Essentials of Compilation
An Incremental Approach
Jeremy G. Siek, Ryan R. Newton
Indiana University
with contributions from: Carl Factora Andre Kuhlenschmidt Michael M. Vitousek Michael Vollmer Ryan Scott Cameron Swords
January 29, 2019
ii
This book is dedicated to the programming language wonks at Indiana University.
iv
Contents
1 Preliminaries 5
1.1 AbstractSyntaxTreesandS-expressions. . . . . . . . . . . . 5 1.2 Grammars……………………….. 7 1.3 PatternMatching……………………. 9 1.4 Recursion ……………………….. 10 1.5 Interpreters ………………………. 12 1.6 ExampleCompiler: aPartialEvaluator . . . . . . . . . . . . 14
2 Integers and Variables 17
2.1 TheR1Language……………………. 17
2.2 Thex86AssemblyLanguage ……………… 20
2.3 Planningthetriptox86viatheC0 language . . . . . . . . . 24
2.3.1 TheC0IntermediateLanguage …………. 27
2.3.2 Thedialectsofx86……………….. 28
2.4 UniquifyVariables …………………… 28
2.5 RemoveComplexOperatorsandOperands . . . . . . . . . . 30
2.6 ExplicateControl……………………. 31
2.7 UncoverLocals …………………….. 32
2.8 SelectInstructions …………………… 32
2.9 AssignHomes……………………… 33
2.10PatchInstructions …………………… 34 2.11Printx86 ……………………….. 35
3 Register Allocation 37
3.1 RegistersandCallingConventions …………… 38 3.2 LivenessAnalysis……………………. 39 3.3 BuildingtheInterferenceGraph…………….. 40 3.4 GraphColoringviaSudoku ………………. 42 3.5 Printx86andConventionsforRegisters . . . . . . . . . . . . 48
v
vi
4
CONTENTS
3.6 Challenge:MoveBiasing∗ ……………….. 48
Booleans and Control Flow 53
4.1 TheR2Language……………………. 54
4.2 TypeCheckingR2Programs ……………… 55
4.3 ShrinktheR2Language………………… 58
4.4 XOR,Comparisons,andControlFlowinx86 . . . . . . . . . 58
4.5 TheC1IntermediateLanguage …………….. 60
4.6 ExplicateControl……………………. 61
4.7 SelectInstructions …………………… 64
4.8 RegisterAllocation…………………… 65
4.8.1 LivenessAnalysis………………… 65
4.8.2 BuildInterference ……………….. 66
4.9 PatchInstructions …………………… 67
4.10AnExampleTranslation………………… 67 4.11Challenge:OptimizeJumps∗ ……………… 67
Tuples and Garbage Collection 71
5.1 TheR3Language……………………. 72
5.2 GarbageCollection…………………… 73 5.2.1 Graph Copying via Cheney’s Algorithm . . . . . . . . 76 5.2.2 DataRepresentation………………. 78 5.2.3 Implementation of the Garbage Collector . . . . . . . 81
5.3 ExposeAllocation …………………… 83
5.4 ExplicateControlandtheC2 language. . . . . . . . . . . . . 84
5.5 UncoverLocals …………………….. 85
5.6 SelectInstructions …………………… 87
5.7 RegisterAllocation…………………… 90
5.8 Printx86 ……………………….. 90
5
6 Functions 93
6.1 TheR4Language……………………. 93
6.2 Functionsinx86 ……………………. 95 6.2.1 CallingConventions ………………. 95 6.2.2 EfficientTailCalls ……………….. 98
6.3 ShrinkR4 ……………………….. 99
6.4 RevealFunctions ……………………. 99
6.5 LimitFunctions…………………….. 99
6.6 Remove Complex Operators and Operands . . . . . . . . . . 100
6.7 ExplicateControlandtheC3 language. . . . . . . . . . . . . 100
CONTENTS vii
6.8 UncoverLocals ……………………..101 6.9 SelectInstructions ……………………101 6.10UncoverLive ………………………103 6.11 Build Interference Graph . . . . . . . . . . . . . . . . . . . . . 103 6.12PatchInstructions ……………………103 6.13Printx86 ………………………..104 6.14 An Example Translation . . . . . . . . . . . . . . . . . . . . . 104
7 Lexically Scoped Functions 107
7.1 TheR5Language…………………….108 7.2 InterpretingR5 ……………………..109 7.3 TypeCheckingR5 ……………………109 7.4 ClosureConversion……………………111 7.5 AnExampleTranslation…………………112
8 Dynamic Typing 115
8.1 TheR6Language:TypedRacket+Any…………118 8.2 ShrinkingR6 ………………………122 8.3 InstructionSelectionforR6 ……………….122 8.4 CompilingR7toR6 …………………..123
9 Gradual Typing 125
10 Parametric Polymorphism 127
11 High-level Optimization 129
12 Appendix 131
12.1Interpreters ……………………….131 12.2UtilityFunctions …………………….131 12.2.1 Testing ……………………..132 12.3 x86InstructionSetQuick-Reference . . . . . . . . . . . . . . 132
