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220 CHAPTER 5 / REDUCIBILITY
machines is an undecidable problem. We could prove it by a reduction from ATM, but we use this opportunity to give an example of an undecidability proof by reduction from ETM. Let
EQTM = {⟨M1,M2⟩| M1 and M2 are TMs and L(M1) = L(M2)}. THEOREM 5.4
EQTM is undecidable.
PROOF IDEA Show that if EQTM were decidable, ETM also would be decid- able by giving a reduction from ETM to EQTM. The idea is simple. ETM is the problem of determining whether the language of a TM is empty. EQTM is the problem of determining whether the languages of two TMs are the same. If one of these languages happens to be ∅, we end up with the problem of determining whether the language of the other machine is empty—that is, the ETM problem. So in a sense, the ETM problem is a special case of the EQTM problem wherein one of the machines is fixed to recognize the empty language. This idea makes giving the reduction easy.
PROOF We let TM R decide EQTM and construct TM S to decide ETM as follows.
S =“Oninput⟨M⟩,whereM isaTM:
1. Run R on input ⟨M, M1⟩, where M1 is a TM that rejects all
inputs.
2. If R accepts, accept; if R rejects, reject.”
If R decides EQTM, S decides ETM. But ETM is undecidable by Theorem 5.2, so EQTM also must be undecidable.
REDUCTIONS VIA COMPUTATION HISTORIES
The computation history method is an important technique for proving that ATM is reducible to certain languages. This method is often useful when the problem to be shown undecidable involves testing for the existence of some- thing. For example, this method is used to show the undecidability of Hilbert’s tenth problem, testing for the existence of integral roots in a polynomial.
The computation history for a Turing machine on an input is simply the se- quence of configurations that the machine goes through as it processes the input. It is a complete record of the computation of this machine.
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5.1 UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 221
DEFINITION 5.5
Let M be a Turing machine and w an input string. An accepting computation history for M on w is a sequence of configurations, C1,C2,...,Cl,whereC1 isthestartconfigurationofM onw,Cl is an accepting configuration of M, and each Ci legally follows from Ci−1 according to the rules of M. A rejecting computation his- tory for M on w is defined similarly, except that Cl is a rejecting configuration.
Computation histories are finite sequences. If M doesn’t halt on w, no accept- ing or rejecting computation history exists for M on w. Deterministic machines have at most one computation history on any given input. Nondeterministic ma- chines may have many computation histories on a single input, corresponding to the various computation branches. For now, we continue to focus on deter- ministic machines. Our first undecidability proof using the computation history method concerns a type of machine called a linear bounded automaton.
DEFINITION 5.6
A linear bounded automaton is a restricted type of Turing machine wherein the tape head isn’t permitted to move off the portion of the tape containing the input. If the machine tries to move its head off either end of the input, the head stays where it is—in the same way that the head will not move off the left-hand end of an ordinary Turing machine’s tape.
A linear bounded automaton is a Turing machine with a limited amount of memory, as shown schematically in the following figure. It can only solve prob- lems requiring memory that can fit within the tape used for the input. Using a tape alphabet larger than the input alphabet allows the available memory to be increased up to a constant factor. Hence we say that for an input of length n, the amount of memory available is linear in n—thus the name of this model.
FIGURE 5.7
Schematic of a linear bounded automaton
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222 CHAPTER 5 / REDUCIBILITY
Despite their memory constraint, linear bounded automata (LBAs) are quite powerful. For example, the deciders for ADFA, ACFG, EDFA, and ECFG all are LBAs. Every CFL can be decided by an LBA. In fact, coming up with a decidable language that can’t be decided by an LBA takes some work. We develop the techniques to do so in Chapter 9.
Here, ALBA is the problem of determining whether an LBA accepts its input. Even though ALBA is the same as the undecidable problem ATM where the Tur- ing machine is restricted to be an LBA, we can show that ALBA is decidable. Let
ALBA = {⟨M, w⟩| M is an LBA that accepts string w}.
Before proving the decidability of ALBA, we find the following lemma useful. It says that an LBA can have only a limited number of configurations when a string of length n is the input.
LEMMA 5.8
Let M be an LBA with q states and g symbols in the tape alphabet. There are
exactly qngn distinct configurations of M for a tape of length n.
PROOF Recall that a configuration of M is like a snapshot in the middle of its computation. A configuration consists of the state of the control, position of the head, and contents of the tape. Here, M has q states. The length of its tape is n, so the head can be in one of n positions, and gn possible strings of tape symbols appear on the tape. The product of these three quantities is the total number of different configurations of M with a tape of length n.
THEOREM 5.9 ALBA is decidable.
PROOF IDEA In order to decide whether LBA M accepts input w, we simulate M on w. During the course of the simulation, if M halts and accepts or rejects, we accept or reject accordingly. The difficulty occurs if M loops on w. We need to be able to detect looping so that we can halt and reject.
The idea for detecting when M is looping is that as M computes on w, it goes from configuration to configuration. If M ever repeats a configuration, it would go on to repeat this configuration over and over again and thus be in a loop. Because M is an LBA, the amount of tape available to it is limited. By Lemma 5.8, M can be in only a limited number of configurations on this amount of tape. Therefore, only a limited amount of time is available to M before it will enter some configuration that it has previously entered. Detecting that M is looping is possible by simulating M for the number of steps given by Lemma 5.8. If M has not halted by then, it must be looping.
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5.1 UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 223 PROOF The algorithm that decides ALBA is as follows.
L = “On input ⟨M,w⟩, where M is an LBA and w is a string:
1. Simulate M on w for qngn steps or until it halts.
2. If M has halted, accept if it has accepted and reject if it has
rejected. If it has not halted, reject.”
If M on w has not halted within qngn steps, it must be repeating a configura- tion according to Lemma 5.8 and therefore looping. That is why our algorithm rejects in this instance.
Theorem 5.9 shows that LBAs and TMs differ in one essential way: For LBAs the acceptance problem is decidable, but for TM s it isn’t. However, certain other problems involving LBAs remain undecidable. One is the emptiness problem ELBA = {⟨M⟩| M is an LBA where L(M) = ∅}. To prove that ELBA is undecid- able, we give a reduction that uses the computation history method.
THEOREM 5.10 ELBA is undecidable.
PROOF IDEA This proof is by reduction from ATM. We show that if ELBA were decidable, ATM would also be. Suppose that ELBA is decidable. How can we use this supposition to decide ATM?
For a TM M and an input w, we can determine whether M accepts w by con- structing a certain LBA B and then testing whether L(B) is empty. The language that B recognizes comprises all accepting computation histories for M on w. If M accepts w, this language contains one string and so is nonempty. If M does not accept w, this language is empty. If we can determine whether B’s language is empty, clearly we can determine whether M accepts w.
Now we describe how to construct B from M and w. Note that we need to show more than the mere existence of B. We have to show how a Turing machine can obtain a description of B, given descriptions of M and w.
As in the previous reductions we’ve given for proving undecidability, we con- struct B only to feed its description into the presumed ELBA decider, but not to run B on some input.
We construct B to accept its input x if x is an accepting computation history for M on w. Recall that an accepting computation history is the sequence of configurations, C1 , C2 , . . . , Cl that M goes through as it accepts some string w. For the purposes of this proof, we assume that the accepting computation history is presented as a single string with the configurations separated from each other by the # symbol, as shown in Figure 5.11.
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224 CHAPTER 5 / REDUCIBILITY
# # # # ··· # # C1 C2 C3 Cl
FIGURE 5.11
A possible input to B
The LBA B works as follows. When it receives an input x, B is supposed to accept if x is an accepting computation history for M on w. First, B breaks up x according to the delimiters into strings C1, C2,..., Cl. Then B determines whether the Ci ’s satisfy the three conditions of an accepting computation history.
1. C1 is the start configuration for M on w. 2. Each Ci+1 legally follows from Ci.
3. Cl is an accepting configuration for M .
The start configuration C1 for M on w is the string q0w1w2 · · · wn, where q0 is the start state for M on w. Here, B has this string directly built in, so it is able to check the first condition. An accepting configuration is one that contains the qaccept state, so B can check the third condition by scanning Cl for qaccept. The second condition is the hardest to check. For each pair of adjacent configurations, B checks on whether Ci+1 legally follows from Ci. This step involves verifying that Ci and Ci+1 are identical except for the positions under and adjacent to the head in Ci. These positions must be updated according to the transition function of M . Then B verifies that the updating was done properly by zig-zagging between corresponding positions of Ci and Ci+1. To keep track of the current positions while zig-zagging, B marks the current position with dots on the tape. Finally, if conditions 1, 2, and 3 are satisfied, B accepts its input.
By inverting the decider’s answer, we obtain the answer to whether M accepts w. Thus we can decide ATM, a contradiction.
PROOF Now we are ready to state the reduction of ATM to ELBA. Suppose that TM R decides ELBA. Construct TM S to decide ATM as follows.
S = “On input ⟨M, w⟩, where M is a TM and w is a string:
1. Construct LBA B from M and w as described in the proof idea.
2. Run R on input ⟨B⟩.
3. If R rejects, accept; if R accepts, reject.”
If R accepts ⟨B⟩, then L(B) = ∅. Thus, M has no accepting computation history on w and M doesn’t accept w. Consequently, S rejects ⟨M, w⟩. Similarly, if R rejects ⟨B⟩, the language of B is nonempty. The only string that B can accept is an accepting computation history for M on w. Thus, M must accept w. Consequently, S accepts ⟨M, w⟩. Figure 5.12 illustrates LBA B.
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5.1 UNDECIDABLE PROBLEMS FROM LANGUAGE THEORY 225
FIGURE 5.12
LBA B checking a TM computation history
We can also use the technique of reduction via computation histories to es- tablish the undecidability of certain problems related to context-free grammars and pushdown automata. Recall that in Theorem 4.8 we presented an algo- rithm to decide whether a context-free grammar generates any strings—that is, whether L(G) = ∅. Now we show that a related problem is undecidable. It is the problem of determining whether a context-free grammar generates all possible strings. Proving that this problem is undecidable is the main step in showing that the equivalence problem for context-free grammars is undecidable. Let
ALLCFG = {⟨G⟩| G is a CFG and L(G) = Σ∗}. THEOREM 5.13
ALLCFG is undecidable.
PROOF This proof is by contradiction. To get the contradiction, we assume that ALLCFG is decidable and use this assumption to show that ATM is decidable. This proof is similar to that of Theorem 5.10 but with a small extra twist: It is a reduction from ATM via computation histories, but we modify the representation of the computation histories slightly for a technical reason that we will explain later.
We now describe how to use a decision procedure for ALLCFG to decide ATM. For a TM M and an input w, we construct a CFG G that generates all strings if and only if M does not accept w. So if M does accept w, G does not generate some particular string. This string is—guess what—the accepting computation history for M on w. That is, G is designed to generate all strings that are not accepting computation histories for M on w.
To make the CFG G generate all strings that fail to be an accepting computa- tion history for M on w, we utilize the following strategy. A string may fail to be an accepting computation history for several reasons. An accepting computation history for M on w appears as #C1 #C2 # · · · #Cl #, where Ci is the configuration of M on the ith step of the computation on w. Then, G generates all strings
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226 CHAPTER 5 / REDUCIBILITY
1. that do not start with C1,
2. that do not end with an accepting configuration, or
3. in which some Ci does not properly yield Ci+1 under the rules of M .
If M does not accept w, no accepting computation history exists, so all strings fail in one way or another. Therefore, G would generate all strings, as desired.
Now we get down to the actual construction of G. Instead of constructing G, we construct a PDA D. We know that we can use the construction given in Theorem 2.20 (page 117) to convert D to a CFG. We do so because, for our purposes, designing a PDA is easier than designing a CFG. In this instance, D will start by nondeterministically branching to guess which of the preceding three conditions to check. One branch checks on whether the beginning of the input string is C1 and accepts if it isn’t. Another branch checks on whether the input string ends with a configuration containing the accept state, qaccept, and accepts if it isn’t.
The third branch is supposed to accept if some Ci does not properly yield Ci+1. It works by scanning the input until it nondeterministically decides that it has come to Ci. Next, it pushes Ci onto the stack until it comes to the end as marked by the # symbol. Then D pops the stack to compare with Ci+1. They are supposed to match except around the head position, where the difference is dictated by the transition function of M. Finally, D accepts if it discovers a mismatch or an improper update.
The problem with this idea is that when D pops Ci off the stack, it is in reverse order and not suitable for comparison with Ci+1. At this point, the twist in the proof appears: We write the accepting computation history differently. Every other configuration appears in reverse order. The odd positions remain written in the forward order, but the even positions are written backward. Thus, an accepting computation history would appear as shown in the following figure.
# −→ # ←− # −→ # ←− # ··· # #
C1C2RC3C4R Cl FIGURE 5.14
Every other configuration written in reverse order
In this modified form, the PDA is able to push a configuration so that when it is popped, the order is suitable for comparison with the next one. We design D to accept any string that is not an accepting computation history in the modified form.
In Exercise 5.1 you can use Theorem 5.13 to show that EQ CFG is undecidable.
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5.2 A SIMPLE UNDECIDABLE PROBLEM 227
5.2
A SIMPLE UNDECIDABLE PROBLEM
In this section we show that the phenomenon of undecidability is not confined to problems concerning automata. We give an example of an undecidable problem concerning simple manipulations of strings. It is called the Post Correspondence Problem, or PCP.
We can describe this problem easily as a type of puzzle. We begin with a col- lection of dominos, each containing two strings, one on each side. An individual
domino looks like
a ab
and a collection of dominos looks like
b , a , ca, abc. ca ab a c
The task is to make a list of these dominos (repetitions permitted) so that the string we get by reading off the symbols on the top is the same as the string of symbols on the bottom. This list is called a match. For example, the following list is a match for this puzzle.
a b ca a abc ab ca a ab c
Reading off the top string we get abcaaabc, which is the same as reading off the bottom. We can also depict this match by deforming the dominos so that the corresponding symbols from top and bottom line up.
For some collections of dominos, finding a match may not be possible. For
example, the collection
abc, ca, acc ab a ba
cannot contain a match because every top string is longer than the corresponding bottom string.
The Post Correspondence Problem is to determine whether a collection of dominos has a match. This problem is unsolvable by algorithms.
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228 CHAPTER 5 / REDUCIBILITY
Before getting to the formal statement of this theorem and its proof, let’s state the problem precisely and then express it as a language. An instance of the PCP is a collection P of dominos t1 t2 tk
P=b,b,...,b, 12k
and a match is a sequence i1,i2,...,il, where ti1ti2 ··· til = bi1bi2 ··· bil. The problem is to determine whether P has a match. Let
PCP = {⟨P ⟩| P is an instance of the Post Correspondence Problem with a match}.
THEOREM 5.15 PCP is undecidable.
PROOF IDEA Conceptually this proof is simple, though it involves many de- tails. The main technique is reduction from ATM via accepting computation histories. We show that from any TM M and input w, we can construct an in- stance P where a match is an accepting computation history for M on w. If we could determine whether the instance has a match, we would be able to deter- mine whether M accepts w.
How can we construct P so that a match is an accepting computation history for M on w? We choose the dominos in P so that making a match forces a simulation of M to occur. In the match, each domino links a position or positions in one configuration with the corresponding one(s) in the next configuration.
Before getting to the construction, we handle three small technical points. (Don’t worry about them too much on your initial reading through this con- struction.) First, for convenience in constructing P, we assume that M on w never attempts to move its head off the left-hand end of the tape. That requires first altering M to prevent this behavior. Second, if w = ε, we use the string ␣ in place of w in the construction. Third, we modify the PCP to require that a match starts with the first domino,
t1 .
b1
Later we show how to eliminate this requirement. We call this problem the Modified Post Correspondence Problem (MPCP). Let
MPCP = {⟨P ⟩| P is an instance of the Post Correspondence Problem with a match that starts with the first domino}.
Now let’s move into the details of the proof and design P to simulate M on w. PROOF We let TM R decide the PCP and construct S deciding ATM. Let
M = (Q, Σ, Γ, δ, q0, qaccept, qreject),
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5.2 A SIMPLE UNDECIDABLE PROBLEM 229
where Q, Σ, Γ, and δ are the state set, input alphabet, tape alphabet, and transi- tion function of M , respectively.
In this case, S constructs an instance of the PCP P that has a match iff M accepts w. To do that, S first constructs an instance P ′ of the MPCP. We de- scribe the construction in seven parts, each of which accomplishes a particular aspect of simulating M on w. To explain what we are doing, we interleave the construction with an example of the construction in action.
Part 1. The construction begins in the following manner.
Put # into P′ as the first domino t1 .
#q0w1w2 · · · wn# b1
Because P ′ is an instance of the MPCP, the match must begin with this domino. Thus, the bottom string begins correctly with C1 = q0 w1 w2 · · · wn , the first configuration in the accepting computation history for M on w, as shown in the following figure.
FIGURE 5.16
Beginning of the MPCP match
In this depiction of the partial match achieved so far, the bottom string con- sists of #q0w1w2 · · · wn# and the top string consists only of #. To get a match, we need to extend the top string to match the bottom string. We provide additional dominos to allow this extension. The additional dominos cause M ’s next config- uration to appear at the extension of the bottom string by forcing a single-step simulation of M .
In parts 2, 3, and 4, we add to P′ dominos that perform the main part of the simulation. Part 2 handles head motions to the right, part 3 handles head motions to the left, and part 4 handles the tape cells not adjacent to the head.
Part 2.
Part 3.
For every a,b ∈ Γ and every q,r ∈ Q where q ̸= qreject, if δ(q,a) = (r,b,R), put qa into P′.
br
For every a,b,c ∈ Γ and every q,r ∈ Q where q ̸= qreject,
if δ(q,a) = (r,b,L), put cqa into P′. rcb
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230 CHAPTER 5 / REDUCIBILITY
Part 4. For every a ∈ Γ,
put a into P′.
a
Now we make up a hypothetical example to illustrate what we have built so far. Let Γ = {0, 1, 2, ␣}. Say that w is the string 0100 and that the start state of M is q0. In state q0, upon reading a 0, let’s say that the transition function dictates that M enters state q7, writes a 2 on the tape, and moves its head to the right. In other words, δ(q0, 0) = (q7, 2, R).
Part 1 places the domino
# = t1 #q0 0100# b1
in P′, and the match begins
In addition, part 2 places the domino
q00 2q7
as δ(q0, 0) = (q7, 2, R) and part 4 places the dominos
0,1,2, and ␣
012 ␣
in P′, as 0, 1, 2, and ␣ are the members of Γ. Together with part 5, that allows us to extend the match to
Thus, the dominos of parts 2, 3, and 4 let us extend the match by adding the second configuration after the first one. We want this process to continue, adding the third configuration, then the fourth, and so on. For it to happen, we need to add one more domino for copying the # symbol.
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Part 5.
Put#and # intoP′. # ␣#
5.2 A SIMPLE UNDECIDABLE PROBLEM 231
The first of these dominos allows us to copy the # symbol that marks the sep- aration of the configurations. In addition to that, the second domino allows us to add a blank symbol ␣ at the end of the configuration to simulate the infinitely many blanks to the right that are suppressed when we write the configuration.
Continuing with the example, let’s say that in state q7, upon reading a 1, M goes to state q5, writes a 0, and moves the head to the right. That is, δ(q7, 1) = (q5, 0, R). Then we have the domino
q71 in P′. 0q5
So the latest partial match extends to
Then, suppose that in state q5, upon reading a 0, M goes to state q9, writes a 2, and moves its head to the left. So δ(q5, 0) = (q9, 2, L). Then we have the
dominos
0q50,1q50,2q50, and ␣q50. q902 q912 q922 q9␣2
The first one is relevant because the symbol to the left of the head is a 0. The preceding partial match extends to
Note that as we construct a match, we are forced to simulate M on input w. This process continues until M reaches a halting state. If the accept state occurs, we want to let the top of the partial match “catch up” with the bottom so that the match is complete. We can arrange for that to happen by adding additional dominos.
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232 CHAPTER 5 / REDUCIBILITY
Part 6. For every a ∈ Γ,
put aqaccept and qaccept a into P′.
qaccept qaccept
This step has the effect of adding “pseudo-steps” of the Turing machine after it has halted, where the head “eats” adjacent symbols until none are left. Con- tinuing with the example, if the partial match up to the point when the machine halts in the accept state is
The dominos we have just added allow the match to continue:
Part 7. Finally, we add the domino
qaccept## #
and complete the match:
That concludes the construction of P′. Recall that P′ is an instance of the MPCP whereby the match simulates the computation of M on w. To finish the proof, we recall that the MPCP differs from the PCP in that the match is required to start with the first domino in the list. If we view P ′ as an instance of
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5.2 A SIMPLE UNDECIDABLE PROBLEM 233
the PCP instead of the MPCP, it obviously has a match, regardless of whether M accepts w. Can you find it? (Hint: It is very short.)
We now show how to convert P ′ to P , an instance of the PCP that still sim- ulates M on w. We do so with a somewhat technical trick. The idea is to take the requirement that the match starts with the first domino and build it directly into the problem instance itself so that it becomes enforced automatically. After that, the requirement isn’t needed. We introduce some notation to implement this idea.
Let u = u1u2 ···un be any string of length n. Define ⋆u, u⋆, and ⋆u⋆ to be the three strings
⋆u = ∗u1∗u2∗u3∗ ··· ∗un u⋆ = u1∗u2∗u3∗ ··· ∗un∗
⋆u⋆ = ∗u1∗u2∗u3∗ ··· ∗un∗.
Here, ⋆u adds the symbol ∗ before every character in u, u⋆ adds one after each character in u, and ⋆u⋆ adds one both before and after each character in u.
To convert P ′ to P , an instance of the PCP, we do the following. If P ′ were the collection
t1 , t2 , t3 , ... ,tk , b1b2b3 bk
we let P be the collection
⋆t1 , ⋆t1 , ⋆t2 , ⋆t3 , ... ,⋆tk , ∗✸.
⋆b1⋆ b1⋆ b2⋆ b3⋆ bk⋆ ✸
Considering P as an instance of the PCP, we see that the only domino that
could possibly start a match is the first one,
⋆t1 , ⋆b1⋆
because it is the only one where both the top and the bottom start with the same symbol—namely, ∗. Besides forcing the match to start with the first domino, the presence of the ∗s doesn’t affect possible matches because they simply interleave with the original symbols. The original symbols now occur in the even positions of the match. The domino
∗✸ ✸
is there to allow the top to add the extra ∗ at the end of the match.
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234 CHAPTER 5 / REDUCIBILITY 5.3
MAPPING REDUCIBILITY
We have shown how to use the reducibility technique to prove that various prob- lems are undecidable. In this section we formalize the notion of reducibility. Doing so allows us to use reducibility in more refined ways, such as for prov- ing that certain languages are not Turing-recognizable and for applications in complexity theory.
The notion of reducing one problem to another may be defined formally in one of several ways. The choice of which one to use depends on the application. Our choice is a simple type of reducibility called mapping reducibility.1
Roughly speaking, being able to reduce problem A to problem B by using a mapping reducibility means that a computable function exists that converts instances of problem A to instances of problem B. If we have such a conversion function, called a reduction, we can solve A with a solver for B. The reason is that any instance of A can be solved by first using the reduction to convert it to an instance of B and then applying the solver for B. A precise definition of mapping reducibility follows shortly.
COMPUTABLE FUNCTIONS
A Turing machine computes a function by starting with the input to the function on the tape and halting with the output of the function on the tape.
EXAMPLE 5.18
All usual, arithmetic operations on integers are computable functions. For ex- ample, we can make a machine that takes input ⟨m, n⟩ and returns m + n, the sum of m and n. We don’t give any details here, leaving them as exercises.
EXAMPLE 5.19
Computable functions may be transformations of machine descriptions. For example, one computable function f takes input w and returns the description of a Turing machine ⟨M′⟩ if w = ⟨M⟩ is an encoding of a Turing machine M.
1It is called many–one reducibility in some other textbooks.
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DEFINITION 5.17
A function f: Σ∗−→Σ∗ is a computable function if some Turing machine M , on every input w, halts with just f (w) on its tape.
5.3 MAPPING REDUCIBILITY 235
The machine M′ is a machine that recognizes the same language as M, but never attempts to move its head off the left-hand end of its tape. The function f accomplishes this task by adding several states to the description of M. The function returns ε if w is not a legal encoding of a Turing machine.
FORMAL DEFINITION OF MAPPING REDUCIBILITY
Now we define mapping reducibility. As usual, we represent computational problems by languages.
The following figure illustrates mapping reducibility.
DEFINITION 5.20
Language A is mapping reducible to language B, written A ≤m B,
if there is a computable function f : Σ∗−→Σ∗, where for every w, w ∈ A ⇐⇒ f ( w ) ∈ B .
The function f is called the reduction from A to B.
FIGURE 5.21
Function f reducing A to B
A mapping reduction of A to B provides a way to convert questions about membership testing in A to membership testing in B. To test whether w ∈ A, we use the reduction f to map w to f(w) and test whether f(w) ∈ B. The term mapping reduction comes from the function or mapping that provides the means of doing the reduction.
If one problem is mapping reducible to a second, previously solved problem, we can thereby obtain a solution to the original problem. We capture this idea in Theorem 5.22.
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236 CHAPTER 5 / REDUCIBILITY THEOREM 5.22
If A ≤m B and B is decidable, then A is decidable.
PROOF We let M be the decider for B and f be the reduction from A to B.
We describe a decider N for A as follows.
N = “On input w:
1. Compute f (w).
2. Run M on input f (w) and output whatever M outputs.”
Clearly, if w ∈ A, then f(w) ∈ B because f is a reduction from A to B. Thus,
M accepts f(w) whenever w ∈ A. Therefore, N works as desired.
The following corollary of Theorem 5.22 has been our main tool for proving
undecidability.
COROLLARY 5.23
If A ≤m B and A is undecidable, then B is undecidable.
Now we revisit some of our earlier proofs that used the reducibility method to get examples of mapping reducibilities.
EXAMPLE 5.24
In Theorem 5.1 we used a reduction from ATM to prove that HALTTM is un- decidable. This reduction showed how a decider for HALT TM could be used to give a decider for ATM. We can demonstrate a mapping reducibility from ATM to HALT TM as follows. To do so, we must present a computable function f that takes input of the form ⟨M,w⟩ and returns output of the form ⟨M′,w′⟩, where
⟨M,w⟩ ∈ ATM if and only if ⟨M′,w′⟩ ∈ HALTTM. The following machine F computes a reduction f .
F = “On input ⟨M, w⟩:
1. Construct the following machine M ′ .
M′ = “On input x:
1. RunMonx.
2. If M accepts, accept.
3. IfMrejects,enteraloop.” 2. Output⟨M′,w⟩.”
A minor issue arises here concerning improperly formed input strings. If TM F determines that its input is not of the correct form as specified in the input line “On input ⟨M,w⟩:” and hence that the input is not in ATM, the TM outputs a
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5.3 MAPPING REDUCIBILITY 237
string not in HALTTM. Any string not in HALTTM will do. In general, when we describe a Turing machine that computes a reduction from A to B, improperly formed inputs are assumed to map to strings outside of B.
EXAMPLE 5.25
The proof of the undecidability of the Post Correspondence Problem in Theo- rem 5.15 contains two mapping reductions. First, it shows that ATM ≤m MPCP and then it shows that MPCP ≤m PCP. In both cases, we can easily ob- tain the actual reduction function and show that it is a mapping reduction. As Exercise 5.6 shows, mapping reducibility is transitive, so these two reductions together imply that ATM ≤m PCP.
EXAMPLE 5.26
A mapping reduction from ETM to EQTM lies in the proof of Theorem 5.4. In thiscase,thereductionf mapstheinput⟨M⟩totheoutput⟨M,M1⟩,whereM1 is the machine that rejects all inputs.
EXAMPLE 5.27
The proof of Theorem 5.2 showing that ETM is undecidable illustrates the dif- ference between the formal notion of mapping reducibility that we have defined in this section and the informal notion of reducibility that we used earlier in this chapter. The proof shows that ETM is undecidable by reducing ATM to it. Let’s see whether we can convert this reduction to a mapping reduction.
From the original reduction, we may easily construct a function f that takes input ⟨M, w⟩ and produces output ⟨M1⟩, where M1 is the Turing machine de- scribed in that proof. But M accepts w iff L(M1) is not empty so f is a mapping reduction from ATM to ETM. It still shows that ETM is undecidable because decidability is not affected by complementation, but it doesn’t give a mapping reduction from ATM to ETM. In fact, no such reduction exists, as you are asked to show in Exercise 5.5.
The sensitivity of mapping reducibility to complementation is important in the use of reducibility to prove nonrecognizability of certain languages. We can also use mapping reducibility to show that problems are not Turing- recognizable. The following theorem is analogous to Theorem 5.22.
THEOREM 5.28
If A ≤m B and B is Turing-recognizable, then A is Turing-recognizable.
The proof is the same as that of Theorem 5.22, except that M and N are recog- nizers instead of deciders.
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238 CHAPTER 5 / REDUCIBILITY COROLLARY 5.29
If A ≤m B and A is not Turing-recognizable, then B is not Turing-recognizable.
In a typical application of this corollary, we let A be ATM, the complement of ATM. We know that ATM is not Turing-recognizable from Corollary 4.23. The definition of mapping reducibility implies that A ≤m B means the same as A ≤m B. To prove that B isn’t recognizable, we may show that ATM ≤m B. We can also use mapping reducibility to show that certain problems are neither Turing-recognizable nor co-Turing-recognizable, as in the following theorem.
THEOREM 5.30
EQTM is neither Turing-recognizable nor co-Turing-recognizable.
PROOF First we show that EQTM is not Turing-recognizable. We do so by showing that ATM is reducible to EQTM. The reducing function f works as follows.
F = “On input ⟨M, w⟩, where M is a TM and w a string:
1. Construct the following two machines, M1 and M2.
M1 = “On any input: 1. Reject.”
M2 = “On any input:
1. RunMonw.Ifitaccepts,accept.”
2. Output ⟨M1 , M2 ⟩.”
Here, M1 accepts nothing. If M accepts w, M2 accepts everything, and so the two machines are not equivalent. Conversely, if M doesn’t accept w, M2 accepts nothing, and they are equivalent. Thus f reduces ATM to EQTM, as desired.
To show that EQTM is not Turing-recognizable, we give a reduction from ATM to the complement of EQ TM —namely, EQ TM . Hence we show that ATM ≤m EQTM. The following TM G computes the reducing function g.
G = “On input ⟨M, w⟩, where M is a TM and w a string:
1. Construct the following two machines, M1 and M2.
M1 = “On any input: 1. Accept.”
M2 = “On any input:
1. RunMonw.
2. If it accepts, accept.” 2. Output ⟨M1 , M2 ⟩.”
The only difference between f and g is in machine M1. In f, machine M1 always rejects, whereas in g it always accepts. In both f and g, M accepts w iff M2 always accepts. In g, M accepts w iff M1 and M2 are equivalent. That is why g is a reduction from ATM to EQTM.
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EXERCISES 239
EXERCISES
5.1 5.2 5.3
5.4 A5.5
A 5.6 A5.7 A 5.8
Show that EQ CFG is undecidable.
Show that EQ CFG is co-Turing-recognizable.
Find a match in the following instance of the Post Correspondence Problem.
ab , b, aba, aa abab a b a
If A ≤m B and B is a regular language, does that imply that A is a regular lan- guage? Why or why not?
Show that ATM is not mapping reducible to ETM. In other words, show that no computable function reduces ATM to ETM. (Hint: Use a proof by contradiction, and facts you already know about ATM and ETM.)
Show that ≤m is a transitive relation.
Show that if A is Turing-recognizable and A ≤m A, then A is decidable.
In the proof of Theorem 5.15, we modified the Turing machine M so that it never tries to move its head off the left-hand end of the tape. Suppose that we did not make this modification to M . Modify the PCP construction to handle this case.
PROBLEMS
5.9 Let T = {⟨M⟩| M is a TM that accepts wR whenever it accepts w}. Show that T is undecidable.
A5.10 Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape when it is run on input w. Formulate this problem as a language and show that it is undecidable.
A5.11 Consider the problem of determining whether a two-tape Turing machine ever writes a nonblank symbol on its second tape during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.
5.12 Consider the problem of determining whether a single-tape Turing machine ever writes a blank symbol over a nonblank symbol during the course of its computation on any input string. Formulate this problem as a language and show that it is undecidable.
5.13 A useless state in a Turing machine is one that is never entered on any input string. Consider the problem of determining whether a Turing machine has any useless states. Formulate this problem as a language and show that it is undecidable.
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240 CHAPTER 5 / REDUCIBILITY
5.14 Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left when its head is on the left-most tape cell. Formulate this problem as a language and show that it is undecidable.
5.15 Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left at any point during its computation on w. Formulate this problem as a language and show that it is decidable.
5.16 Let Γ = {0, 1, ␣} be the tape alphabet for all TM s in this problem. Define the busy beaver function BB: N−→N as follows. For each value of k, consider all k-state TMs that halt when started with a blank tape. Let BB(k) be the maximum number of 1s that remain on the tape among all of these machines. Show that BB is not a computable function.
5.17 Show that the Post Correspondence Problem is decidable over the unary alphabet Σ = {1}.
5.18 Show that the Post Correspondence Problem is undecidable over the binary alpha- bet Σ = {0,1}.
5.19 In the silly Post Correspondence Problem, SPCP, the top string in each pair has the same length as the bottom string. Show that the SPCP is decidable.
5.20 Prove that there exists an undecidable subset of {1}∗.
5.21 Let AMBIGCFG = {⟨G⟩| G is an ambiguous CFG}. Show that AMBIGCFG is unde-
cidable. (Hint: Use a reduction from PCP. Given an instance P =t1, t2, ... ,tk
b1b2 bk
of the Post Correspondence Problem, construct a CFG G with the rules
S→T|B
T→t1Ta1|···|tkTak |t1a1|···|tkak B→b1Ba1|···|bkBak |b1a1|···|bkak,
wherea1,...,ak arenewterminalsymbols.Provethatthisreductionworks.)
5.22 Show that A is Turing-recognizable iff A ≤m ATM.
5.23 Show that A is decidable iff A ≤m 0∗ 1∗ .
5.24 LetJ={w|eitherw=0xforsomex∈ATM,orw=1yforsomey∈ATM}. Show that neither J nor J is Turing-recognizable.
5.25 Give an example of an undecidable language B, where B ≤m B.
5.26 Define a two-headed finite automaton (2DFA) to be a deterministic finite automa- ton that has two read-only, bidirectional heads that start at the left-hand end of the input tape and can be independently controlled to move in either direction. The tape of a 2DFA is finite and is just large enough to contain the input plus two ad- ditional blank tape cells, one on the left-hand end and one on the right-hand end, that serve as delimiters. A 2DFA accepts its input by entering a special accept state. For example, a 2DFA can recognize the language {anbncn| n ≥ 0}.
a. Let A2DFA = {⟨M, x⟩| M is a 2DFA and M accepts x}. Show that A2DFA is decidable.
b. Let E2DFA = {⟨M⟩| M is a 2DFA and L(M) = ∅}. Show that E2DFA is not decidable.
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5.27 A two-dimensional finite automaton (2DIM-DFA) is defined as follows. The input is an m × n rectangle, for any m, n ≥ 2. The squares along the boundary of the rectangle contain the symbol # and the internal squares contain symbols over the input alphabet Σ. The transition function δ : Q × (Σ ∪ {#})−→ Q × {L, R, U, D} indicates the next state and the new head position (Left, Right, Up, Down). The machine accepts when it enters one of the designated accept states. It rejects if it tries to move off the input rectangle or if it never halts. Two such machines are equivalent if they accept the same rectangles. Consider the problem of determin- ing whether two of these machines are equivalent. Formulate this problem as a language and show that it is undecidable.
A⋆5.28 Rice’s theorem. Let P be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine’s language has property P is undecidable.
In more formal terms, let P be a language consisting of Turing machine descrip- tions where P fulfills two conditions. First, P is nontrivial—it contains some, but not all, TM descriptions. Second, P is a property of the TM’s language—whenever L(M1) = L(M2), we have ⟨M1⟩ ∈ P iff ⟨M2⟩ ∈ P. Here, M1 and M2 are any TMs. Prove that P is an undecidable language.
5.29 Show that both conditions in Problem 5.28 are necessary for proving that P is undecidable.
5.30 Use Rice’s theorem, which appears in Problem 5.28, to prove the undecidability of each of the following languages.
Aa. INFINITETM = {⟨M⟩| M is a TM and L(M) is an infinite language}.
b. {⟨M⟩| M is a TM and 1011 ∈ L(M)}.
c. ALLTM ={⟨M⟩|M isaTMandL(M)=Σ∗}.
5.31 Let
3x+1 foroddx f(x) = x/2 for even x
for any natural number x. If you start with an integer x and iterate f , you obtain a sequence, x, f (x), f (f (x)), . . . . Stop if you ever hit 1. For example, if x = 17, you get the sequence 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Extensive computer tests have shown that every starting point between 1 and a large positive integer gives a sequence that ends in 1. But the question of whether all positive starting points end up at 1 is unsolved; it is called the 3x + 1 problem.
Suppose that ATM were decidable by a TM H. Use H to describe a TM that is guaranteed to state the answer to the 3x + 1 problem.
5.32 Prove that the following two languages are undecidable.
a. OVERLAPCFG = {⟨G, H⟩| G and H are CFGs where L(G) ∩ L(H) ̸= ∅}.
(Hint: Adapt the hint in Problem 5.21.)
b. PREFIX-FREECFG = {⟨G⟩| G is a CFG where L(G) is prefix-free}.
5.33 Consider the problem of determining whether a PDA accepts some string of the form {ww| w ∈ {0,1}∗} . Use the computation history method to show that this problem is undecidable.
5.34 Let X = {⟨M, w⟩| M is a single-tape TM that never modifies the portion of the tape that contains the input w}. Is X decidable? Prove your answer.
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PROBLEMS 241
242
CHAPTER 5 / REDUCIBILITY
5.35
⋆ 5.36
Say that a variable A in CFG G is necessary if it appears in every derivation of some string w ∈ G. Let NECESSARY CFG = {⟨G, A⟩| A is a necessary variable in G}.
a. Show that NECESSARY CFG is Turing-recognizable.
b. Show that NECESSARY CFG is undecidable.
Say that a CFG is minimal if none of its rules can be removed without changing the language generated. Let MINCFG = {⟨G⟩| G is a minimal CFG}.
a. Show that MINCFG is T-recognizable. b. Show that MINCFG is undecidable.
SELECTED SOLUTIONS
5.5 Suppose for a contradiction that ATM ≤m ETM via reduction f. It follows from the definition of mapping reducibility that ATM ≤m ETM via the same reduction function f. However, ETM is Turing-recognizable (see the solution to Exercise 4.5) and ATM is not Turing-recognizable, contradicting Theorem 5.28.
