CS 332: Theory of Computation Prof. Mark Bun Boston University March 15, 2021 Homework 6 – Due *Friday*, March 19, 2021 at 11:59 PM Reminder Collaboration is permitted, but you must write the solutions by yourself without as- sistance, and be ready to explain them orally to the course staff if asked. You must also […]
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Microsoft PowerPoint – CS332-Lec22-ann BU CS 332 – Theory of Computation Lecture 22: • Nondeterministic time, NP • NP‐completeness Reading: Sipser Ch 7.3‐7.4 Mark Bun April 14, 2021 Big‐Oh, formally means: There exist constants such that for every Example: Show that 4/14/2021 CS332 ‐ Theory of Computation 2 Big‐Oh, formally means: There exist constants such that for every Example: Show that 4/14/2021 CS332 ‐ Theory of Computation 3 Nondeterministic time Let A NTM runs in time if on every input , halts on within at most steps on every computational branch is a class (i.e., set) of languages: A language if there exists an NTM that 1) Decides
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PowerPoint Presentation BU CS 332 – Theory of Computation Lecture 16: • Examples of Reductions • Test 2 Review Reading: Sipser Ch 5.1 Mark Bun March 15, 2021 Reductions A reduction from problem 𝐴𝐴 to problem 𝐵𝐵 is an algorithm for problem 𝐴𝐴 which uses an algorithm for problem 𝐵𝐵 as a subroutine If such
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#include #include #include #include #include #include “err.h” using namespace std; /// Run the AES symmetric encryption/decryption algorithm on a buffer of bytes. /// Note that this will do either encryption or decryption, depending on how the /// provided CTX has been configured. After calling, the CTX cannot be used /// again until it is reset.
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PowerPoint Presentation BU CS 332 – Theory of Computation Lecture 20: • Time/Space Hierarchies • Complexity Class P Reading: Sipser Ch 9.1, 7.2 Mark Bun April 7, 2021 Last Time • Asymptotic notation • Analyzing time / space usage of Turing machines (algorithms) • Multi-tape TMs can solve problems faster than single- tape TMs 4/7/2021
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Microsoft PowerPoint – CS332-Lec19-ann BU CS 332 – Theory of Computation Lecture 19: • Asymptotic Notation • Time/Space Complexity Reading: Sipser Ch 7.1, 8.0 Mark Bun April 5, 2021 Where we are in CS 332 4/5/2021 CS332 ‐ Theory of Computation 2 Automata Computability Complexity Previous unit: Computability theory What kinds of problems can / can’t computers solve? Final unit: Complexity theory What kinds of problems can / can’t computers solve under constraints on their computational resources? Time and space complexity Today: Start answering the basic questions 1. How do we measure complexity? (as in CS 330) 2. Asymptotic notation (as in CS 330) 3. How robust is the TM model when we care about measuring complexity? 4. How do we mathematically capture our intuitive notion of “efficient algorithms”? 4/5/2021 CS332 ‐ Theory of Computation 3 Time and space complexity Time complexity of a TM = Running time of an algorithm = Max number of steps as a function of input length Space complexity of a TM = Memory usage of algorithm = Max number of tape cells as a function of input length 4/5/2021
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CS 332: Theory of Computation Prof. Mark Bun Boston University February 9, 2021 Limitations of Finite Automata So far in this course, we’ve studied a number of properties of regular languages. First, they are quite robust, with different characterizations based on DFAs, NFAs, and regular expressions. Second, they enjoy strong closure properties: unions, concatenations, stars,
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PowerPoint Presentation BU CS 332 – Theory of Computation Lecture 21: • Complexity Class P • Nondeterministic time, NP Reading: Sipser Ch 7.2, 7.3 Mark Bun April 12, 2021 Complexity class P Definition: P is the class of languages decidable in polynomial time on a basic single-tape (deterministic) TM P = ⋃𝑘𝑘=1 ∞ TIME(𝑛𝑛𝑘𝑘) •
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Microsoft PowerPoint – CS332-Lec15-ann BU CS 332 – Theory of Computation Lecture 15: • Review mid‐semester feedback • Reductions Reading: Sipser Ch 5.1 Mark Bun March 15, 2021 What helps you learn best? • Lectures in general (13) • In‐class examples / walkthroughs (11) • Interaction in lecture, polls (7) • Discussion sections in general (7) • Gradescope check‐ins (5) • Homework – useful, appropriate length/difficulty (5) • Automata Tutor, TM simulator (4) • Office hours (4) • Course organization, perspective (3) • Annotated slides (3) • Piazza use (3) • Homework feedback • Reading 3/15/2021 CS332 ‐ Theory of Computation 2 What hinders your learning? • Remote format, COVID / Zoom fatigue (7)
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CS 332: Theory of Computation Prof. Mark Bun Boston University April 7, 2021 Homework 7 – Due Friday, April 16, 2021 at 11:59 AM Reminder Collaboration is permitted, but you must write the solutions by yourself without as- sistance, and be ready to explain them orally to the course staff if asked. You must also
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