viii CONTENTS
List of Figures
1.1 The syntax of R0, a language of integer arithmetic. . . . . . . 9
1.2 InterpreterfortheR0language……………… 12
1.3 ApartialevaluatorforR0 expressions. . . . . . . . . . . . . . 15
2.1 The syntax of R1, a language of integers and variables. . . . . 18
2.2 InterpreterfortheR1language……………… 19
2.3 A subset of the x86 assembly language (AT&T syntax).
2.4 Anx86programequivalentto(+1032). . . . . . . . . .
2.5 Anx86programequivalentto(+52(-10)). . . . . . . .
2.6 Memorylayoutofaframe………………… 23
2.7 Abstractsyntaxforx860assembly……………. 24
2.8 OverviewofthepassesforcompilingR1. . . . . . . . . . . . 27
2.9 TheC0intermediatelanguage. …………….. 28
2.10Skeletonfortheuniquifypass……………… 29
3.1 An example program for register allocation. . . . . . . . . . . 38
3.2 An example block annotated with live-after sets. . . . . . . . 40
3.3 The interference graph of the example program. . . . . . . . . 42
3.4 A Sudoku game board and the corresponding colored graph. . 43
3.5 The saturation-based greedy graph coloring algorithm. . . . . 44
3.6 Diagram of the passes for R1 with register allocation. . . . . . 47
4.1 The syntax of R2, extending R1 (Figure 2.1) with Booleans andconditionals. ……………………. 54
4.2 InterpreterfortheR2language……………… 56
4.3 SkeletonofatypecheckerfortheR2 language. . . . . . . . . 57
4.4 The x861 language (extends x860 of Figure 2.7). . . . . . . . . 59
4.5 The C1 language, extending C0 with Booleans and conditionals. 60
4.6 Example translation from R2 to C1 via the explicate-control. 62
4.7 Example compilation of an if expression to x86. . . . . . . . 68
ix
… 21 . . . 21 . . . 22
x
4.8
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10
LIST OF FIGURES
Diagram of the passes for R2, a language with conditionals. . 69
Example program that creates tuples and reads from them. . 72 The syntax of R3, extending R2 (Figure 4.1) with tuples. . . 72 InterpreterfortheR3language……………… 74 TypecheckerfortheR3language. …………… 75 Acopyingcollectorinaction. ……………… 77 Depiction of the Cheney algorithm copying the live tuples. . . 79 Maintaining a root stack to facilitate garbage collection. . . . 80 Representationfortuplesintheheap. . . . . . . . . . . . . . 81 The compiler’s interface to the garbage collector. . . . . . . . 82 Output of the expose-allocation pass, minus all of the has-typeforms. ……………………. 84 The C2 language, extending C1 (Figure 4.5) with vectors. . . 85 Output of uncover-locals for the running example. . . . . . 86 The x862 language (extends x861 of Figure 4.4). . . . . . . . . 88 Output of the select-instructions pass. . . . . . . . . . . 89
5.11
5.12
5.13
5.14
5.15Outputoftheprint-x86pass. …………….. 91 5.16 Diagram of the passes for R3, a language with tuples. . . . . 92
6.1 Syntax of R4, extending R3 (Figure 5.2) with functions. . . . 94
6.2 ExampleofusingfunctionsinR4…………….. 94
6.3 InterpreterfortheR4language……………… 96
6.4 Memorylayoutofcallerandcalleeframes. . . . . . . . . . . . 97
6.5 The F1 language, an extension of R4 (Figure 6.1). . . . . . . 100
6.6 The C3 language, extending C2 (Figure 5.11) with functions. 101
6.7 The x863 language (extends x862 of Figure 5.13). . . . . . . . 102
6.8 Example compilation of a simple function to x86. . . . . . . . 105
6.9 Diagram of the passes for R4, a language with functions. . . . 106
7.1 Exampleofalexicallyscopedfunction.. . . . . . . . . . . . . 107
7.2 Syntax of R5, extending R4 (Figure 6.1) with lambda. . . . . 108
7.3 Example closure representation for the lambda’s in Figure 7.1. 109
7.4 InterpreterforR5. ……………………110
7.5 Typecheckingthelambda’sinR5. ……………110
7.6 Exampleofclosureconversion. ……………..112
7.7 Diagram of the passes for R5, a language with lexically-scoped
functions. ………………………..113
8.1 Syntax of R7, an untyped language (a subset of Racket). . . . 116