5.6 Suppose A ≤m B and B ≤m C. Then there are computable functions f and gsuchthatx∈A⇐⇒f(x)∈Bandy∈B⇐⇒g(y)∈C. Considerthe composition function h(x) = g(f(x)). We can build a TM that computes h as follows: First, simulate a TM for f (such a TM exists because we assumed that f is computable) on input x and call the output y. Then simulate a TM for g on y. The output is h(x) = g(f(x)). Therefore, h is a computable function. Moreover, x ∈ A ⇐⇒ h(x) ∈ C. Hence A ≤m C via the reduction function h.
5.7 Suppose that A ≤m A. Then A ≤m A via the same mapping reduction. Because A is Turing-recognizable, Theorem 5.28 implies that A is Turing-recognizable, and then Theorem 4.22 implies that A is decidable.
5.8 You need to handle the case where the head is at the leftmost tape cell and attempts to move left. To do so, add dominos
#qa #rb
for every q, r ∈ Q and a, b ∈ Γ, where δ(q, a) = (r, b, L). Additionally, replace the first domino with
# ##q0w1w2 · · · wn
to handle the case where the head attempts to move left in the very first move.
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5.10 Let B = {⟨M, w⟩| M is a two-tape TM that writes a nonblank symbol on its second tape when it is run on w}. Show that ATM reduces to B. Assume for the sake of contradiction that TM R decides B. Then construct a TM S that uses R to decide ATM .
S = “On input ⟨M, w⟩:
1. Use M to construct the following two-tape TM T .
T = “On input x:
1. SimulateMonxusingthefirsttape.
2. IfthesimulationshowsthatMaccepts,writeanon-
blank symbol on the second tape.”
2. Run R on ⟨T,w⟩ to determine whether T on input w writes a
nonblank symbol on its second tape.
3. If R accepts, M accepts w, so accept . Otherwise, reject .”
5.11 Let C = {⟨M⟩| M is a two-tape TM that writes a nonblank symbol on its second tape when it is run on some input}. Show that ATM reduces to C. Assume for the sake of contradiction that TM R decides C. Construct a TM S that uses R to decide ATM .
S = “On input ⟨M, w⟩:
1. Use M and w to construct the following two-tape TM Tw.
Tw = “On any input:
1. SimulateMonwusingthefirsttape.
2. IfthesimulationshowsthatMaccepts,writeanon-
blank symbol on the second tape.”
2. Run R on ⟨Tw ⟩ to determine whether Tw ever writes a nonblank
symbol on its second tape.
3. If R accepts, M accepts w, so accept . Otherwise, reject .”
5.28 Assume for the sake of contradiction that P is a decidable language satisfying the properties and let RP be a TM that decides P . We show how to decide ATM using RP by constructing TM S. First, let T∅ be a TM that always rejects, so L(T∅) = ∅. You may assume that ⟨T∅⟩ ̸∈ P without loss of generality because you could pro- ceed with P instead of P if ⟨T∅⟩ ∈ P . Because P is not trivial, there exists a TM T with ⟨T ⟩ ∈ P . Design S to decide ATM using RP ’s ability to distinguish between T∅ andT.
S = “On input ⟨M, w⟩:
1. Use M and w to construct the following TM Mw .
Mw = “On input x:
1. SimulateMonw.Ifithaltsandrejects,reject.
If it accepts, proceed to stage 2.
2. SimulateTonx.Ifitaccepts,accept.”
2. Use TM RP to determine whether ⟨Mw⟩ ∈ P. If YES, accept. If NO, reject.”
TM Mw simulates T if M accepts w. Hence L(Mw) equals L(T) if M accepts w and ∅ otherwise. Therefore, ⟨Mw⟩ ∈ P iff M accepts w.
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SELECTED SOLUTIONS 243
244 CHAPTER 5 / REDUCIBILITY
5.30 (a) INFINITETM is a language of TM descriptions. It satisfies the two conditions of Rice’s theorem. First, it is nontrivial because some TMs have infinite languages and others do not. Second, it depends only on the language. If two TMs recognize the same language, either both have descriptions in INFINITETM or neither do. Consequently, Rice’s theorem implies that INFINITETM is undecidable.
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6
ADVANCED TOPICS IN COMPUTABILITY THEORY
In this chapter we delve into four deeper aspects of computability theory: (1) the recursion theorem, (2) logical theories, (3) Turing reducibility, and (4) descrip- tive complexity. The topic covered in each section is mainly independent of the others, except for an application of the recursion theorem at the end of the sec- tion on logical theories. Part Three of this book doesn’t depend on any material from this chapter.
6.1
THE RECURSION THEOREM
The recursion theorem is a mathematical result that plays an important role in advanced work in the theory of computability. It has connections to mathemati- cal logic, the theory of self-reproducing systems, and even computer viruses.
To introduce the recursion theorem, we consider a paradox that arises in the study of life. It concerns the possibility of making machines that can construct replicas of themselves. The paradox can be summarized in the following manner.
245
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246 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
1. Living things are machines.
2. Living things can self-reproduce. 3. Machines cannot self-reproduce.
Statement 1 is a tenet of modern biology. We believe that organisms operate in a mechanistic way. Statement 2 is obvious. The ability to self-reproduce is an essential characteristic of every biological species. For statement 3, we make the following argument that machines cannot self-reproduce. Consider a machine that constructs other machines, such as an automated factory that produces cars. Raw materials go in at one end, the manufacturing robots follow a set of instructions, and then completed vehicles come out the other end.
We claim that the factory must be more complex than the cars produced, in the sense that designing the factory would be more difficult than designing a car. This claim must be true because the factory itself has the car’s design within it, in addition to the design of all the manufacturing robots. The same reasoning applies to any machine A that constructs a machine B: A must be more complex than B. But a machine cannot be more complex than itself. Consequently, no machine can construct itself, and thus self-reproduction is impossible.
How can we resolve this paradox? The answer is simple: Statement 3 is in- correct. Making machines that reproduce themselves is possible. The recursion theorem demonstrates how.
SELF-REFERENCE
Let’s begin by making a Turing machine that ignores its input and prints out a copy of its own description. We call this machine SELF. To help describe SELF , we need the following lemma.
LEMMA 6.1
There is a computable function q: Σ∗−→Σ∗, where if w is any string, q(w) is
the description of a Turing machine Pw that prints out w and then halts.
PROOF Once we understand the statement of this lemma, the proof is easy. Obviously, we can take any string w and construct from it a Turing machine that has w built into a table so that the machine can simply output w when started. The following TM Q computes q(w).
Q = “On input string w:
1. Construct the following Turing machine Pw.
Pw = “On any input: 1. Eraseinput.
2. Writewonthetape.
3. Halt.”
2. Output ⟨Pw ⟩.”
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6.1 THE RECURSION THEOREM 247
The Turing machine SELF is in two parts: A and B. We think of A and B as being two separate procedures that go together to make up SELF . We want SELF to print out ⟨SELF ⟩ = ⟨AB⟩.
Part A runs first and upon completion passes control to B. The job of A is to print out a description of B, and conversely the job of B is to print out a description of A. The result is the desired description of SELF . The jobs are similar, but they are carried out differently. We show how to get part A first.
For A we use the machine P⟨B⟩, described by q⟨B⟩, which is the result of applying the function q to ⟨B⟩. Thus, part A is a Turing machine that prints out ⟨B⟩. Our description of A depends on having a description of B. So we can’t complete the description of A until we construct B.
Now for part B. We might be tempted to define B with q ⟨A⟩ , but that doesn’t make sense! Doing so would define B in terms of A, which in turn is defined in terms of B. That would be a circular definition of an object in terms of itself, a logical transgression. Instead, we define B so that it prints A by using a different strategy: B computes A from the output that A produces.
We defined ⟨A⟩ to be q⟨B⟩. Now comes the tricky part: If B can obtain ⟨B⟩, it can apply q to that and obtain ⟨A⟩. But how does B obtain ⟨B⟩? It was left on the tape when A finished! So B only needs to look at the tape to obtain ⟨B⟩. Then after B computes q⟨B⟩ = ⟨A⟩, it combines A and B into a single machine and writes its description ⟨AB⟩ = ⟨SELF⟩ on the tape. In summary, we have:
A = P⟨B⟩, and
B = “On input ⟨M⟩, where M is a portion of a TM:
1. Compute q⟨M ⟩.
2. Combine the result with ⟨M ⟩ to make a complete TM.
3. Print the description of this TM and halt.”
This completes the construction of SELF , for which a schematic diagram is presented in the following figure.
FIGURE 6.2
Schematic of SELF , a TM that prints its own description
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248 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY If we now run SELF , we observe the following behavior.
1. FirstAruns.Itprints⟨B⟩onthetape.
2. Bstarts.Itlooksatthetapeandfindsitsinput,⟨B⟩.
3. B calculates q⟨B⟩ = ⟨A⟩ and combines that with ⟨B⟩ into a
TM description, ⟨SELF ⟩.
4. Bprintsthisdescriptionandhalts.
We can easily implement this construction in any programming language to obtain a program that outputs a copy of itself. We can even do so in plain En- glish. Suppose that we want to give an English sentence that commands the reader to print a copy of the same sentence. One way to do so is to say:
Print out this sentence.
This sentence has the desired meaning because it directs the reader to print a copy of the sentence itself. However, it doesn’t have an obvious translation into a programming language because the self-referential word “this” in the sentence usually has no counterpart. But no self-reference is needed to make such a sen- tence. Consider the following alternative.
Print out two copies of the following, the second one in quotes: “Print out two copies of the following, the second one in quotes:”
In this sentence, the self-reference is replaced with the same construction used to make the TM SELF . Part B of the construction is the clause:
Print out two copies of the following, the second one in quotes:
Part A is the same, with quotes around it. A provides a copy of B to B so B can process that copy as the TM does.
The recursion theorem provides the ability to implement the self-referential this into any programming language. With it, any program has the ability to refer to its own description, which has certain applications, as you will see. Before getting to that, we state the recursion theorem itself. The recursion theorem extends the technique we used in constructing SELF so that a program can obtain its own description and then go on to compute with it, instead of merely printing it out.
THEOREM 6.3
Recursion theorem Let T be a Turing machine that computes a function t : Σ∗ × Σ∗ −→ Σ∗ . There is a Turing machine R that computes a function r: Σ∗−→Σ∗, where for every w,
r(w) = t⟨R⟩, w.
The statement of this theorem seems a bit technical, but it actually represents something quite simple. To make a Turing machine that can obtain its own description and then compute with it, we need only make a machine, called T
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6.1 THE RECURSION THEOREM 249
in the statement, that receives the description of the machine as an extra input. Then the recursion theorem produces a new machine R, which operates exactly as T does but with R’s description filled in automatically.
PROOF The proof is similar to the construction of SELF . We construct a TM R in three parts, A, B, and T , where T is given by the statement of the theorem; a schematic diagram is presented in the following figure.
FIGURE 6.4 Schematic of R
Here, A is the Turing machine P⟨BT⟩ described by q⟨BT⟩. To preserve the input w, we redesign q so that P⟨BT ⟩ writes its output following any string preexisting on the tape. After A runs, the tape contains w⟨BT ⟩.
Again, B is a procedure that examines its tape and applies q to its contents. The result is ⟨A⟩. Then B combines A, B, and T into a single machine and ob- tains its description ⟨ABT ⟩ = ⟨R⟩. Finally, it encodes that description together with w, places the resulting string ⟨R, w⟩ on the tape, and passes control to T .
TERMINOLOGY FOR THE RECURSION THEOREM
The recursion theorem states that Turing machines can obtain their own de- scription and then go on to compute with it. At first glance, this capability may seem to be useful only for frivolous tasks such as making a machine that prints a copy of itself. But, as we demonstrate, the recursion theorem is a handy tool for solving certain problems concerning the theory of algorithms.
You can use the recursion theorem in the following way when designing Tur- ing machine algorithms. If you are designing a machine M , you can include the phrase “obtain own description ⟨M⟩” in the informal description of M’s algo- rithm. Upon having obtained its own description, M can then go on to use it as it would use any other computed value. For example, M might simply print out ⟨M ⟩ as happens in the TM SELF , or it might count the number of states in ⟨M ⟩, or possibly even simulate ⟨M⟩. To illustrate this method, we use the recursion theorem to describe the machine SELF .
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250 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
SELF = “On any input:
1. Obtain, via the recursion theorem, own description ⟨SELF ⟩.
2. Print ⟨SELF ⟩.”
The recursion theorem shows how to implement the “obtain own descrip- tion” construct. To produce the machine SELF, we first write the following machine T .
T = “On input ⟨M, w⟩:
1. Print ⟨M ⟩ and halt.”
The TM T receives a description of a TM M and a string w as input, and it prints the description of M . Then the recursion theorem shows how to obtain a TM R, which on input w operates like T on input ⟨R, w⟩. Thus, R prints the description of R—exactly what is required of the machine SELF .
APPLICATIONS
A computer virus is a computer program that is designed to spread itself among computers. Aptly named, it has much in common with a biological virus. Com- puter viruses are inactive when standing alone as a piece of code. But when placed appropriately in a host computer, thereby “infecting” it, they can become activated and transmit copies of themselves to other accessible machines. Vari- ous media can transmit viruses, including the Internet and transferable disks. In order to carry out its primary task of self-replication, a virus may contain the construction described in the proof of the recursion theorem.
Let’s now consider three theorems whose proofs use the recursion theorem. An additional application appears in the proof of Theorem 6.17 in Section 6.2.
First we return to the proof of the undecidability of ATM. Recall that we ear- lier proved it in Theorem 4.11, using Cantor’s diagonal method. The recursion theorem gives us a new and simpler proof.
THEOREM 6.5 ATM is undecidable.
PROOF We assume that Turing machine H decides ATM, for the purpose of obtaining a contradiction. We construct the following machine B.
B = “On input w:
1. Obtain, via the recursion theorem, own description ⟨B⟩.
2. Run H on input ⟨B, w⟩.
3. Do the opposite of what H says. That is, accept if H rejects and
reject if H accepts.”
Running B on input w does the opposite of what H declares it does. Therefore,
H cannot be deciding ATM. Done!
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6.1 THE RECURSION THEOREM 251 The following theorem concerning minimal Turing machines is another ap-
plication of the recursion theorem.
DEFINITION 6.6
If M is a Turing machine, then we say that the length of the descrip- tion ⟨M ⟩ of M is the number of symbols in the string describing M . Say that M is minimal if there is no Turing machine equivalent to M that has a shorter description. Let
MIN TM = {⟨M⟩| M is a minimal TM}.
THEOREM 6.7
MIN TM is not Turing-recognizable.
PROOF Assume that some TM E enumerates MIN TM and obtain a contradic- tion. We construct the following TM C.
C = “On input w:
1. Obtain, via the recursion theorem, own description ⟨C⟩.
2. Run the enumerator E until a machine D appears with a longer
description than that of C.
3. Simulate D on input w.”
BecauseMINTM isinfinite,E’slistmustcontainaTMwithalongerdescrip- tion than C’s description. Therefore, step 2 of C eventually terminates with some TM D that is longer than C. Then C simulates D and so is equivalent to it. Because C is shorter than D and is equivalent to it, D cannot be minimal. But D appears on the list that E produces. Thus, we have a contradiction.
Our final application of the recursion theorem is a type of fixed-point theo- rem. A fixed point of a function is a value that isn’t changed by application of the function. In this case, we consider functions that are computable transforma- tions of Turing machine descriptions. We show that for any such transformation, some Turing machine exists whose behavior is unchanged by the transformation. This theorem is called the fixed-point version of the recursion theorem.
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252 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY THEOREM 6.8
Let t : Σ∗ −→ Σ∗ be a computable function. Then there is a Turing machine F for which t⟨F⟩ describes a Turing machine equivalent to F. Here we’ll assume that if a string isn’t a proper Turing machine encoding, it describes a Turing machine that always rejects immediately.
In this theorem, t plays the role of the transformation, and F is the fixed point. PROOF Let F be the following Turing machine.
F = “On input w:
1. Obtain, via the recursion theorem, own description ⟨F ⟩.
2. Compute t⟨F ⟩ to obtain the description of a TM G.
3. Simulate G on w.”
Clearly, ⟨F ⟩ and t⟨F ⟩ = ⟨G⟩ describe equivalent Turing machines because F simulates G.
6.2
DECIDABILITY OF LOGICAL THEORIES
Mathematical logic is the branch of mathematics that investigates mathematics itself. It addresses questions such as: What is a theorem? What is a proof? What is truth? Can an algorithm decide which statements are true? Are all true state- ments provable? We’ll touch on a few of these topics in our brief introduction to this rich and fascinating subject.
We focus on the problem of determining whether mathematical statements are true or false and investigate the decidability of this problem. The answer depends on the domain of mathematics from which the statements are drawn. We examine two domains: one for which we can give an algorithm to decide truth, and another for which this problem is undecidable.
First, we need to set up a precise language to formulate these problems. Our intention is to be able to consider mathematical statements such as
1. ∀q ∃p ∀x,y p>q ∧ (x,y>1 → xy̸=p) ,
2. ∀a,b,c,n (a,b,c>0 ∧ n>2) → an+bn̸=cn , and
3. ∀q ∃p ∀x,y p>q ∧ (x,y>1 → (xy̸=p ∧ xy̸=p+2)) .
Statement 1 says that infinitely many prime numbers exist, which has been known to be true since the time of Euclid, about 2,300 years ago. Statement 2 is Fermat’s last theorem, which has been known to be true only since Andrew Wiles proved it in 1994. Finally, statement 3 says that infinitely many prime pairs1 exist. Known as the twin prime conjecture, it remains unsolved.
1Prime pairs are primes that differ by 2.
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6.2 DECIDABILITY OF LOGICAL THEORIES 253
To consider whether we could automate the process of determining which of these statements are true, we treat such statements merely as strings and define a language consisting of those statements that are true. Then we ask whether this language is decidable.
To make this a bit more precise, let’s describe the form of the alphabet of this language:
{∧,∨,¬,(,),∀,∃,x,R1,…,Rk}.
The symbols ∧, ∨, and ¬ are called Boolean operations; “(” and “)” are the parentheses; the symbols ∀ and ∃ are called quantifiers; the symbol x is used to denote variables;2 and the symbols R1 , . . . , Rk are called relations.
A formula is a well-formed string over this alphabet. For completeness, we’ll sketch the technical but obvious definition of a well-formed formula here, but feel free to skip this part and go on to the next paragraph. A string of the form Ri(x1, . . . , xk) is an atomic formula. The value j is the arity of the relation symbol Ri . All appearances of the same relation symbol in a well-formed formula must have the same arity. Subject to this requirement, a string φ is a formula if it
1. is an atomic formula,
2.hastheformφ1 ∧φ2 orφ1 ∨φ2 or¬φ1,whereφ1 andφ2 aresmaller
formulas, or
3. has the form ∃xi [φ1 ] or ∀xi [φ1 ], where φ1 is a smaller formula.
A quantifier may appear anywhere in a mathematical statement. Its scope is the fragment of the statement appearing within the matched pair of parentheses or brackets following the quantified variable. We assume that all formulas are in prenex normal form, where all quantifiers appear in the front of the formula. A variable that isn’t bound within the scope of a quantifier is called a free variable. A formula with no free variables is called a sentence or statement.
EXAMPLE 6.9
Among the following examples of formulas, only the last one is a sentence.
1. R1(x1) ∧ R2(x1, x2, x3)
2. ∀x1 R1(x1) ∧ R2(x1, x2, x3)
3. ∀x1 ∃x2 ∃x3 R1(x1) ∧ R2(x1, x2, x3)
Having established the syntax of formulas, let’s discuss their meanings. The Boolean operations and the quantifiers have their usual meanings. But to deter- mine the meaning of the variables and relation symbols, we need to specify two items. One is the universe over which the variables may take values. The other
2If we need to write several variables in a formula, we use the symbols w, y, z, or x1, x2, x3, and so on. We don’t list all the infinitely many possible variables in the alphabet to keep the alphabet finite. Instead, we list only the variable symbol x, and use strings of x’s to indicate other variables, as in xx for x2, xxx for x3, and so on.
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254 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
is an assignment of specific relations to the relation symbols. As we described in Section 0.2 (page 9), a relation is a function from k-tuples over the universe to {TRUE, FALSE}. The arity of a relation symbol must match that of its assigned relation.
A universe together with an assignment of relations to relation symbols is called a model.3 Formally, we say that a model M is a tuple (U, P1, . . . , Pk), where U is the universe and P1 through Pk are the relations assigned to symbols R1 through Rk. We sometimes refer to the language of a model to be the collection of formulas that use only the relation symbols the model assigns, and that use each relation symbol with the correct arity. If φ is a sentence in the language of a model, φ is either true or false in that model. If φ is true in a model M, we say that M is a model of φ.
If you feel overwhelmed by these definitions, concentrate on our objective in stating them. We want to set up a precise language of mathematical statements so that we can ask whether an algorithm can determine which are true and which are false. The following two examples should be helpful.
EXAMPLE 6.10
Letφbethesentence∀x∀yR1(x,y)∨R1(y,x).LetmodelM1 =(N,≤)be the model whose universe is the natural numbers and that assigns the “less than or equal” relation to the symbol R1. Obviously, φ is true in model M1 because either a ≤ b or b ≤ a for any two natural numbers a and b. However, if M1 assigned “less than” instead of “less than or equal” to R1, then φ would not be true because it fails when x and y are equal.
If we know in advance which relation will be assigned to Ri, we may use the customary symbol for that relation in place of Ri with infix notation rather than prefix notation if customary for that symbol. Thus, with model M1 in mind, we could write φ as ∀x ∀y x≤y ∨ y≤x .
EXAMPLE 6.11
Now let M2 be the model whose universe is the real numbers R and that assigns the relation PLUS to R1, where PLUS(a,b,c) = TRUE whenever a + b = c. Then M2 is a model of ψ = ∀y ∃x R1(x, x, y) . However, if N were used for the universe instead of R in M2, the sentence would be false.
As in Example 6.10, we may write ψ as ∀y∃x x+x = y in place of ∀y ∃x R1(x, x, y) when we know in advance that we will be assigning the ad- dition relation to R1.
As Example 6.11 illustrates, we can represent functions such as the addition function by relations. Similarly, we can represent constants such as 0 and 1 by relations.
3A model is also variously called an interpretation or a structure.
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6.2 DECIDABILITY OF LOGICAL THEORIES 255
Now we give one final definition in preparation for the next section. If M is a model, we let the theory of M, written Th(M), be the collection of true sentences in the language of that model.
A DECIDABLE THEORY
Number theory is one of the oldest branches of mathematics and also one of its most difficult. Many innocent-looking statements about the natural num- bers with the plus and times operations have confounded mathematicians for centuries, such as the twin prime conjecture mentioned earlier.
In one of the celebrated developments in mathematical logic, Alonzo Church, building on the work of Kurt Go ̈del, showed that no algorithm can decide in general whether statements in number theory are true or false. Formally, we write (N , +, ×) to be the model whose universe is the natural numbers4 with the usual + and × relations. Church showed that Th(N , +, ×), the theory of this model, is undecidable.
Before looking at this undecidable theory, let’s examine one that is decidable. Let (N , +) be the same model, without the × relation. Its theory is Th(N , +). For example, the formula ∀x ∃y x + x = y is true and is therefore a member of Th(N , +), but the formula ∃y∀x x + x = y is false and is therefore not a member.
THEOREM 6.12 Th(N , +) is decidable.
PROOF IDEA This proof is an interesting and nontrivial application of the theory of finite automata that we presented in Chapter 1. One fact about finite automata that we use appears in Problem 1.32, (page 88) where you were asked to show that they are capable of doing addition if the input is presented in a special form. The input describes three numbers in parallel by representing one bit of each number in a single symbol from an eight-symbol alphabet. Here we use a generalization of this method to present i-tuples of numbers in parallel using an alphabet with 2i symbols.
We give an algorithm that can determine whether its input, a sentence φ in the language of (N , +), is true in that model. Let
φ = Q1x1 Q2x2 ··· Qlxl ψ,
where Q1, . . . , Ql each represents either ∃ or ∀ and ψ is a formula without quan-
tifiers that has variables x1, . . . , xl. For each i from 0 to l, define formula φi as φi =Qi+1xi+1Qi+2xi+2 ···Qlxl ψ.
Thusφ0 =φandφl =ψ.
4For convenience in this chapter, we change our usual definition of N to be {0, 1, 2, . . .}.
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256 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
Formula φi has i free variables. For a1,…,ai ∈ N, write φi(a1,…,ai) to be the sentence obtained by substituting the constants a1 , . . . , ai for the variables x1,…,xi in φi.
For each i from 0 to l, the algorithm constructs a finite automaton Ai that recognizes the collection of strings representing i-tuples of numbers that make φi true. The algorithm begins by constructing Al directly, using a generalization of the method in the solution to Problem 1.32. Then, for each i from l down to 1, it uses Ai to construct Ai−1. Finally, once the algorithm has A0, it tests whether A0 accepts the empty string. If it does, φ is true and the algorithm accepts.
PROOF For i > 0, define the alphabet
Σi =0.,0.,0.,0.,…,1..
0011 1 0101 1
Hence Σi contains all size i columns of 0s and 1s. A string over Σi represents i binary integers (reading across the rows). We also define Σ0 = {[ ]}, where [ ] is a symbol.
We now present an algorithm that decides Th(N,+). On input φ, where φ is a sentence, the algorithm operates as follows. Write φ and define φi for each i from 0 to l, as in the proof idea. For each such i, construct a finite automaton Ai from φi that accepts strings over Σi corresponding to i-tuples a1,…,ai whenever φi(a1,…,ai) is true, as follows.
To construct the first machine Al, observe that φl = ψ is a Boolean combi- nation of atomic formulas. An atomic formula in the language of Th(N , +) is a single addition. Finite automata can be constructed to compute any of these in- dividual relations corresponding to a single addition and then combined to give the automaton Al. Doing so involves the use of the regular language closure constructions for union, intersection, and complementation to compute Boolean combinations of the atomic formulas.
Next, we show how to construct Ai from Ai+1. If φi = ∃xi+1 φi+1, we con- struct Ai to operate as Ai+1 operates, except that it nondeterministically guesses the value of ai+1 instead of receiving it as part of the input.
More precisely, Ai contains a state for each Ai+1 state and a new start state. Every time Ai reads a symbol
⎡b1 ⎤ ⎣ . ⎦
bi−1 bi
z ∈ {0,1} and simulates Ai+1 on the input symbol ⎡b1 ⎤
.
,
where every bj ∈ {0,1} is a bit of the number aj , it nondeterministically guesses
⎢ . ⎥ ⎣bi−1 ⎦
bi z
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6.2 DECIDABILITY OF LOGICAL THEORIES 257
Initially, Ai nondeterministically guesses the leading bits of ai+1 corresponding to suppressed leading 0s in a1 through ai by nondeterministically branching using ε-transitions from its new start state to all states that Ai+1 could reach from its start state with input strings of the symbols
0.,0. 00
01
in Σi+1. Clearly, Ai accepts its input (a1, . . . , ai) if some ai+1 exists where Ai+1 accepts (a1, . . . , ai+1).
If φi = ∀xi+1 φi+1, it is equivalent to ¬∃xi+1¬ φi+1. Thus, we can construct the finite automaton that recognizes the complement of the language of Ai+1, then apply the preceding construction for the ∃ quantifier, and finally apply com- plementation once again to obtain Ai.
Finite automaton A0 accepts any input iff φ0 is true. So the final step of the algorithm tests whether A0 accepts ε. If it does, φ is true and the algorithm accepts; otherwise, it rejects.
AN UNDECIDABLE THEORY
As we mentioned earlier, Th(N , +, ×) is an undecidable theory. No algorithm exists for deciding the truth or falsity of mathematical statements, even when re- stricted to the language of (N , +, ×). This theorem has great importance philo- sophically because it demonstrates that mathematics cannot be mechanized. We state this theorem, but give only a brief sketch of its proof.
THEOREM 6.13
Th(N , +, ×) is undecidable.
Although it contains many details, the proof of this theorem is not difficult conceptually. It follows the pattern of the other proofs of undecidability pre- sented in Chapter 4. We show that Th(N , +, ×) is undecidable by reducing ATM to it, using the computation history method as previously described (page 220). The existence of the reduction depends on the following lemma.
LEMMA 6.14
Let M be a Turing machine and w a string. We can construct from M and w a formula φM,w in the language of (N , +, ×) that contains a single free variable x, whereby the sentence ∃x φM,w is true iff M accepts w.
PROOF IDEA Formula φM,w “says” that x is a (suitably encoded) accepting computation history of M on w. Of course, x actually is just a rather large integer, but it represents a computation history in a form that can be checked by using the + and × operations.
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258 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
The actual construction of φM,w is too complicated to present here. It ex- tracts individual symbols in the computation history with the + and × operations to check that the start configuration for M on w is correct, that each configura- tion legally follows from the one preceding it, and that the last configuration is accepting.
PROOF OF THEOREM 6.13 We give a mapping reduction from ATM to Th(N , +, ×). The reduction constructs the formula φM,w from the input ⟨M, w⟩ by using Lemma 6.14. Then it outputs the sentence ∃x φM,w.
Next, we sketch the proof of Kurt Go ̈ del’s celebrated incompleteness theorem. Informally, this theorem says that in any reasonable system of formalizing the notion of provability in number theory, some true statements are unprovable.
Loosely speaking, the formal proof π of a statement φ is a sequence of state- ments, S1, S2, . . . , Sl, where Sl = φ. Each Si follows from the preceding state- ments and certain basic axioms about numbers, using simple and precise rules of implication. We don’t have space to define the concept of proof; but for our purposes, assuming the following two reasonable properties of proofs will be enough.
1. The correctness of a proof of a statement can be checked by machine. Formally, {⟨φ, π⟩| π is a proof of φ} is decidable.
2. The system of proofs is sound. That is, if a statement is provable (i.e., has a proof), it is true.
If a system of provability satisfies these two conditions, the following three the- orems hold.
THEOREM 6.15
The collection of provable statements in Th(N , +, ×) is Turing-recognizable.
PROOF The following algorithm P accepts its input φ if φ is provable. Al- gorithm P tests each string as a candidate for a proof π of φ, using the proof checker assumed in provability property 1. If it finds that any of these candi- dates is a proof, it accepts.
Now we can use the preceding theorem to prove our version of the incom- pleteness theorem.
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6.2 DECIDABILITY OF LOGICAL THEORIES 259
THEOREM 6.16
Some true statement in Th(N , +, ×) is not provable.
PROOF We give a proof by contradiction. We assume to the contrary that all true statements are provable. Using this assumption, we describe an algorithm D that decides whether statements are true, contradicting Theorem 6.13.
On input φ, algorithm D operates by running algorithm P given in the proof of Theorem 6.15 in parallel on inputs φ and ¬φ. One of these two statements is true and thus by our assumption is provable. Therefore, P must halt on one of the two inputs. By provability property 2, if φ is provable, then φ is true; and if ¬φ is provable, then φ is false. So algorithm D can decide the truth or falsity of φ.
In the final theorem of this section, we use the recursion theorem to give an explicit sentence in the language of (N , +, ×) that is true but not provable. In Theorem 6.16 we demonstrated the existence of such a sentence but didn’t actually describe one, as we do now.
THEOREM 6.17
The sentence ψunprovable, as described in the proof, is unprovable.
PROOF IDEA Construct a sentence that says “This sentence is not provable,” using the recursion theorem to obtain the self-reference.
PROOF Let S be a TM that operates as follows.
S = “On any input:
1. Obtain own description ⟨S⟩ via the recursion theorem.
2. Construct the sentence ψ = ¬∃c φS,0 , using Lemma 6.14.
3. Run algorithm P from the proof of Theorem 6.15 on input ψ.
4. If stage 3 accepts, accept .”
Let ψunprovable be the sentence ψ described in stage 2 of algorithm S. That sentence is true iff S doesn’t accept 0 (the string 0 was selected arbitrarily).
If S finds a proof of ψunprovable, S accepts 0, and the sentence would thus be false. A false sentence cannot be provable, so this situation cannot occur. The only remaining possibility is that S fails to find a proof of ψunprovable and so S doesn’t accept 0. But then ψunprovable is true, as we claimed.
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260 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY 6.3
TURING REDUCIBILITY
We introduced the reducibility concept in Chapter 5 as a way of using a solution to one problem to solve other problems. Thus, if A is reducible to B, and we find a solution to B, we can obtain a solution to A. Subsequently, we described mapping reducibility, a specific form of reducibility. But does mapping reducibility capture our intuitive concept of reducibility in the most general way? It doesn’t.
For example, consider the two languages ATM and ATM. Intuitively, they are reducible to one another because a solution to either could be used to solve the other by simply reversing the answer. However, we know that ATM is not mapping reducible to ATM because ATM is Turing-recognizable but ATM isn’t. Here we present a very general form of reducibility, called Turing reducibility, which captures our intuitive concept of reducibility more closely.
DEFINITION 6.18
An oracle for a language B is an external device that is capable of reporting whether any string w is a member of B. An oracle Turing machine is a modified Turing machine that has the additional ca- pability of querying an oracle. We write MB to describe an oracle Turing machine that has an oracle for language B.
We aren’t concerned with the way the oracle determines its responses. We use the term oracle to connote a magical ability and consider oracles for languages that aren’t decidable by ordinary algorithms, as the following example shows.
EXAMPLE 6.19
Consider an oracle for ATM. An oracle Turing machine with an oracle for ATM can decide more languages than an ordinary Turing machine can. Such a ma- chine can (obviously) decide ATM itself, by querying the oracle about the input. It can also decide ETM , the emptiness testing problem for TM s with the following procedure called T ATM .
TATM = “On input ⟨M⟩, where M is a TM:
1. Construct the following TM N .
N = “On any input:
1. RunMinparallelonallstringsinΣ∗.
2. IfMacceptsanyofthesestrings,accept.”
2. Query the oracle to determine whether ⟨N, 0⟩ ∈ ATM.
3. If the oracle answers NO, accept; if YES, reject.”
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6.4 A DEFINITION OF INFORMATION 261
If M’s language isn’t empty, N will accept every input and, in particular, in- put 0. Hence the oracle will answer YES, and TATM will reject. Conversely, if M’s language is empty, TATM will accept. Thus TATM decides ETM. We say that ETM is decidable relative to ATM. That brings us to the definition of Turing reducibility.
DEFINITION 6.20
Language A is Turing reducible to language B, written A ≤T B, if A is decidable relative to B.
Example 6.19 shows that ETM is Turing reducible to ATM. Turing reducibility satisfies our intuitive concept of reducibility as shown by the following theorem.
THEOREM 6.21
If A ≤T B and B is decidable, then A is decidable.
PROOF If B is decidable, then we may replace the oracle for B by an actual procedure that decides B. Thus, we may replace the oracle Turing machine that decides A by an ordinary Turing machine that decides A.
Turing reducibility is a generalization of mapping reducibility. If A ≤m B, then A ≤T B because the mapping reduction may be used to give an oracle Turing machine that decides A relative to B.
An oracle Turing machine with an oracle for ATM is very powerful. It can solve many problems that are not solvable by ordinary Turing machines. But even such a powerful machine cannot decide all languages (see Exercise 6.4).
6.4
A DEFINITION OF INFORMATION
The concepts algorithm and information are fundamental in computer science. While the Church–Turing thesis gives a universally applicable definition of al- gorithm, no equally comprehensive definition of information is known. Instead of a single, universal definition of information, several definitions are used— depending upon the application. In this section we present one way of defining information, using computability theory.
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262 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
We start with an example. Consider the information content of the following
two binary sequences.
A = 0101010101010101010101010101010101010101 B = 1110010110100011101010000111010011010111
Intuitively, sequence A contains little information because it is merely a repe- tition of the pattern 01 twenty times. In contrast, sequence B appears to contain more information.
We can use this simple example to illustrate the idea behind the definition of information that we present. We define the quantity of information contained in an object to be the size of that object’s smallest representation or description. By a description of an object, we mean a precise and unambiguous characterization of the object so that we may recreate it from the description alone. Thus, se- quence A contains little information because it has a small description, whereas sequence B apparently contains more information because it seems to have no concise description.
Why do we consider only the shortest description when determining an ob- ject’s quantity of information? We may always describe an object, such as a string, by placing a copy of the object directly into the description. Thus, we can obviously describe the preceding string B with a table that is 40 bits long containing a copy of B. This type of description is never shorter than the object itself and doesn’t tell us anything about its information quantity. However, a de- scription that is significantly shorter than the object implies that the information contained within it can be compressed into a small volume, and so the amount of information can’t be very large. Hence the size of the shortest description determines the amount of information.
Now we formalize this intuitive idea. Doing so isn’t difficult, but we must do some preliminary work. First, we restrict our attention to objects that are binary strings. Other objects can be represented as binary strings, so this restriction doesn’t limit the scope of the theory. Second, we consider only descriptions that are themselves binary strings. By imposing this requirement, we may easily compare the length of the object with the length of its description. In the next section, we consider the type of description that we allow.
MINIMAL LENGTH DESCRIPTIONS
Many types of description language can be used to define information. Selecting which language to use affects the characteristics of the definition. Our descrip- tion language is based on algorithms.
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6.4 A DEFINITION OF INFORMATION 263
One way to use algorithms to describe strings is to construct a Turing machine that prints out the string when it is started on a blank tape and then represent that Turing machine itself as a string. Thus, the string representing the Turing machine is a description of the original string. A drawback to this approach is that a Turing machine cannot represent a table of information concisely with its transition function. To represent a string of n bits, you might use n states and n rows in the transition function table. That would result in a description that is excessively long for our purpose. Instead, we use the following more concise description language.
We describe a binary string x with a Turing machine M and a binary input w to M. The length of the description is the combined length of representing M and w. We write this description with our usual notation for encoding sev- eral objects into a single binary string ⟨M, w⟩. But here we must pay additional attention to the encoding operation ⟨· , ·⟩ because we need to produce a concise result. We define the string ⟨M,w⟩ to be ⟨M⟩w, where we simply concatenate the binary string w onto the end of the binary encoding of M. The encoding ⟨M ⟩ of M may be done in any standard way, except for the subtlety that we de- scribe in the next paragraph. (Don’t worry about this subtle point on your first reading of this material. For now, skip past the next paragraph and the following figure.)
When concatenating w onto the end of ⟨M⟩ to yield a description of x, you might run into trouble if the point at which ⟨M⟩ ends and w begins is not dis- cernible from the description itself. Otherwise, several ways of partitioning the description ⟨M⟩w into a syntactically correct TM and an input may occur, and then the description would be ambiguous and hence invalid. We avoid this prob- lem by ensuring that we can locate the separation between ⟨M ⟩ and w in ⟨M ⟩w. One way to do so is to write each bit of ⟨M⟩ twice, writing 0 as 00 and 1 as 11, and then follow it with 01 to mark the separation point. We illustrate this idea in the following figure, depicting the description ⟨M, w⟩ of some string x.
⟨M,w⟩ = 11001111001100···1100 01 01101011···010
delimiter
w Example of the format of the description ⟨M, w⟩ of some string x
Now that we have fixed our description language, we are ready to define our measure of the quantity of information in a string.
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FIGURE 6.22
⟨M⟩
264 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
DEFINITION 6.23
Let x be a binary string. The minimal description of x, written d(x), is the shortest string ⟨M,w⟩ where TM M on input w halts with x on its tape. If several such strings exist, select the lexi- cographically first among them. The descriptive complexity5of x, written K(x), is
K(x) = |d(x)|.
In other words, K(x) is the length of the minimal description of x. The definition of K(x) is intended to capture our intuition for the amount of infor- mation in the string x. Next we establish some simple results about descriptive complexity.
THEOREM 6.24
∃c ∀x K(x) ≤ |x| + c
This theorem says that the descriptive complexity of a string is at most a fixed constant more than its length. The constant is a universal one, not dependent on the string.
PROOF To prove an upper bound on K(x) as this theorem claims, we need only demonstrate some description of x that is no longer than the stated bound. Then the minimal description of x may be shorter than the demonstrated de- scription, but not longer.
Consider the following description of the string x. Let M be a Turing ma- chine that halts as soon as it is started. This machine computes the identity function—its output is the same as its input. A description of x is simply ⟨M ⟩x. Letting c be the length of ⟨M ⟩ completes the proof.
Theorem 6.24 illustrates how we use the input to the Turing machine to rep- resent information that would require a significantly larger description if stored instead by using the machine’s transition function. It conforms to our intuition that the amount of information contained by a string cannot be (substantially) more than its length. Similarly, intuition says that the information contained by the string xx is not significantly more than the information contained by x. The following theorem verifies this fact.
5Descriptive complexity is called Kolmogorov complexity or Kolmogorov–Chaitin com- plexity in some treatments.
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6.4 A DEFINITION OF INFORMATION 265
THEOREM 6.25
∃c ∀x K(xx) ≤ K(x) + c
PROOF Consider the following Turing machine M, which expects an input of the form ⟨N, w⟩, where N is a Turing machine and w is an input for it.
M =“Oninput⟨N,w⟩,whereN isaTMandwisastring:
1. Run N on w until it halts and produces an output string s.
2. Output the string ss.”
A description of xx is ⟨M ⟩d(x). Recall that d(x) is a minimal description of x. The length of this description is |⟨M ⟩| + |d(x)|, which is c + K(x) where c is the length of ⟨M⟩.
Next we examine how the descriptive complexity of the concatenation xy of two strings x and y is related to their individual complexities. Theorem 6.24 might lead us to believe that the complexity of the concatenation is at most the sum of the individual complexities (plus a fixed constant), but the cost of com- bining two descriptions leads to a greater bound, as described in the following theorem.
THEOREM 6.26
∃c ∀x,y K(xy) ≤ 2K(x) + K(y) + c
PROOF We construct a TM M that breaks its input w into two separate de- scriptions. The bits of the first description d(x) are all doubled and terminated with string 01 before the second description d(y) appears, as described in the text preceding Figure 6.22. Once both descriptions have been obtained, they are run to obtain the strings x and y and the output xy is produced.
The length of this description of xy is clearly twice the complexity of x plus the complexity of y plus a fixed constant for describing M . This sum is
2K(x) + K(y) + c,
and the proof is complete.
We may improve this theorem somewhat by using a more efficient method of indicating the separation between the two descriptions. One way avoids dou- bling the bits of d(x). Instead we prepend the length of d(x) as a binary integer that has been doubled to differentiate it from d(x). The description still contains enough information to decode it into the two descriptions of x and y, and it now has length at most
2 log2(K(x)) + K(x) + K(y) + c.
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266 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY Further small improvements are possible. However, as Problem 6.26 asks you to
show, we cannot reach the bound K(x) + K(y) + c. OPTIMALITY OF THE DEFINITION
Now that we have established some of the elementary properties of descriptive complexity and you have had a chance to develop some intuition, we discuss some features of the definitions.
Our definition of K(x) has an optimality property among all possible ways of defining descriptive complexity with algorithms. Suppose that we consider a general description language to be any computable function p: Σ∗−→Σ∗ and define the minimal description of x with respect to p, written dp(x), to be the first string s where p(s) = x, in the standard string order. Thus, s is lexico- graphically first among the shortest descriptions of x. Define Kp(x) = |dp(x)|.
For example, consider a programming language such as Python (encoded into binary) as the description language. Then dPython(x) would be the minimal Python program that outputs x, and KPython(x) would be the length of the min- imal program.
The following theorem shows that any description language of this type is not significantly more concise than the language of Turing machines and inputs that we originally defined.
THEOREM 6.27
For any description language p, a fixed constant c exists that depends only on p,
where
∀x K(x) ≤ Kp(x) + c .
PROOF IDEA We illustrate the idea of this proof by using the Python exam- ple. Suppose that x has a short description w in Python. Let M be a TM that can interpret Python and use the Python program for x as M’s input w. Then ⟨M, w⟩ is a description of x that is only a fixed amount larger than the Python description of x. The extra length is for the Python interpreter M .
PROOF Take any description language p and consider the following Turing machine M.
M = “On input w:
1. Output p(w).”
Then ⟨M⟩dp(x) is a description of x whose length is at most a fixed constant greater than Kp(x). The constant is the length of ⟨M⟩.
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6.4 A DEFINITION OF INFORMATION 267 INCOMPRESSIBLE STRINGS AND RANDOMNESS
Theorem 6.24 shows that a string’s minimal description is never much longer than the string itself. Of course for some strings, the minimal description may be much shorter if the information in the string appears sparsely or redundantly. Do some strings lack short descriptions? In other words, is the minimal de- scription of some strings actually as long as the string itself? We show that such strings exist. These strings can’t be described any more concisely than simply writing them out explicitly.
DEFINITION 6.28
Let x be a string. Say that x is c-compressible if
K(x) ≤ |x| − c.
If x is not c-compressible, we say that x is incompressible by c. If x is incompressible by 1, we say that x is incompressible.
In other words, if x has a description that is c bits shorter than its length, x is c-compressible. If not, x is incompressible by c. Finally, if x doesn’t have any description shorter than itself, x is incompressible. We first show that in- compressible strings exist, and then we discuss their interesting properties. In particular, we show that incompressible strings look like strings that are obtained from random coin tosses.
THEOREM 6.29
Incompressible strings of every length exist.
PROOF IDEA The number of strings of length n is greater than the number of descriptions of length less than n. Each description describes at most one string. Therefore, some string of length n is not described by any description of length less than n. That string is incompressible.
PROOF The number of binary strings of length n is 2n. Each description is a binary string, so the number of descriptions of length less than n is at most the sum of the number of strings of each length up to n − 1, or
2i =1+2+4+8+···+2n−1 =2n −1. 0≤i≤n−1
The number of short descriptions is less than the number of strings of length n. Therefore, at least one string of length n is incompressible.
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268 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY COROLLARY 6.30
At least 2n − 2n−c+1 + 1 strings of length n are incompressible by c.
PROOF We extend the proof of Theorem 6.29. Every c-compressible string has a description of length at most n − c. No more than 2n−c+1 − 1 such descriptions can occur. Therefore, at most 2n−c+1 − 1 of the 2n strings of length n may have such descriptions. The remaining strings, numbering at least 2n − (2n−c+1 − 1), are incompressible by c.
Incompressible strings have many properties that we would expect to find in randomly chosen strings. For example, we can show that any incompressible string of length n has roughly an equal number of 0s and 1s, and that the length of its longest run of 0s is approximately log2 n, as we would expect to find in a random string of that length. Proving such statements would take us too far afield into combinatorics and probability, but we will prove a theorem that forms the basis for these statements.
That theorem shows that any computable property that holds for “almost all” strings also holds for all sufficiently long incompressible strings. As we men- tioned in Section 0.2, a property of strings is simply a function f that maps strings to {TRUE, FALSE}. We say that a property holds for almost all strings if the fraction of strings of length n on which it is FALSE approaches 0 as n grows large. A randomly chosen long string is likely to satisfy a computable property that holds for almost all strings. Therefore, random strings and incompressible strings share such properties.
THEOREM 6.31
Let f be a computable property that holds for almost all strings. Then, for any b > 0, the property f is FALSE on only finitely many strings that are incompress- ible by b.
PROOF Let M be the following algorithm.
M = “On input i, a binary integer:
1. Find the ith string s where f(s) = FALSE, in the standard string
order.
2. Output string s.”
We can use M to obtain short descriptions of strings that fail to have property f as follows. For any such string x, let ix be the position or index of x on a list of all strings that fail to have property f , in the standard string order (i.e., by length and lexicographically within each length). Then ⟨M,ix⟩ is a description of x. The length of this description is |ix| + c, where c is the length of ⟨M⟩. Because few strings fail to have property f, the index of x is small and its description is correspondingly short.
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6.4 A DEFINITION OF INFORMATION 269
Fix any number b > 0. Select n such that at most a 1/2b+c+1 fraction of strings of length n or less fail to have property f. All sufficiently large n satisfy this condition because f holds for almost all strings. Let x be a string of length n that fails to have property f . We have 2n+1 − 1 strings of length n or less, so
ix ≤ 2n+1 − 1 ≤ 2n−b−c. 2b+c+1
Therefore, |ix| ≤ n−b−c, so the length of ⟨M, ix⟩ is at most (n−b−c)+c = n−b, which implies that
K(x) ≤ n − b.
Thus every sufficiently long x that fails to have property f is compressible by b. Hence only finitely many strings that fail to have property f are incompressible by b, and the theorem is proved.
At this point, exhibiting some examples of incompressible strings would be appropriate. However, as Problem 6.23 asks you to show, the K measure of complexity is not computable. Furthermore, no algorithm can decide in general whether strings are incompressible, by Problem 6.24. Indeed, by Problem 6.25, no infinite subset of them is Turing-recognizable. So we have no way to ob- tain long incompressible strings and would have no way to determine whether a string is incompressible even if we had one. The following theorem describes certain strings that are nearly incompressible, although it doesn’t provide a way to exhibit them explicitly.
THEOREM 6.32
For some constant b, for every string x, the minimal description d(x) of x is
incompressible by b.
PROOF Consider the following TM M :
M =“Oninput⟨R,y⟩,whereRisaTMandyisastring:
1. Run R on y and reject if its output is not of the form ⟨S, z⟩.
2. Run S on z and halt with its output on the tape.”
Let b be |⟨M⟩| + 1. We show that b satisfies the theorem. Suppose to the contrary that d(x) is b-compressible for some string x. Then
|d(d(x))| ≤ |d(x)| − b.
But then ⟨M ⟩d(d(x)) is a description of x whose length is at most
|⟨M ⟩| + |d(d(x))| ≤ (b − 1) + (|d(x)| − b) = |d(x)| − 1.
This description of x is shorter than d(x), contradicting the latter’s minimality.
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270 CHAPTER 6 / ADVANCED TOPICS IN COMPUTABILITY THEORY
EXERCISES
6.1
6.2 A6.3 6.4
A 6.5
Give an example in the spirit of the recursion theorem of a program in a real pro- gramming language (or a reasonable approximation thereof ) that prints itself out.
Show that any infinite subset of MIN TM is not Turing-recognizable. ShowthatifA≤T BandB≤T C,thenA≤T C.
Let ATM′ = {⟨M, w⟩| M is an oracle TM and MATM accepts w}. Show that ATM′ is undecidable relative to ATM.
Is the statement ∃x ∀y x+y=y a member of Th(N , +)? Why or why not? What about the statement ∃x ∀y x+y=x ?
PROBLEMS
6.6 6.7
⋆6.8 A 6.9
A 6.10
⋆ 6.11
A 6.12
Describe two different Turing machines, M and N, where M outputs ⟨N⟩ and N outputs ⟨M ⟩, when started on any input.
In the fixed-point version of the recursion theorem (Theorem 6.8), let the trans- formation t be a function that interchanges the states qaccept and qreject in Turing machine descriptions. Give an example of a fixed point for t.
Show that EQTM ̸≤m EQTM.
Use the recursion theorem to give an alternative proof of Rice’s theorem in Prob-
lem 5.28.
Give a model of the sentence
φeq = ∀xR1(x,x)
∧ ∀x,y R1(x, y) ↔ R1(y, x)
∧ ∀x,y,z (R1(x, y) ∧ R1(y, z)) → R1(x, z) .
Let φeq be defined as in Problem 6.10. Give a model of the sentence
φlt = φeq
∧∀x,y R1(x,y) → ¬R2(x,y)
∧ ∀x,y ¬R1(x, y) → (R2(x, y) ⊕ R2(y, x)) ∧ ∀x,y,z (R2(x, y) ∧ R2(y, z)) → R2(x, z) ∧∀x∃y R2(x,y).
Let (N , <) be the model with universe N and the “less than” relation. Show that Th(N , <) is decidable.
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6.13
6.14 6.15
⋆ 6.16 ⋆6.17
6.18 6.19
6.20 6.21 6.22
6.23 6.24 6.25
⋆ 6.26 6.27
6.28
For each m > 1 let Zm = {0,1,2,…,m − 1}, and let Fm = (Zm,+,×) be the model whose universe is Zm and that has relations corresponding to the + and × relations computed modulo m. Show that for each m, the theory Th(Fm) is decidable.
Show that for any two languages A and B, a language J exists, where A ≤T J and B ≤T J.
Show that for any language A, a language B exists, where A ≤T B and B ̸≤T A. Prove that there exist two languages A and B that are Turing-incomparable—that
is,whereA̸≤T BandB̸≤T A.
Let A and B be two disjoint languages. Say that language C separates A and B if A ⊆ C and B ⊆ C. Describe two disjoint Turing-recognizable languages that aren’t separable by any decidable language.
Show that EQTM is recognizable by a Turing machine with an oracle for ATM.
In Corollary 4.18, we showed that the set of all languages is uncountable. Use this result to prove that languages exist that are not recognizable by an oracle Turing machine with an oracle for ATM.
Recall the Post Correspondence Problem that we defined in Section 5.2 and its associated language PCP. Show that PCP is decidable relative to ATM.
Show how to compute the descriptive complexity of strings K(x) with an oracle for ATM.
Use the result of Problem 6.21 to give a function f that is computable with an oracle for ATM , where for each n, f (n) is an incompressible string of length n.
Show that the function K(x) is not a computable function. Show that the set of incompressible strings is undecidable.
Show that the set of incompressible strings contains no infinite subset that is Turing-recognizable.
Show that for any c, some strings x and y exist, where K(xy) > K(x) + K(y) + c. LetS={⟨M⟩|MisaTMandL(M)={⟨M⟩}}.ShowthatneitherSnorSis
Turing-recognizable.
Let R ⊆ Nk be a k-ary relation. Say that R is definable in Th(N,+) if we can give a formula φ with k free variables x1,…,xk such that for all a1,…,ak ∈ N, φ(a1,…,ak)istrueexactlywhena1,…,ak ∈ R. Showthateachofthefollowing relations is definable in Th(N , +).
Aa. R0={0}
b. R1 = {1}
c. R= ={(a,a)|a∈N}
d. R< ={(a,b)|a,b∈N anda
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Output x.”
Algorithm R solves RELPRIME, using E as a subroutine.
290 CHAPTER 7 / TIME COMPLEXITY
executed, x > y. If x/2 ≥ y, then x mod y < y ≤ x/2 and x drops by at least half. If x/2 < y, then x mod y = x − y < x/2 and x drops by at least half.
The values of x and y are exchanged every time stage 3 is executed, so each of the original values of x and y are reduced by at least half every other time through the loop. Thus, the maximum number of times that stages 2 and 3 are executed is the lesser of 2 log2 x and 2 log2 y. These logarithms are proportional to the lengths of the representations, giving the number of stages executed as O(n). Each stage of E uses only polynomial time, so the total running time is polynomial.
The final example of a polynomial time algorithm shows that every context- free language is decidable in polynomial time.
THEOREM 7.16
Every context-free language is a member of P.
PROOF IDEA In Theorem 4.9, we proved that every CFL is decidable. To do so, we gave an algorithm for each CFL that decides it. If that algorithm runs in polynomial time, the current theorem follows as a corollary. Let’s recall that algorithm and find out whether it runs quickly enough.
Let L be a CFL generated by CFG G that is in Chomsky normal form. From Problem 2.26, any derivation of a string w has 2n − 1 steps, where n is the length of w because G is in Chomsky normal form. The decider for L works by trying all possible derivations with 2n − 1 steps when its input is a string of length n. If any of these is a derivation of w, the decider accepts; if not, it rejects.
A quick analysis of this algorithm shows that it doesn’t run in polynomial time. The number of derivations with k steps may be exponential in k, so this algorithm may require exponential time.
Togetapolynomialtimealgorithm,weintroduceapowerfultechniquecalled dynamic programming. This technique uses the accumulation of information about smaller subproblems to solve larger problems. We record the solution to any subproblem so that we need to solve it only once. We do so by making a table of all subproblems and entering their solutions systematically as we find them.
In this case, we consider the subproblems of determining whether each vari- able in G generates each substring of w. The algorithm enters the solution to this subproblem in an n × n table. For i ≤ j, the (i, j)th entry of the table con- tains the collection of variables that generate the substring wiwi+1 ···wj. For i > j, the table entries are unused.
The algorithm fills in the table entries for each substring of w. First it fills in the entries for the substrings of length 1, then those of length 2, and so on.
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It uses the entries for the shorter lengths to assist in determining the entries for the longer lengths.
For example, suppose that the algorithm has already determined which vari- ables generate all substrings up to length k. To determine whether a variable A generates a particular substring of length k+1, the algorithm splits that substring into two nonempty pieces in the k possible ways. For each split, the algorithm examines each rule A → BC to determine whether B generates the first piece and C generates the second piece, using table entries previously computed. If both B and C generate the respective pieces, A generates the substring and so is added to the associated table entry. The algorithm starts the process with the strings of length 1 by examining the table for the rules A → b.
PROOF The following algorithm D implements the proof idea. Let G be a CFG in Chomsky normal form generating the CFL L. Assume that S is the start variable. (Recall that the empty string is handled specially in a Chomsky normal form grammar. The algorithm handles the special case in which w = ε in stage 1.) Comments appear inside double brackets.
D = “On input w = w1 · · · wn:
1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11.
12.
Forw=ε,ifS→εisarule,accept;else,reject. [w=εcase] For i = 1 to n: [ examine each substring of length 1 ]
For each variable A:
Test whether A → b is a rule, where b = wi. If so, place A in table (i, i).
For l = 2 to n: [l is the length of the substring] Fori=1ton−l+1: [iisthestartpositionofthesubstring] Let j = i + l − 1. [ j is the end position of the substring ]
For k = i to j − 1:
For each rule A → BC:
[ k is the split position ]
If table(i, k) contains B and table(k + 1, j) contains
C, put A in table(i, j).
If S is in table (1, n), accept ; else, reject .”
Now we analyze D. Each stage is easily implemented to run in polynomial time. Stages 4 and 5 run at most nv times, where v is the number of variables in G and is a fixed constant independent of n; hence these stages run O(n) times. Stage 6 runs at most n times. Each time stage 6 runs, stage 7 runs at most n times. Each time stage 7 runs, stages 8 and 9 run at most n times. Each time stage 9 runs, stage 10 runs r times, where r is the number of rules of G and is another fixed constant. Thus stage 11, the inner loop of the algorithm, runs O(n3) times. Summing the total shows that D executes O(n3) stages.
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7.2 THE CLASS P 291
292 CHAPTER 7 / TIME COMPLEXITY 7.3
THE CLASS NP
As we observed in Section 7.2, we can avoid brute-force search in many problems and obtain polynomial time solutions. However, attempts to avoid brute force in certain other problems, including many interesting and useful ones, haven’t been successful, and polynomial time algorithms that solve them aren’t known to exist.
Why have we been unsuccessful in finding polynomial time algorithms for these problems? We don’t know the answer to this important question. Perhaps these problems have as yet undiscovered polynomial time algorithms that rest on unknown principles. Or possibly some of these problems simply cannot be solved in polynomial time. They may be intrinsically difficult.
One remarkable discovery concerning this question shows that the complex- ities of many problems are linked. A polynomial time algorithm for one such problem can be used to solve an entire class of problems. To understand this phenomenon, let’s begin with an example.
A Hamiltonian path in a directed graph G is a directed path that goes through each node exactly once. We consider the problem of testing whether a directed graph contains a Hamiltonian path connecting two specified nodes, as shown in the following figure. Let
HAMPATH ={⟨G,s,t⟩|Gisadirectedgraph
with a Hamiltonian path from s to t}.
FIGURE 7.17
A Hamiltonian path goes through every node exactly once
We can easily obtain an exponential time algorithm for the HAMPATH prob- lem by modifying the brute-force algorithm for PATH given in Theorem 7.14. We need only add a check to verify that the potential path is Hamiltonian. No one knows whether HAMPATH is solvable in polynomial time.
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The HAMPATH problem has a feature called polynomial verifiability that is important for understanding its complexity. Even though we don’t know of a fast (i.e., polynomial time) way to determine whether a graph contains a Hamiltonian path, if such a path were discovered somehow (perhaps using the exponential time algorithm), we could easily convince someone else of its existence simply by presenting it. In other words, verifying the existence of a Hamiltonian path may be much easier than determining its existence.
Another polynomially verifiable problem is compositeness. Recall that a nat- ural number is composite if it is the product of two integers greater than 1 (i.e., a composite number is one that is not a prime number). Let
COMPOSITES = {x| x = pq, for integers p, q > 1}.
We can easily verify that a number is composite—all that is needed is a divisor of that number. Recently, a polynomial time algorithm for testing whether a number is prime or composite was discovered, but it is considerably more com- plicated than the preceding method for verifying compositeness.
Some problems may not be polynomially verifiable. For example, take HAMPATH, the complement of the HAMPATH problem. Even if we could determine (somehow) that a graph did not have a Hamiltonian path, we don’t know of a way for someone else to verify its nonexistence without using the same exponential time algorithm for making the determination in the first place. A formal definition follows.
7.3 THE CLASS NP 293
DEFINITION 7.18
A verifier for a language A is an algorithm V , where
A = {w| V accepts ⟨w, c⟩ for some string c}.
We measure the time of a verifier only in terms of the length of w, so a polynomial time verifier runs in polynomial time in the length of w. A language A is polynomially verifiable if it has a polynomial time verifier.
A verifier uses additional information, represented by the symbol c in Defini- tion 7.18, to verify that a string w is a member of A. This information is called a certificate, or proof, of membership in A. Observe that for polynomial verifiers, the certificate has polynomial length (in the length of w) because that is all the verifier can access in its time bound. Let’s apply this definition to the languages HAMPATH and COMPOSITES.
For the HAMPATH problem, a certificate for a string ⟨G, s, t⟩ ∈ HAMPATH simply is a Hamiltonian path from s to t. For the COMPOSITES problem, a certificate for the composite number x simply is one of its divisors. In both cases, the verifier can check in polynomial time that the input is in the language when it is given the certificate.
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294 CHAPTER 7 / TIME COMPLEXITY
DEFINITION 7.19
NP is the class of languages that have polynomial time verifiers.
The class NP is important because it contains many problems of practical in- terest. From the preceding discussion, both HAMPATH and COMPOSITES are members of NP. As we mentioned, COMPOSITES is also a member of P, which is a subset of NP; but proving this stronger result is much more difficult. The term NP comes from nondeterministic polynomial time and is derived from an alternative characterization by using nondeterministic polynomial time Turing machines. Problems in NP are sometimes called NP-problems.
The following is a nondeterministic Turing machine (NTM) that decides the HAMPATH problem in nondeterministic polynomial time. Recall that in Def- inition 7.9, we defined the time of a nondeterministic machine to be the time used by the longest computation branch.
N1 = “On input ⟨G, s, t⟩, where G is a directed graph with nodes s and t:
1. Write a list of m numbers, p1,…,pm, where m is the number of nodes in G. Each number in the list is nondeterministically
selected to be between 1 and m.
2. Check for repetitions in the list. If any are found, reject.
3. Check whether s = p1 and t = pm. If either fail, reject.
4. For each i between 1 and m − 1, check whether (pi, pi+1) is an
edge of G. If any are not, reject. Otherwise, all tests have been passed, so accept .”
To analyze this algorithm and verify that it runs in nondeterministic poly- nomial time, we examine each of its stages. In stage 1, the nondeterministic selection clearly runs in polynomial time. In stages 2 and 3, each part is a simple check, so together they run in polynomial time. Finally, stage 4 also clearly runs in polynomial time. Thus, this algorithm runs in nondeterministic polynomial time.
THEOREM 7.20
A language is in NP iff it is decided by some nondeterministic polynomial time
Turing machine.
PROOF IDEA We show how to convert a polynomial time verifier to an equivalent polynomial time NTM and vice versa. The NTM simulates the ver- ifier by guessing the certificate. The verifier simulates the NTM by using the accepting branch as the certificate.
PROOF For the forward direction of this theorem, let A ∈ NP and show that A is decided by a polynomial time NTM N . Let V be the polynomial time verifier for A that exists by the definition of NP. Assume that V is a TM that runs in time nk and construct N as follows.
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N = “On input w of length n:
1. Nondeterministically select string c of length at most nk.
2. Run V on input ⟨w, c⟩.
3. If V accepts, accept ; otherwise, reject .”
To prove the other direction of the theorem, assume that A is decided by a polynomial time NTM N and construct a polynomial time verifier V as follows.
V = “On input ⟨w, c⟩, where w and c are strings:
1. Simulate N on input w, treating each symbol of c as a descrip-
tion of the nondeterministic choice to make at each step (as in
the proof of Theorem 3.16).
2. If this branch of N ’s computation accepts, accept ; otherwise,
reject .”
We define the nondeterministic time complexity class NTIME(t(n)) as anal- ogous to the deterministic time complexity class TIME(t(n)).
COROLLARY 7.22 NP = k NTIME(nk).
The class NP is insensitive to the choice of reasonable nondeterministic com- putational model because all such models are polynomially equivalent. When describing and analyzing nondeterministic polynomial time algorithms, we fol- low the preceding conventions for deterministic polynomial time algorithms. Each stage of a nondeterministic polynomial time algorithm must have an obvi- ous implementation in nondeterministic polynomial time on a reasonable non- deterministic computational model. We analyze the algorithm to show that every branch uses at most polynomially many stages.
EXAMPLES OF PROBLEMS IN NP
A clique in an undirected graph is a subgraph, wherein every two nodes are connected by an edge. A k-clique is a clique that contains k nodes. Figure 7.23 illustrates a graph with a 5-clique.
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7.3 THE CLASS NP 295
DEFINITION 7.21
NTIME(t(n)) = {L| L is a language decided by an O(t(n)) time nondeterministic Turing machine}.
296 CHAPTER 7 / TIME COMPLEXITY
FIGURE 7.23
A graph with a 5-clique
The clique problem is to determine whether a graph contains a clique of a specified size. Let
CLIQUE = {⟨G, k⟩| G is an undirected graph with a k-clique}. THEOREM 7.24
CLIQUE is in NP.
PROOF IDEA The clique is the certificate.
PROOF The following is a verifier V for CLIQUE.
V =“Oninput⟨⟨G,k⟩,c⟩:
1. Test whether c is a subgraph with k nodes in G.
2. Test whether G contains all edges connecting nodes in c.
3. If both pass, accept; otherwise, reject.”
ALTERNATIVE PROOF If you prefer to think of NP in terms of nonde- terministic polynomial time Turing machines, you may prove this theorem by giving one that decides CLIQUE. Observe the similarity between the two proofs.
N = “On input ⟨G, k⟩, where G is a graph:
1. Nondeterministically select a subset c of k nodes of G.
2. Test whether G contains all edges connecting nodes in c.
3. If yes, accept ; otherwise, reject .”
Next, we consider the SUBSET-SUM problem concerning integer arithmetic. We are given a collection of numbers x1, . . . , xk and a target number t. We want to determine whether the collection contains a subcollection that adds up to t.
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Thus,
SUBSET-SUM = {⟨S,t⟩| S = {x1,…,xk}, and for some
{y1,…,yl} ⊆ {x1,…,xk}, we have Σyi = t}.
For example, ⟨{4, 11, 16, 21, 27}, 25⟩ ∈ SUBSET-SUM because 4 + 21 = 25. Note that {x1,…,xk} and {y1,…,yl} are considered to be multisets and so allow repetition of elements.
THEOREM 7.25 SUBSET-SUM is in NP.
PROOF IDEA The subset is the certificate.
PROOF The following is a verifier V for SUBSET-SUM.
V =“Oninput⟨⟨S,t⟩,c⟩:
1. Test whether c is a collection of numbers that sum to t.
2. Test whether S contains all the numbers in c.
3. If both pass, accept; otherwise, reject.”
ALTERNATIVE PROOF We can also prove this theorem by giving a nonde- terministic polynomial time Turing machine for SUBSET-SUM as follows.
N = “On input ⟨S, t⟩:
1. Nondeterministically select a subset c of the numbers in S.
2. Test whether c is a collection of numbers that sum to t.
3. If the test passes, accept ; otherwise, reject .”
Observe that the complements of these sets, CLIQUE and SUBSET-SUM, are not obviously members of NP. Verifying that something is not present seems to be more difficult than verifying that it is present. We make a separate com- plexity class, called coNP, which contains the languages that are complements of languages in NP. We don’t know whether coNP is different from NP.
THE P VERSUS NP QUESTION
As we have been saying, NP is the class of languages that are solvable in polyno- mial time on a nondeterministic Turing machine; or, equivalently, it is the class of languages whereby membership in the language can be verified in polynomial
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7.3 THE CLASS NP 297
298 CHAPTER 7 / TIME COMPLEXITY
time. P is the class of languages where membership can be tested in polyno- mial time. We summarize this information as follows, where we loosely refer to polynomial time solvable as solvable “quickly.”
P = the class of languages for which membership can be decided quickly. NP = the class of languages for which membership can be verified quickly.
We have presented examples of languages, such as HAMPATH and CLIQUE, that are members of NP but that are not known to be in P. The power of polyno- mial verifiability seems to be much greater than that of polynomial decidability. But, hard as it may be to imagine, P and NP could be equal. We are unable to prove the existence of a single language in NP that is not in P.
The question of whether P = NP is one of the greatest unsolved problems in theoretical computer science and contemporary mathematics. If these classes were equal, any polynomially verifiable problem would be polynomially decid- able. Most researchers believe that the two classes are not equal because people have invested enormous effort to find polynomial time algorithms for certain problems in NP, without success. Researchers also have tried proving that the classes are unequal, but that would entail showing that no fast algorithm exists to replace brute-force search. Doing so is presently beyond scientific reach. The following figure shows the two possibilities.
FIGURE 7.26
One of these two possibilities is correct
The best deterministic method currently known for deciding languages in NP uses exponential time. In other words, we can prove that
NP ⊆ EXPTIME = TIME(2nk ), k
but we don’t know whether NP is contained in a smaller deterministic time com- plexity class.
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7.4
NP-COMPLETENESS
One important advance on the P versus NP question came in the early 1970s with the work of Stephen Cook and Leonid Levin. They discovered certain problems in NP whose individual complexity is related to that of the entire class. If a polynomial time algorithm exists for any of these problems, all problems in NP would be polynomial time solvable. These problems are called NP-complete. The phenomenon of NP-completeness is important for both theoretical and practical reasons.
On the theoretical side, a researcher trying to show that P is unequal to NP may focus on an NP-complete problem. If any problem in NP requires more than polynomial time, an NP-complete one does. Furthermore, a researcher attempting to prove that P equals NP only needs to find a polynomial time al- gorithm for an NP-complete problem to achieve this goal.
On the practical side, the phenomenon of NP-completeness may prevent wasting time searching for a nonexistent polynomial time algorithm to solve a particular problem. Even though we may not have the necessary mathematics to prove that the problem is unsolvable in polynomial time, we believe that P is unequal to NP. So proving that a problem is NP-complete is strong evidence of its nonpolynomiality.
The first NP-complete problem that we present is called the satisfiability problem. Recall that variables that can take on the values TRUE and FALSE are called Boolean variables (see Section 0.2). Usually, we represent TRUE by 1 and FALSE by 0. The Boolean operations AND, OR, and NOT, represented by the symbols ∧, ∨, and ¬, respectively, are described in the following list. We use the overbar as a shorthand for the ¬ symbol, so x means ¬ x.
0∧0=0 0∨0=0 0=1 0∧1=0 0∨1=1 1=0 1∧0=0 1∨0=1
1∧1=1 1∨1=1
A Boolean formula is an expression involving Boolean variables and opera- tions. For example,
x∧y) ∨ (x∧z)
is a Boolean formula. A Boolean formula is satisfiable if some assignment of 0s and 1s to the variables makes the formula evaluate to 1. The preceding formula is satisfiable because the assignment x = 0, y = 1, and z = 0 makes φ evaluate to 1. We say the assignment satisfies φ. The satisfiability problem is to test whether a Boolean formula is satisfiable. Let
SAT = {⟨φ⟩| φ is a satisfiable Boolean formula}.
Now we state a theorem that links the complexity of the SAT problem to the
complexities of all problems in NP.
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7.4 NP-COMPLETENESS 299
φ=(
300 CHAPTER 7 / TIME COMPLEXITY THEOREM 7.27
SAT ∈ P iff P = NP.
Next, we develop the method that is central to the proof of this theorem.
POLYNOMIAL TIME REDUCIBILITY
In Chapter 5, we defined the concept of reducing one problem to another. When problem A reduces to problem B, a solution to B can be used to solve A. Now we define a version of reducibility that takes the efficiency of computation into account. When problem A is efficiently reducible to problem B, an efficient solution to B can be used to solve A efficiently.
DEFINITION 7.28
A function f : Σ∗ −→ Σ∗ is a polynomial time computable function if some polynomial time Turing machine M exists that halts with just f (w) on its tape, when started on any input w.
DEFINITION 7.29
Language A is polynomial time mapping reducible,1or simply poly- nomial time reducible, to language B, written A ≤P B, if a polyno- mial time computable function f : Σ∗ −→ Σ∗ exists, where for every w,
w ∈ A ⇐⇒ f ( w ) ∈ B .
The function f is called the polynomial time reduction of A to B.
Polynomial time reducibility is the efficient analog to mapping reducibility as defined in Section 5.3. Other forms of efficient reducibility are available, but polynomial time reducibility is a simple form that is adequate for our purposes so we won’t discuss the others here. Figure 7.30 illustrates polynomial time reducibility.
1It is called polynomial time many–one reducibility in some other textbooks.
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7.4 NP-COMPLETENESS 301
FIGURE 7.30
Polynomial time function f reducing A to B
As with an ordinary mapping reduction, a polynomial time reduction of A to B provides a way to convert membership testing in A to membership testing in B—but now the conversion is done efficiently. To test whether w ∈ A, we use the reduction f to map w to f(w) and test whether f(w) ∈ B.
If one language is polynomial time reducible to a language already known to have a polynomial time solution, we obtain a polynomial time solution to the original language, as in the following theorem.
THEOREM 7.31
IfA≤P BandB∈P,thenA∈P.
PROOF Let M be the polynomial time algorithm deciding B and f be the polynomial time reduction from A to B. We describe a polynomial time algo- rithm N deciding A as follows.
N = “On input w:
1. Compute f (w).
2. Run M on input f (w) and output whatever M outputs.”
We have w ∈ A whenever f(w) ∈ B because f is a reduction from A to B. Thus, M accepts f(w) whenever w ∈ A. Moreover, N runs in polynomial time because each of its two stages runs in polynomial time. Note that stage 2 runs in polynomial time because the composition of two polynomials is a polynomial.
Before demonstrating a polynomial time reduction, we introduce 3SAT, a special case of the satisfiability problem whereby all formulas are in a special
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302 CHAPTER 7 / TIME COMPLEXITY
form. A literal is a Boolean variable or a negated Boolean variable, as in x or x. A clause is several literals connected with ∨s, as in (x1 ∨ x2 ∨ x3 ∨ x4 ). A Boolean formula is in conjunctive normal form, called a cnf-formula, if it comprises several clauses connected with ∧s, as in
(x1 ∨x2 ∨x3 ∨x4) ∧ (x3 ∨x5 ∨x6) ∧ (x3 ∨x6). It is a 3cnf-formula if all the clauses have three literals, as in
(x1 ∨x2 ∨x3) ∧ (x3 ∨x5 ∨x6) ∧ (x3 ∨x6 ∨x4) ∧ (x4 ∨x5 ∨x6).
Let 3SAT = {⟨φ⟩| φ is a satisfiable 3cnf-formula}. If an assignment satisfies a cnf-formula, each clause must contain at least one literal that evaluates to 1.
The following theorem presents a polynomial time reduction from the 3SAT problem to the CLIQUE problem.
THEOREM 7.32
3SAT is polynomial time reducible to CLIQUE.
PROOF IDEA The polynomial time reduction f that we demonstrate from 3SAT to CLIQUE converts formulas to graphs. In the constructed graphs, cliques of a specified size correspond to satisfying assignments of the formula. Structures within the graph are designed to mimic the behavior of the variables and clauses.
PROOF Let φ be a formula with k clauses such as
φ=(a1 ∨b1 ∨c1) ∧ (a2 ∨b2 ∨c2) ∧ ··· ∧ (ak ∨bk ∨ck).
The reduction f generates the string ⟨G, k⟩, where G is an undirected graph defined as follows.
The nodes in G are organized into k groups of three nodes each called the triples, t1 , . . . , tk . Each triple corresponds to one of the clauses in φ, and each node in a triple corresponds to a literal in the associated clause. Label each node of G with its corresponding literal in φ.
The edges of G connect all but two types of pairs of nodes in G. No edge is present between nodes in the same triple, and no edge is present between two nodes with contradictory labels, as in x2 and x2. Figure 7.33 illustrates this
construction when φ = (x1 ∨ x1 ∨ x2) ∧ (
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x1 ∨x2 ∨x2) ∧ (
x1 ∨x2 ∨x2).
FIGURE 7.33
The graph that the reduction produces from
φ = (x1 ∨ x1 ∨ x2) ∧ (
Now we demonstrate why this construction works. We show that φ is satisfi- able iff G has a k-clique.
Suppose that φ has a satisfying assignment. In that satisfying assignment, at least one literal is true in every clause. In each triple of G, we select one node corresponding to a true literal in the satisfying assignment. If more than one literal is true in a particular clause, we choose one of the true literals arbitrarily. The nodes just selected form a k-clique. The number of nodes selected is k because we chose one for each of the k triples. Each pair of selected nodes is joined by an edge because no pair fits one of the exceptions described previously. They could not be from the same triple because we selected only one node per triple. They could not have contradictory labels because the associated literals were both true in the satisfying assignment. Therefore, G contains a k-clique.
Suppose that G has a k-clique. No two of the clique’s nodes occur in the same triple because nodes in the same triple aren’t connected by edges. Therefore, each of the k triples contains exactly one of the k clique nodes. We assign truth values to the variables of φ so that each literal labeling a clique node is made true. Doing so is always possible because two nodes labeled in a contradictory way are not connected by an edge and hence both can’t be in the clique. This assignment to the variables satisfies φ because each triple contains a clique node and hence each clause contains a literal that is assigned TRUE. Therefore, φ is satisfiable.