8.2 Interpreter for the R7 language. UPDATE ME -Jeremy . . . 117
LIST OF FIGURES xi
8.3 Syntax of R6, extending R5 (Figure 7.2) with Any. . . . . . . 118 8.4 TypecheckerforpartsoftheR6 language.. . . . . . . . . . . 120 8.5 InterpreterforR6. ……………………121 8.6 CompilingR7toR6……………………124
xii LIST OF FIGURES
Preface
The tradition of compiler writing at Indiana University goes back to research and courses about programming languages by Daniel Friedman in the 1970’s and 1980’s. Dan had conducted research on lazy evaluation [Friedman and Wise, 1976] in the context of Lisp [McCarthy, 1960] and then studied con- tinuations [Felleisen and Friedman, 1986] and macros [Kohlbecker et al., 1986] in the context of the Scheme [Sussman and Jr., 1975], a dialect of Lisp. One of the students of those courses, Kent Dybvig, went on to build Chez Scheme [Dybvig, 2006], a production-quality and efficient compiler for Scheme. After completing his Ph.D. at the University of North Carolina, Kent returned to teach at Indiana University. Throughout the 1990’s and 2000’s, Kent continued development of Chez Scheme and taught the com- piler course.
The compiler course evolved to incorporate novel pedagogical ideas while also including elements of effective real-world compilers. One of Dan’s ideas was to split the compiler into many small “passes” so that the code for each pass would be easy to understood in isolation. (In contrast, most compilers of the time were organized into only a few monolithic passes for reasons of compile-time efficiency.) Kent, with later help from his students Dipanwita Sarkar and Andrew Keep, developed infrastructure to support this approach and evolved the course, first to use micro-sized passes and then into even smaller nano passes [Sarkar et al., 2004, Keep, 2012]. Jeremy Siek was a student in this compiler course in the early 2000’s, as part of his Ph.D. studies at Indiana University. Needless to say, Jeremy enjoyed the course immensely!
One of Jeremy’s classmates, Abdulaziz Ghuloum, observed that the front-to-back organization of the course made it difficult for students to understand the rationale for the compiler design. Abdulaziz proposed an incremental approach in which the students build the compiler in stages; they start by implementing a complete compiler for a very small subset of the input language, then in each subsequent stage they add a feature to the
1
2 LIST OF FIGURES
input language and add or modify passes to handle the new feature [Ghu- loum, 2006]. In this way, the students see how the language features motivate aspects of the compiler design.
After graduating from Indiana University in 2005, Jeremy went on to teach at the University of Colorado. He adapted the nano pass and incre- mental approaches to compiling a subset of the Python language [Siek and Chang, 2012]. Python and Scheme are quite different on the surface but there is a large overlap in the compiler techniques required for the two lan- guages. Thus, Jeremy was able to teach much of the same content from the Indiana compiler course. He very much enjoyed teaching the course orga- nized in this way, and even better, many of the students learned a lot and got excited about compilers.
Jeremy returned to teach at Indiana University in 2013. In his absence the compiler course had switched from the front-to-back organization to a back-to-front organization. Seeing how well the incremental approach worked at Colorado, he started porting and adapting the structure of the Colorado course back into the land of Scheme. In the meantime Indiana had moved on from Scheme to Racket, so the course is now about compiling a subset of Racket (and Typed Racket) to the x86 assembly language. The compiler is implemented in Racket 7.1 [Flatt and PLT, 2014].