Theorems 7.31 and 7.32 tell us that if CLIQUE is solvable in polynomial time, so is 3SAT . At first glance, this connection between these two problems appears quite remarkable because, superficially, they are rather different. But polynomial time reducibility allows us to link their complexities. Now we turn to a definition that will allow us similarly to link the complexities of an entire class of problems.
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x1 ∨x2 ∨x2) ∧ (
7.4 NP-COMPLETENESS 303
x1 ∨x2 ∨x2)
304 CHAPTER 7 / TIME COMPLEXITY DEFINITION OF NP-COMPLETENESS
DEFINITION 7.34
A language B is NP-complete if it satisfies two conditions:
1. B is in NP, and
2. every A in NP is polynomial time reducible to B.
THEOREM 7.35
If B is NP-complete and B ∈ P, then P = NP.
PROOF This theorem follows directly from the definition of polynomial time reducibility.
THEOREM 7.36
If B is NP-complete and B ≤P C for C in NP, then C is NP-complete.
PROOF We already know that C is in NP, so we must show that every A in NP is polynomial time reducible to C. Because B is NP-complete, every lan- guage in NP is polynomial time reducible to B, and B in turn is polynomial time reducible to C. Polynomial time reductions compose; that is, if A is poly- nomial time reducible to B and B is polynomial time reducible to C, then A is polynomial time reducible to C. Hence every language in NP is polynomial time reducible to C.
THE COOK—LEVIN THEOREM
Once we have one NP-complete problem, we may obtain others by polynomial time reduction from it. However, establishing the first NP-complete problem is more difficult. Now we do so by proving that SAT is NP-complete.
THEOREM 7.37 SAT is NP-complete.2
This theorem implies Theorem 7.27.
2An alternative proof of this theorem appears in Section 9.3.
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PROOF IDEA Showing that SAT is in NP is easy, and we do so shortly. The hard part of the proof is showing that any language in NP is polynomial time reducible to SAT.
To do so, we construct a polynomial time reduction for each language A in NP to SAT . The reduction for A takes a string w and produces a Boolean formula φ that simulates the NP machine for A on input w. If the machine accepts, φ has a satisfying assignment that corresponds to the accepting computation. If the machine doesn’t accept, no assignment satisfies φ. Therefore, w is in A if and only if φ is satisfiable.
Actually constructing the reduction to work in this way is a conceptually simple task, though we must cope with many details. A Boolean formula may contain the Boolean operations AND, OR, and NOT, and these operations form the basis for the circuitry used in electronic computers. Hence the fact that we can design a Boolean formula to simulate a Turing machine isn’t surprising. The details are in the implementation of this idea.
PROOF First, we show that SAT is in NP. A nondeterministic polynomial time machine can guess an assignment to a given formula φ and accept if the assignment satisfies φ.
Next, we take any language A in NP and show that A is polynomial time reducible to SAT. Let N be a nondeterministic Turing machine that decides A in nk time for some constant k. (For convenience, we actually assume that N runs in time nk − 3; but only those readers interested in details should worry about this minor point.) The following notion helps to describe the reduction.
A tableau for N on w is an nk × nk table whose rows are the configurations of a branch of the computation of N on input w, as shown in the following figure.
7.4 NP-COMPLETENESS 305
FIGURE 7.38
A tableau is an nk × nk table of configurations
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306 CHAPTER 7 / TIME COMPLEXITY
For convenience later, we assume that each configuration starts and ends with a # symbol. Therefore, the first and last columns of a tableau are all #s. The first row of the tableau is the starting configuration of N on w, and each row follows the previous one according to N’s transition function. A tableau is accepting if any row of the tableau is an accepting configuration.
Every accepting tableau for N on w corresponds to an accepting computation branch of N on w. Thus, the problem of determining whether N accepts w is equivalent to the problem of determining whether an accepting tableau for N on w exists.
Now we get to the description of the polynomial time reduction f from A to SAT. On input w, the reduction produces a formula φ. We begin by describing the variables of φ. Say that Q and Γ are the state set and tape alphabet of N, respectively. Let C = Q ∪ Γ ∪ {#}. For each i and j between 1 and nk and for each s in C, we have a variable, xi,j,s.
Each of the (nk)2 entries of a tableau is called a cell. The cell in row i and column j is called cell[i,j] and contains a symbol from C. We represent the contents of the cells with the variables of φ. If xi,j,s takes on the value 1, it means that cell[i, j] contains an s.
Now we design φ so that a satisfying assignment to the variables does corre- spond to an accepting tableau for N on w. The formula φ is the AND of four parts: φcell ∧ φstart ∧ φmove ∧ φaccept. We describe each part in turn.
As we mentioned previously, turning variable xi,j,s on corresponds to placing symbol s in cell [i, j]. The first thing we must guarantee in order to obtain a cor- respondence between an assignment and a tableau is that the assignment turns on exactly one variable for each cell. Formula φcell ensures this requirement by expressing it in terms of Boolean operations:
φcell = xi,j,s ∧ (
xi,j,s ∨ xi,j,t). 1≤i,j≤nk s∈C s,t∈C
s̸=t
The symbols and stand for iterated AND and OR. For example, the
is shorthand for
expression in the preceding formula
xi,j,s s∈C
xi,j,s1 ∨xi,j,s2 ∨···∨xi,j,sl
where C = {s1, s2, . . . , sl}. Hence φcell is actually a large expression that con- tains a fragment for each cell in the tableau because i and j range from 1 to nk. The first part of each fragment says that at least one variable is turned on in the corresponding cell. The second part of each fragment says that no more than one variable is turned on (literally, it says that in each pair of variables, at least one is turned off) in the corresponding cell. These fragments are connected by ∧ operations.
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The first part of φcell inside the brackets stipulates that at least one variable that is associated with each cell is on, whereas the second part stipulates that no more than one variable is on for each cell. Any assignment to the variables that satisfies φ (and therefore φcell) must have exactly one variable on for every cell. Thus, any satisfying assignment specifies one symbol in each cell of the table. Parts φstart, φmove, and φaccept ensure that these symbols actually correspond to an accepting tableau as follows.
Formula φstart ensures that the first row of the table is the starting configu- ration of N on w by explicitly stipulating that the corresponding variables are on:
φstart = x1,1,# ∧ x1,2,q0 ∧
x1,3,w1 ∧x1,4,w2 ∧…∧x1,n+2,wn∧
␣∧x x1,n+3,␣∧…∧x1,n −1, 1,n ,#
Formula φaccept guarantees that an accepting configuration occurs in the tableau. It ensures that qaccept, the symbol for the accept state, appears in one of the cells of the tableau by stipulating that one of the corresponding variables is on:
φaccept = xi,j,qaccept . 1≤i,j≤nk
Finally, formula φmove guarantees that each row of the tableau corresponds to a configuration that legally follows the preceding row’s configuration according to N’s rules. It does so by ensuring that each 2 × 3 window of cells is legal. We say that a 2 × 3 window is legal if that window does not violate the actions specified by N’s transition function. In other words, a window is legal if it might appear when one configuration correctly follows another.3
For example, say that a, b, and c are members of the tape alphabet, and q1 and q2 are states of N . Assume that when in state q1 with the head reading an a, N writes a b, stays in state q1, and moves right; and that when in state q1 with the head reading a b, N nondeterministically either
1. writes a c, enters q2, and moves to the left, or 2. writes an a, enters q2, and moves to the right.
Expressed formally, δ(q1, a) = {(q1,b,R)} and δ(q1, b) = {(q2,c,L), (q2,a,R)}. Examples of legal windows for this machine are shown in Figure 7.39.
3We could give a precise definition of legal window here, in terms of the transition func- tion. But doing so is quite tedious and would distract us from the main thrust of the proof argument. Anyone desiring more precision should refer to the related analysis in the proof of Theorem 5.15, the undecidability of the Post Correspondence Problem.
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7.4 NP-COMPLETENESS 307
k
k
.
308
CHAPTER 7 / TIME COMPLEXITY
a
q1
b
q2
a
c
a
q1
b
a
a
q2
a
a
q1
a
a
b
(a)
(d)
FIGURE 7.39 Examples of legal windows
(b) (c)
(e) (f )
#
b
a
#
b
a
a
b
a
a
b
q2
b
b
b
c
b
b
In Figure 7.39, windows (a) and (b) are legal because the transition function allows N to move in the indicated way. Window (c) is legal because, with q1 appearing on the right side of the top row, we don’t know what symbol the head is over. That symbol could be an a, and q1 might change it to a b and move to the right. That possibility would give rise to this window, so it doesn’t violate N’s rules. Window (d) is obviously legal because the top and bottom are identical, which would occur if the head weren’t adjacent to the location of the window. Note that # may appear on the left or right of both the top and bottom rows in a legal window. Window (e) is legal because state q1 reading a b might have been immediately to the right of the top row, and it would then have moved to the left in state q2 to appear on the right-hand end of the bottom row. Finally, window (f ) is legal because state q1 might have been immediately to the left of the top row, and it might have changed the b to a c and moved to the left.
The windows shown in the following figure aren’t legal for machine N .
(a) (b) (c)
FIGURE 7.40
Examples of illegal windows
In window (a), the central symbol in the top row can’t change because a state wasn’t adjacent to it. Window (b) isn’t legal because the transition function spec- ifies that the b gets changed to a c but not to an a. Window (c) isn’t legal because two states appear in the bottom row.
CLAIM 7.41
If the top row of the tableau is the start configuration and every window in the tableau is legal, each row of the tableau is a configuration that legally follows the preceding one.
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a
b
a
a
a
a
a
q1
b
q2
a
a
b
q1
b
q2
b
q2
We prove this claim by considering any two adjacent configurations in the tableau, called the upper configuration and the lower configuration. In the up- per configuration, every cell that contains a tape symbol and isn’t adjacent to a state symbol is the center top cell in a window whose top row contains no states. Therefore, that symbol must appear unchanged in the center bottom of the window. Hence it appears in the same position in the bottom configuration.
The window containing the state symbol in the center top cell guarantees that the corresponding three positions are updated consistently with the transition function. Therefore, if the upper configuration is a legal configuration, so is the lower configuration, and the lower one follows the upper one according to N’s rules. Note that this proof, though straightforward, depends crucially on our choice of a 2 × 3 window size, as Problem 7.41 shows.
Now we return to the construction of φmove. It stipulates that all the windows in the tableau are legal. Each window contains six cells, which may be set in a fixed number of ways to yield a legal window. Formula φmove says that the settings of those six cells must be one of these ways, or
φmove = the (i, j)-window is legal. 1≤i
Consider the following scheduling problem. You are given a list of final exams F1,…,Fk tobescheduled,andalistofstudentsS1,…,Sl. Eachstudentistaking some specified subset of these exams. You must schedule these exams into slots so that no student is required to take two exams in the same slot. The problem is to determine if such a schedule exists that uses only h slots. Formulate this problem as a language and show that this language is NP-complete.
This problem is inspired by the single-player game Minesweeper, generalized to an arbitrary graph. Let G be an undirected graph, where each node either contains a single, hidden mine or is empty. The player chooses nodes, one by one. If the player chooses a node containing a mine, the player loses. If the player chooses an empty node, the player learns the number of neighboring nodes containing mines. (A neighboring node is one connected to the chosen node by an edge.) The player wins if and when all empty nodes have been so chosen.
In the mine consistency problem, you are given a graph G along with numbers labeling some of G’s nodes. You must determine whether a placement of mines on the remaining nodes is possible, so that any node v that is labeled m has exactly m neighboring nodes containing mines. Formulate this problem as a language and show that it is NP-complete.
In the following solitaire game, you are given an m × m board. On each of its m2 positions lies either a blue stone, a red stone, or nothing at all. You play by removing stones from the board until each column contains only stones of a sin- gle color and each row contains at least one stone. You win if you achieve this objective. Winning may or may not be possible, depending upon the initial con- figuration. Let SOLITAIRE = {⟨G⟩| G is a winnable game configuration}. Prove that SOLITAIRE is NP-complete.
Recall, in our discussion of the Church–Turing thesis, that we introduced the lan- guage D = {⟨p⟩| p is a polynomial in several variables having an integral root}. We stated, but didn’t prove, that D is undecidable. In this problem, you are to prove a different property of D—namely, that D is NP-hard. A problem is NP-hard if all problems in NP are polynomial time reducible to it, even though it may not be in NP itself. So you must show that all problems in NP are polynomial time reducible to D.
A subset of the nodes of a graph G is a dominating set if every other node of G is adjacent to some node in the subset. Let
DOMINATING-SET = {⟨G, k⟩| G has a dominating set with k nodes}. Show that it is NP-complete by giving a reduction from VERTEX-COVER.
Show that the following problem is NP-complete. You are given a set of states Q = {q0,q1,…,ql} and a collection of pairs {(s1,r1),…,(sk,rk)} where the si are distinct strings over Σ = {0, 1}, and the ri are (not necessarily distinct) members of Q. Determine whether a DFA M = (Q,Σ,δ,q0,F) exists where δ(q0,si) = ri for each i. Here, δ(q, s) is the state that M enters after reading s, starting at state q. (Note that F is irrelevant here.)
A7.33
7.34
7.35
⋆ 7.36
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7.37 ⋆ 7.38
⋆ 7.39 A⋆ 7.40
7.41 ⋆7.42
PROBLEMS 327 Let U = {⟨M,x,#t⟩| NTM M accepts x within t steps on at least one branch}.
Note that M isn’t required to halt on all branches. Show that U is NP-complete.
Show that if P = NP, a polynomial time algorithm exists that produces a satisfying assignment when given a satisfiable Boolean formula. (Note: The algorithm you are asked to provide computes a function; but NP contains languages, not func- tions. The P = NP assumption implies that SAT is in P, so testing satisfiability is solvable in polynomial time. But the assumption doesn’t say how this test is done, and the test may not reveal satisfying assignments. You must show that you can find them anyway. Hint: Use the satisfiability tester repeatedly to find the assignment bit-by-bit.)
Show that if P = NP, you can factor integers in polynomial time. (See the note in Problem 7.38.)
Show that if P = NP, a polynomial time algorithm exists that takes an undirected graph as input and finds a largest clique contained in that graph. (See the note in Problem 7.38.)
In the proof of the Cook–Levin theorem, a window is a 2 × 3 rectangle of cells. Show why the proof would have failed if we had used 2 × 2 windows instead.
Consider the algorithm MINIMIZE, which takes a DFA M as input and outputs DFA M′.
MINIMIZE = “On input ⟨M ⟩, where M = (Q, Σ, δ, q0 , A) is a DFA:
1. Remove all states of M that are unreachable from the start state.
2. Construct the following undirected graph G whose nodes are
the states of M.
3. Place an edge in G connecting every accept state with every
nonaccept state. Add additional edges as follows.
4. Repeat until no new edges are added to G:
5. For every pair of distinct states q and r of M and every a ∈ Σ:
6. Add the edge (q,r) to G if (δ(q,a),δ(r,a)) is an edge of G.
7. For each state q, let [q] be the collection of states
[q]={r∈Q| noedgejoinsqandrinG}.
8. Form a new DFA M′ = (Q′, Σ, δ′, q0′, A′) where
Q′ ={[q]|q∈Q} (if[q]=[r],onlyoneofthemisinQ′), δ′([q],a) = [δ(q,a)] for every q ∈ Q and a ∈ Σ,
q0′ = [q0], and
A′ ={[q]|q∈A}.
9. Output ⟨M ′ ⟩.”
a. Show that M and M ′ are equivalent.
b. Show that M′ is minimal—that is, no DFA with fewer states recognizes the same language. You may use the result of Problem 1.52 without proof.
c. Show that MINIMIZE operates in polynomial time.
For a cnf-formula φ with m variables and c clauses, show that you can construct in polynomial time an NFA with O(cm) states that accepts all nonsatisfying assign- ments, represented as Boolean strings of length m. Conclude that P ̸= NP implies that NFAs cannot be minimized in polynomial time.
A 2cnf-formula is an AND of clauses, where each clause is an OR of at most two literals. Let 2SAT = {⟨φ⟩| φ is a satisfiable 2cnf-formula}. Show that 2SAT ∈ P.
7.43
⋆ 7.44
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328
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7.45 7.46
7.47
Modify the algorithm for context-free language recognition in the proof of The- orem 7.16 to give a polynomial time algorithm that produces a parse tree for a string, given the string and a CFG, if that grammar generates the string.
Say that two Boolean formulas are equivalent if they have the same set of variables and are true on the same set of assignments to those variables (i.e., they describe the same Boolean function). A Boolean formula is minimal if no shorter Boolean formula is equivalent to it. Let MIN-FORMULA be the collection of minimal Boolean formulas. Show that if P = NP, then MIN-FORMULA ∈ P.
The difference hierarchy DiP is defined recursively as a. D1P = NP and
b. DiP={A|A=B\CforBinNPandCinDi−1P}. (Here B \ C = B ∩ C.)
For example, a language in D2P is the difference of two NP languages. Sometimes D2P is called DP (and may be written DP). Let
Z={⟨G1,k1,G2,k2⟩|G1 hasak1-cliqueandG2 doesn’thaveak2-clique}. Show that Z is complete for DP. In other words, show that Z is in DP and every
language in DP is polynomial time reducible to Z.
Let MAX-CLIQUE = {⟨G, k⟩| a largest clique in G is of size exactly k}. Use the
result of Problem 7.47 to show that MAX-CLIQUE is DP-complete.
Let f : N −→ N be any function where f (n) = o(n log n). Show that TIME(f (n))
contains only the regular languages.
Call a regular expression star-free if it does not contain any star operations. Then, let EQSF−REX = {⟨R,S⟩| R and S are equivalent star-free regular expressions}. Show that EQSF−REX is in coNP. Why does your argument fail for general regular expressions?
This problem investigates resolution, a method for proving the unsatisfiability of cnf-formulas.Letφ=C1∧C2∧···∧Cm beaformulaincnf,wheretheCi areits clauses. Let C = {Ci| Ci is a clause of φ}. In a resolution step, we take two clauses Ca and Cb in C, which both have some variable x occurring positively in one of the clauses and negatively in the other. Thus, Ca = (x ∨ y1 ∨ y2 ∨ ··· ∨ yk) and Cb =(x∨z1∨z2∨···∨zl),wheretheyiandziareliterals.Weformthenew clause(y1 ∨y2 ∨···∨yk ∨ z1 ∨z2 ∨···∨zl)andremoverepeatedliterals.Add this new clause to C. Repeat the resolution steps until no additional clauses can be obtained. If the empty clause ( ) is in C, then declare φ unsatisfiable.
Say that resolution is sound if it never declares satisfiable formulas to be unsatisfi- able. Say that resolution is complete if all unsatisfiable formulas are declared to be unsatisfiable.
a. Show that resolution is sound and complete.
b. Use part (a) to show that 2SAT ∈ P.
Show that P is closed under homomorphism iff P = NP.
Let A ⊆ 1∗ be any unary language. Show that if A is NP-complete, then P = NP. (Hint: Consider a polynomial time reduction f from SAT to A. For a formula φ, let φ0100 be the reduced formula where variables x1, x2, x3, and x4 in φ are set to the values 0, 1, 0, and 0, respectively. What happens when you apply f to all of these exponentially many reduced formulas?)
⋆ 7.48 ⋆ 7.49 ⋆ 7.50
⋆ 7.51
⋆ 7.52 ⋆7.53
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7.54 In a directed graph, the indegree of a node is the number of incoming edges and the outdegree is the number of outgoing edges. Show that the following problem is NP-complete. Given an undirected graph G and a designated subset C of G’s nodes, is it possible to convert G to a directed graph by assigning directions to each of its edges so that every node in C has indegree 0 or outdegree 0, and every other node in G has indegree at least 1?
SELECTED SOLUTIONS
7.1 (c) FALSE; (d) TRUE.
7.2 (c) TRUE; (d) TRUE.
7.16 Let A ∈ NP. Construct NTM M to decide A∗ in nondeterministic polynomial time.
M = “On input w:
1. Nondeterministically divide w into pieces w = x1 x2 · · · xk .
2. For each xi, nondeterministically guess the certificates that
show xi ∈ A.
3. Verify all certificates if possible, then accept .
Otherwise, if verification fails, reject .”
7.23 We give a polynomial time mapping reduction from CLIQUE to HALF-CLIQUE. The input to the reduction is a pair ⟨G, k⟩ and the reduction produces the graph ⟨H⟩ as output where H is as follows. If G has m nodes and k = m/2, then H =G. If k < m/2, then H is the graph obtained from G by adding j nodes, each con- nected to every one of the original nodes and to each other, where j = m − 2k. Thus, H has m + j = 2m − 2k nodes. Observe that G has a k-clique iff H has a cliqueofsizek+j =m−k,andso⟨G,k⟩∈CLIQUE iff ⟨H⟩∈HALF-CLIQUE. If k > m/2, then H is the graph obtained by adding j nodes to G without any additional edges, where j = 2k − m. Thus, H has m + j = 2k nodes, and so G has a k-clique iff H has a clique of size k. Therefore, ⟨G, k⟩ ∈ CLIQUE iff ⟨H⟩ ∈ HALF-CLIQUE. We also need to show HALF-CLIQUE ∈ NP. The certifi- cate is simply the clique.
7.33 First, SOLITAIRE ∈ NP because we can verify that a solution works, in polynomial time. Second, we show that 3SAT ≤P SOLITAIRE. Given φ with m variables x1,…,xm and k clauses c1,…,ck, construct the following k × m game G. We assume that φ has no clauses that contain both xi and xi because such clauses may be removed without affecting satisfiability.
If xi is in clause cj, put a blue stone in row cj, column xi. If xi is in clause cj, put a red stone in row cj , column xi . We can make the board square by repeating a row or adding a blank column as necessary without affecting solvability. We show that φ is satisfiable iff G has a solution.
(→) Take a satisfying assignment. If xi is true (false), remove the red (blue) stones from the corresponding column. So stones corresponding to true literals remain. Because every clause has a true literal, every row has a stone.
(←) Take a game solution. If the red (blue) stones were removed from a column, set the corresponding variable true (false). Every row has a stone remaining, so every clause has a true literal. Therefore, φ is satisfied.
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SELECTED SOLUTIONS 329
330
CHAPTER 7 / TIME COMPLEXITY
7.40
If you assume that P = NP, then CLIQUE ∈ P, and you can test whether G con- tains a clique of size k in polynomial time, for any value of k. By testing whether G contains a clique of each size, from 1 to the number of nodes in G, you can deter- mine the size t of a maximum clique in G in polynomial time. Once you know t, you can find a clique with t nodes as follows. For each node x of G, remove x and calculate the resulting maximum clique size. If the resulting size decreases, replace x and continue with the next node. If the resulting size is still t, keep x perma- nently removed and continue with the next node. When you have considered all nodes in this way, the remaining nodes are a t-clique.
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8
SPACE COMPLEXITY
In this chapter, we consider the complexity of computational problems in terms of the amount of space, or memory, that they require. Time and space are two of the most important considerations when we seek practical solutions to many computational problems. Space complexity shares many of the features of time complexity and serves as a further way of classifying problems according to their computational difficulty.
As we did with time complexity, we need to select a model for measuring the space used by an algorithm. We continue with the Turing machine model for the same reason that we used it to measure time. Turing machines are mathe- matically simple and close enough to real computers to give meaningful results.
DEFINITION 8.1
Let M be a deterministic Turing machine that halts on all inputs. The space complexity of M is the function f : N −→ N , where f (n) is the maximum number of tape cells that M scans on any input of length n. If the space complexity of M is f(n), we also say that M runs in space f (n).
If M is a nondeterministic Turing machine wherein all branches halt on all inputs, we define its space complexity f(n) to be the maximum number of tape cells that M scans on any branch of its computation for any input of length n.
331
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332 CHAPTER 8 / SPACE COMPLEXITY
We typically estimate the space complexity of Turing machines by using
asymptotic notation.
DEFINITION 8.2
Let f : N −→ R+ be a function. The space complexity classes,
SPACE(f(n)) and NSPACE(f(n)), are defined as follows.
SPACE(f (n)) = {L| L is a language decided by an O(f (n)) space deterministic Turing machine}.
NSPACE(f (n)) = {L| L is a language decided by an O(f (n)) space nondeterministic Turing machine}.
EXAMPLE 8.3
In Chapter 7, we introduced the NP-complete problem SAT. Here, we show that SAT can be solved with a linear space algorithm. We believe that SAT cannot be solved with a polynomial time algorithm, much less with a linear time algorithm, because SAT is NP-complete. Space appears to be more powerful than time because space can be reused, whereas time cannot.
M1 = “On input ⟨φ⟩, where φ is a Boolean formula:
1. 2. 3.
For each truth assignment to the variables x1 , . . . , xm of φ: Evaluate φ on that truth assignment.
If φ ever evaluated to 1, accept; if not, reject.”
Machine M1 clearly runs in linear space because each iteration of the loop can reuse the same portion of the tape. The machine needs to store only the current truth assignment, and that can be done with O(m) space. The number of variables m is at most n, the length of the input, so this machine runs in space O(n).
EXAMPLE 8.4
Here, we illustrate the nondeterministic space complexity of a language. In the next section, we show how determining the nondeterministic space complex- ity can be useful in determining its deterministic space complexity. Consider
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8.1 SAVITCH’S THEOREM 333 the problem of testing whether a nondeterministic finite automaton accepts all
strings. Let
ALLNFA = {⟨A⟩| A is an NFA and L(A) = Σ∗}.
We give a nondeterministic linear space algorithm that decides the complement of this language, ALLNFA. The idea behind this algorithm is to use nondeter- minism to guess a string that is rejected by the NFA, and to use linear space to keep track of which states the NFA could be in at a particular time. Note that this language is not known to be in NP or in coNP.
N = “On input ⟨M⟩, where M is an NFA:
1. Place a marker on the start state of the NFA.
2. Repeat 2q times, where q is the number of states of M :
3. Nondeterministically select an input symbol and change the
positions of the markers on M’s states to simulate reading
that symbol.
4. Accept if stages 2 and 3 reveal some string that M rejects; that
is, if at some point none of the markers lie on accept states of M. Otherwise, reject.”
If M rejects any strings, it must reject one of length at most 2q because in any longer string that is rejected, the locations of the markers described in the preceding algorithm would repeat. The section of the string between the rep- etitions can be removed to obtain a shorter rejected string. Hence N decides ALLNFA. (Note that N accepts improperly formed inputs, too.)
The only space needed by this algorithm is for storing the location of the markers and the repeat loop counter, and that can be done with linear space. Hence the algorithm runs in nondeterministic space O(n). Next, we prove a theorem that provides information about the deterministic space complexity of ALLNFA .
8.1
SAVITCH’S THEOREM
Savitch’s theorem is one of the earliest results concerning space complexity. It shows that deterministic machines can simulate nondeterministic machines by using a surprisingly small amount of space. For time complexity, such a simu- lation seems to require an exponential increase in time. For space complexity, Savitch’s theorem shows that any nondeterministic TM that uses f(n) space can be converted to a deterministic TM that uses only f2(n) space.
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334 CHAPTER 8 / SPACE COMPLEXITY
THEOREM 8.5
Savitch’s theorem For any1 function f : N −→ R+ , where f (n) ≥ n,
NSPACE(f(n)) ⊆ SPACE(f2(n)).
PROOF IDEA We need to simulate an f(n) space NTM deterministically. A naive approach is to proceed by trying all the branches of the NTM’s computation, one by one. The simulation needs to keep track of which branch it is currently trying so that it is able to go on to the next one. But a branch that uses f(n) space may run for 2O(f (n)) steps and each step may be a nondeterministic choice. Exploring the branches sequentially would require recording all the choices used on a particular branch in order to be able to find the next branch. Therefore, this approach may use 2O(f(n)) space, exceeding our goal of O(f2(n)) space.
Instead, we take a different approach by considering the following more gen- eral problem. We are given two configurations of the NTM, c1 and c2, together with a number t, and we test whether the NTM can get from c1 to c2 within t steps using only f(n) space. We call this problem the yieldability problem. By solv- ing the yieldability problem, where c1 is the start configuration, c2 is the accept configuration, and t is the maximum number of steps that the nondeterministic machine can use, we can determine whether the machine accepts its input.
We give a deterministic, recursive algorithm that solves the yieldability prob- lem. It operates by searching for an intermediate configuration cm, and recur- sively testing whether (1) c1 can get to cm within t/2 steps, and (2) whether cm can get to c2 within t/2 steps. Reusing the space for each of the two recursive tests allows a significant savings of space.
This algorithm needs space for storing the recursion stack. Each level of the recursion uses O(f(n)) space to store a configuration. The depth of the recur- sion is log t, where t is the maximum time that the nondeterministic machine may use on any branch. We have t = 2O(f(n)), so logt = O(f(n)). Hence the deterministic simulation uses O(f2(n)) space.
PROOF Let N be an NTM deciding a language A in space f (n). We construct a deterministic TM M deciding A. Machine M uses the procedure CANYIELD, which tests whether one of N’s configurations can yield another within a speci- fied number of steps. This procedure solves the yieldability problem described in the proof idea.
Let w be a string considered as input to N. For configurations c1 and c2 of N, and integer t, CANYIELD(c1,c2,t) outputs accept if N can go from config- uration c1 to configuration c2 in t or fewer steps along some nondeterministic
1On page 351, we show that Savitch’s theorem also holds whenever f (n) ≥ log n.
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8.1 SAVITCH’S THEOREM 335 path. If not, CANYIELD outputs reject. For convenience, we assume that t is a
power of 2.
CANYIELD = “On input c1, c2, and t:
1.
2. 3. 4. 5. 6.
If t = 1, then test directly whether c1 = c2 or whether c1 yields c2 in one step according to the rules of N . Accept if either test succeeds; reject if both fail.
If t > 1, then for each configuration cm of N using space f (n):
Run CANYIELD(c1, cm, t ). 2
Run CANYIELD(cm, c2, t ). 2
If steps 3 and 4 both accept, then accept. If haven’t yet accepted, reject .”
Now we define M to simulate N as follows. We first modify N so that when it accepts, it clears its tape and moves the head to the leftmost cell—thereby entering a configuration called caccept. We let cstart be the start configuration of N o n w . W e s e l e c t a c o n s t a n t d s o t h a t N h a s n o m o r e t h a n 2 df ( n ) c o n fi g u r a t i o n s using f(n) tape, where n is the length of w. Then we know that 2df(n) provides an upper bound on the running time of any branch of N on w.
M = “On input w:
1. Output the result of CANYIELD(cstart,caccept,2df(n)).”
Algorithm CANYIELD obviously solves the yieldability problem, and hence M correctly simulates N. We need to analyze it to verify that M works within O(f2(n)) space.
Whenever CANYIELD invokes itself recursively, it stores the current stage number and the values of c1, c2, and t on a stack so that these values may be restored upon return from the recursive invocation. Each level of the recursion thus uses O(f(n)) additional space. Furthermore, each level of the recursion divides the size of t in half. Initially t starts out equal to 2df(n), so the depth of the recursion is O(log2df(n)) or O(f(n)). Therefore, the total space used is O(f2(n)), as claimed.
One technical difficulty arises in this argument because algorithm M needs to know the value of f(n) when it calls CANYIELD. We can handle this difficulty by modifying M so that it tries f(n) = 1,2,3,… . For each value f(n) = i, the modified algorithm uses CANYIELD to determine whether the accept configu- ration is reachable. In addition, it uses CANYIELD to determine whether N uses at least space i + 1 by testing whether N can reach any of the configurations of length i + 1 from the start configuration. If the accept configuration is reachable, M accepts; if no configuration of length i + 1 is reachable, M rejects; and other- wise, M continues with f (n) = i + 1. (We could have handled this difficulty in another way by assuming that M can compute f(n) within O(f(n)) space, but then we would need to add that assumption to the statement of the theorem.)
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336 CHAPTER 8 / SPACE COMPLEXITY 8.2
THE CLASS PSPACE
By analogy with the class P, we define the class PSPACE for space complexity.
DEFINITION 8.6
PSPACE is the class of languages that are decidable in polynomial
space on a deterministic Turing machine. In other words,
PSPACE = SPACE(nk ). k
We define NPSPACE, the nondeterministic counterpart to PSPACE, in terms of the NSPACE classes. However, PSPACE = NPSPACE by virtue of Savitch’s theorem because the square of any polynomial is still a polynomial.
In Examples 8.3 and 8.4, we showed that SAT is in SPACE(n) and that ALLNFA is in coNSPACE(n) and hence, by Savitch’s theorem, in SPACE(n2) because the deterministic space complexity classes are closed under complement. Therefore, both languages are in PSPACE.
Let’s examine the relationship of PSPACE with P and NP. We observe that P ⊆ PSPACE because a machine that runs quickly cannot use a great deal of space. More precisely, for t(n) ≥ n, any machine that operates in time t(n) can use at most t(n) space because a machine can explore at most one new cell at each step of its computation. Similarly, NP ⊆ NPSPACE, and so NP ⊆ PSPACE.
Conversely, we can bound the time complexity of a Turing machine in terms
of its space complexity. For f (n) ≥ n, a TM that uses f (n) space can have at most
f (n) 2O(f (n)) different configurations, by a simple generalization of the proof of
Lemma 5.8 on page 222. A TM computation that halts may not repeat a configu- 2 O(f(n))
ration. Therefore, a TM that uses space f (n) must run in time f (n) 2 , so k
PSPACE ⊆ EXPTIME = k TIME(2n ).
We summarize our knowledge of the relationships among the complexity
classes defined so far in the series of containments
P ⊆ NP ⊆ PSPACE = NPSPACE ⊆ EXPTIME.
We don’t know whether any of these containments is actually an equality. Someone may yet discover a simulation like the one in Savitch’s theorem that merges some of these classes into the same class. However, in Chapter 9 we prove that P ̸= EXPTIME. Therefore, at least one of the preceding contain- ments is proper, but we are unable to say which! Indeed, most researchers
2The requirement here that f(n) ≥ n is generalized later to f(n) ≥ logn when we introduce TMs that use sublinear space on page 350.
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8.3 PSPACE-COMPLETENESS 337 believe that all the containments are proper. The following diagram depicts
the relationships among these classes, assuming that all are different.
FIGURE 8.7
Conjectured relationships among P, NP, PSPACE, and EXPTIME
8.3
PSPACE-COMPLETENESS
In Section 7.4, we introduced the category of NP-complete languages as rep- resenting the most difficult languages in NP. Demonstrating that a language is NP-complete provides strong evidence that the language is not in P. If it were, P and NP would be equal. In this section, we introduce the analogous notion PSPACE-completeness for the class PSPACE.
DEFINITION 8.8
A language B is PSPACE-complete if it satisfies two conditions:
1. B is in PSPACE, and
2. every A in PSPACE is polynomial time reducible to B.
If B merely satisfies condition 2, we say that it is PSPACE-hard.
In defining PSPACE-completeness, we use polynomial time reducibility as given in Definition 7.29. Why don’t we define a notion of polynomial space reducibility and use that instead of polynomial time reducibility? To understand the answer to this important question, consider our motivation for defining com- plete problems in the first place.
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338 CHAPTER 8 / SPACE COMPLEXITY
Complete problems are important because they are examples of the most difficult problems in a complexity class. A complete problem is most difficult because any other problem in the class is easily reduced into it. So if we find an easy way to solve the complete problem, we can easily solve all other problems in the class. The reduction must be easy, relative to the complexity of typical prob- lems in the class, for this reasoning to apply. If the reduction itself were difficult to compute, an easy solution to the complete problem wouldn’t necessarily yield an easy solution to the problems reducing to it.
Therefore, the rule is: Whenever we define complete problems for a com- plexity class, the reduction model must be more limited than the model used for defining the class itself.
THE TQBF PROBLEM
Our first example of a PSPACE-complete problem involves a generalization of the satisfiability problem. Recall that a Boolean formula is an expression that contains Boolean variables, the constants 0 and 1, and the Boolean operations ∧, ∨, and ¬. We now introduce a more general type of Boolean formula.
The quantifiers ∀ (for all) and ∃ (there exists) make frequent appearances in mathematical statements. Writing the statement ∀x φ means that for every value for the variable x, the statement φ is true. Similarly, writing the statement ∃x φ means that for some value of the variable x, the statement φ is true. Sometimes, ∀ is referred to as the universal quantifier and ∃ as the existential quantifier. We say that the variable x immediately following the quantifier is bound to the quantifier.
For example, considering the natural numbers, the statement ∀x [x + 1 > x] means that the successor x + 1 of every natural number x is greater than the number itself. Obviously, this statement is true. However, the statement ∃y [y + y = 3] obviously is false. When interpreting the meaning of statements involving quantifiers, we must consider the universe from which the values are drawn. In the preceding cases, the universe comprised the natural numbers; but if we took the real numbers instead, the existentially quantified statement would become true.
Statements may contain several quantifiers, as in ∀x ∃y [y > x]. For the uni- verse of the natural numbers, this statement says that every natural number has another natural number larger than it. The order of the quantifiers is impor- tant. Reversing the order, as in the statement ∃y ∀x [y > x], gives an entirely different meaning—namely, that some natural number is greater than all others. Obviously, the first statement is true and the second statement is false.
A quantifier may appear anywhere in a mathematical statement. It applies to the fragment of the statement appearing within the matched pair of parentheses or brackets following the quantified variable. This fragment is called the scope of the quantifier. Often, it is convenient to require that all quantifiers appear at the beginning of the statement and that each quantifier’s scope is everything fol- lowing it. Such statements are said to be in prenex normal form. Any statement
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8.3 PSPACE-COMPLETENESS 339
may be put into prenex normal form easily. We consider statements in this form only, unless otherwise indicated.
Boolean formulas with quantifiers are called quantified Boolean formulas. For such formulas, the universe is {0, 1}. For example,
φ = ∀x ∃y (x ∨ y) ∧ (
is a quantified Boolean formula. Here, φ is true, but it would be false if the quantifiers ∀x and ∃y were reversed.
When each variable of a formula appears within the scope of some quantifier, the formula is said to be fully quantified. A fully quantified Boolean formula is sometimes called a sentence and is always either true or false. For example, the preceding formula φ is fully quantified. However, if the initial part, ∀x, of φ were removed, the formula would no longer be fully quantified and would be neither true nor false.
The TQBF problem is to determine whether a fully quantified Boolean for- mula is true or false. We define the language
TQBF = {⟨φ⟩| φ is a true fully quantified Boolean formula}. THEOREM 8.9
TQBF is PSPACE-complete.
PROOF IDEA To show that TQBF is in PSPACE, we give a straightforward algorithm that assigns values to the variables and recursively evaluates the truth of the formula for those values. From that information, the algorithm can deter- mine the truth of the original quantified formula.
To show that every language A in PSPACE reduces to TQBF in polynomial time, we begin with a polynomial space-bounded Turing machine for A. Then we give a polynomial time reduction that maps a string to a quantified Boolean formula φ that encodes a simulation of the machine on that input. The formula is true iff the machine accepts.