This is the textbook for the incremental version of the compiler course at Indiana University (Spring 2016 – present) and it is the first open textbook for an Indiana compiler course. With this book we hope to make the Indiana compiler course available to people that have not had the chance to study in Bloomington in person. Many of the compiler design decisions in this book are drawn from the assignment descriptions of Dybvig and Keep [2010]. We have captured what we think are the most important topics from Dybvig and Keep [2010] but we have omitted topics that we think are less interesting conceptually and we have made simplifications to reduce complexity. In this way, this book leans more towards pedagogy than towards the absolute efficiency of the generated code. Also, the book differs in places where we saw the opportunity to make the topics more fun, such as in relating register allocation to Sudoku (Chapter 3).
Prerequisites
The material in this book is challenging but rewarding. It is meant to prepare students for a lifelong career in programming languages. We do not recommend this book for students who want to dabble in programming
LIST OF FIGURES 3
languages.
The book uses the Racket language both for the implementation of the
compiler and for the language that is compiled, so a student should be proficient with Racket (or Scheme) prior to reading this book. There are many other excellent resources for learning Scheme and Racket [Dybvig, 1987, Abelson and Sussman, 1996, Friedman and Felleisen, 1996, Felleisen et al., 2001, 2013, Flatt et al., 2014]. It is helpful but not necessary for the student to have prior exposure to x86 (or x86-64) assembly language [Intel, 2015], as one might obtain from a computer systems course [Bryant and O’Hallaron, 2005, 2010]. This book introduces the parts of x86-64 assembly language that are needed.
Acknowledgments
Many people have contributed to the ideas, techniques, organization, and teaching of the materials in this book. We especially thank the following people.
• Bor-Yuh Evan Chang • Kent Dybvig
• Daniel P. Friedman
• Ronald Garcia
• Abdulaziz Ghuloum • Jay McCarthy
• Dipanwita Sarkar
• Andrew Keep
• Oscar Waddell
• Michael Wollowski
Jeremy G. Siek
http://homes.soic.indiana.edu/jsiek
4 LIST OF FIGURES
1 Preliminaries
In this chapter, we review the basic tools that are needed for implementing a compiler. We use abstract syntax trees (ASTs), which refer to data struc- tures in the compilers memory, rather than programs as they are stored on disk, in concrete syntax. ASTs can be represented in many different ways, de- pending on the programming language used to write the compiler. Because this book uses Racket (http://racket-lang.org), a descendant of Lisp, we use S-expressions to represent programs (Section 1.1). We use gram- mars to define programming languages (Section 1.2) and pattern matching to inspect individual nodes in an AST (Section 1.3). We use recursion to construct and deconstruct entire ASTs (Section 1.4). This chapter provides an brief introduction to these ideas.
1.1 Abstract Syntax Trees and S-expressions
The primary data structure that is commonly used for representing pro- grams is the abstract syntax tree (AST). When considering some part of a program, a compiler needs to ask what kind of part it is and what sub-parts it has. For example, the program on the left, represented by an S-expression, corresponds to the AST on the right.
5
6
1. PRELIMINARIES
(+ (read) (- 8))
+
8
(1.1)
read
–
We shall use the standard terminology for trees: each circle above is called a node. The arrows connect a node to its children (which are also nodes). The top-most node is the root. Every node except for the root has a parent (the node it is the child of). If a node has no children, it is a leaf node. Otherwise it is an internal node.
Recall that an symbolic expression (S-expression) is either
1. an atom, or
2. a pair of two S-expressions, written (e1.e2), where e1 and e2 are each an S-expression.
An atom can be a symbol, such as ‘hello, a number, the null value ’(), etc. We can create an S-expression in Racket simply by writing a backquote (called a quasi-quote in Racket). followed by the textual representation of the S-expression. It is quite common to use S-expressions to represent a list, such as a, b, c in the following way:
‘(a . (b . (c . ())))
Each element of the list is in the first slot of a pair, and the second slot is either the rest of the list or the null value, to mark the end of the list. Such lists are so common that Racket provides special notation for them that removes the need for the periods and so many parenthesis:
‘(a b c)
For another example, an S-expression to represent the AST (1.1) is created by the following Racket expression:
‘(+ (read) (- 8))
When using S-expressions to represent ASTs, the convention is to represent each AST node as a list and to put the operation symbol at the front of the list. The rest of the list contains the children. So in the above case, the root
1.2. GRAMMARS 7
AST node has operation ‘+ and its two children are ‘(read) and ‘(- 8), just as in the diagram (1.1).
To build larger S-expressions one often needs to splice together sev- eral smaller S-expressions. Racket provides the comma operator to splice an S-expression into a larger one. For example, instead of creating the S-expression for AST (1.1) all at once, we could have first created an S- expression for AST (1.5) and then spliced that into the addition S-expression.