As a first attempt at this construction, let’s try to imitate the proof of the Cook–Levin theorem, Theorem 7.37. We can construct a formula φ that simu- lates M on an input w by expressing the requirements for an accepting tableau. A tableau for M on w has width O(nk), the space used by M, but its height is exponential in nk because M can run for exponential time. Thus, if we were to represent the tableau with a formula directly, we would end up with a formula of exponential size. However, a polynomial time reduction cannot produce an exponential-size result, so this attempt fails to show that A ≤P TQBF.
Instead, we use a technique related to the proof of Savitch’s theorem to con- struct the formula. The formula divides the tableau into halves and employs the universal quantifier to represent each half with the same part of the formula. The result is a much shorter formula.
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x∨y)
340 CHAPTER 8 / SPACE COMPLEXITY
PROOF First, we give a polynomial space algorithm deciding TQBF.
T = “On input ⟨φ⟩, a fully quantified Boolean formula:
1. If φ contains no quantifiers, then it is an expression with only
constants, so evaluate φ and accept if it is true; otherwise, reject .
2. If φ equals ∃x ψ, recursively call T on ψ, first with 0 substituted for x and then with 1 substituted for x. If either result is accept,
then accept; otherwise, reject.
3. If φ equals ∀x ψ, recursively call T on ψ, first with 0 substituted
for x and then with 1 substituted for x. If both results are ac- cept, then accept; otherwise, reject.”
Algorithm T obviously decides TQBF. To analyze its space complexity, we observe that the depth of the recursion is at most the number of variables. At each level we need only store the value of one variable, so the total space used is O(m), where m is the number of variables that appear in φ. Therefore, T runs in linear space.
Next, we show that TQBF is PSPACE-hard. Let A be a language decided by a TM M in space nk for some constant k. We give a polynomial time reduction from A to TQBF.
The reduction maps a string w to a quantified Boolean formula φ that is true iff M accepts w. To show how to construct φ, we solve a more general problem. Using two collections of variables denoted c1 and c2 representing two configu- rations and a number t > 0, we construct a formula φc1,c2,t. If we assign c1 and c2 to actual configurations, the formula is true iff M can go from c1 to c2 in at most t steps. Then we can let φ be the formula φcstart,caccept,h, where h = 2df(n) for a constant d, chosen so that M has no more than 2df(n) possible configurations on an input of length n. Here, let f(n) = nk. For convenience, we assume that t is a power of 2.
The formula encodes the contents of configuration cells as in the proof of the Cook–Levin theorem. Each cell has several variables associated with it, one for each tape symbol and state, corresponding to the possible settings of that cell. Each configuration has nk cells and so is encoded by O(nk) variables.
If t = 1, we can easily construct φc1,c2,t. We design the formula to say that either c1 equals c2, or c2 follows from c1 in a single step of M. We express the equality by writing a Boolean expression saying that each of the variables representing c1 contains the same Boolean value as the corresponding variable representing c2. We express the second possibility by using the technique pre- sented in the proof of the Cook–Levin theorem. That is, we can express that c1 yields c2 in a single step of M by writing Boolean expressions stating that the contents of each triple of c1’s cells correctly yields the contents of the corre- sponding triple of c2’s cells.
If t > 1, we construct φc1,c2,t recursively. As a warm-up, let’s try one idea that doesn’t quite work and then fix it. Let
φc1,c2,t = ∃m1 φc1,m1, t ∧ φm1,c2, t . 22
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8.3 PSPACE-COMPLETENESS 341
The symbol m1 represents a configuration of M . Writing ∃m1 is shorthand for
∃x1,…,xl, where l = O(nk) and x1,…,xl are the variables that encode m1.
So this construction of φc1 ,c2 ,t says that M can go from c1 to c2 in at most t steps
if some intermediate configuration m1 exists, whereby M can go from c1 to m1
in at most t steps and then from m1 to c2 in at most t steps. Then we construct 22
thetwoformulasφc1,m1,t andφm1,c2,t recursively. 22
The formula φc1,c2,t has the correct value; that is, it is TRUE whenever M can go from c1 to c2 within t steps. However, it is too big. Every level of the recursion involved in the construction cuts t in half but roughly doubles the size of the formula. Hence we end up with a formula of size roughly t. Initially t = 2df(n), so this method gives an exponentially large formula.
To reduce the size of the formula, we use the ∀ quantifier in addition to the ∃ quantifier. Let
φc1,c2,t =∃m1∀(c3,c4)∈{(c1,m1),(m1,c2)}φc3,c4,t . 2
The introduction of the new variables representing the configurations c3 and c4
allows us to “fold” the two recursive subformulas into a single subformula, while
preserving the original meaning. By writing ∀(c3,c4) ∈ {(c1,m1),(m1,c2)}, we
indicate that the variables representing the configurations c3 and c4 may take the
values of the variables of c1 and m1 or of m1 and c2, respectively, and that the
∀x ∈ {y,z} […] with the equivalent construct ∀x [(x = y ∨ x = z) → …] to obtain a syntactically correct quantified Boolean formula. Recall that in Sec- tion 0.2, we showed that Boolean implication (→) and Boolean equality (=) can be expressed in terms of AND and NOT. Here, for clarity, we use the symbol = for Boolean equality instead of the equivalent symbol ↔ used in Section 0.2.
To calculate the size of the formula φcstart,caccept,h, where h = 2df(n), we note that each level of the recursion adds a portion of the formula that is linear in the size of the configurations and is thus of size O(f(n)). The number of levels of the recursion is log(2df(n)), or O(f(n)). Hence the size of the resulting formula is O(f2(n)).
WINNING STRATEGIES FOR GAMES
For the purposes of this section, a game is loosely defined to be a competition in which opposing parties attempt to achieve some goal according to prespec- ified rules. Games appear in many forms, from board games such as chess to economic and war games that model corporate or societal conflict.
Games are closely related to quantifiers. A quantified statement has a corre- sponding game; conversely, a game often has a corresponding quantified state- ment. These correspondences are helpful in several ways. For one, expressing a mathematical statement that uses many quantifiers in terms of the correspond- ing game may give insight into the statement’s meaning. For another, expressing a game in terms of a quantified statement aids in understanding the complexity of the game. To illustrate the correspondence between games and quantifiers, we turn to an artificial game called the formula game.
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resulting formula φc3,c4, t is true in either case. We may replace the construct 2
342 CHAPTER 8 / SPACE COMPLEXITY
Letφ=∃x1∀x2∃x3 ···Qxk [ψ]beaquantifiedBooleanformulainprenex normal form. Here, Q represents either a ∀ or an ∃ quantifier. We associate a game with φ as follows. Two players, called Player A and Player E, take turns selecting the values of the variables x1,…,xk. Player A selects values for the variables that are bound to ∀ quantifiers, and Player E selects values for the variables that are bound to ∃ quantifiers. The order of play is the same as that of the quantifiers at the beginning of the formula. At the end of play, we use the values that the players have selected for the variables and declare that Player E has won the game if ψ, the part of the formula with the quantifiers stripped off, is now TRUE. Player A has won if ψ is now FALSE.
EXAMPLE 8.10
Say that φ1 is the formula
∃x1 ∀x2 ∃x3 (x1 ∨x2)∧(x2 ∨x3)∧(
In the formula game for φ1, Player E picks the value of x1, then Player A picks the value of x2, and finally Player E picks the value of x3.
Toillustrateasampleplayofthisgame,webeginbyrepresentingtheBoolean value TRUE with 1 and FALSE with 0, as usual. Let’s say that Player E picks x1 = 1, then Player A picks x2 = 0, and finally Player E picks x3 = 1. With these values for x1, x2, and x3, the subformula
(x1 ∨x2)∧(x2 ∨x3)∧(
is 1, so Player E has won the game. In fact, Player E may always win this game by selecting x1 = 1 and then selecting x3 to be the negation of whatever Player A selects for x2. We say that Player E has a winning strategy for this game. A player has a winning strategy for a game if that player wins when both sides play optimally.
Now let’s change the formula slightly to get a game in which Player A has a winning strategy. Let φ2 be the formula
∃x1 ∀x2 ∃x3 (x1 ∨x2)∧(x2 ∨x3)∧(x2 ∨x3).
Player A now has a winning strategy because no matter what Player E selects for x1, Player A may select x2 = 0, thereby falsifying the part of the formula appearing after the quantifiers, whatever Player E’s last move may be.
We next consider the problem of determining which player has a winning strategy in the formula game associated with a particular formula. Let
FORMULA-GAME = {⟨φ⟩| Player E has a winning strategy in the formula game associated with φ}.
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x2 ∨x3)
x2 ∨ x3).
8.3 PSPACE-COMPLETENESS 343
THEOREM 8.11
FORMULA-GAME is PSPACE-complete.
PROOF IDEA FORMULA-GAME is PSPACE-complete for a simple reason. It is the same as TQBF. To see that FORMULA-GAME = TQBF, observe that a formula is TRUE exactly when Player E has a winning strategy in the associated formula game. The two statements are different ways of saying the same thing.
PROOF The formula φ = ∃x1 ∀x2 ∃x3 ··· [ψ] is TRUE when some setting for x1 exists such that for any setting of x2, a setting of x3 exists such that, and so on …, where ψ is TRUE under the settings of the variables. Similarly, Player E has a winning strategy in the game associated with φ when Player E can make some assignment to x1 such that for any setting of x2, Player E can make an assignment to x3 such that, and so on . . . , where ψ is TRUE under these settings of the variables.
The same reasoning applies when the formula doesn’t alternate between ex- istential and universal quantifiers. If φ has the form ∀x1 , x2 , x3 ∃x4 , x5 ∀x6 [ ψ ], Player A would make the first three moves in the formula game to assign values to x1, x2, and x3; then Player E would make two moves to assign x4 and x5; and finally Player A would assign a value x6.
Hence φ ∈ TQBF exactly when φ ∈ FORMULA-GAME, and the theorem follows from Theorem 8.9.
GENERALIZED GEOGRAPHY
Now that we know that the formula game is PSPACE-complete, we can es- tablish the PSPACE-completeness or PSPACE-hardness of some other games more easily. We’ll begin with a generalization of the game geography and later discuss games such as chess, checkers, and GO.
Geography is a child’s game in which players take turns naming cities from anywhere in the world. Each city chosen must begin with the same letter that ended the previous city’s name. Repetition isn’t permitted. The game starts with some designated starting city and ends when some player loses because he or she is unable to continue. For example, if the game starts with Peoria, then Amherst might legally follow (because Peoria ends with the letter a, and Amherst begins with the letter a), then Tucson, then Nashua, and so on until one player gets stuck and thereby loses.
We can model this game with a directed graph whose nodes are the cities of the world. We draw an arrow from one city to another if the first can lead to the second according to the game rules. In other words, the graph contains an edge from a city X to a city Y if city X ends with the same letter that begins city Y. We illustrate a portion of the geography graph in Figure 8.12.
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344 CHAPTER 8 / SPACE COMPLEXITY
FIGURE 8.12
Portion of the graph representing the geography game
When the rules of geography are interpreted for this graphic representation, one player starts by selecting the designated start node and then the players take turns picking nodes that form a simple path in the graph. The requirement that the path be simple (i.e., doesn’t use any node more than once) corresponds to the requirement that a city may not be repeated. The first player unable to extend the path loses the game.
In generalized geography, we take an arbitrary directed graph with a des- ignated start node instead of the graph associated with the actual cities. For example, the following graph is an example of a generalized geography game.
FIGURE 8.13
A sample generalized geography game
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8.3 PSPACE-COMPLETENESS 345
Say that Player I is the one who moves first and Player II second. In this example, Player I has a winning strategy as follows. Player I starts at node 1, the designated start node. Node 1 points only at nodes 2 and 3, so Player I’s first move must be one of these two choices. He chooses 3. Now Player II must move, but node 3 points only to node 5, so she is forced to select node 5. Then Player I selects 6, from choices 6, 7, and 8. Now Player II must play from node 6, but it points only to node 3, and 3 was previously played. Player II is stuck and thus Player I wins.
If we change the example by reversing the direction of the edge between nodes 3 and 6, Player II has a winning strategy. Can you see it? If Player I starts out with node 3 as before, Player II responds with 6 and wins immediately, so Player I’s only hope is to begin with 2. In that case, however, Player II responds with 4. If Player I now takes 5, Player II wins with 6. If Player I takes 7, Player II wins with 9. No matter what Player I does, Player II can find a way to win, so Player II has a winning strategy.
The problem of determining which player has a winning strategy in a gener- alized geography game is PSPACE-complete. Let
GG = {⟨G, b⟩| Player I has a winning strategy for the generalized geography game played on graph G starting at node b}.
THEOREM 8.14
GG is PSPACE-complete.
PROOF IDEA A recursive algorithm similar to the one used for TQBF in Theorem 8.9 determines which player has a winning strategy. This algorithm runs in polynomial space and so GG ∈ PSPACE.
To prove that GG is PSPACE-hard, we give a polynomial time reduction from FORMULA-GAME to GG. This reduction converts a formula game to a generalized geography graph so that play on the graph mimics play in the formula game. In effect, the players in the generalized geography game are really playing an encoded form of the formula game.
PROOF The following algorithm decides whether Player I has a winning strategy in instances of generalized geography; in other words, it decides GG. We show that it runs in polynomial space.
M = “On input ⟨G,b⟩, where G is a directed graph and b is a node of G:
1. If b has outdegree 0, reject because Player I loses immediately.
2. Remove node b and all connected arrows to get a new graph G′.
3. For each of the nodes b1, b2, . . . , bk that b originally pointed at,
recursively call M on ⟨G′ , bi ⟩.
4. If all of these accept, Player II has a winning strategy in the
original game, so reject. Otherwise, Player II doesn’t have a winning strategy, so Player I must; therefore, accept.”
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346 CHAPTER 8 / SPACE COMPLEXITY
The only space required by this algorithm is for storing the recursion stack. Each level of the recursion adds a single node to the stack, and at most m levels occur, where m is the number of nodes in G. Hence the algorithm runs in linear space.
To establish the PSPACE-hardness of GG, we show that FORMULA-GAME is polynomial time reducible to GG. The reduction maps the formula
φ = ∃x1 ∀x2 ∃x3 · · · Qxk [ ψ ]
to an instance ⟨G, b⟩ of generalized geography. Here we assume for simplicity that φ’s quantifiers begin and end with ∃, and that they strictly alternate between ∃ and ∀. A formula that doesn’t conform to this assumption may be converted to a slightly larger one that does by adding extra quantifiers binding otherwise unused or “dummy” variables. We assume also that ψ is in conjunctive normal form (see Problem 8.12).
The reduction constructs a geography game on a graph G where optimal play mimics optimal play of the formula game on φ. Player I in the geography game takes the role of Player E in the formula game, and Player II takes the role of Player A.
The structure of graph G is partially shown in the following figure. Play starts at node b, which appears at the top left-hand side of G. Underneath b, a sequence of diamond structures appears, one for each of the variables of φ. Before getting to the right-hand side of G, let’s see how play proceeds on the left-hand side.
FIGURE 8.15
Partial structure of the geography game simulating the formula game
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8.3 PSPACE-COMPLETENESS 347
Play starts at b. Player I must select one of the two edges going from b. These edges correspond to Player E’s possible choices at the beginning of the formula game. The left-hand choice for Player I corresponds to TRUE for Player E in the formula game and the right-hand choice to FALSE. After Player I has selected one of these edges—say, the left-hand one—Player II moves. Only one outgoing edge is present, so this move is forced. Similarly, Player I’s next move is forced and play continues from the top of the second diamond. Now two edges again are present, but Player II gets the choice. This choice corresponds to Player A’s first move in the formula game. As play continues in this way, Players I and II choose a rightward or leftward path through each of the diamonds.
After play passes through all the diamonds, the head of the path is at the bottom node in the last diamond, and it is Player I’s turn because we assumed that the last quantifier is ∃. Player I’s next move is forced. Then they are at node c in Figure 8.15 and Player II makes the next move.
This point in the geography game corresponds to the end of play in the formula game. The chosen path through the diamonds corresponds to an as- signment to φ’s variables. Under that assignment, if ψ is TRUE, Player E wins the formula game; and if ψ is FALSE, Player A wins. The structure on the right- hand side of the following figure guarantees that Player I can win if Player E has won, and that Player II can win if Player A has won.
FIGURE 8.16
Full structure of the geography game simulating the formula game, where
φ=∃x1∀x2 ···∃xk[(x1∨x2∨x3)∧(
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x2 ∨x3 ∨···)∧ ··· ∧( )]
348 CHAPTER 8 / SPACE COMPLEXITY
At node c, Player II may choose a node corresponding to one of ψ’s clauses. Then Player I may choose a node corresponding to a literal in that clause. The nodes corresponding to unnegated literals are connected to the left-hand (TRUE) sides of the diamond for associated variables, and similarly for negated literals and right-hand (FALSE) sides as shown in Figure 8.16.
If ψ is FALSE, Player II may win by selecting the unsatisfied clause. Any literal that Player I may then pick is FALSE and is connected to the side of the diamond that hasn’t yet been played. Thus Player II may play the node in the diamond, but then Player I is unable to move and loses. If ψ is TRUE, any clause that Player II picks contains a TRUE literal. Player I selects that literal after Player II’s move. Because the literal is TRUE, it is connected to the side of the diamond that has already been played, so Player II is unable to move and loses.
In Theorem 8.14, we showed that no polynomial time algorithm exists for optimal play in generalized geography unless P = PSPACE. We’d like to prove a similar theorem regarding the difficulty of computing optimal play in board games such as chess, but an obstacle arises. Only a finite number of different game positions may occur on the standard 8 × 8 chess board. In principle, all these positions may be placed in a table, along with a best move in each position. The table would be too large to fit inside our galaxy but, being finite, could be stored in the control of a Turing machine (or even that of a finite automaton!). Thus, the machine would be able to play optimally in linear time, using table lookup. Perhaps at some time in the future, methods that can quantify the com- plexity of finite problems will be developed. But current methods are asymptotic and hence apply only to the rate of growth of the complexity as the problem size increases—not to any fixed size. Nevertheless, we can give some evidence for the difficulty of computing optimal play for many board games by generaliz- ing them to an n × n board. Such generalizations of chess, checkers, and GO have been shown to be PSPACE-hard or hard for even larger complexity classes, depending on the details of the generalization.
8.4
THE CLASSES L AND NL
Until now, we have considered only time and space complexity bounds that are at least linear—that is, bounds where f (n) is at least n. Now we examine smaller, sublinear space bounds. In time complexity, sublinear bounds are insufficient for reading the entire input, so we don’t consider them here. In sublinear space com- plexity, the machine is able to read the entire input but it doesn’t have enough space to store the input. To consider this situation meaningfully, we must modify our computational model.
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8.4 THE CLASSES L AND NL 349
We introduce a Turing machine with two tapes: a read-only input tape, and a read/write work tape. On the read-only tape, the input head can detect symbols but not change them. We provide a way for the machine to detect when the head is at the left-hand and right-hand ends of the input. The input head must remain on the portion of the tape containing the input. The work tape may be read and written in the usual way. Only the cells scanned on the work tape contribute to the space complexity of this type of Turing machine.
Think of a read-only input tape as a CD-ROM, a device used for input on many personal computers. Often, the CD-ROM contains more data than the computer can store in its main memory. Sublinear space algorithms allow the computer to manipulate the data without storing all of it in main memory.
For space bounds that are at least linear, the two-tape TM model is equivalent to the standard one-tape model (see Exercise 8.1). For sublinear space bounds, we use only the two-tape model.
DEFINITION 8.17
L is the class of languages that are decidable in logarithmic space
on a deterministic Turing machine. In other words,
L = SPACE(log n).
NL is the class of languages that are decidable in logarithmic space
on a nondeterministic Turing machine. In other words,
NL = NSPACE(log n).
We focus on log n space instead of, say, √n or log2 n space, for several rea- sons that are similar to those for our selection of polynomial time and space bounds. Logarithmic space is just large enough to solve a number of interesting computational problems, and it has attractive mathematical properties such as robustness even when the machine model and input encoding method change. Pointers into the input may be represented in logarithmic space, so one way to think about the power of log space algorithms is to consider the power of a fixed number of input pointers.
EXAMPLE 8.18
The language A = {0k1k| k ≥ 0} is a member of L. In Section 7.1, on page 275 we described a Turing machine that decides A by zig-zagging back and forth across the input, crossing off the 0s and 1s as they are matched. That algorithm uses linear space to record which positions have been crossed off, but it can be modified to use only log space.
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350 CHAPTER 8 / SPACE COMPLEXITY
The log space TM for A cannot cross off the 0s and 1s that have been matched on the input tape because that tape is read-only. Instead, the machine counts the number of 0s and, separately, the number of 1s in binary on the work tape. The only space required is that used to record the two counters. In binary, each counter uses only logarithmic space and hence the algorithm runs in O(log n) space. Therefore, A ∈ L.
EXAMPLE 8.19 Recall the language
PATH = {⟨G, s, t⟩| G is a directed graph that has a directed path from s to t}
defined in Section 7.2. Theorem 7.14 shows that PATH is in P, but that the algorithm given uses linear space. We don’t know whether PATH can be solved in logarithmic space deterministically, but we do know a nondeterministic log space algorithm for PATH.
The nondeterministic log space Turing machine deciding PATH operates by starting at node s and nondeterministically guessing the nodes of a path from s to t. The machine records only the position of the current node at each step on the work tape, not the entire path (which would exceed the logarithmic space requirement). The machine nondeterministically selects the next node from among those pointed at by the current node. It repeats this action until it reaches node t and accepts, or until it has gone on for m steps and rejects, where m is the number of nodes in the graph. Thus, PATH is in NL.
Our earlier claim that any f(n) space bounded Turing machine also runs in time 2O(f(n)) is no longer true for very small space bounds. For example, a Turing machine that uses O(1) (i.e., constant) space may run for n steps. To obtain a bound on the running time that applies for every space bound f (n), we give the following definition.
DEFINITION 8.20
If M is a Turing machine that has a separate read-only input tape and w is an input, a configuration of M on w is a setting of the state, the work tape, and the positions of the two tape heads. The input w is not a part of the configuration of M on w.
If M runs in f (n) space and w is an input of length n, the number of configu- rations of M on w is n2O(f(n)). To explain this result, let’s say that M has c states and g tape symbols. The number of strings that can appear on the work tape is
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gf(n). The input head can be in one of n positions, and the work tape head can
be in one of f (n) positions. Therefore, the total number of configurations of M
on w, which is an upper bound on the running time of M on w, is cnf(n)gf(n), or n2O(f (n)).
We focus almost exclusively on space bounds f (n) that are at least log n. Our earlier claim that the time complexity of a machine is at most exponential in its space complexity remains true for such bounds because n2O(f(n)) is 2O(f(n)) when f (n) ≥ log n.
Recall that Savitch’s theorem shows that we can convert nondeterministic TM s to deterministic TM s and increase the space complexity f (n) by only a squaring, provided that f(n) ≥ n. We can extend Savitch’s theorem to hold for sublinear space bounds down to f (n) ≥ log n. The proof is identical to the original one we gave on page 334, except that we use Turing machines with a read-only input tape; and instead of referring to configurations of N , we refer to configurations of N on w. Storing a configuration of N on w uses log(n2O(f(n))) = logn + O(f(n)) space. If f(n) ≥ logn, the storage used is O(f(n)) and the remainder of the proof remains the same.
8.5
NL-COMPLETENESS
As we mentioned in Example 8.19, the PATH problem is known to be in NL but isn’t known to be in L. We believe that PATH doesn’t belong to L, but we don’t know how to prove this conjecture. In fact, we don’t know of any problem in NL that can be proven to be outside L. Analogous to the question of whether P = NP, we have the question of whether L = NL.
As a step toward resolving the L versus NL question, we can exhibit certain languages that are NL-complete. As with complete languages for other com- plexity classes, the NL-complete languages are examples of languages that are, in a certain sense, the most difficult languages in NL. If L and NL are different, all NL-complete languages don’t belong to L.
As with our previous definitions of completeness, we define an NL-complete language to be one that is in NL and to which any other language in NL is reducible. However, we don’t use polynomial time reducibility here because, as you will see, all problems in NL are solvable in polynomial time. Therefore, every two problems in NL except ∅ and Σ∗ are polynomial time reducible to one another (see the discussion of polynomial time reducibility in the definition of PSPACE-completeness on page 337). Hence polynomial time reducibility is too strong to differentiate problems in NL from one another. Instead we use a new type of reducibility called log space reducibility.
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8.5 NL-COMPLETENESS 351
352 CHAPTER 8 / SPACE COMPLEXITY
DEFINITION 8.21
A log space transducer is a Turing machine with a read-only input tape, a write-only output tape, and a read/write work tape. The head on the output tape cannot move leftward, so it cannot read what it has written. The work tape may contain O(log n) symbols. A log space transducer M computes a function f : Σ∗−→Σ∗, where f (w) is the string remaining on the output tape after M halts when it is started with w on its input tape. We call f a log space com- putable function. Language A is log space reducible to language B, written A ≤L B, if A is mapping reducible to B by means of a log space computable function f .
Now we are ready to define NL-completeness.
DEFINITION 8.22
A language B is NL-complete if
1. B ∈NL,and
2. every A in NL is log space reducible to B.
If one language is log space reducible to another language already known to be in L, the original language is also in L, as the following theorem demonstrates.
THEOREM 8.23
IfA≤L BandB∈L,thenA∈L.
PROOF A tempting approach to the proof of this theorem is to follow the model presented in Theorem 7.31, the analogous result for polynomial time re- ducibility. In that approach, a log space algorithm for A first maps its input w to f(w), using the log space reduction f, and then applies the log space algorithm for B. However, the storage required for f(w) may be too large to fit within the log space bound, so we need to modify this approach.
Instead, A’s machine MA computes individual symbols of f(w) as requested by B’s machine MB. In the simulation, MA keeps track of where MB’s input head would be on f(w). Every time MB moves, MA restarts the computation of f on w from the beginning and ignores all the output except for the desired location of f(w). Doing so may require occasional recomputation of parts of f(w) and so is inefficient in its time complexity. The advantage of this method is that only a single symbol of f(w) needs to be stored at any point, in effect trading time for space.
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COROLLARY 8.24
If any NL-complete language is in L, then L = NL.
SEARCHING IN GRAPHS
THEOREM 8.25 PATH is NL-complete.
PROOF IDEA Example 8.19 shows that PATH is in NL, so we only need to show that PATH is NL-hard. In other words, we must show that every language A in NL is log space reducible to PATH.
The idea behind the log space reduction from A to PATH is to construct a graph that represents the computation of the nondeterministic log space Turing machine for A. The reduction maps a string w to a graph whose nodes cor- respond to the configurations of the NTM on input w. One node points to a second node if the corresponding first configuration can yield the second con- figuration in a single step of the NTM. Hence the machine accepts w whenever some path from the node corresponding to the start configuration leads to the node corresponding to the accepting configuration.
PROOF We show how to give a log space reduction from any language A in NL to PATH. Let’s say that NTM M decides A in O(logn) space. Given an input w, we construct ⟨G, s, t⟩ in log space, where G is a directed graph that contains a path from s to t if and only if M accepts w.
The nodes of G are the configurations of M on w. For configurations c1 and c2 ofM onw,thepair(c1,c2)isanedgeofGifc2 isoneofthepossiblenext configurations of M starting from c1. More precisely, if M’s transition function indicates that c1’s state together with the tape symbols under its input and work tape heads can yield the next state and head actions to make c1 into c2, then (c1, c2) is an edge of G. Node s is the start configuration of M on w. Machine M is modified to have a unique accepting configuration, and we designate this configuration to be node t.
This mapping reduces A to PATH because whenever M accepts its input, some branch of its computation accepts, which corresponds to a path from the start configuration s to the accepting configuration t in G. Conversely, if some path exists from s to t in G, some computation branch accepts when M runs on input w, and M accepts w.
To show that the reduction operates in log space, we give a log space trans- ducer that outputs ⟨G, s, t⟩ on input w. We describe G by listing its nodes and edges. Listing the nodes is easy because each node is a configuration of M on w and can be represented in c log n space for some constant c. The transducer se- quentially goes through all possible strings of length c log n, tests whether each
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8.5 NL-COMPLETENESS 353
354 CHAPTER 8 / SPACE COMPLEXITY
is a legal configuration of M on w, and outputs those that pass the test. The transducer lists the edges similarly. Log space is sufficient for verifying that a configuration c1 of M on w can yield configuration c2 because the transducer only needs to examine the actual tape contents under the head locations given in c1 to determine that M’s transition function would give configuration c2 as a result. The transducer tries all pairs (c1,c2) in turn to find which qualify as edges of G. Those that do are added to the output tape.
One immediate spinoff of Theorem 8.25 is the following corollary, which states that NL is a subset of P.
COROLLARY 8.26 NL ⊆ P.
PROOF Theorem 8.25 shows that any language in NL is log space reducible to PATH. Recall that a Turing machine that uses space f(n) runs in time n2O(f(n)), so a reducer that runs in log space also runs in polynomial time. Therefore, any language in NL is polynomial time reducible to PATH, which in turn is in P, by Theorem 7.14. We know that every language that is polyno- mial time reducible to a language in P is also in P, so the proof is complete.
Though log space reducibility appears to be highly restrictive, it is adequate for most reductions in complexity theory because these are usually computa- tionally simple. For example, in Theorem 8.9 we showed that every PSPACE problem is polynomial time reducible to TQBF. The highly repetitive formu- las that these reductions produce may be computed using only log space, and therefore we may conclude that TQBF is PSPACE-complete with respect to log space reducibility. This conclusion is important because Corollary 9.6 shows that NL PSPACE. This separation and log space reducibility imply that TQBF ̸∈ NL.
8.6
NL EQUALS CONL
This section contains one of the most surprising results known concerning the relationships among complexity classes. The classes NP and coNP are generally believed to be different. At first glance, the same appears to hold for the classes NL and coNL. The fact that NL equals coNL, as we are about to prove, shows that our intuition about computation still has many gaps in it.
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THEOREM 8.27 NL = coNL.
PROOF IDEA We show that PATH is in NL, and thereby establish that ev- ery problem in coNL is also in NL, because PATH is NL-complete. The NL algorithm M that we present for PATH must have an accepting computation whenever the input graph G does not contain a path from s to t.
First, let’s tackle an easier problem. Let c be the number of nodes in G that are reachable from s. We assume that c is provided as an input to M and show how to use c to solve PATH. Later we show how to compute c.
Given G, s, t, and c, the machine M operates as follows. One by one, M goes through all the m nodes of G and nondeterministically guesses whether each one is reachable from s. Whenever a node u is guessed to be reachable, M attempts to verify this guess by guessing a path of length m or less from s to u. If a computation branch fails to verify this guess, it rejects. In addition, if a branch guesses that t is reachable, it rejects. Machine M counts the number of nodes that have been verified to be reachable. When a branch has gone through all of G’s nodes, it checks that the number of nodes that it verified to be reachable from s equals c, the number of nodes that actually are reachable, and rejects if not. Otherwise, this branch accepts.
In other words, if M nondeterministically selects exactly c nodes reachable from s, not including t, and proves that each is reachable from s by guessing the path, M knows that the remaining nodes, including t, are not reachable, so it can accept.
Next, we show how to calculate c, the number of nodes reachable from s. We describe a nondeterministic log space procedure whereby at least one computa- tion branch has the correct value for c and all other branches reject.
For each i from 0 to m, we define Ai to be the collection of nodes that are at a distance of i or less from s (i.e., that have a path of length at most i from s). So A0 = {s}, each Ai ⊆ Ai+1, and Am contains all nodes that are reachable from s. Let ci be the number of nodes in Ai. We next describe a procedure that calculates ci+1 from ci. Repeated application of this procedure yields the desired value of c = cm.
We calculate ci+1 from ci, using an idea similar to the one presented earlier in this proof sketch. The algorithm goes through all the nodes of G, determines whether each is a member of Ai+1, and counts the members.
To determine whether a node v is in Ai+1, we use an inner loop to go through all the nodes of G and guess whether each node is in Ai. Each positive guess is verified by guessing the path of length at most i from s. For each node u verified to be in Ai, the algorithm tests whether (u, v) is an edge of G. If it is an edge, v is in Ai+1. Additionally, the number of nodes verified to be in Ai is counted. At the completion of the inner loop, if the total number of nodes verified to be in Ai is not ci, all Ai have not been found, so this computation branch rejects. If the count equals ci and v has not yet been shown to be in Ai+1, we conclude that it isn’t in Ai+1. Then we go on to the next v in the outer loop.
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8.6 NL EQUALS CONL 355
356 CHAPTER 8 / SPACE COMPLEXITY
PROOF Here is an algorithm for PATH. Let m be the number of nodes of G.
M = “On input ⟨G, s, t⟩: 1. Letc0 =1.
[A0 ={s}has1node] [computeci+1 fromci] [ ci+1 counts nodes in Ai+1 ] [ check if v ∈ Ai+1 ]
5. Letd=0. [dre-countsAi]
6. ForeachnodeuinG: [checkifu∈Ai]
7. Nondeterministically either perform or skip these steps:
8. Nondeterministically follow a path of length at most i
from s and reject if it doesn’t end at u.
9. Increment d. [ verified that u ∈ Ai ]
10. If (u,v) is an edge of G, increment ci+1 and go to stage 5 with the next v. [ verified that v ∈ Ai+1 ]
2. Fori=0tom−1:
3. Let ci+1 = 1.
4. For each node v ̸= s in G:
11. If d ̸= ci, then reject.
12. Let d = 0.
13. ForeachnodeuinG:
14. Nondeterministically either perform or skip these steps:
15. Nondeterministically follow a path of length at most m
from s and reject if it doesn’t end at u.
16. If u = t, then reject.
17. Increment d.
18. If d ̸= cm , then reject .
Otherwise, accept.”
[found path from s to t] [ verified that u ∈ Am ] [ check whether found all of Am ]
This algorithm only needs to store m, u, v, ci, ci+1, d, i, and a pointer to the head of a path at any given time. Hence it runs in log space. (Note that M accepts improperly formed inputs, too.)
We summarize our present knowledge of the relationships among several complexity classes as follows:
L ⊆ NL = coNL ⊆ P ⊆ NP ⊆ PSPACE.
We don’t know whether any of these containments are proper, although we prove NL PSPACE in Corollary 9.6. Consequently, either coNL P or P PSPACE must hold, but we don’t know which one does! Most researchers conjecture that all these containments are proper.
[ check whether found all Ai ] [ cm now known; d re-counts Am ] [checkifu∈Am]
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EXERCISES
8.1
8.2
Show that for any function f : N −→ R+ , where f (n) ≥ n, the space complexity class SPACE(f(n)) is the same whether you define the class by using the single- tape TM model or the two-tape read-only input TM model.
Consider the following position in the standard tic-tac-toe game.
×
⃝ ×
Let’s say that it is the ×-player’s turn to move next. Describe a winning strategy for this player. (Recall that a winning strategy isn’t merely the best move to make in the current position. It also includes all the responses that this player must make in order to win, however the opponent moves.)
Consider the following generalized geography game wherein the start node is the one with the arrow pointing in from nowhere. Does Player I have a winning strat- egy? Does Player II? Give reasons for your answers.
EXERCISES 357
⃝
8.3
8.4
A8.5 8.6 A 8.7
Show that PSPACE is closed under the operations union, complementation, and star.
Show that ADFA ∈ L.
Show that any PSPACE-hard language is also NP-hard.
Show that NL is closed under the operations union, concatenation, and star.
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358 CHAPTER 8 / SPACE COMPLEXITY PROBLEMS
8.8 8.9
8.10
Let EQREX = {⟨R, S⟩| R and S are equivalent regular expressions}. Show that EQREX ∈ PSPACE.
A ladder is a sequence of strings s1 , s2 , . . . , sk , wherein every string differs from the preceding one by exactly one character. For example, the following is a ladder of English words, starting with “head” and ending with “free”:
head, hear, near, fear, bear, beer, deer, deed, feed, feet, fret, free.
Let LADDERDFA = {⟨M, s, t⟩| M is a DFA and L(M) contains a ladder of strings,
starting with s and ending with t}. Show that LADDERDFA is in PSPACE.
The Japanese game go-moku is played by two players, “X” and “O,” on a 19 × 19 grid. Players take turns placing markers, and the first player to achieve five of her markers consecutively in a row, column, or diagonal is the winner. Consider this game generalized to an n × n board. Let
GM = {⟨B⟩| B is a position in generalized go-moku, where player “X” has a winning strategy}.
By a position we mean a board with markers placed on it, such as may occur in the middle of a play of the game, together with an indication of which player moves next. Show that GM ∈ PSPACE.
Show that if every NP-hard language is also PSPACE-hard, then PSPACE = NP. Show that TQBF restricted to formulas where the part following the quantifiers is
in conjunctive normal form is still PSPACE-complete.
Define ALBA = {⟨M, w⟩| M is an LBA that accepts input w}. Show that ALBA is
PSPACE-complete.
The cat-and-mouse game is played by two players, “Cat” and “Mouse,” on an arbi- trary undirected graph. At a given point, each player occupies a node of the graph. The players take turns moving to a node adjacent to the one that they currently occupy. A special node of the graph is called “Hole.” Cat wins if the two players ever occupy the same node. Mouse wins if it reaches the Hole before the preceding happens. The game is a draw if a situation repeats (i.e., the two players simultane- ously occupy positions that they simultaneously occupied previously, and it is the same player’s turn to move).
HAPPY-CAT = {⟨G, c, m, h⟩| G, c, m, h are respectively a graph, and
positions of the Cat, Mouse, and Hole, such that
Cat has a winning strategy if Cat moves first}.
Show that HAPPY-CAT is in P. (Hint: The solution is not complicated and doesn’t depend on subtle details in the way the game is defined. Consider the entire game tree. It is exponentially big, but you can search it in polynomial time.)
8.11 8.12
8.13 ⋆ 8.14
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8.15
8.16
8.17 ⋆ 8.18 ⋆8.19
Consider the following two-person version of the language PUZZLE that was de- scribed in Problem 7.28. Each player starts with an ordered stack of puzzle cards. The players take turns placing the cards in order in the box and may choose which side faces up. Player I wins if all hole positions are blocked in the final stack, and Player II wins if some hole position remains unblocked. Show that the problem of determining which player has a winning strategy for a given starting configuration of the cards is PSPACE-complete.
Read the definition of MIN-FORMULA in Problem 7.46.
a. Show that MIN-FORMULA ∈ PSPACE.
b. Explain why this argument fails to show that MIN-FORMULA ∈ coNP: If φ ̸∈ MIN-FORMULA, then φ has a smaller equivalent formula. An NTM can verify that φ ∈ MIN-FORMULA by guessing that formula.
Let A be the language of properly nested parentheses. For example, (()) and (()(()))() are in A, but )( is not. Show that A is in L.
Let B be the language of properly nested parentheses and brackets. For example, ([()()]()[]) is in B but ([)] is not. Show that B is in L.
The game of Nim is played with a collection of piles of sticks. In one move, a player may remove any nonzero number of sticks from a single pile. The players alternately take turns making moves. The player who removes the very last stick loses. Say that we have a game position in Nim with k piles containing s1 , . . . , sk sticks. Call the position balanced if each column of bits contains an even number of 1s when each of the numbers si is written in binary, and the binary numbers are written as rows of a matrix aligned at the low order bits. Prove the following two facts.
a. Starting in an unbalanced position, a single move exists that changes the position into a balanced one.
b. Starting in a balanced position, every single move changes the position into an unbalanced one.
Let NIM = {⟨s1, . . . , sk⟩| each si is a binary number and Player I has a winning strategy in the Nim game starting at this position}. Use the preceding facts about balanced positions to show that NIM ∈ L.
Let MULT = {a#b#c| a, b, c are binary natural numbers and a × b = c}. Show that MULT ∈ L.
For any positive integer x, let xR be the integer whose binary representation is the reverse of the binary representation of x. (Assume no leading 0s in the binary representation of x.) Define the function R+ : N −→ N where R+ (x) = x + xR .
a. LetA2={⟨x,y⟩|R+(x)=y}.ShowA2∈L.
b. Let A3 = {⟨x, y⟩| R+(R+(x)) = y}. Show A3 ∈ L.
a. LetADD={⟨x,y,z⟩|x,y,z>0arebinaryintegersandx+y=z}.Show that ADD ∈ L.
b. Let PAL-ADD = {⟨x,y⟩| x,y > 0 are binary integers where x + y is an integer whose binary representation is a palindrome}. (Note that the binary representation of the sum is assumed not to have leading zeros. A palin- drome is a string that equals its reverse.) Show that PAL-ADD ∈ L.
PROBLEMS 359
8.20 8.21
8.22
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360
CHAPTER 8 / SPACE COMPLEXITY
⋆ 8.23 ⋆8.24
8.25
8.26
8.27
8.28
8.29
8.30 ⋆ 8.31 8.32
⋆ 8.33 A⋆ 8.34
Define UCYCLE = {⟨G⟩| G is an undirected graph that contains a simple cycle}. Show that UCYCLE ∈ L. (Note: G may be a graph that is not connected.)
For each n, exhibit two regular expressions, R and S, of length poly(n), where L(R) ̸= L(S), but where the first string on which they differ is exponentially long. In other words, L(R) and L(S) must be different, yet agree on all strings of length up to 2εn for some constant ε > 0.
An undirected graph is bipartite if its nodes may be divided into two sets so that all edges go from a node in one set to a node in the other set. Show that a graph is bipartite if and only if it doesn’t contain a cycle that has an odd number of nodes. Let BIPARTITE = {⟨G⟩| G is a bipartite graph}. Show that BIPARTITE ∈ NL.
Define UPATH to be the counterpart of PATH for undirected graphs. Show that BIPARTITE ≤L UPATH. (Note: In fact, we can prove UPATH ∈ L, and therefore BIPARTITE ∈ L, but the algorithm [62] is too difficult to present here.)
Recall that a directed graph is strongly connected if every two nodes are connected by a directed path in each direction. Let
STRONGLY-CONNECTED = {⟨G⟩| G is a strongly connected graph}. Show that STRONGLY-CONNECTED is NL-complete.
LetBOTHNFA ={⟨M1,M2⟩|M1 andM2 areNFAswhereL(M1)∩L(M2)̸=∅}. Show that BOTH NFA is NL-complete.
Show that ANFA is NL-complete. Show that EDFA is NL-complete. Show that 2SAT is NL-complete.
Let CNFH1 = {⟨φ⟩| φ is a satisfiable cnf-formula where each clause contains any number of positive literals and at most one negated literal. Furthermore, each negated literal has at most one occurrence in φ}. Show that CNFH1 is NL- complete.
Give an example of an NL-complete context-free language.
Define CYCLE = {⟨G⟩| G is a directed graph that contains a directed cycle}. Show
that CYCLE is NL-complete.
SELECTED SOLUTIONS
8.5 Construct a TM M to decide ADFA. When M receives input ⟨A,w⟩, a DFA and a string, M simulates A on w by keeping track of A’s current state and its current head location, and updating them appropriately. The space required to carry out this simulation is O(log n) because M can record each of these values by storing a pointer into its input.
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8.7 Let A1 and A2 be languages that are decided by NL-machines N1 and N2. Con- struct three Turing machines: N∪ deciding A1 ∪ A2; N◦ deciding A1 ◦ A2; and N∗ deciding A∗1. Each of these machines operates as follows.
Machine N∪ nondeterministically branches to simulate N1 or to simulate N2. In either case, N∪ accepts if the simulated machine accepts.
Machine N◦ nondeterministically selects a position on the input to divide it into two substrings. Only a pointer to that position is stored on the work tape— insufficient space is available to store the substrings themselves. Then N◦ simulates N1 on the first substring, branching nondeterministically to simulate N1’s nonde- terminism. On any branch that reaches N1’s accept state, N◦ simulates N2 on the second substring. On any branch that reaches N2’s accept state, N◦ accepts. Machine N∗ has a more complex algorithm, so we describe its stages.
N∗ = “On input w:
1. Initialize two input position pointers p1 and p2 to 0, the position
immediately preceding the first input symbol.
2. Accept if no input symbols occur after p2.
3. Move p2 forward to a nondeterministically selected position.
4. Simulate N1 on the substring of w from the position following
p1 to the position at p2, branching nondeterministically to sim-
ulate N1’s nondeterminism.
5. If this branch of the simulation reaches N1’s accept state, copy
p2 to p1 and go to stage 2. If N1 rejects on this branch, reject .”
8.34 Reduce PATH to CYCLE. The idea behind the reduction is to modify the PATH problem instance ⟨G, s, t⟩ by adding an edge from t to s in G. If a path exists from s to t in G, a directed cycle will exist in the modified G. However, other cycles may exist in the modified G because they may already be present in G. To handle that problem, first change G so that it contains no cycles. A leveled directed graph is one where the nodes are divided into groups, A1 , A2 , . . . , Ak , called levels, and only edges from one level to the next higher level are permitted. Observe that a leveled graph is acyclic. The PATH problem for leveled graphs is still NL-complete, as the following reduction from the unrestricted PATH problem shows. Given a graph G with two nodes s and t, and m nodes in total, produce the leveled graph G′ whose levels are m copies of G’s nodes. Draw an edge from node i at each level to node j in the next level if G contains an edge from i to j. Additionally, draw an edge from node i in each level to node i in the next level. Let s′ be the node s in the first level and let t′ be the node t in the last level. Graph G contains a path from s to t iff G′ contains a path from s′ to t′. If you modify G′ by adding an edge from t′ to s′, you obtain a reduction from PATH to CYCLE. The reduction is computationally sim- ple, and its implementation in logspace is routine. Furthermore, a straightforward procedure shows that CYCLE ∈ NL. Hence CYCLE is NL-complete.
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SELECTED SOLUTIONS 361
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9
INTRACTABILITY
Certain computational problems are solvable in principle, but the solutions re- quire so much time or space that they can’t be used in practice. Such problems are called intractable.
In Chapters 7 and 8, we introduced several problems thought to be intractable but none that have been proven to be intractable. For example, most people believe the SAT problem and all other NP-complete problems are intractable, although we don’t know how to prove that they are. In this chapter, we give examples of problems that we can prove to be intractable.
In order to present these examples, we develop several theorems that relate the power of Turing machines to the amount of time or space available for computation. We conclude the chapter with a discussion of the possibility of proving that problems in NP are intractable and thereby solving the P versus NP question. First, we introduce the relativization technique and use it to argue that certain methods won’t allow us to achieve this goal. Then, we discuss cir- cuit complexity theory, an approach taken by researchers that has shown some promise.
363
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364 CHAPTER 9 / INTRACTABILITY 9.1
HIERARCHY THEOREMS
Common sense suggests that giving a Turing machine more time or more space should increase the class of problems that it can solve. For example, Turing machines should be able to decide more languages in time n3 than they can in time n2. The hierarchy theorems prove that this intuition is correct, subject to certain conditions described below. We use the term hierarchy theorem because these theorems prove that the time and space complexity classes aren’t all the same—they form a hierarchy whereby the classes with larger bounds contain more languages than do the classes with smaller bounds.
The hierarchy theorem for space complexity is slightly simpler than the one for time complexity, so we present it first. We begin with the following technical definition.
DEFINITION 9.1
A function f: N−→N, where f(n) is at least O(logn), is called space constructible if the function that maps the string 1n to the binary representation of f (n) is computable in space O(f (n)).1
In other words, f is space constructible if some O(f(n)) space TM exists that always halts with the binary representation of f(n) on its tape when started on
input 1n. Fractional functions such as n log2 n and √
n are rounded down to the next lower integer for the purposes of time and space constructibility.
EXAMPLE 9.2
All commonly occurring functions that are at least O(logn) are space con- structible, including the functions log2 n, n log2 n, and n2.
For example, n2 is space constructible because a machine may take its input 1n, obtain n in binary by counting the number of 1s, and output n2 by using any standard method for multiplying n by itself. The total space used is O(n), which is certainly O(n2).
When showing functions f (n) that are o(n) to be space constructible, we use a separate read-only input tape, as we did when we defined sublinear space com- plexity in Section 8.4. For example, such a machine can compute the function that maps 1n to the binary representation of log2 n as follows. It first counts the number of 1s in its input in binary, using its work tape as it moves its head along the input tape. Then, with n in binary on its work tape, it can compute log2 n by counting the number of bits in the binary representation of n.
1Recall that 1n means a string of n 1s.
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9.1 HIERARCHY THEOREMS 365
The role of space constructibility in the space hierarchy theorem may be un- derstood from the following situation. If f(n) and g(n) are two space bounds, where f(n) is asymptotically larger than g(n), we would expect a machine to be able to decide more languages in f(n) space than in g(n) space. However, sup- pose that f(n) exceeds g(n) by only a very small and hard to compute amount. Then, the machine may not be able to use the extra space profitably because even computing the amount of extra space may require more space than is available. In this case, a machine may not be able to compute more languages in f (n) space than it can in g(n) space. Stipulating that f(n) is space constructible avoids this situation and allows us to prove that a machine can compute more than it would be able to in any asymptotically smaller bound, as the following theorem shows.
THEOREM 9.3
Space hierarchy theorem For any space constructible function f : N −→ N ,
a language A exists that is decidable in O(f (n)) space but not in o(f (n)) space.
PROOF IDEA We must demonstrate a language A that has two properties. The first says that A is decidable in O(f(n)) space. The second says that A isn’t decidable in o(f (n)) space.
We describe A by giving an algorithm D that decides it. Algorithm D runs in O(f(n)) space, thereby ensuring the first property. Furthermore, D guarantees that A is different from any language that is decidable in o(f (n)) space, thereby ensuring the second property. Language A is different from languages we have discussed previously in that it lacks a nonalgorithmic definition. Therefore, we cannot offer a simple mental picture of A.
In order to ensure that A not be decidable in o(f(n)) space, we design D to implement the diagonalization method that we used to prove the unsolvability of the acceptance problem ATM in Theorem 4.11 on page 202. If M is a TM that decides a language in o(f(n)) space, D guarantees that A differs from M’s language in at least one place. Which place? The place corresponding to a description of M itself.
Let’s look at the way D operates. Roughly speaking, D takes its input to be the description of a TM M. (If the input isn’t the description of any TM, then D’s action is inconsequential on this input, so we arbitrarily make D reject.) Then, D runs M on the same input—namely, ⟨M⟩—within the space bound f(n). If M halts within that much space, D accepts iff M rejects. If M doesn’t halt, D just rejects. So if M runs within space f(n), D has enough space to ensure that its language is different from M’s. If not, D doesn’t have enough space to figure out what M does. But fortunately D has no requirement to act differently from machines that don’t run in o(f(n)) space, so D’s action on this input is inconsequential.
This description captures the essence of the proof but omits several impor- tant details. If M runs in o(f(n)) space, D must guarantee that its language is
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366 CHAPTER 9 / INTRACTABILITY
different from M ’s language. But even when M runs in o(f (n)) space, it may use more than f(n) space for small n, when the asymptotic behavior hasn’t “kicked in” yet. Possibly, D might not have enough space to run M to completion on input ⟨M ⟩, and hence D will miss its one opportunity to avoid M ’s language. So, if we aren’t careful, D might end up deciding the same language that M decides, and the theorem wouldn’t be proved.
We can fix this problem by modifying D to give it additional opportunities to avoid M ’s language. Instead of running M only when D receives input ⟨M ⟩, it runs M whenever it receives an input of the form ⟨M⟩10∗; that is, an input of the form ⟨M⟩ followed by a 1 and some number of 0s. Then, if M really is running in o(f(n)) space, D will have enough space to run it to completion on input ⟨M⟩10k for some large value of k because the asymptotic behavior must eventually kick in.
One last technical point arises. When D runs M on some string, M may get into an infinite loop while using only a finite amount of space. But D is supposed to be a decider, so we must ensure that D doesn’t loop while simulating M . Any machine that runs in space o(f (n)) uses only 2o(f (n)) time. We modify D so that it counts the number of steps used in simulating M. If this count ever exceeds 2f(n), then D rejects.
PROOF The following O(f (n)) space algorithm D decides a language A that is not decidable in o(f (n)) space.
D = “On input w:
1. Let n be the length of w.
2. Compute f(n) using space constructibility and mark off this
much tape. If later stages ever attempt to use more, reject.
3. If w is not of the form ⟨M⟩10∗ for some TM M, reject.
4. Simulate M on w while counting the number of steps used in
the simulation. If the count ever exceeds 2f(n), reject.
5. If M accepts, reject . If M rejects, accept .”
In stage 4, we need to give additional details of the simulation in order to determine the amount of space used. The simulated TM M has an arbitrary tape alphabet and D has a fixed tape alphabet, so we represent each cell of M’s tape with several cells on D’s tape. Therefore, the simulation introduces a constant factor overhead in the space used. In other words, if M runs in g(n) space, then D uses d g(n) space to simulate M for some constant d that depends on M .
Machine D is a decider because each of its stages can run for a limited time. Let A be the language that D decides. Clearly, A is decidable in space O(f(n)) because D does so. Next, we show that A is not decidable in o(f (n)) space.
Assume to the contrary that some Turing machine M decides A in space g(n), where g(n) is o(f(n)). As mentioned earlier, D can simulate M, using space dg(n) for some constant d. Because g(n) is o(f(n)), some constant n0 exists, where dg(n) < f(n) for all n ≥ n0. Therefore, D’s simulation of M will run to completion so long as the input has length n0 or more. Consider what happens
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9.1 HIERARCHY THEOREMS 367
when D is run on input ⟨M⟩10n0 . This input is longer than n0, so the simulation in stage 4 will complete. Therefore, D will do the opposite of M on the same in- put. Hence M doesn’t decide A, which contradicts our assumption. Therefore, A is not decidable in o(f (n)) space.
COROLLARY 9.4
For any two functions f1,f2: N−→N, where f1(n) is o(f2(n)) and f2 is space
constructible, SPACE(f1(n)) SPACE(f2(n)).2
This corollary allows us to separate various space complexity classes. For example, we can show that the function nc is space constructible for any natu- ral number c. Hence for any two natural numbers c1 < c2, we can prove that SPACE(nc1 ) SPACE(nc2 ). With a bit more work, we can show that nc is space constructible for any rational number c > 0 and thereby extend the pre- ceding containment to hold for any rational numbers 0 ≤ c1 < c2. Observing that two rational numbers c1 and c2 always exist between any two real numbers ε1 < ε2 such that ε1 < c1 < c2 < ε2, we obtain the following additional corollary demonstrating a fine hierarchy within the class PSPACE.
COROLLARY 9.5
For any two real numbers 0 ≤ ε1 < ε2,
SPACE(nε1 ) SPACE(nε2 ).
We can also use the space hierarchy theorem to separate two space complexity
classes we previously encountered.
COROLLARY 9.6 NL PSPACE.
PROOF Savitch’s theorem shows that NL ⊆ SPACE(log2 n), and the space hierarchy theorem shows that SPACE(log2 n) SPACE(n). Hence the corol- lary follows.
As we observed on page 354, this separation shows that TQBF ̸∈ NL because TQBF is PSPACE-complete with respect to log space reducibility.
2Recall that A B means A is a proper (i.e., not equal) subset of B.
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368 CHAPTER 9 / INTRACTABILITY
Now we establish the main objective of this chapter: proving the existence of problems that are decidable in principle but not in practice—that is, problems that are decidable but intractable. Each of the SPACE(nk) classes is contained within the class SPACE(nlog n), which in turn is strictly contained within the
class SPACE(2n). Therefore, we obtainthe following additional corollary sepa- k
rating PSPACE from EXPSPACE = k SPACE(2n ). COROLLARY 9.7
PSPACE EXPSPACE.
This corollary establishes the existence of decidable problems that are in- tractable, in the sense that their decision procedures must use more than poly- nomial space. The languages themselves are somewhat artificial—interesting only for the purpose of separating complexity classes. We use these languages to prove the intractability of other, more natural, languages after we discuss the time hierarchy theorem.
DEFINITION 9.8
A function t: N−→N, where t(n) is at least O(nlogn), is called time constructible if the function that maps the string 1n to the binary representation of t(n) is computable in time O(t(n)).
In other words, t is time constructible if some O(t(n)) time TM exists that always halts with the binary representation of t(n) on its tape when started on input 1n.
EXAMPLE 9.9
All commonly occurring functions that are at least n log n are time constructible,
including the functions n log n, n√
used for each of the n input positions. Then, we compute ⌊n√
n, n2, and 2n.
n is time constructible, we first design a TM
For example, to show that n√
to count the number of 1s in binary. To do so, the TM moves a binary counter along the tape, incrementing it by 1 for every input position, until it reaches the end of the input. This part uses O(n log n) steps because O(log n) steps are
n⌋ in binary from the binary representation of n. Any reasonable method of doing so will work in
O(n log n) time because the length of the numbers involved is O(log n).
The time hierarchy theorem is an analog for time complexity to Theorem 9.3.
For technical reasons that will appear in its proof, the time hierarchy theorem
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9.1 HIERARCHY THEOREMS 369
is slightly weaker than the one we proved for space. Whereas any space con- structible asymptotic increase in the space bound enlarges the class of languages decidable therein, for time we must further increase the time bound by a log- arithmic factor in order to guarantee that we can obtain additional languages. Conceivably, a tighter time hierarchy theorem is true; but at present, we don’t know how to prove it. This aspect of the time hierarchy theorem arises because we measure time complexity with single-tape Turing machines. We can prove tighter time hierarchy theorems for other models of computation.
THEOREM 9.10
Time hierarchy theorem For any time constructible function t: N−→N, a language A exists that is decidable in O(t(n)) time but not decidable in time o(t(n)/ log t(n)).
PROOF IDEA This proof is similar to the proof of Theorem 9.3. We con- struct a TM D that decides a language A in time O(t(n)), whereby A cannot be decided in o(t(n)/ log t(n)) time. Here, D takes an input w of the form ⟨M ⟩10∗ and simulates M on input w, making sure not to use more than t(n) time. If M halts within that much time, D gives the opposite output.
The important difference in the proof concerns the cost of simulating M while, at the same time, counting the number of steps that the simulation is us- ing. Machine D must perform this timed simulation efficiently so that D runs in O(t(n)) time while accomplishing the goal of avoiding all languages decid- able in o(t(n)/ log t(n)) time. For space complexity, the simulation introduced a constant factor overhead, as we observed in the proof of Theorem 9.3. For time complexity, the simulation introduces a logarithmic factor overhead. The larger overhead for time is the reason for the appearance of the 1/ log t(n) factor in the statement of this theorem. If we had a way of simulating a single-tape TM by another single-tape TM for a prespecified number of steps, using only a constant factor overhead in time, we would be able to strengthen this theorem by changing o(t(n)/ log t(n)) to o(t(n)). No such efficient simulation is known.
PROOF The following O(t(n)) time algorithm D decides a language A that is not decidable in o(t(n)/ log t(n)) time.
D = “On input w:
1. Let n be the length of w.
2. Compute t(n) using time constructibility and store the value
⌈t(n)/ log t(n)⌉ in a binary counter. Decrement this counter before each step used to carry out stages 4 and 5. If the counter ever hits 0, reject.
3. If w is not of the form ⟨M⟩10∗ for some TM M, reject.
4. Simulate M on w.
5. If M accepts, then reject . If M rejects, then accept .”
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370 CHAPTER 9 / INTRACTABILITY
We examine each of the stages of this algorithm to determine the running time. Stages 1, 2, and 3 can be performed within O(t(n)) time.
In stage 4, every time D simulates one step of M, it takes M’s current state together with the tape symbol under M ’s tape head and looks up M ’s next action in its transition function so that it can update M’s tape appropriately. All three of these objects (state, tape symbol, and transition function) are stored on D’s tape somewhere. If they are stored far from each other, D will need many steps to gather this information each time it simulates one of M’s steps. Instead, D always keeps this information close together.
We can think of D’s single tape as organized into tracks. One way to get two tracks is by storing one track in the odd positions and the other in the even posi- tions. Alternatively, the two-track effect may be obtained by enlarging D’s tape alphabet to include each pair of symbols: one from the top track and the second from the bottom track. We can get the effect of additional tracks similarly. Note that multiple tracks introduce only a constant factor overhead in time, provided that only a fixed number of tracks are used. Here, D has three tracks.
One of the tracks contains the information on M ’s tape, and a second contains its current state and a copy of M’s transition function. During the simulation, D keeps the information on the second track near the current position of M’s head on the first track. Every time M’s head position moves, D shifts all the information on the second track to keep it near the head. Because the size of the information on the second track depends only on M and not on the length of the input to M, the shifting adds only a constant factor to the simulation time. Furthermore, because the required information is kept close together, the cost of looking up M’s next action in its transition function and updating its tape is only a constant. Hence if M runs in g(n) time, D can simulate it in O(g(n)) time.
At every step in stage 4, D must decrement the step counter it originally set in stage 2. Here, D can do so without adding excessively to the simulation time by keeping the counter in binary on a third track and moving it to keep it near the present head position. This counter has a magnitude of about t(n)/ log t(n), so its length is log(t(n)/ log t(n)), which is O(log t(n)). Hence the cost of updating and moving it at each step adds a log t(n) factor to the simulation time, thus bringing the total running time to O(t(n)). Therefore, A is decidable in time O(t(n)).
To show that A is not decidable in o(t(n)/ log t(n)) time, we use an argument similar to one used in the proof of Theorem 9.3. Assume to the contrary that TM M decides A in time g(n), where g(n) is o(t(n)/ log t(n)). Here, D can sim- ulate M , using time d g(n) for some constant d. If the total simulation time (not counting the time to update the step counter) is at most t(n)/ log t(n), the sim- ulation will run to completion. Because g(n) is o(t(n)/ log t(n)), some constant n0 exists where d g(n) < t(n)/ log t(n) for all n ≥ n0. Therefore, D’s simula- tion of M will run to completion as long as the input has length n0 or more. Consider what happens when we run D on input ⟨M ⟩10n0 . This input is longer than n0, so the simulation in stage 4 will complete. Therefore, D will do the
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9.1 HIERARCHY THEOREMS 371 opposite of M on the same input. Hence M doesn’t decide A, which contradicts
our assumption. Therefore, A is not decidable in o(t(n)/ log t(n)) time.
We establish analogs to Corollaries 9.4, 9.5, and 9.7 for time complexity.
COROLLARY 9.11
For any two functions t1,t2: N−→N, where t1(n) is o(t2(n)/logt2(n)) and
t2 is time constructible, TIME(t1(n)) TIME(t2(n)). COROLLARY 9.12
For any two real numbers 1 ≤ ε1 < ε2, we have TIME(nε1 ) TIME(nε2 ). COROLLARY 9.13
P EXPTIME.
EXPONENTIAL SPACE COMPLETENESS
We can use the preceding results to demonstrate that a specific language is ac- tually intractable. We do so in two steps. First, the hierarchy theorems tell us that a Turing machine can decide more languages in EXPSPACE than it can in PSPACE. Then, we show that a particular language concerning generalized regular expressions is complete for EXPSPACE and hence can’t be decided in polynomial time or even in polynomial space.
Before getting to their generalization, let’s briefly review the way we intro- duced regular expressions in Definition 1.52. They are built up from the atomic expressions ∅, ε, and members of the alphabet, by using the regular operations union, concatenation, and star, denoted ∪ , ◦, and ∗, respectively. From Prob- lem 8.8, we know that we can test the equivalence of two regular expressions in polynomial space.
We show that by allowing regular expressions with more operations than the usual regular operations, the complexity of analyzing the expressions may grow dramatically. Let ↑ be the exponentiation operation. If R is a regular expression and k is a nonnegative integer, writing R ↑ k is equivalent to the concatenation of R with itself k times. We also write Rk as shorthand for R ↑ k. In other words,
k
k
R =R↑k=R◦R◦···◦R.
Generalized regular expressions allow the exponentiation operation in addition
to the usual regular operations. Obviously, these generalized regular expressions
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372 CHAPTER 9 / INTRACTABILITY
still generate the same class of regular languages as do the standard regular ex- pressions because we can eliminate the exponentiation operation by repeating the base expression. Let
EQ REX↑ = {⟨Q, R⟩| Q and R are equivalent regular expressions with exponentiation}.
To show that EQ REX↑ is intractable, we demonstrate that it is complete for the class EXPSPACE. Any EXPSPACE-complete problem cannot be in PSPACE, much less in P. Otherwise, EXPSPACE would equal PSPACE, contradicting Corollary 9.7.
THEOREM 9.15
EQREX↑ is EXPSPACE-complete.
PROOF IDEA In measuring the complexity of deciding EQREX↑, we assume that all exponents are written as binary integers. The length of an expression is the total number of symbols that it contains.
We sketch an EXPSPACE algorithm for EQREX↑. To test whether two ex- pressions with exponentiation are equivalent, we first use repetition to eliminate exponentiation, then convert the resulting expressions to NFAs. Finally, we use an NFA equivalence testing procedure similar to the one used for deciding the complement of ALLNFA in Example 8.4.
To show that a language A in EXPSPACE is polynomial time reducible to EQREX↑, we utilize the technique of reductions via computation histories that we introduced in Section 5.1. The construction is similar to the construction given in the proof of Theorem 5.13.
Given a TM M for A, we design a polynomial time reduction mapping an in- put w to a pair of expressions, R1 and R2, that are equivalent exactly when M accepts w. The expressions R1 and R2 simulate the computation of M on w. Ex- pression R1 simply generates all strings over the alphabet consisting of symbols that may appear in computation histories. Expression R2 generates all strings that are not rejecting computation histories. So if the TM accepts its input, no rejecting computation histories exist, and expressions R1 and R2 generate the same language. Recall that a rejecting computation history is the sequence of configurations that the machine enters in a rejecting computation on the input. See page 220 in Section 5.1 for a review of computation histories.
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DEFINITION 9.14
A language B is EXPSPACE-complete if
1. B ∈ EXPSPACE, and
2. every A in EXPSPACE is polynomial time reducible to B.
9.1 HIERARCHY THEOREMS 373
The difficulty in this proof is that the size of the expressions constructed must be polynomial in n (so that the reduction can run in polynomial time), whereas the simulated computation may have exponential length. The exponentiation operation is useful here to represent the long computation with a relatively short expression.
PROOF First, we present a nondeterministic algorithm for testing whether two NFAs are inequivalent.
N =“Oninput⟨N1,N2⟩,whereN1 andN2 areNFAs:
1. Place a marker on each of the start states of N1 and N2.
2. Repeat 2q1 +q2 times, where q1 and q2 are the numbers of states
in N1 and N2:
3. Nondeterministically select an input symbol and change the
positions of the markers on the states of N1 and N2 to simu-
late reading that symbol.
4. If at any point a marker was placed on an accept state of one
of the finite automata and not on any accept state of the other finite automaton, accept. Otherwise, reject.”
If automata N1 and N2 are equivalent, N clearly rejects because it only ac- cepts when it determines that one machine accepts a string that the other does not accept. If the automata are not equivalent, some string is accepted by one machine and not by the other. Some such string must be of length at most 2q1 +q2 . Otherwise, consider using the shortest such string as the sequence of nondeter- ministic choices. Only 2q1 +q2 different ways exist to place markers on the states of N1 and N2; so in a longer string, the positions of the markers would repeat. By removing the portion of the string between the repetitions, a shorter such string would be obtained. Hence algorithm N would guess this string among its nondeterministic choices and would accept. Thus, N operates correctly.
Algorithm N runs in nondeterministic linear space. Thus, Savitch’s theorem provides a deterministic O(n2) space algorithm for this problem. Next, we use the deterministic form of this algorithm to design the following algorithm E that decides EQREX↑.
E = “On input ⟨R1,R2⟩, where R1 and R2 are regular expressions with exponentiation:
1. Convert R1 and R2 to equivalent regular expressions B1 and B2 that use repetition instead of exponentiation.
2. Convert B1 and B2 to equivalent NFAs N1 and N2, using the conversion procedure given in the proof of Lemma 1.55.
3. Use the deterministic version of algorithm N to determine whether N1 and N2 are equivalent.”
Algorithm E obviously is correct. To analyze its space complexity, we ob- serve that using repetition to replace exponentiation may increase the length of an expression by a factor of 2l, where l is the sum of the lengths of the ex-
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374 CHAPTER 9 / INTRACTABILITY
ponents. Thus, expressions B1 and B2 have a length of at most n2n, where n is the input length. The conversion procedure of Lemma 1.55 increases the size linearly, and hence NFAs N1 and N2 have at most O(n2n) states. Thus, with input size O(n2n), the deterministic version of algorithm N uses space O((n2n)2) = O(n222n). Hence EQREX↑ is decidable in exponential space.
Next, we show that EQREX↑ is EXPSPACE-hard. Let A be a language that
is decided by TM M running in space 2(nk) for some constant k. The reduction maps an input w to a pair of regular expressions, R1 and R2. Expression R1 is ∆∗ where if Γ and Q are M ’s tape alphabet and states, ∆ = Γ∪Q∪{#} is the alphabet consisting of all symbols that may appear in a computation history. We construct expression R2 to generate all strings that aren’t rejecting computation histories of M on w. Of course, M accepts w iff M on w has no rejecting computation histories. Therefore, the two expressions are equivalent iff M accepts w. The construction is as follows.
A rejecting computation history for M on w is a sequence of configura- tions separated by # symbols. We use our standard encoding of configurations whereby a symbol corresponding to the current state is placed to the left of the current head position. We assume that all configurations have length 2(nk) and are padded on the right by blank symbols if they otherwise would be too short. The first configuration in a rejecting computation history is the start configu- ration of M on w. The last configuration is a rejecting configuration. Each configuration must follow from the preceding one according to the rules speci- fied in the transition function.
A string may fail to be a rejecting computation in several ways: It may fail to start or end properly, or it may be incorrect somewhere in the middle. Ex- pression R2 equals Rbad-start ∪ Rbad-window ∪ Rbad-reject, where each subexpression corresponds to one of the three ways a string may fail.
We construct expression Rbad-start to generate all strings that fail to start with the start configuration C1 of M on w, as follows. Configuration C1 looks like q0w1w2 · · · wn␣ ␣ · · · ␣ #. We write Rbad-start as the union of several subexpres- sions to handle each part of C1:
Rbad-start =S0 ∪S1 ∪ ··· ∪Sn ∪Sb ∪S#.
Expression S0 generates all strings that don’t start with q0. We let S0 be the expression ∆−q0 ∆∗. The notation ∆−q0 is shorthand for writing the union of all symbols in ∆ except q0.
Expression S1 generates all strings that don’t contain w1 in the second po- sition. We let S1 be ∆∆−w1 ∆∗. In general, for 1 ≤ i ≤ n, expression Si is ∆i ∆−wi ∆∗. Thus, Si generates all strings that contain any symbols in the first i positions, any symbol except wi in position i + 1, and any string of symbols fol- lowing position i + 1. Note that we have used the exponentiation operation here. Actually, at this point, exponentiation is more of a convenience than a necessity because we could have instead repeated the symbol ∆ i times without exces- sively increasing the length of the expression. But in the next subexpression, exponentiation is crucial to keeping the size polynomial.
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9.1 HIERARCHY THEOREMS 375
Expression Sb generates all strings that fail to contain a blank symbol in some position n + 2 through 2(nk). We could introduce subexpressions Sn+2 through S2(nk) for this purpose, but then expression Rbad-start would have exponential length. Instead, we let
Sb = ∆n+1 (∆ ∪ ε)2(nk)−n−2 ∆−␣ ∆∗.
Thus, Sb generates strings that contain any symbols in the first n + 1 positions, any symbols in the next t positions, where t can range from 0 to 2(nk ) − n − 2, and any symbol except blank in the next position.
Finally,S# generatesallstringsthatdon’thavea#symbolinposition2(nk)+1.
Let S# be ∆(2(nk)) ∆−# ∆∗.
Now that we have completed the construction of Rbad-start, we turn to the
next piece, Rbad-reject. It generates all strings that don’t end properly; that is, strings that fail to contain a rejecting configuration. Any rejecting configuration contains the state qreject, so we let
Rbad-reject = ∆∗−qreject .
Thus, Rbad-reject generates all strings that don’t contain qreject.
Finally, we construct Rbad-window, the expression that generates all strings whereby one configuration does not properly lead to the next configuration. Recall that in the proof of the Cook–Levin theorem, we determined that one configuration legally yields another whenever every three consecutive symbols in the first configuration correctly yield the corresponding three symbols in the second configuration according to the transition function. Hence, if one config- uration fails to yield another, the error will be apparent from an examination of
the appropriate six symbols. We use this idea to construct Rbad-window: Rbad-window = ∆∗ abc ∆(2(nk)−2) def ∆∗,
bad(abc,def )
where bad(abc, def ) means that abc doesn’t yield def according to the transition function. The union is taken only over such symbols a, b, c, d, e, and f in ∆. The following figure illustrates the placement of these symbols in a computation history.
FIGURE 9.16
Corresponding places in adjacent configurations
To calculate the length of R2, we determine the length of the exponents that appear in it. Several exponents of magnitude roughly 2(nk) appear, and their total length in binary is O(nk ). Therefore, the length of R2 is polynomial in n.
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376 CHAPTER 9 / INTRACTABILITY 9.2
RELATIVIZATION
The proof that EQ REX↑ is intractable rests on the diagonalization method. Why don’t we show that SAT is intractable in the same way? Possibly we could use diagonalization to show that a nondeterministic polynomial time TM can decide a language that is provably not in P. In this section, we introduce the method of relativization to give strong evidence against the possibility of solving the P versus NP question by using a proof by diagonalization.
In the relativization method, we modify our model of computation by giv- ing the Turing machine certain information essentially for “free.” Depending on which information is actually provided, the TM may be able to solve some problems more easily than before.
For example, suppose that we grant the TM the ability to solve the satisfiability problem in a single step, for any size Boolean formula. Never mind how this feat is accomplished—imagine an attached “black box” that gives the machine this capability. We call the black box an oracle to emphasize that it doesn’t necessarily correspond to any physical device. Obviously, the machine could use the oracle to solve any NP problem in polynomial time, regardless of whether P equals NP, because every NP problem is polynomial time reducible to the satisfiability problem. Such a TM is said to be computing relative to the satisfiability problem; hence the term relativization.
In general, an oracle can correspond to any particular language, not just the satisfiability problem. The oracle allows the TM to test membership in the lan- guage without actually having to compute the answer itself. We formalize this notion shortly. You may recall that we introduced oracles in Section 6.3. There, we defined them for the purpose of classifying problems according to the de- gree of unsolvability. Here, we use oracles to understand better the power of the diagonalization method.
DEFINITION 9.17
An oracle for a language A is a device that is capable of reporting whether any string w is a member of A. An oracle Turing machine M A is a modified Turing machine that has the additional capability of querying an oracle for A. Whenever MA writes a string on a special oracle tape, it is informed whether that string is a member of A in a single computation step.
Let PA be the class of languages decidable with a polynomial time oracle Turing machine that uses oracle A. Define the class NPA similarly.
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EXAMPLE 9.18
As we mentioned earlier, polynomial time computation relative to the satisfia- bility problem contains all of NP. In other words, NP ⊆ PSAT . Furthermore, coNP ⊆ PSAT because PSAT , being a deterministic complexity class, is closed under complementation.
EXAMPLE 9.19
Just as PSAT contains languages that we believe are not in P, the class NPSAT contains languages that we believe are not in NP. The complement of the lan- guage MIN-FORMULA that we defined in Problem 7.46 on page 328 provides one such example.
MIN-FORMULA doesn’t seem to be in NP (though whether it actually be- longs to NP is not known). However, MIN-FORMULA is in NPSAT because a nondeterministic polynomial time oracle Turing machine with a SAT oracle can test whether φ is a member, as follows. First, the inequivalence problem for two Boolean formulas is solvable in NP, and hence the equivalence problem is in coNP because a nondeterministic machine can guess the assignment on which the two formulas have different values. Then, the nondeterministic oracle ma- chine for MIN-FORMULA nondeterministically guesses the smaller equivalent formula, tests whether it actually is equivalent, using the SAT oracle, and accepts if it is.
LIMITS OF THE DIAGONALIZATION METHOD
The next theorem demonstrates oracles A and B for which PA and NPA are provably different, and PB and NPB are provably equal. These two oracles are important because their existence indicates that we are unlikely to resolve the P versus NP question by using the diagonalization method.
At its core, the diagonalization method is a simulation of one Turing machine by another. The simulation is done so that the simulating machine can deter- mine the behavior of the other machine and then behave differently. Suppose that both of these Turing machines were given identical oracles. Then, whenever the simulated machine queries the oracle, so can the simulator; and therefore, the simulation can proceed as before. Consequently, any theorem proved about Turing machines by using only the diagonalization method would still hold if both machines were given the same oracle.
In particular, if we could prove that P and NP were different by diagonaliz- ing, we could conclude that they are different relative to any oracle as well. But PB and NPB are equal, so that conclusion is false. Hence diagonalization isn’t sufficient to separate these two classes. Similarly, no proof that relies on a sim- ple simulation could show that the two classes are the same because that would
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9.2 RELATIVIZATION 377
378 CHAPTER 9 / INTRACTABILITY
show that they are the same relative to any oracle; but in fact, PA and NPA are
different.
THEOREM 9.20
1. An oracle A exists whereby PA ̸= NPA.
2. An oracle B exists whereby PB = NPB .
PROOF IDEA Exhibiting oracle B is easy. Let B be any PSPACE-complete problem such as TQBF.
We exhibit oracle A by construction. We design A so that a certain language LA in NPA provably requires brute-force search, and so LA cannot be in PA. Hence we can conclude that PA ̸= NPA. The construction considers every polynomial time oracle machine in turn and ensures that each fails to decide the language LA.
PROOF Let B be TQBF. We have the series of containments NPTQBF ⊆1 NPSPACE ⊆2 PSPACE ⊆3 PTQBF.
Containment 1 holds because we can convert the nondeterministic polynomial time oracle TM to a nondeterministic polynomial space machine that computes the answers to queries regarding TQBF instead of using the oracle. Contain- ment 2 follows from Savitch’s theorem. Containment 3 holds because TQBF is PSPACE-complete. Hence we conclude that PTQBF = NPTQBF .