(define ast1.4 ‘(- 8))
(define ast1.1 ‘(+ (read) ,ast1.4))
In general, the Racket expression that follows the comma (splice) can be any expression that computes an S-expression.
When deciding how to compile program (1.1), we need to know that the operation associated with the root node is addition and that it has two children: read and a negation. The AST data structure directly supports these queries, as we shall see in Section 1.3, and hence is a good choice for use in compilers. In this book, we will often write down the S-expression representation of a program even when we really have in mind the AST because the S-expression is more concise. We recommend that, in your mind, you always think of programs as abstract syntax trees.
1.2 Grammars
A programming language can be thought of as a set of programs. The set is typically infinite (one can always create larger and larger programs), so one cannot simply describe a language by listing all of the programs in the language. Instead we write down a set of rules, a grammar, for building programs. We shall write our rules in a variant of Backus-Naur Form (BNF) [Backus et al., 1960, Knuth, 1964]. As an example, we describe a small language, named R0, of integers and arithmetic operations. The first rule says that any integer is an expression, exp, in the language:
exp ::= int (1.2)
Each rule has a left-hand-side and a right-hand-side. The way to read a rule is that if you have all the program parts on the right-hand-side, then you can create an AST node and categorize it according to the left-hand-side. A name such as exp that is defined by the grammar rules is a non-terminal. The name int is a also a non-terminal, however, we do not define int be- cause the reader already knows what an integer is. Further, we make the
8 1. PRELIMINARIES
simplifying design decision that all of the languages in this book only handle machine-representable integers. On most modern machines this corresponds to integers represented with 64-bits, i.e., the in range −263 to 263 − 1. How- ever, we restrict this range further to match the Racket fixnum datatype, which allows 63-bit integers on a 64-bit machine.
The second grammar rule is the read operation that receives an input integer from the user of the program.
exp ::= (read) (1.3) The third rule says that, given an exp node, you can build another exp
node by negating it.
exp ::= (- exp) (1.4)
Symbols such as – in typewriter font are terminal symbols and must literally appear in the program for the rule to be applicable.
We can apply the rules to build ASTs in the R0 language. For example, by rule (1.2), 8 is an exp, then by rule (1.4), the following AST is an exp.
(- 8)
–
8
(1.5)
The following grammar rule defines addition expressions: exp ::= (+ exp exp)
(1.6)
Now we can see that the AST (1.1) is an exp in R0. We know that (read) is an exp by rule (1.3) and we have shown that (- 8) is an exp, so we can apply rule (1.6) to show that (+ (read) (- 8)) is an exp in the R0 language.
If you have an AST for which the above rules do not apply, then the AST is not in R0. For example, the AST (- (read) (+ 8)) is not in R0 because there are no rules for + with only one argument, nor for – with two arguments. Whenever we define a language with a grammar, we implicitly mean for the language to be the smallest set of programs that are justified by the rules. That is, the language only includes those programs that the rules allow.
The last grammar rule for R0 states that there is a program node to mark the top of the whole program:
R0 ::= (program exp)
1.3. PATTERN MATCHING 9
Figure 1.1: The syntax of R0, a language of integer arithmetic.
The read-program function provided in utilities.rkt reads programs in from a file (the sequence of characters in the concrete syntax of Racket) and parses them into the abstract syntax tree. The concrete syntax does not include a program form; that is added by the read-program function as it creates the AST. See the description of read-program in Appendix 12.2 for more details.
It is common to have many rules with the same left-hand side, such as exp in the grammar for R0, so there is a vertical bar notation for gathering several rules, as shown in Figure 1.1. Each clause between a vertical bar is called an alternative.
1.3 Pattern Matching
As mentioned above, one of the operations that a compiler needs to perform on an AST is to access the children of a node. Racket provides the match form to access the parts of an S-expression. Consider the following example and the output on the right.
exp ::= int | (read) | (- exp) | (+ exp exp) R0 ::= (program exp)
(match ast1.1
[‘(,op ,child1 ,child2)
(print op) (newline) (print child1) (newline) (print child2)])
’+ ’(read) ’(- 8)
The match form takes AST (1.1) and binds its parts to the three variables op, child1, and child2. In general, a match clause consists of a pattern and a body. The pattern is a quoted S-expression that may contain pattern- variables (each one preceded by a comma). The pattern is not the same thing as a quasiquote expression used to construct ASTs, however, the similarity is intentional: constructing and deconstructing ASTs uses similar syntax. While the pattern uses a restricted syntax, the body of the match clause may contain any Racket code whatsoever.