Next, we show how to construct oracle A. For any oracle A, let LA be the collection of all strings for which a string of equal length appears in A. Thus,
LA ={w|∃x∈A[|x|=|w|]}.
Obviously, for any A, the language LA is in NPA.
To show LA is not in PA, we design A as follows. Let M1, M2, ... be a list of
all polynomial time oracle TMs. We may assume for simplicity that Mi runs in time ni. The construction proceeds in stages, where stage i constructs a part of A, which ensures that MiA doesn’t decide LA. We construct A by declaring that certain strings are in A and others aren’t in A. Each stage determines the status of only a finite number of strings. Initially, we have no information about A. We begin with stage 1.
Stage i. So far, a finite number of strings have been declared to be in or out of A. We choose n greater than the length of any such string and large enough that 2n is greater than ni, the running time of Mi. We show how to extend our information about A so that MiA accepts 1n whenever that string is not in LA.
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We run Mi on input 1n and respond to its oracle queries as follows. If Mi queries a string y whose status has already been determined, we respond consis- tently. If y’s status is undetermined, we respond NO to the query and declare y to be out of A. We continue the simulation of Mi until it halts.
Now consider the situation from Mi’s perspective. If it finds a string of length n in A, it should accept because it knows that 1n is in LA. If Mi de- termines that all strings of length n aren’t in A, it should reject because it knows that 1n is not in LA. However, it doesn’t have enough time to ask about all strings of length n, and we have answered NO to each of the queries it has made. Hence when Mi halts and must decide whether to accept or reject, it doesn’t have enough information to be sure that its decision is correct.
Our objective is to ensure that its decision is not correct. We do so by observ- ing its decision and then extending A so that the reverse is true. Specifically, if Mi accepts 1n, we declare all the remaining strings of length n to be out of A and so determine that 1n is not in LA. If Mi rejects 1n, we find a string of length n that Mi hasn’t queried and declare that string to be in A to guarantee that 1n is in LA. Such a string must exist because Mi runs for ni steps, which is fewer than 2n, the total number of strings of length n. Either way, we have ensured that MiA doesn’t decide LA.
We finish stage i by arbitrarily declaring that any string of length at most n, whose status remains undetermined at this point, is out of A. Stage i is com- pleted and we proceed with stage i + 1.
We have shown that no polynomial time oracle TM decides LA with oracle A, thereby proving the theorem.
In summary, the relativization method tells us that to solve the P versus NP question, we must analyze computations, not just simulate them. In Section 9.3, we introduce one approach that may lead to such an analysis.
9.3
CIRCUIT COMPLEXITY
Computers are built from electronic devices wired together in a design called a digital circuit. We can also simulate theoretical models, such as Turing machines, with the theoretical counterpart to digital circuits, called Boolean circuits. Two purposes are served by establishing the connection between TMs and Boolean circuits. First, researchers believe that circuits provide a convenient compu- tational model for attacking the P versus NP and related questions. Second, circuits provide an alternative proof of the Cook–Levin theorem that SAT is NP-complete. We cover both topics in this section.
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9.3 CIRCUIT COMPLEXITY 379
380 CHAPTER 9 / INTRACTABILITY
DEFINITION 9.21
A Boolean circuit is a collection of gates and inputs connected by wires. Cycles aren’t permitted. Gates take three forms: AND gates, OR gates, and NOT gates, as shown schematically in the following figure.
FIGURE 9.22
An AND gate, an OR gate, and a NOT gate
The wires in a Boolean circuit carry the Boolean values 0 and 1. The gates are simple processors that compute the Boolean functions AND, OR, and NOT. The AND function outputs 1 if both of its inputs are 1 and outputs 0 otherwise. The OR function outputs 0 if both of its inputs are 0 and outputs 1 otherwise. The NOT function outputs the opposite of its input; in other words, it outputs a 1 if its input is 0 and a 0 if its input is 1. The inputs are labeled x1,...,xn. One of the gates is designated the output gate. The following figure depicts a Boolean circuit.
FIGURE 9.23
An example of a Boolean circuit
A Boolean circuit computes an output value from a setting of the inputs by propagating values along the wires and computing the function associated with the respective gates until the output gate is assigned a value. The following figure shows a Boolean circuit computing a value from a setting of its inputs.
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9.3 CIRCUIT COMPLEXITY 381
FIGURE 9.24
An example of a Boolean circuit computing
We use functions to describe the input/output behavior of Boolean cir- cuits. To a Boolean circuit C with n input variables, we associate a function fC : {0,1}n−→{0,1}, where if C outputs b when its inputs x1, . . . , xn are set to a1,...,an, we write fC(a1,...,an) = b. We say that C computes the function fC . We sometimes consider Boolean circuits that have multiple output gates. A function with k output bits computes a function whose range is {0,1}k.
EXAMPLE 9.25
The n-input parity function parityn : {0,1}n−→{0,1} outputs 1 if an odd num- ber of 1s appear in the input variables. The circuit in Figure 9.26 computes parity4, the parity function on 4 variables.
FIGURE 9.26
A Boolean circuit that computes the parity function on 4 variables
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382 CHAPTER 9 / INTRACTABILITY
We plan to use circuits to test membership in languages once they have been suitably encoded into {0,1}. One problem that occurs is that any particular circuit can handle only inputs of some fixed length, whereas a language may contain strings of different lengths. So instead of using a single circuit to test language membership, we use an entire family of circuits, one for each input length, to perform this task. We formalize this notion in the following definition.
DEFINITION 9.27
A circuit family C is an infinite list of circuits, (C0,C1,C2,...), where Cn has n input variables. We say that C decides a language A over {0,1} if for every string w,
w∈A iff Cn(w)=1, where n is the length of w.
The size of a circuit is the number of gates that it contains. Two circuits are equivalent if they have the same input variables and output the same value on every input assignment. A circuit is size minimal if no smaller circuit is equivalent to it. The problem of minimizing circuits has obvious engineering applications but is very difficult to solve in general. Even the problem of testing whether a particular circuit is minimal does not appear to be solvable in P or in NP. A circuit family is minimal if every Ci on the list is a minimal circuit. The size complexity of a circuit family (C0,C1,C2,...) is the function f: N−→N, where f(n) is the size of Cn. We may simply refer to the complexity of a circuit family, instead of the size complexity, when it is clear that we are speaking about size.
The depth of a circuit is the length (number of wires) of the longest path from an input variable to the output gate. We define depth minimal circuits and circuit families, and the depth complexity of circuit families, as we did with circuit size. Circuit depth complexity is of particular interest in Section 10.5 concerning parallel computation.
DEFINITION 9.28
The circuit complexity of a language is the size complexity of a min- imal circuit family for that language. The circuit depth complexity of a language is defined similarly, using depth instead of size.
EXAMPLE 9.29
We can easily generalize Example 9.25 to give circuits that compute the parity function on n variables with O(n) gates. One way to do so is to build a binary tree of gates that compute the XOR function, where the XOR function is the
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same as the parity2 function, and then implement each XOR gate with two NOTs, two ANDs, and one OR, as we did in that earlier example.
Let A be the language of strings that contain an odd number of 1s. Then A has circuit complexity O(n).
The circuit complexity of a language is related to its time complexity. Any language with small time complexity also has small circuit complexity, as the following theorem shows.
THEOREM 9.30
Let t: N−→N be a function, where t(n) ≥ n. If A ∈ TIME(t(n)), then A has
circuit complexity O(t2(n)).
This theorem gives an approach to proving that P ̸= NP whereby we attempt
to show that some language in NP has more than polynomial circuit complexity.
PROOF IDEA Let M be a TM that decides A in time t(n). (For simplicity, we ignore the constant factor in O(t(n)), the actual running time of M.) For each n, we construct a circuit Cn that simulates M on inputs of length n. The gates of Cn are organized in rows, one for each of the t(n) steps in M’s computation on an input of length n. Each row of gates represents the configuration of M at the corresponding step. Each row is wired into the previous row so that it can calculate its configuration from the previous row’s configuration. We modify M so that the input is encoded into {0,1}. Moreover, when M is about to accept, it moves its head onto the leftmost tape cell and writes the ␣ symbol on that cell prior to entering the accept state. That way, we can designate a gate in the final row of the circuit to be the output gate.
PROOF Let M = (Q, Σ, Γ, δ, q0, qaccept, qreject) decide A in time t(n), and let w be an input of length n to M . Define a tableau for M on w to be a t(n) × t(n) table whose rows are configurations of M . The top row of the tableau contains the start configuration of M on w. The ith row contains the configuration at the ith step of the computation.
For convenience, we modify the representation format for configurations in this proof. Instead of the old format, described on page 168, where the state appears to the left of the symbol that the head is reading, we represent both the state and the tape symbol under the tape head by a single composite character. For example, if M is in state q and its tape contains the string 1011 with the head reading the second symbol from the left, the old format would be 1q011 and the new format would be 1 11—where the composite character represents both q, the state, and 0, the symbol under the head.
Each entry of the tableau can contain a tape symbol (member of Γ) or a com- bination of a state and a tape symbol (member of Q × Γ). The entry at the ith row and jth column of the tableau is cell[i,j]. The top row of the tableau then is cell[1,1],...,cell[1,t(n)] and contains the starting configuration.
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9.3 CIRCUIT COMPLEXITY 383
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384 CHAPTER 9 / INTRACTABILITY
We make two assumptions about TM M in defining the notion of a tableau. First, as we mentioned in the proof idea, M accepts only when its head is on the leftmost tape cell and that cell contains the ␣ symbol. Second, once M has halted, it stays in the same configuration for all future time steps. So by looking at the leftmost cell in the final row of the tableau, cell [t(n), 1], we can determine whether M has accepted. The following figure shows part of a tableau for M on the input 0010.
FIGURE 9.31
A tableau for M on input 0010
The content of each cell is determined by certain cells in the preceding row. If we know the values at cell[i − 1,j − 1], cell[i − 1,j], and cell[i − 1,j + 1], we can obtain the value at cell[i,j] with M’s transition function. For example, the following figure magnifies a portion of the tableau in Figure 9.31. The three top symbols, 0, 0, and 1, are tape symbols without states, so the middle symbol must remain a 0 in the next row, as shown.
Now we can begin to construct the circuit Cn. It has several gates for each cell in the tableau. These gates compute the value at a cell from the values of the three cells that affect it.
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To make the construction easier to describe, we add lights that show the out- put of some of the gates in the circuit. The lights are for illustrative purposes only and don’t affect the operation of the circuit.
Let k be the number of elements in Γ ∪ (Q × Γ). We create k lights for each cell in the tableau—one light for each member of Γ, and one light for each member of (Q × Γ)—or a total of kt2(n) lights. We call these lights light[i, j, s], where1≤i,j≤t(n)ands ∈ Γ∪(Q×Γ). Theconditionofthelightsina cell indicates the contents of that cell. If light[i,j,s] is on, cell[i,j] contains the symbol s. Of course, if the circuit is constructed properly, only one light would be on per cell.
Let’s pick one of the lights—say, light[i,j,s] in cell[i,j]. This light should be on if that cell contains the symbol s. We consider the three cells that can affect cell[i,j] and determine which of their settings cause cell[i,j] to contain s. This determination can be made by examining the transition function δ.
Supposethatifthecells cell[i−1,j−1], cell[i−1,j], andcell[i−1,j+1] contain a, b, and c, respectively, cell[i,j] contains s, according to δ. We wire the circuit so that if light[i − 1,j − 1,a], light[i − 1,j,b], and light[i − 1,j + 1,c] are on, then so is light[i,j,s]. We do so by connecting the three lights at the i−1leveltoanAND gatewhoseoutputisconnectedtolight[i,j,s].
In general, several different settings (a1, b1, c1), (a2, b2, c2), . . . , (al, bl, cl) of cell[i−1,j −1], cell[i−1,j], and cell[i−1,j +1] may cause cell[i,j] to contain s. In this case, we wire the circuit so that for each setting ai, bi, ci, the respective lights are connected with an AND gate, and all the AND gates are connected with an OR gate. This circuitry is illustrated in the following figure.
9.3 CIRCUIT COMPLEXITY 385
FIGURE 9.32 Circuitry for one light
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386 CHAPTER 9 / INTRACTABILITY
The circuitry just described is repeated for each light, with a few exceptions at the boundaries. Each cell at the left boundary of the tableau—that is, cell[i, 1] for 1 ≤ i ≤ t(n)—has only two preceding cells that affect its contents. The cells at the right boundary are similar. In these cases, we modify the circuitry to simulate the behavior of TM M in this situation.
The cells in the first row have no predecessors and are handled in a special way. These cells contain the start configuration and their lights are wired to the input variables. Thus, light [1, 1, ] is connected to input w1 because the start configuration begins with the start state symbol q0 and the head starts over w1. Similarly, light[1,1, ] is connected through a NOT gate to input w1. Furthermore, light[1,2,1],...,light[1,n,1] are connected to inputs w2,...,wn, and light[1,2,0],...,light[1,n,0] are connected through NOT gates to inputs w2 , . . . , wn because the input string w determines these values. Additionally, light [1, n + 1, ␣], . . . , light [1, t(n), ␣] are on because the remaining cells in the first row correspond to positions on the tape that initially are blank (␣). Finally, all other lights in the first row are off.
So far, we have constructed a circuit that simulates M through its t(n)th step. All that remains to be done is to assign one of the gates to be the output gate of the circuit. We know that M accepts w if it is in an accept state qaccept on a cell containing ␣ at the left-hand end of the tape at step t(n). So we designate the output gate to be the one attached to light [t(n), 1, qaccept␣ ]. This completes the proof of the theorem.
Besides linking circuit complexity and time complexity, Theorem 9.30 yields an alternative proof of Theorem 7.27, the Cook–Levin theorem, as follows. We say that a Boolean circuit is satisfiable if some setting of the inputs causes the circuit to output 1. The circuit-satisfiability problem tests whether a circuit is satisfiable. Let
CIRCUIT-SAT = {⟨C⟩| C is a satisfiable Boolean circuit}.
Theorem 9.30 shows that Boolean circuits are capable of simulating Turing ma-
chines. We use that result to show that CIRCUIT-SAT is NP-complete. THEOREM 9.33
CIRCUIT-SAT is NP-complete.
PROOF To prove this theorem, we must show that CIRCUIT-SAT is in NP, and that any language A in NP is reducible to CIRCUIT-SAT . The first is obvi- ous. To do the second, we must give a polynomial time reduction f that maps
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q01
q00
strings to circuits, where implies that
f(w) = ⟨C⟩
w ∈ A ⇐⇒ Boolean circuit C is satisfiable.
Because A is in NP, it has a polynomial time verifier V whose input has the form ⟨x, c⟩, where c may be the certificate showing that x is in A. To construct f, we obtain the circuit simulating V using the method in Theorem 9.30. We fill in the inputs to the circuit that correspond to x with the symbols of w. The only remaining inputs to the circuit correspond to the certificate c. We call this circuit C and output it.
If C is satisfiable, a certificate exists, so w is in A. Conversely, if w is in A, a certificate exists, so C is satisfiable.
To show that this reduction runs in polynomial time, we observe that in the proof of Theorem 9.30, the construction of the circuit can be done in time that is polynomial in n. The running time of the verifier is nk for some k, so the size of the circuit constructed is O(n2k). The structure of the circuit is quite simple (actually, it is highly repetitious), so the running time of the reduction is O(n2k).
Now we show that 3SAT is NP-complete, completing the alternative proof of the Cook–Levin theorem.
THEOREM 9.34 3SAT is NP-complete.
PROOF IDEA 3SAT is obviously in NP. We show that all languages in NP reduce to 3SAT in polynomial time. We do so by reducing CIRCUIT-SAT to 3SAT in polynomial time. The reduction converts a circuit C to a formula φ, whereby C is satisfiable iff φ is satisfiable. The formula contains one variable for each variable and each gate in the circuit.
Conceptually, the formula simulates the circuit. A satisfying assignment for φ contains a satisfying assignment to C. It also contains the values at each of C’s gates in C’s computation on its satisfying assignment. In effect, φ’s satisfying assignment “guesses” C’s entire computation on its satisfying assignment, and φ’s clauses check the correctness of that computation. In addition, φ contains a clause stipulating that C’s output is 1.
PROOF We give a polynomial time reduction f from CIRCUIT-SAT to 3SAT. Let C be a circuit containing inputs x1,...,xl and gates g1,...,gm. The
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9.3 CIRCUIT COMPLEXITY 387
388 CHAPTER 9 / INTRACTABILITY
reduction builds from C a formula φ with variables x1, . . . , xl, g1, . . . , gm. Each of φ’s variables corresponds to a wire in C. The xi variables correspond to the input wires, and the gi variables correspond to the wires at the gate outputs. We r e l a b e l φ ’s v a r i a b l e s a s w 1 , . . . , w l + m .
Now we describe φ’s clauses. We write φ’s clauses more intuitively using im- plications. Recall that we can convert the implication operation (P → Q) to the clause (P ∨ Q). Each NOT gate in C with input wire wi and output wire wj is equivalent to the expression
wi → wj) ∧ (wi → wj), which in turn yields the two clauses
(wi ∨wj) ∧ (
Observe that both clauses are satisfied iff an assignment is made to the variables wi and wj corresponding to the correct functioning of the NOT gate.
Each AND gate in C with inputs wi and wj and output wk is equivalent to
(wi ∨wj ∨wk) ∧ (wi ∨wj ∨wk) ∧ (
Similarly, each OR gate in C with inputs wi and wj and output wk is equivalent to
(wi ∨wj ∨wk) ∧ (wi ∨wj ∨wk) ∧ (
In each case, all four clauses are satisfied when an assignment is made to the variables wi, wj, and wk, corresponding to the correct functioning of the gate. Additionally, we add the clause (wm) to φ, where wm is C’s output gate.
Some of the clauses described contain fewer than three literals. We expand such clauses to the desired size by repeating literals. For example, we expand the clause (wm) to the equivalent clause (wm ∨ wm ∨ wm). That completes the construction.
We briefly argue that the construction works. If a satisfying assignment for C exists, we obtain a satisfying assignment for φ by assigning the gi variables according to C’s computation on this assignment. Conversely, if a satisfying as- signment for φ exists, it gives an assignment for C because it describes C’s entire computation where the output value is 1. The reduction can be done in poly- nomial time because it is simple to compute and the output size is polynomial (actually linear) in the size of the input.
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(
wi ∨wj).
wi ∧wj) → wk) ∧ ((wi ∧wj) → wk) ∧ ((wi ∧wj) → wk), which in turn yields the four clauses
((
wi ∧wj) → wk) ∧ ((
wi ∨wj ∨wk) ∧ (
wi ∨wj ∨wk).
wi ∧wj) → wk) ∧ ((wi ∧wj) → wk) ∧ ((wi ∧wj) → wk), which in turn yields the four clauses
((
wi ∧wj) → wk) ∧ ((
wi ∨wj ∨wk) ∧ (
wi ∨wj ∨wk).
EXERCISES
A 9.1 A 9.2 A 9.3
9.4 9.5
9.6 9.7
Prove that TIME(2n ) = TIME(2n+1 ).
Prove that TIME(2n ) TIME(22n ).
Prove that NTIME(n) PSPACE.
Show how the circuit depicted in Figure 9.26 computes on input 0110 by showing the values computed by all of the gates, as we did in Figure 9.24.
Give a circuit that computes the parity function on three input variables and show how it computes on input 011.
ProvethatifA∈P,thenPA =P.
Give regular expressions with exponentiation that generate the following languages
over the alphabet {0,1}.
Aa. All strings of length 500
Ab. All strings of length 500 or less
Ac. All strings of length 500 or more
Ad. All strings of length different than 500
e. All strings that contain exactly 500 1s
f. All strings that contain at least 500 1s
g. All strings that contain at most 500 1s
h. All strings of length 500 or more that contain a 0 in the 500th position
i. All strings that contain two 0s that have at least 500 symbols between them
If R is a regular expression, let R{m,n} represent the expression Rm ∪Rm+1 ∪ ··· ∪Rn.
Show how to implement the R{m,n} operator, using the ordinary exponentiation operator, but without “· · · ”.
Show that if NP = PSAT, then NP = coNP.
Problem 8.13 showed that ALBA is PSPACE-complete.
a. Do we know whether ALBA ∈ NL? Explain your answer. b. Do we know whether ALBA ∈ P? Explain your answer.
Show that the language MAX-CLIQUE from Problem 7.48 is in PSAT.
9.8
9.9 9.10
9.11
EXERCISES 389
PROBLEMS
9.12 Describe the error in the following fallacious “proof” that P ̸= NP. Assume that P = NP and obtain a contradiction. If P = NP, then SAT ∈ P and so for some k, SAT ∈ TIME(nk ). Because every language in NP is polynomial time reducible to SAT, you have NP ⊆ TIME(nk). Therefore, P ⊆ TIME(nk). But by the time hierarchy theorem, TIME(nk+1) contains a language that isn’t in TIME(nk), which contradicts P ⊆ TIME(nk). Therefore, P ̸= NP.
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390
CHAPTER 9 / INTRACTABILITY
9.13
Consider the function pad : Σ∗ × N −→ Σ∗ #∗ that is defined as follows. Let pad (s, l) = s#j , where j = max(0, l − m) and m is the length of s. Thus, pad (s, l) simply adds enough copies of the new symbol # to the end of s so that the length of the result is at least l. For any language A and function f : N −→ N , define the language pad (A, f ) as
pad(A, f) = {pad(s, f(m))| where s ∈ A and m is the length of s}. Prove that if A ∈ TIME(n6 ), then pad (A, n2 ) ∈ TIME(n3 ).
Prove that if NEXPTIME ̸= EXPTIME, then P ̸= NP. You may find the func- tion pad , defined in Problem 9.13, to be helpful.
Define pad as in Problem 9.13.
a. Prove that for every A and natural number k, A ∈ P iff pad(A, nk) ∈ P.
b. Prove that P ̸= SPACE(n). Prove that TQBF ̸∈ SPACE(n1/3 ).
Read the definition of a 2DFA (two-headed finite automaton) given in Prob- lem 5.26. Prove that P contains a language that is not recognizable by a 2DFA.
Let EREX↑ = {⟨R⟩| R is a regular expression with exponentiation and L(R) = ∅}. Show that EREX↑ ∈ P.
Define the unique-sat problem to be
USAT = {⟨φ⟩| φ is a Boolean formula that has a single satisfying assignment}.
Show that USAT ∈ PSAT .
Prove that an oracle C exists for which NPC ̸= coNPC .
A k-query oracle Turing machine is an oracle Turing machine that is permitted to make at most k queries on each input. A k-query oracle Turing machine M with an oracle for A is written M A,k. Define PA,k to be the collection of languages that are decidable by polynomial time k-query oracle Turing machines with an oracle for A.
a. Show that NP ∪ coNP ⊆ PSAT,1.
b. Assume that NP ̸= coNP. Show that NP ∪ coNP PSAT,1.
Suppose that A and B are two oracles. One of them is an oracle for TQBF, but you don’t know which. Give an algorithm that has access to both A and B, and that is guaranteed to solve TQBF in polynomial time.
Recall that you may consider circuits that output strings over {0,1} by designating several output gates. Let add n : {0,1}2n −→ {0,1}n+1 take two n bit binary inte- gers and produce the n + 1 bit sum. Show that you can compute the add n function with O(n) size circuits.
9.14 A 9.15
9.16 ⋆9.17
9.18 9.19
9.20 9.21
9.22
9.23
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9.24
SELECTED SOLUTIONS 391 Define the function majorityn : {0,1}n−→{0,1} as
0 xi
t = 1 + qe−1. Expanding tp using the binomial theorem, we obtain
tp = (1 + qe−1)p = 1 + p · qe−1 + multiples of higher powers of qe−1,
which is equivalent to 1 mod p. Hence t is a stage 4 witness because if tp−1 ≡ 1 (mod p), then tp ≡ t ̸≡ 1 (mod p). As in the previous case, we use this one witness to get many others. If d is a nonwitness, we have dp−1 ≡ 1 (mod p), but then dt mod p is a witness. Moreover, if d1 and d2 are distinct nonwitnesses, then d1t mod p ̸= d2t mod p. Otherwise,
d1 =d1 ·t·tp−1 modp=d2 ·t·tp−1 modp=d2.
Thus, the number of witnesses must be as large as the number of nonwitnesses
and the proof is complete.
The preceding algorithm and its analysis establishes the following theorem. Let PRIMES = {n| n is a prime number in binary}.
THEOREM 10.9 PRIMES ∈ BPP.
Note that the probabilistic primality algorithm has one-sided error. When the algorithm outputs reject , we know that the input must be composite. When the output is accept, we know only that the input could be prime or composite. Thus, an incorrect answer can only occur when the input is a composite number. The one-sided error feature is common to many probabilistic algorithms, so the special complexity class RP is designated for it.
DEFINITION 10.10
RP is the class of languages that are decided by probabilistic poly-
nomial time Turing machines where inputs in the language are
accepted with a probability of at least 1 , and inputs not in the lan- 2
guage are rejected with a probability of 1.
We can make the error probability exponentially small and maintain a poly- nomial running time by using a probability amplification technique similar to (actually simpler than) the one we used in Lemma 10.5. Our earlier algorithm shows that COMPOSITES ∈ RP.
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404 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY READ-ONCE BRANCHING PROGRAMS
A branching program is a model of computation used in complexity theory and in certain practical areas such as computer-aided design. This model represents a decision process that queries the values of input variables and determines how to proceed based on the answers to those queries. We represent this decision process as a graph whose nodes correspond to the particular variable queried at that point in the process.
In this section, we investigate the complexity of testing whether two branch- ing programs are equivalent. In general, that problem is coNP-complete. If we place a certain natural restriction on the class of branching programs, we can give a probabilistic polynomial time algorithm for testing equivalence. This algorithm is especially interesting for two reasons. First, no polynomial time algorithm is known for this problem, so it provides an example of probabilism apparently expanding the class of languages whereby membership can be tested efficiently. Second, this algorithm introduces the technique of assigning non- Boolean values to normally Boolean variables in order to analyze the behavior of some Boolean function of those variables. That technique is used to great effect in interactive proof systems, as we show in Section 10.4.
DEFINITION 10.11
A branching program is a directed acyclic2 graph where all nodes are labeled by variables, except for two output nodes labeled 0 or 1. The nodes that are labeled by variables are called query nodes. Every query node has two outgoing edges: one labeled 0 and the other labeled 1. Both output nodes have no outgoing edges. One of the nodes in a branching program is designated the start node.
A branching program determines a Boolean function as follows. Take any assignment to the variables appearing on its query nodes and, beginning at the start node, follow the path determined by taking the outgoing edge from each query node according to the value assigned to the indicated variable until one of the output nodes is reached. The output is the label of that output node. Figure 10.12 gives two examples of branching programs.
Branching programs are related to the class L in a way that is analogous to the relationship between Boolean circuits and the class P. Problem 10.17 asks you to show that a branching program with polynomially many nodes can test membership in any language over {0,1} that is in L.
2A directed graph is acyclic if it has no directed cycles.
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10.2 PROBABILISTIC ALGORITHMS 405
FIGURE 10.12
Two read-once branching programs
Two branching programs are equivalent if they determine equal functions. Problem 10.21 asks you to show that the problem of testing equivalence for branching programs is coNP-complete. Here we consider a restricted form of branching programs. A read-once branching program is one that can query each variable at most one time on every directed path from the start node to an output node. Both branching programs in Figure 10.12 have the read-once feature. Let
EQROBP = {⟨B1, B2⟩|B1 and B2 are equivalent read-once branching programs}. THEOREM 10.13
EQROBP is in BPP.
PROOF IDEA First, let’s try assigning random values to the variables x1 through xm that appear in B1 and B2, and evaluate these branching programs on that setting. We accept if B1 and B2 agree on the assignment and reject oth- erwise. However, this strategy doesn’t work because two inequivalent read-once branching programs may disagree only on a single assignment out of the 2m possible Boolean assignments to the variables. The probability that we would select that assignment is exponentially small. Hence we would accept with high probability even when B1 and B2 are not equivalent, and that is unsatisfactory.
Instead, we modify this strategy by randomly selecting a non-Boolean assign- ment to the variables, and evaluate B1 and B2 in a suitably defined manner. We can then show that if B1 and B2 are not equivalent, the random evaluations will likely be unequal.
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406 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY
PROOF We assign polynomials over x1 , . . . , xm to the nodes and to the edges of a read-once branching program B as follows. The constant function 1 is assigned to the start node. If a node labeled x has been assigned polynomial p, assign the polynomial xp to its outgoing 1-edge, and assign the polynomial (1 − x)p to its outgoing 0-edge. If the edges incoming to some node have been assigned polynomials, assign the sum of those polynomials to that node. Fi- nally, the polynomial that has been assigned to the output node labeled 1 is also assigned to the branching program itself. Now we are ready to present the prob- abilistic polynomial time algorithm for EQROBP. Let F be a finite field with at least 3m elements.
D = “On input ⟨B1, B2⟩, two read-once branching programs:
1. Select elements a1 through am at random from F.
2. Evaluate the assigned polynomials p1 and p2 at a1 through am.
3. If p1(a1, . . . , am) = p2(a1, . . . , am), accept; otherwise, reject.”
This algorithm runs in polynomial time because we can evaluate the polyno-
mial corresponding to a branching program without actually constructing the
polynomial. We show that the algorithm decides EQROBP with an error proba-
bility of at most 1 . 3
Let’s examine the relationship between a read-once branching program B and its assigned polynomial p. Observe that for any Boolean assignment to B’s variables, all polynomials assigned to its nodes evaluate to either 0 or 1. The polynomials that evaluate to 1 are those on the computation path for that as- signment. Hence B and p agree when the variables take on Boolean values. Similarly, because B is read-once, we may write p as a sum of product terms y1y2 · · · ym, where each yi is xi, (1 − xi), or 1, and where each product term corresponds to a path in B from the start node to the output node labeled 1. The case of yi = 1 occurs when a path doesn’t contain variable xi.
Take each such product term of p containing a yi that is 1 and split it into the sum of two product terms, one where yi = xi and the other where yi = (1 − xi ). Doing so yields an equivalent polynomial because 1 = xi + (1 − xi). Continue splitting product terms until each yi is either xi or (1 − xi). The end result is an equivalent polynomial q that contains a product term for each assignment on which B evaluates to 1. Now we are ready to analyze the behavior of the algorithm D.
First, we show that if B1 and B2 are equivalent, D always accepts. If the branching programs are equivalent, they evaluate to 1 on exactly the same assign- ments. Consequently, the polynomials q1 and q2 are equal because they contain identical product terms. Therefore, p1 and p2 are equal on every assignment.
Second, we show that if B1 and B2 aren’t equivalent, D rejects with a proba- bility of at least 2 . This conclusion follows immediately from Lemma 10.15.
3
The preceding proof relies on the following lemmas concerning the proba- bility of randomly finding a root of a polynomial as a function of the number of variables it has, the degrees of its variables, and the size of the underlying field.
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10.2 PROBABILISTIC ALGORITHMS 407
LEMMA 10.14
For every d ≥ 0, a degree-d polynomial p on a single variable x either has at
most d roots, or is everywhere equal to 0. PROOF We use induction on d.
Basis: Prove for d = 0. A polynomial of degree 0 is constant. If that constant is not 0, the polynomial clearly has no roots.
Induction step: Assume true for d − 1 and prove true for d. If p is a nonzero polynomial of degree d with a root at a, the polynomial x − a divides p evenly. Then p/(x−a) is a nonzero polynomial of degree d−1, and it has at most d−1 roots by virtue of the induction hypothesis.
LEMMA 10.15
Let F be a finite field with f elements and let p be a nonzero polynomial on the variables x1 through xm , where each variable has degree at most d. If a1 through am are selected randomly in F, then Prp(a1,…,am) = 0 ≤ md/f.
PROOF We use induction on m.
Basis: Prove for m = 1. By Lemma 10.14, p has at most d roots, so the proba-
bility that a1 is one of them is at most d/f .
Induction step: Assume true for m − 1 and prove true for m. Let x1 be one of p’s variables. For each i ≤ d, let pi be the polynomial comprising the terms of p containing xi1, but where xi1 has been factored out. Then
p = p 0 + x 1 p 1 + x 21 p 2 + · · · + x d1 p d .
If p(a1, . . . , am) = 0, one of two cases arises. Either all pi evaluate to 0, or some pi doesn’t evaluate to 0 and a1 is a root of the single variable polynomial obtained by evaluating p0 through pd on a2 through am.
To bound the probability that the first case occurs, observe that one of the pj must be nonzero because p is nonzero. Then the probability that all pi evaluate to 0 is at most the probability that pj evaluates to 0. By the induction hypothesis, that is at most (m − 1)d/f because pj has at most m − 1 variables.
To bound the probability that the second case occurs, observe that if some pi doesn’t evaluate to 0, then on the assignment of a2 through am, p reduces to a nonzero polynomial in the single variable x1. The basis already shows that a1 is a root of such a polynomial with a probability of at most d/f .
Therefore, the probability that a1 through am is a root of the polynomial is atmost(m−1)d/f+d/f = md/f.
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408 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY
We conclude this section with one important point concerning the use of randomness in probabilistic algorithms. In our analyses, we assume that these algorithms are implemented using true randomness. True randomness may be difficult (or impossible) to obtain, so it is usually simulated with pseudorandom generators, which are deterministic algorithms whose output appears random. Although the output of any deterministic procedure can never be truly random, some of these procedures generate results that have certain characteristics of randomly generated results. Algorithms that are designed to use randomness may work equally well with these pseudorandom generators, but proving that they do is generally more difficult. Indeed, sometimes probabilistic algorithms may not work well with certain pseudorandom generators. Sophisticated pseu- dorandom generators have been devised that produce results indistinguishable from truly random results by any test that operates in polynomial time, under the assumption that a one-way function exists. (See Section 10.6 for a discussion of one-way functions.)
10.3
ALTERNATION
Alternation is a generalization of nondeterminism that has proven to be useful in understanding relationships among complexity classes, and in classifying specific problems according to their complexity. Using alternation, we may simplify various proofs in complexity theory and exhibit a surprising connection between the time and space complexity measures.
An alternating algorithm may contain instructions to branch a process into multiple child processes, just as in a nondeterministic algorithm. The difference between the two lies in the mode of determining acceptance. A nondeterministic computation accepts if any one of the initiated processes accepts. When an alter- nating computation divides into multiple processes, two possibilities arise. The algorithm can designate that the current process accepts if any of the children accept, or it can designate that the current process accepts if all of the children accept.
Picture the difference between alternating and nondeterministic computation with trees that represent the branching structure of the spawned processes. Each node represents a configuration in a process. In a nondeterministic computa- tion, each node computes the OR operation of its children. That corresponds to the usual nondeterministic acceptance mode whereby a process is accepting if any of its children are accepting. In an alternating computation, the nodes may compute the AND or OR operations as determined by the algorithm. That corresponds to the alternating acceptance mode whereby a process is accepting
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10.3 ALTERNATION 409 if all or any of its children accept. We define an alternating Turing machine as
follows.
DEFINITION 10.16
An alternating Turing machine is a nondeterministic Turing ma- chine with an additional feature. Its states, except for qaccept and qreject, are divided into universal states and existential states. When we run an alternating Turing machine on an input string, we label each node of its nondeterministic computation tree with ∧ or ∨, depending on whether the corresponding configuration contains a universal state or an existential state. We designate a node to be accepting if it is labeled with ∧ and all of its children are accepting, or if it is labeled with ∨ and any of its children are accepting. The input is accepted if the start node is designated accepting.
The following figure shows nondeterministic and alternating computation trees. We label the nodes of the alternating computation tree with ∧ or ∨ to indicate which function of their children they compute.
FIGURE 10.17
Nondeterministic and alternating computation trees
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410 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY ALTERNATING TIME AND SPACE
We define the time and space complexity of these machines in the same way that we did for nondeterministic Turing machines: by taking the maximum time or space used by any computation branch. We define the alternating time and space complexity classes as follows.
DEFINITION 10.18
ATIME(t(n)) = {L| L is decided by an O(t(n)) time alternating Turing machine}.
ASPACE(f(n)) = {L| L is decided by an O(f (n)) space alternating Turing machine}.
We define AP, APSPACE, and AL to be the classes of languages that are decided by alternating polynomial time, alternating polynomial space, and alter- nating logarithmic space Turing machines, respectively.
EXAMPLE 10.19
A tautology is a Boolean formula that evaluates to 1 on every assignment to its variables. Let TAUT = {⟨φ⟩| φ is a tautology}. The following alternating algorithm shows that TAUT is in AP.
“On input ⟨φ⟩:
1. Universally select all assignments to the variables of φ.
2. For a particular assignment, evaluate φ.
3. If φ evaluates to 1, accept; otherwise, reject.”
Stage 1 of this algorithm nondeterministically selects every assignment to φ’s variables with universal branching. That requires all branches to accept in order for the entire computation to accept. Stages 2 and 3 deterministically check whether the assignment that was selected on a particular computation branch satisfies the formula. Hence this algorithm accepts its input if it determines that all assignments are satisfying.
Observe that TAUT is a member of coNP. In fact, any problem in coNP can easily be shown to be in AP by using an algorithm similar to the preceding one.
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EXAMPLE 10.20
This example features a language in AP that isn’t known to be in NP or in coNP. Recall the language MIN-FORMULA that we defined in Problem 7.46 on page 328. The following algorithm shows that MIN-FORMULA is in AP.
“On input ⟨φ⟩:
1. Universally select all formulas ψ that are shorter than φ.
2. Existentially select an assignment to the variables of φ.
3. Evaluate both φ and ψ on this assignment.
4. Accept if the formulas evaluate to different values.
Reject if they evaluate to the same value.”
This algorithm starts with universal branching to select all shorter formulas in stage 1 and then switches to existential branching to select an assignment in stage 2. The term alternation stems from the ability to alternate, or switch, between universal and existential branching.
Alternation allows us to make a remarkable connection between the time and space measures of complexity. Roughly speaking, the following theorem demonstrates an equivalence between alternating time and deterministic space for polynomially related bounds, and another equivalence between alternating space and deterministic time when the time bound is exponentially more than the space bound.
THEOREM 10.21
For f(n) ≥ n, we have ATIME(f(n)) ⊆ SPACE(f(n)) ⊆ ATIME(f2(n)). For f(n) ≥ logn, we have ASPACE(f(n)) = TIME(2O(f(n))).
Consequently, AL = P, AP = PSPACE, and APSPACE = EXPTIME. The proof of this theorem is in the following four lemmas.
LEMMA 10.22
For f(n) ≥ n, we have ATIME(f(n)) ⊆ SPACE(f(n)).
PROOF We convert an alternating time O(f(n)) machine M to a determin- istic space O(f(n)) machine S that simulates M as follows. On input w, the simulator S performs a depth-first search of M’s computation tree to determine which nodes in the tree are accepting. Then S accepts if it determines that the root of the tree, corresponding to M’s starting configuration, is accepting.
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10.3 ALTERNATION 411
412 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY
Machine S requires space for storing the recursion stack that is used in the depth-first search. Each level of the recursion stores one configuration. The recursion depth is M’s time complexity. Each configuration uses O(f(n)) space, and M’s time complexity is O(f(n)). Hence S uses O(f2(n)) space.
We can improve the space complexity by observing that S does not need to store the entire configuration at each level of the recursion. Instead it records only the nondeterministic choice that M made to reach that configuration from its parent. Then S can recover this configuration by replaying the computation from the start and following the recorded “signposts.” Making this change re- duces the space usage to a constant at each level of the recursion. The total used now is thus O(f(n)).
LEMMA 10.23
For f(n) ≥ n, we have SPACE(f(n)) ⊆ ATIME(f2(n)).
PROOF We start with a deterministic space O(f(n)) machine M and con- struct an alternating machine S that uses time O(f2(n)) to simulate it. The approach is similar to that used in the proof of Savitch’s theorem (Theorem 8.5), where we constructed a general procedure for the yieldability problem.
In the yieldability problem, we are given configurations c1 and c2 of M and a number t. We must test whether M can get from c1 to c2 within t steps. An alternating procedure for this problem first branches existentially to guess a configuration cm midway between c1 and c2. Then it branches universally into two processes: one that recursively tests whether c1 can get to cm within t/2 steps, and the other whether cm can get to c2 within t/2 steps.
Machine S uses this recursive alternating procedure to test whether the start configuration can reach an accepting configuration within 2df(n) steps. Here, d is selected so that M has no more than 2df(n) configurations within its space bound.
The maximum time used on any branch of this alternating procedure is O(f(n)) to write a configuration at each level of the recursion, times the depth of the recursion, which is log2df(n) = O(f(n)). Hence this algorithm runs in alternating time O(f2(n)).
LEMMA 10.24
For f(n) ≥ logn, we have ASPACE(f(n)) ⊆ TIME(2O(f(n))).
PROOF We construct a deterministic time 2O(f(n)) machine S to simulate an alternating space O(f(n)) machine M. On input w, the simulator S constructs the following graph of the computation of M on w. The nodes are the con- figurations of M on w that use at most df(n) space, where d is the appropriate constant factor for M . Edges go from a configuration to those configurations it
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can yield in a single move of M . After constructing the graph, S repeatedly scans it and marks certain configurations as accepting. Initially, only the actual accept- ing configurations of M are marked this way. A configuration that performs universal branching is marked accepting if all of its children are so marked, and an existential configuration is marked if any of its children are marked. Machine S continues scanning and marking until no additional nodes are marked on a scan. Finally, S accepts if the start configuration of M on w is marked.
The number of configurations of M on w is 2O(f (n)) because f (n) ≥ log n. Therefore, the size of the configuration graph is 2O(f (n)) and constructing it may be done in 2O(f(n)) time. Scanning the graph once takes roughly the same time. The total number of scans is at most the number of nodes in the graph because each scan except for the final one marks at least one additional node. Hence the total time used is 2O(f (n)).
LEMMA 10.25
For f(n) ≥ logn, we have ASPACE(f(n)) ⊇ TIME(2O(f(n))).
PROOF We show how to simulate a deterministic time 2O(f(n)) machine M by an alternating Turing machine S that uses space O(f(n)). This simulation is tricky because the space available to S is so much less than the size of M’s computation. In this case, S has only enough space to store pointers into a tableau for M on w, as depicted in the following figure.
10.3 ALTERNATION 413
FIGURE 10.26
A tableau for M on w
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414 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY
We use the representation for configurations as given in the proof of Theo- rem 9.30, whereby a single symbol may represent both the state of the machine and the content of the tape cell under the head. The contents of cell d in Fig- ure 10.26 are then determined by the contents of its parents a, b, and c. (A cell on the left or right boundary has only two parents.)
Simulator S operates recursively to guess and then verify the contents of the individual cells of the tableau. To verify the contents of a cell d outside the first row, simulator S existentially guesses the contents of the parents, checks whether their contents would yield d’s contents according to M’s transition function, and then universally branches to verify these guesses recursively. If d were in the first row, S verifies the answer directly because it knows M’s starting configuration. We assume that M moves its head to the left-hand end of the tape on acceptance, so S can determine whether M accepts w by checking the contents of the lower leftmost cell of the tableau. Hence S never needs to store more than a single pointer to a cell in the tableau, so it uses space log 2O(f (n)) = O(f (n)).
THE POLYNOMIAL TIME HIERARCHY
Alternating machines provide a way to define a natural hierarchy of classes within the class PSPACE.
DEFINITION 10.27
Let i be a natural number. A Σi-alternating Turing machine is an alternating Turing machine that on every input and on every computation branch contains at most i runs of universal or existen- tial steps, starting with existential steps. A Πi-alternating Turing machine is similar except that it starts with universal steps.
Define ΣiTIME(f(n)) to be the class of languages that a Σi-alternating TM can decide in O(f(n)) time. Similarly, define the class ΠiTIME(f(n)) for Πi-alternating Turing machines, and define the classes ΣiSPACE(f(n)) and ΠiSPACE(f(n)) for space bounded alternating Turing machines. We define the polynomial time hierarchy to be the collection of classes
ΣiP= ΣiTIME(nk) and
ΠiTIME(nk). Additionally, MIN-FORMULA ∈ Π2P.
k k
ΠiP =
Define class PH = i ΣiP = i ΠiP. Clearly, NP = Σ1P and coNP = Π1P.
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10.4 INTERACTIVE PROOF SYSTEMS 415
10.4
INTERACTIVE PROOF SYSTEMS
Interactive proof systems provide a way to define a probabilistic analog of the class NP, much like probabilistic polynomial time algorithms provide a prob- abilistic analog to P. The development of interactive proof systems has pro- foundly affected complexity theory and has led to important advances in the fields of cryptography and approximation algorithms. To get a feel for this new concept, let’s revisit our intuition about NP.
The languages in NP are those whose members all have short certificates of membership that can be easily checked. If you need to, go back to page 294 and review this formulation of NP. Let’s rephrase this formulation by creating two entities: a Prover that finds the proofs of membership, and a Verifier that checks them. Think of the Prover as if it were convincing the Verifier of w’s membership in A. We require the Verifier to be a polynomial time bounded machine; otherwise, it could figure out the answer itself. We don’t impose any computational bound on the Prover because finding the proof may be time- consuming.
Take the SAT problem, for example. A Prover can convince a polynomial time Verifier that a formula φ is satisfiable by supplying a satisfying assignment. Can a Prover similarly convince a computationally limited Verifier that a for- mula is not satisfiable? The complement of SAT is not known to be in NP, so we can’t rely on the certificate idea. Nonetheless, the surprising answer is yes, provided we give the Prover and Verifier two additional features. First, they are permitted to engage in a two-way dialog. Second, the Verifier may be a prob- abilistic polynomial time machine that reaches the correct answer with a high degree of, but not absolute, certainty. Such a Prover and Verifier constitute an interactive proof system.
GRAPH NONISOMORPHISM
We illustrate the interactive proof concept through the elegant example of the graph isomorphism problem. Call graphs G and H isomorphic if the nodes of G may be reordered so that it is identical to H. Let
ISO = {⟨G, H ⟩| G and H are isomorphic graphs}.
Although ISO is obviously in NP, extensive research has so far failed to demon- strate either a polynomial time algorithm for this problem or a proof that it is NP-complete. It is one of a relatively small number of naturally occurring lan- guages in NP that haven’t been placed in either category.
Here, we consider the language that is complementary to ISO—namely, the language NONISO = {⟨G, H ⟩| G and H are not isomorphic graphs}. NONISO is not known to be in NP because we don’t know how to provide short certificates that graphs aren’t isomorphic. Nonetheless, when two graphs aren’t isomorphic, a Prover can convince a Verifier of this fact, as we will show.
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416 CHAPTER 10 / ADVANCED TOPICS IN COMPLEXITY THEORY
Suppose that we have two graphs: G1 and G2. If they are isomorphic, the Prover can convince the Verifier of this fact by presenting the isomorphism or reordering. But if they aren’t isomorphic, how can the Prover convince the Verifier of that fact? Don’t forget: The Verifier doesn’t necessarily trust the Prover, so it isn’t enough for the Prover to declare that they aren’t isomorphic. The Prover must convince the Verifier. Consider the following short protocol.
The Verifier randomly selects either G1 or G2 and then randomly reorders its nodes to obtain a graph H . The Verifier sends H to the Prover. The Prover must respond by declaring whether G1 or G2 was the source of H. That concludes the protocol.
If G1 and G2 were indeed nonisomorphic, the Prover could always carry out the protocol because the Prover could identify whether H came from G1 or G2. However, if the graphs were isomorphic, H might have come from either G1 or G2. So even with unlimited computational power, the Prover would have no better than a 50–50 chance of getting the correct answer. Thus, if the Prover is able to answer correctly consistently (say in 100 repetitions of the protocol), the Verifier has convincing evidence that the graphs are actually nonisomorphic.
DEFINITION OF THE MODEL
To define the interactive proof system model formally, we describe the Verifier, the Prover, and their interaction. You’ll find it helpful to keep the graph non- isomorphism example in mind. We define the Verifier to be a function V that computes its next transmission to the Prover from the message history sent so far. The function V has three inputs:
1. Input string. The objective is to determine whether this string is a mem- ber of some language. In the NONISO example, the input string encoded the two graphs.
2. Random input. For convenience in making the definition, we provide the Verifier with a randomly chosen input string instead of the equivalent capability to make probabilistic moves during its computation.
3. Partial message history. A function has no memory of the dialog that has been sent so far, so we provide the memory externally via a string representing the exchange of messages up to the present point. We use the notation m1#m2# · · · #mi to represent the exchange of messages m1 through mi.
The Verifier’s output is either the next message mi+1 in the sequence or accept or reject, designating the conclusion of the interaction. Thus, V has the func- t i o n a l f o r m V : Σ ∗ × Σ ∗ × Σ ∗ −→ Σ ∗ ∪ { a c c e p t , r e j e c t } .
V(w,r,m1#···#mi) = mi+1 means that the input string is w, the random input is r, the current message history is m1 through mi, and the Verifier’s next message to the Prover is mi+1.
The Prover is a party with unlimited computational ability. We define it to be a function P with two inputs:
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10.4 INTERACTIVE PROOF SYSTEMS 417
1. Input string
2. Partial message history
The Prover’s output is the next message to the Verifier. Formally, P has the form P : Σ ∗ × Σ ∗ −→ Σ ∗ .
P (w, m1# · · · #mi) = mi+1 means that the Prover sends mi+1 to the Verifier after having exchanged messages m1 through mi so far.
Next, we define the interaction between the Prover and the Verifier. For par- ticular strings w and r, we write (V ↔P )(w, r) = accept if a message sequence m1 through mk exists for some k whereby
1. for0≤i
10.17 Prove that if A is a language in L, a family of branching programs (B1 , B2 , . . .) exists wherein each Bn accepts exactly the strings in A of length n and is bounded in size by a polynomial in n.
10.18 Prove that if A is a regular language, a family of branching programs (B1 , B2 , . . .) exists wherein each Bn accepts exactly the strings in A of length n and is bounded in size by a constant times n.
10.19 ShowthatifNP⊆BPP,thenNP=RP.
10.20 Define a ZPP-machine to be a probabilistic Turing machine that is permitted
three types of output on each of its branches: accept, reject, and ?. A ZPP-machine
M decides a language A if M outputs the correct answer on every input string w
(accept if w ∈ A and reject if w ̸∈ A) with probability at least 2 , and M never 3
outputs the wrong answer. On every input, M may output ? with probability at most 1 . Furthermore, the average running time over all branches of M on w must
3
be bounded by a polynomial in the length of w. Show that RP ∩ coRP = ZPP, where ZPP is the collection of languages that are recognized by ZPP-machines.
10.21 Let EQ BP = {⟨B1 , B2 ⟩| B1 and B2 are equivalent branching programs}. Show that EQBP is coNP-complete.
10.22 Let BPL be the collection of languages that are decided by probabilistic log space Turing machines with error probability 1 . Prove that BPL ⊆ P.
3
10.23 Let CNFH = {⟨φ⟩| φ is a satisfiable cnf-formula where each clause contains any number of literals, but at most one negated literal}. Problem 7.25 asked you to show that CNFH ∈ P. Now give a log-space reduction from CIRCUIT-VALUE to CNFH to conclude that CNFH is P-complete.
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SELECTED SOLUTIONS
10.7 If M is a probabilistic TM that runs in polynomial time, we can modify M so that it makes exactly nr coin tosses on each branch of its computation, for some con- stant r. Thus, the problem of determining the probability that M accepts its input string reduces to counting how many branches are accepting and comparing this number with 2 2(nr ) . This count can be performed by using polynomial space.
SELECTED SOLUTIONS 441
3
10.16 Call a a witness if it fails the Fermat test for p; that is, if ap−1 ̸≡ 1 (mod p).
Let Zp∗ be all numbers in {1,…,p − 1} that are relatively prime to p. If p isn’t
pseudoprime, it has a witness a in Zp∗.
Use a to get many more witnesses. Find a unique witness in Zp∗ for each non- witness. If d ∈ Zp∗ is a nonwitness, you have dp−1 ≡ 1 (mod p). Hence (da mod p)p−1 ̸≡ 1 (mod p) and so da mod p is a witness. If d1 and d2 are distinct nonwitnesses in Zp∗, then d1a mod p ̸= d2a mod p. Otherwise, (d1 − d2)a ≡ 0 (mod p), and thus (d1 − d2)a = cp for some integer c. But d1 and d2 are in Zp∗, and thus (d1 − d2) < p, so a = cp/(d1 − d2) and p have a factor greater than 1 in common, which is impossible because a and p are relatively prime. Thus, the number of witnesses in Zp∗ must be as large as the number of nonwitnesses in Zp∗ , and consequently at least half of the members of Zp∗ are witnesses.
Next, show that every member b of Zp+ that is not relatively prime to p is a witness. If b and p share a factor, then be and p share that factor for any e > 0. Hence bp−1 ̸≡ 1 (mod p). Therefore, you can conclude that at least half of the members of Zp+ are witnesses.
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447
N (natural numbers), 4, 255
R (real numbers), 185, 205
R+ (nonnegative real numbers), 277 ∅ (empty set), 4
∈ (element), 4
̸∈ (not element), 4
⊆ (subset), 4
(proper subset), 4
⇐⇒ (logical equivalence), 18
◦ (concatenation operation), 44
∗ (star operation), 44
+
Σ (alphabet), 53
Σε (Σ ∪ {ε}), 53
⟨·⟩ (encoding), 185, 287
␣ (blank), 168
≤m (mapping reduction), 235
≤T (Turing reduction), 261
≤L (log space reduction), 352
≤P (polynomial time reduction), 300 d(x) (minimal description), 264 Th(M) (theory of model), 255
K(x) (descriptive complexity), 264
∀ (universal quantifier), 338
∃ (existential quantifier), 338
↑ (exponentiation), 371
O(f(n)) (big-O notation), 277–278 o(f(n)) (small-o notation), 278
∪ (union operation), 4, 44
∩ (intersection operation), 4
× (Cartesian or cross product), 6 Z (integers), 4
ε (empty string), 14
wR (reverse of w), 14
¬ (negation operation), 14
∧ (conjunction operation), 14
∨ (disjunction operation), 14
⊕ (exclusive O R operation), 15 → (implication operation), 15
↔ (equality operation), 15
⇐ (reverse implication), 18
⇒ (implication), 18
Index
448
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(plus operation), 65 P(Q) (power set), 53
Accept state, 34, 35 Acceptance problem
for CFG, 198 for DFA, 194 for LBA, 222 for NFA, 195 for TM, 202
Accepting computation history, 221 Accepting configuration, 169 Accepts a language, meaning of, 36 ACFG, 198
Acyclic graph, 404
ADFA, 194
Adjacency matrix, 287 Adleman, Leonard M., 443, 446 Agrawal, Manindra, 443
Aho, Alfred V., 443, 447 Akl, Selim G., 443 ALBA, 222
Algorithm
complexity analysis, 276–281 decidability and undecidability,
193–210 defined, 182–184
describing, 184–187 Euclidean, 289 polynomial time, 284–291 running time, 276
ALLCFG, 225
Allen, Robin W., 444
Alon, Noga, 443
Alphabet, defined, 13 Alternating Turing machine, 409 Alternation, 408–414
Ambiguity, 107–108
Ambiguous
NFA, 212
grammar, 107, 240 Amplification lemma, 397
AND operation, 14
ANFA, 195
Angluin, Dana, 443
Anti-clique, 28
Approximation algorithm, 393–395 AREX, 196
Argument, 8
Arithmetization, 422
Arity, 8, 253
Arora, Sanjeev, 443 ASPACE(f(n)), 410
Asymptotic analysis, 276 Asymptotic notation
big-O notation, 277–278
small-o notation, 278 Asymptotic upper bound, 277 ATIME(t(n)), 410
ATM, 202
Atomic formula, 253 Automata theory, 3, see also
Context-free language;
Regular language Average-case analysis, 276
Baase, Sara, 443
Babai, Laszlo, 443
Bach, Eric, 443
Balca ́zar, Jose ́ Luis, 444 Basis of induction, 23 Beame, Paul W., 444 Big-O notation, 276–278 Bijective function, 203 Binary function, 8 Binary operation, 44 Binary relation, 9 Bipartite graph, 360 Blank symbol ␣, 168 Blum, Manuel, 444 Boolean circuit, 379–387
depth, 428
gate, 380
size, 428
uniform family, 428 wire, 380
Boolean formula, 299, 338 minimal, 328, 377, 411, 414 quantified, 339
Boolean logic, 14–15
Boolean matrix multiplication, 429 Boolean operation, 14, 253, 299 Boolean variable, 299
Bound variable, 338
Branching program, 404
read-once, 405 Brassard, Gilles, 444
Bratley, Paul, 444
Breadth-first search, 284
Brute-force search, 285, 288, 292, 298
Cantor, Georg, 202 Carmichael number, 400
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INDEX 449
450 INDEX
Carmichael, R. D., 444 Cartesian product, 6, 46 CD-ROM, 349 Certificate, 293
CFG, see Context-free grammar CFL, see Context-free language Chaitin, Gregory J., 264 Chandra, Ashok, 444 Characteristic sequence, 206 Checkers, game of, 348 Chernoff bound, 398
Chess, game of, 348
Chinese remainder theorem, 401 Chomsky normal form, 108–111, 157,
198, 291 Chomsky, Noam, 444
Church, Alonzo, 3, 183, 255 Church–Turing thesis, 183–184, 281 CIRCUIT-SAT, 386 Circuit-satisfiability problem, 386 CIRCUIT-VALUE, 432
Circular definition, 65 Clause, 302
Clique, 28, 296 CLIQUE, 296
Closed under, 45
Closure under complementation
context-free languages, non-, 154 deterministic context-free
languages, 133 P, 322
regular languages, 85 Closure under concatenation
context-free languages, 156 NP, 322
P, 322
regular languages, 47, 60
Closure under intersection context-free languages, non-, 154 regular languages, 46
Closure under star
context-free languages, 156 NP, 323
P, 323
regular languages, 62
Closure under union context-free languages, 156 NP, 322
P, 322
regular languages, 45, 59
CNF-formula, 302 Co-Turing-recognizable language, 209 Cobham, Alan, 444
Coefficient, 183
Coin-flip step, 396
Complement operation, 4
Completed rule, 140
Complexity class
ASPACE(f(n)), 410 ATIME(t(n)), 410 BPP, 397
coNL, 354
coNP, 297 EXPSPACE, 368 EXPTIME, 336
IP, 417
L, 349
NC, 430
NL, 349
NP, 292–298 NPSPACE, 336 NSPACE(f(n)), 332 NTIME(f(n)), 295 P, 284–291, 297–298 PH, 414
PSPACE, 336
RP, 403 SPACE(f(n)), 332 TIME(f(n)), 279 ZPP, 440
Complexity theory, 2 Composite number, 293, 399 Compositeness witness, 401 COMPOSITES, 293 Compressible string, 267 Computability theory, 3
decidability and undecidability, 193–210
recursion theorem, 245–252 reducibility, 215–239 Turing machines, 165–182
Computable function, 234 Computation history
context-free languages, 225–226 defined, 220
linear bounded automata,
221–225
Post Correspondence Problem,
227–233 reducibility, 220–233
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Computational model, 31
Computer virus, 250
Concatenation of strings, 14 Concatenation operation, 44, 47, 60–61 Configuration, 168, 169, 350 Conjunction operation, 14
Conjunctive normal form, 302 coNL, 354
Connected graph, 12, 185 coNP, 297
Context-free grammar ambiguous, 107, 240
defined, 104 Context-free language
decidability, 198–200 defined, 103
deterministic, 131
efficient decidability, 290–291 inherently ambiguous, 108 pumping lemma, 125–130
Cook, Stephen A., 299, 387, 430, 444 Cook–Levin theorem, 299–388 Cormen, Thomas, 444
Corollary, 17
Correspondence, 203 Countable set, 203 Counterexample, 18 Counting problem, 420 Cross product, 6 Cryptography, 433–439 Cut edge, 395
Cut, in a graph, 325, 395 Cycle, 12
Davis, Martin, 183
DCFG, see Deterministic context-free
grammar
Decidability, see also Undecidability
context-free language, 198–200 of ACFG, 198
of ADFA, 194
of AREX, 196
of ECFG, 199
of EQDFA, 197
regular language, 194–198
Decidable language, 170 Decider
deterministic, 170
nondeterministic, 180 Decision problem, 394
Definition, 17
Degree of a node, 10
DeMorgan’s laws, example of proof, 20 Depth complexity, 428
Derivation, 102
leftmost, 108 Derives, 104
Descriptive complexity, 264 Deterministic computation, 47 Deterministic context-free grammar,
139
Deterministic context-free language
defined, 131
properties, 133 Deterministic finite automaton
acceptance problem, 194 defined, 35
emptiness testing, 196 minimization, 327
Deterministic pushdown automaton, 131
defined, 130
DFA, see Deterministic finite automaton Diagonalization method, 202–209 D ́ıaz, Josep, 444
Difference hierarchy, 328
Digital signatures, 435
Directed graph, 12
Directed path, 13
Disjunction operation, 14
Distributive law, 15
DK-test, 143
DK1-test, 152
Domain of a function, 7
Dotted rule, 140
DPDA, see Deterministic pushdown
automaton Dynamic programming, 290
ECFG , 199
EDFA,196
Edge of a graph, 10
Edmonds, Jack, 444
ELBA , 223
Element distinctness problem, 175 Element of a set, 3
Emptiness testing
for CFG, 199 for DFA, 196 for LBA, 223
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INDEX 451
452 INDEX
for TM, 217 Empty set, 4
Empty string, 14 Encoding, 185, 287 Enderton, Herbert B., 444 Endmarked language, 134 Enumerator, 180–181 EQCFG, 200
EQDFA, 197
EQREX↑, 372
EQTM
Turing-unrecognizability, 238
undecidability, 220 Equality operation, 15 Equivalence relation, 9 Equivalent machines, 54 Erdo ̈ s, Paul, 443
Error probability, 397
ETM, 211
ETM, undecidability, 217
Euclidean algorithm, 289
Even, Shimon, 444
EXCLUSIVE OR operation, 15 Existential state, 409
Exponential bound, 278 Exponential, versus polynomial, 285 EXPSPACE, 368 EXPSPACE-completeness, 371–376 EXPTIME, 336
Factor of a number, 399 Feller, William, 444 Fermat test, 400
Fermat’s little theorem, 399 Feynman, Richard P., 444 Final state, 35
Finite automaton
automatic door example, 32 computation of, 40 decidability, 194–198 defined, 35
designing, 41–44
transition function, 35 two-dimensional, 241 two-headed, 240
Finite state machine, see Finite automaton
Finite state transducer, 87 Fixed point theorem, 251 Forced handle, 138
Formal proof, 258 Formula, 253, 299 FORMULA-GAME, 342 Fortnow, Lance, 446 Free variable, 253
FST, see Finite state transducer Function, 7–10
argument, 8
binary, 8
computable, 234
domain, 7
one-to-one, 203
one-way, 436
onto, 7, 203
polynomial time computable, 300 range, 7
space constructible, 364 time constructible, 368 transition, 35
unary, 8
Gabarro ́ , Joaquim, 444
Gadget in a completeness proof, 311 Game, 341
Garey, Michael R., 444
Gate in a Boolean circuit, 380 Generalized geography, 344 Generalized nondeterministic finite
automaton, 70–76 converting to a regular
expression, 71 defined, 70, 73
Geography game, 343
GG (generalized geography), 345 Gill, John T., 444
GNFA, see Generalized
nondeterministic finite
automaton GO, game of, 348
Go-moku, game of, 358
Go ̈ del, Kurt, 3, 255, 258, 444 Goemans, Michel X., 444 Goldwasser, Shafi, 445 Graph
acyclic, 404 coloring, 325 cycle in, 12 degree, 10 directed, 12 edge, 10
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isomorphism problem, 323, 415 k-regular, 21
labeled, 11
node, 10
strongly connected, 13 sub-, 11
undirected, 10
vertex, 10
Greenlaw, Raymond, 445
Halting configuration, 169 Halting problem, 216–217
unsolvability of, 216 HALTTM, 216
Hamiltonian path problem, 292 exponential time algorithm, 292 NP-completeness of, 314–319 polynomial time verifier, 293
HAMPATH, 292, 314 Handle, 136
forced, 138 Harary, Frank, 445
Hartmanis, Juris, 445
Hey, Anthony J. G., 444 Hierarchy theorem, 364–371
space, 365
time, 369
High-level description of a Turing
machine, 185 Hilbert, David, 182, 445
Hofstadter, Douglas R., 445 Hoover, H. James, 444, 445 Hopcroft, John E., 443, 445, 447 Huang, Ming-Deh A., 443
iff, 18
Immerman, Neil, 445 Implementation description of a
Turing machine, 185 Implication operation, 15
Incompleteness theorem, 258 Incompressible string, 267 Indegree of a node, 12 Independent set, 28 Induction
basis, 23
proof by, 22–25 step, 23
Induction hypothesis, 23 Inductive definition, 65
Infinite set, 4
Infix notation, 8
Inherent ambiguity, 108 Inherently ambiguous context-free
language, 108 Injective function, 203
Integers, 4
Interactive proof system, 415–427 Interpretation, 254
Intersection operation, 4
IP, 417
ISO, 415
Isomorphic graphs, 323
Johnson, David S., 444, 445
k-ary function, 8
k-ary relation, 9
k-clique, 295
k-optimal approximation algorithm,
395 k-tuple, 6
Karloff, Howard, 446
Karp, Richard M., 445 Kayal, Neeraj, 443
Knuth, Donald E., 139, 445 Kolmogorov, Andrei N., 264
L, 349
Labeled graph, 11 Ladder, 358 Language
co-Turing-recognizable, 209 context-free, 103
decidable, 170
defined, 14
deterministic context-free, 131 endmarked, 134
of a grammar, 103
recursively enumerable, 170 regular, 40
Turing-decidable, 170 Turing-recognizable, 170 Turing-unrecognizable, 209
Lawler, Eugene L., 445
LBA, see Linear bounded automaton Leaf in a tree, 12
Leeuwen, Jan van, 447
Leftmost derivation, 108
Leighton, F. Thomson, 445
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INDEX 453
454 INDEX
Leiserson, Charles E., 444 Lemma, 17
Lenstra, Jan Karel, 445 Leveled graph, 361
Levin, Leonid A., 299, 387, 445 Lewis, Harry, 445
Lexical analyzer, 66 Lexicographic order, 14
Li, Ming, 445
Lichtenstein, David, 446
Linear bounded automaton, 221–225 Linear time, 281
Lipton, Richard J., 445
LISP, 182
Literal, 302
Log space computable function, 352 Log space reduction, 352, 432
Log space transducer, 352 Lookahead, 152
LR(k) grammar, 152
Luby, Michael, 446
Lund, Carsten, 443, 446
Majority function, 391 Many–one reducibility, 234 Mapping, 7
Mapping reducibility, 234–239
polynomial time, 300 Markov chain, 33
Match, 227
Matijasevic ̆, Yuri, 183 MAX-CLIQUE, 328, 389 MAX-CUT, 325
Maximization problem, 395
Member of a set, 3
Micali, Silvio, 445
Miller, Gary L., 446 MIN-FORMULA, 328, 359, 377, 411,
414 Minesweeper, 326
Minimal description, 264
Minimal formula, 328, 359, 377, 411,
414 Minimization of a DFA, 327
Minimization problem, 394 Minimum pumping length, 91 MIN TM , 251, 270
Model, 254
MODEXP, 323 Modulo operation, 8
Motwani, Rajeev, 443
Multiset, 4, 297
Multitape Turing machine, 176–178 Myhill–Nerode theorem, 91
Natural numbers, 4
NC, 430
Negation operation, 14
NFA, see Nondeterministic finite
automaton Nim, game of, 359 Nisan, Noam, 446
Niven, Ivan, 446
NL, 349
NL-complete problem
PATH, 350 NL-completeness defined, 352
Node of a graph, 10 degree, 10
indegree, 12
outdegree, 12 Nondeterministic computation, 47 Nondeterministic finite automaton,
47–58 computation by, 48
defined, 53
equivalence with deterministic
finite automaton, 55 equivalence with regular
expression, 66 Nondeterministic polynomial time, 294
Nondeterministic Turing machine, 178–180
space complexity of, 332
time complexity of, 283 NONISO, 415
NOT operation, 14 NP, 292–298 NP-complete problem
3SAT, 302, 387 CIRCUIT-SAT, 386 HAMPATH, 314 SUBSET-SUM, 320 3COLOR, 325 UHAMPATH, 319 VERTEX-COVER, 312
NP-completeness, 299–322 defined, 304
NP-hard, 326
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NP-problem, 294
NPA, 376
NPSPACE, 336
NSPACE(f(n)), 332 NTIME(f(n)), 295
NTM, see Nondeterministic Turing
machine
o(f (n)) (small-o notation), 278 One-sided error, 403 One-time pad, 434 One-to-one function, 203 One-way function, 436 One-way permutation, 436 Onto function, 7, 203
Optimal solution, 394 Optimization problem, 393 OR operation, 14
Oracle, 260, 376
Oracle tape, 376
Ordered pair, 6
Outdegree of a node, 12
P, 284–291, 297–298 P-complete problem
CIRCUIT-VALUE, 432 P-completeness, 432
PA, 376 Pair
ordered, 6
unordered, 4 Palindrome, 90, 155
Papadimitriou, Christos H., 445, 446 Parallel computation, 427–432 Parallel random access machine, 428 Parity function, 381
Parse tree, 102 Parser, 101 Pascal, 182 Path
Hamiltonian, 292 in a graph, 12 simple, 12
PATH, 287, 350
PCP, see Post Correspondence Problem PDA, see Pushdown automaton
Perfect shuffle operation, 89, 158
PH, 414
Pigeonhole principle, 78, 79, 126 Pippenger, Nick, 430
Polynomial, 182 Polynomial bound, 278 Polynomial time
algorithm, 284–291 computable function, 300 hierarchy, 414
verifier, 293
Polynomial verifiability, 293 Polynomial, versus exponential, 285 Polynomially equivalent models, 285 Pomerance, Carl, 443, 446
Popping a symbol, 112
Post Correspondence Problem (PCP),
227–233 modified, 228
Power set, 6, 53
PRAM, 428
Pratt, Vaughan R., 446
Prefix notation, 8
Prefix of a string, 14, 89 Prefix-free language, 14, 212 Prenex normal form, 253, 339 Prime number, 293, 324, 399 Private-key cryptosystem, 435 Probabilistic algorithm, 396–408 Probabilistic function, 436 Probabilistic Turing machine, 396 Processor complexity, 428 Production, 102
Proof, 17
by construction, 21
by contradiction, 21–22 by induction, 22–25 finding, 17–20 necessity for, 77
Proper subset, 4
Prover, 416
Pseudoprime, 400 PSPACE, 336 PSPACE-complete problem
FORMULA-GAME, 342 GG, 345
TQBF, 339
PSPACE-completeness, 337–348 defined, 337
PSPACE-hard, 337 Public-key cryptosystem, 435 Pumping lemma
for context-free languages, 125–130
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INDEX 455
456 INDEX
for regular languages, 77–82
Pumping length, 77, 91, 125 Pushdown automaton, 111–124
context-free grammars, 117–124 defined, 113
deterministic, 131
examples, 114–116
schematic of, 112 Pushing a symbol, 112 Putnam, Hilary, 183 PUZZLE, 325, 359
Quantified Boolean formula, 339 Quantifier, 338
in a logical sentence, 253 Query node in a branching program,
404
Rabin, Michael O., 446
Rackoff, Charles, 445
Ramsey’s theorem, 28
Range of a function, 7
Read-once branching program, 405 Real number, 204
Recognizes a language, meaning of, 36, 40
Recursion theorem, 245–252 fixed-point version, 251 terminology for, 249
Recursive language, see Decidable language
Recursively enumerable, see Turing-recognizable
Recursively enumerable language, 170 Reduce step, 135
Reducibility, 215–239
mapping, 234–239 polynomial time, 300
via computation histories,
220–233 Reducing string, 135
Reduction
between problems, 215 function, 235
mapping, 235
reversed derivation, 135 Turing, 261
Reflexive relation, 9 Regular expression, 63–76
defined, 64
equivalence to finite automaton, 66–76
examples of, 65 Regular language, 31–82
closure under concatenation, 47, 60
closure under intersection, 46 closure under star, 62
closure under union, 45, 59 decidability, 194–198 defined, 40
Regular operation, 44 REGULARTM, 218
Reingold, Omer, 446
Rejecting computation history, 221 Rejecting configuration, 169 Relation, 9, 253
binary, 9
Relatively prime, 288
Relativization, 376–379
RELPRIME, 289
Reverse of a string, 14
Rice’s theorem, 219, 241, 243, 270, 272 Rinooy Kan, A. H. G., 445
Rivest, Ronald L., 444, 446
Robinson, Julia, 183
Roche, Emmanuel, 446
Root
in a tree, 12
of a polynomial, 183
Rule in a context-free grammar, 102,
104
Rumely, Robert S., 443
Ruzzo, Walter L., 445
SAT, 304, 336
#SAT, 420
Satisfiability problem, 299 Satisfiable formula, 299 Savitch’s theorem, 333–335 Saxena, Nitin, 443 Schabes, Yves, 446 Schaefer, Thomas J., 446 Scope, 338
Scope, of a quantifier, 253 Secret key, 433
Sedgewick, Robert, 446 Self-loop, 10 Self-reference, 246 Sentence, 339
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Sequence, 6
Sequential computer, 427 Set, 3
countable, 203
uncountable, 204 Sethi, Ravi, 443
Shallit, Jeffrey, 443 Shamir, Adi, 446
Shen, Alexander, 446 Shmoys, David B., 445 Shor, Peter W., 446 Shortlex order, 14
Shuffle operation, 89, 158 Simple path, 12 Singleton set, 4
Sipser, Michael, 446, 447 Size complexity, 428 Small-o notation, 278 SPACE(f(n)), 332
Space complexity, 331–361 Space complexity class, 332 Space complexity of
nondeterministic Turing machine, 332
Space constructible function, 364 Space hierarchy theorem, 365 Spencer, Joel H., 443
Stack, 111
Star operation, 44, 62–63, 323 Start configuration, 169
Start state, 34
Start variable, in a context-free
grammar, 102, 104 State diagram
finite automaton, 34 pushdown automaton, 114 Turing machine, 172
Stearns, Richard E., 445 Steiglitz, Kenneth, 446 Stinson, Douglas R., 447 String, 14
String order, 14
Strongly connected graph, 13, 360 Structure, 254
Subgraph, 11
Subset of a set, 4
SUBSET-SUM, 296, 320 Substitution rule, 102
Substring, 14
Sudan, Madhu, 443
Surjective function, 203 Symmetric difference, 197 Symmetric relation, 9 Synchronizing sequence, 92 Szegedy, Mario, 443 Szelepcze ́ nyi, Ro ́ bert, 447
Tableau, 383
Tarjan, Robert E., 447
Tautology, 410
Term, in a polynomial, 182 Terminal, 102
Terminal in a context-free grammar,
104 Th(M), 255 Theorem, 17
Theory, of a model, 255 3COLOR, 325
3SAT , 302, 387 Tic-tac-toe, game of, 357 TIME(f(n)), 279
Time complexity, 275–322 analysis of, 276–281
of nondeterministic Turing machine, 283
Time complexity class, 295 Time constructible function, 368 Time hierarchy theorem, 369 TM, see Turing machine
TQBF, 339
Transducer
finite state, 87
log space, 352 Transition, 34
Transition function, 35 Transitive closure, 429 Transitive relation, 9 Trapdoor function, 438 Tree, 12
leaf, 12 parse, 102 root, 12
Triangle in a graph, 323 Tuple, 6
Turing machine, 165–182
alternating, 409 comparison with finite
automaton, 166 defined, 168
describing, 184–187
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INDEX 457
458 INDEX examples of, 170–175
marking tape symbols, 174 multitape, 176–178 nondeterministic, 178–180 oracle, 260, 376
schematic of, 166
universal, 202
Turing reducibility, 260–261 Turing, Alan M., 3, 165, 183, 447 Turing-decidable language, 170 Turing-recognizable language, 170 Turing-unrecognizable language,
209–210 EQTM, 238
Two-dimensional finite automaton, 241 Two-headed finite automaton, 240 2DFA, see Two-headed finite automaton 2DIM-DFA, see Two-dimensional finite
automaton 2SAT, 327
Ullman, Jeffrey D., 443, 445, 447 Unary
alphabet, 52, 82, 240 function, 8
notation, 287, 323 operation, 44
Uncountable set, 204 Undecidability
diagonalization method, 202–209 of ATM, 202
of ELBA, 223
of EQTM, 220
of ETM, 217
of HALTTM, 216
of REGULARTM, 219
of EQCFG, 200
of Post Correspondence Problem,
228
via computation histories,
220–233 Undirected graph, 10
Union operation, 4, 44, 45, 59–60 Unit rule, 109
Universal quantifier, 338 Universal state, 409
Universal Turing machine, 202 Universe, 253, 338
Unordered pair, 4
Useless state
in PDA, 212 in TM, 239
Valiant, Leslie G., 443 Valid string, 136 Variable
Boolean, 299
bound, 338
in a context-free grammar, 102,
104 start, 102, 104
Venn diagram, 4 Verifier, 293, 416 Vertex of a graph, 10 VERTEX-COVER, 312 Virus, 250
Vitanyi, Paul, 445
Wegman, Mark, 444 Well-formed formula, 253 Williamson, David P., 444 Window, in a tableau, 307 Winning strategy, 342
Wire in a Boolean circuit, 380 Worst-case analysis, 276
XOR operation, 15, 383
Yannakakis, Mihalis, 446 Yields
for configurations, 169
for context-free grammars, 104
ZPP, 440
Zuckerman, Herbert S., 446
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