A match form may contain several clauses, as in the following function leaf? that recognizes when an R0 node is a leaf. The match proceeds
10 1. PRELIMINARIES
through the clauses in order, checking whether the pattern can match the input S-expression. The body of the first clause that matches is executed. The output of leaf? for several S-expressions is shown on the right. In the below match, we see another form of pattern: the (? fixnum?) applies the predicate fixnum? to the input S-expression to see if it is a machine- representable integer.
(define (leaf? arith)
(match arith
[(? fixnum?) #t] [‘(read) #t]
[‘(- ,c1) #f]
[‘(+ ,c1 ,c2) #f]))
(leaf? ‘(read)) #t (leaf? ‘(- 8)) #f (leaf? ‘(+ (read) (- 8))) #f
1.4 Recursion
Programs are inherently recursive in that an R0 expression (exp) is made up of smaller expressions. Thus, the natural way to process an entire program is with a recursive function. As a first example of such a function, we define exp? below, which takes an arbitrary S-expression, sexp, and determines whether or not sexp is an R0 expression. Note that each match clause corresponds to one grammar rule the body of each clause makes a recursive call for each child node. This pattern of recursive function is so common that it has a name, structural recursion. In general, when a recursive function is defined using a sequence of match clauses that correspond to a grammar, and each clause body makes a recursive call on each child node, then we say the function is defined by structural recursion. Below we also define a second function, named R0?, determines whether an S-expression is an R0 program.
1.4. RECURSION
(define (exp? sexp) (match sexp
[(? fixnum?) #t] [‘(read) #t]
[‘(- ,e) (exp? e)] [‘(+ ,e1 ,e2)
(and (exp? e1) (exp? e2))] [else #f]))
(define (R0? sexp)
(match sexp
[‘(program ,e) (exp? e)] [else #f]))
(R0? ‘(program (+ (read) (- 8)))) (R0? ‘(program (- (read) (+ 8))))
11
Indeed, the structural recursion follows the grammar itself. We can generally expect to write a recursive function to handle each non-terminal in the grammar.1
You may be tempted to write the program with just one function, like this:
(define (R0? sexp)
(match sexp
[(? fixnum?) #t]
[‘(read) #t]
[‘(- ,e) (R0? e)]
[‘(+ ,e1 ,e2) (and (R0? e1) (R0? e2))] [‘(program ,e) (R0? e)]
[else #f]))
Sometimes such a trick will save a few lines of code, especially when it comes to the program wrapper. Yet this style is generally not recommended because it can get you into trouble. For instance, the above function is sub- tly wrong: (R0? ‘(program (program 3))) will return true, when it should return false.
1This principle of structuring code according to the data definition is advocated in the book How to Design Programs http://www.ccs.neu.edu/home/matthias/HtDP2e/.
#t #f
12
1. PRELIMINARIES
(define (interp-exp e)
(match e
[(? fixnum?) e] [‘(read)
(let ([r (read)])
(cond [(fixnum? r) r]
[else (error ’interp-R0 “input␣not␣an␣integer” r)]))] [‘(- ,e1) (fx- 0 (interp-exp e1))]
[‘(+ ,e1 ,e2) (fx+ (interp-exp e1) (interp-exp e2))]
))
(define (interp-R0 p)
(match p
[‘(program ,e) (interp-exp e)]))
Figure 1.2: Interpreter for the R0 language. 1.5 Interpreters
The meaning, or semantics, of a program is typically defined in the spec- ification of the language. For example, the Scheme language is defined in the report by Sperber et al. [2009]. The Racket language is defined in its reference manual [Flatt and PLT, 2014]. In this book we use an interpreter to define the meaning of each language that we consider, following Reynold’s advice in this regard [Reynolds, 1972]. Here we warm up by writing an inter- preter for the R0 language, which serves as a second example of structural recursion. The interp-R0 function is defined in Figure 1.2. The body of the function is a match on the input program p and then a call to the interp-exp helper function, which in turn has one match clause per grammar rule for R0 expressions.
Let us consider the result of interpreting a few R0 programs. The fol- lowing program simply adds two integers.
(+ 10 32)
The result is 42, as you might have expected. Here we have written the program in concrete syntax, whereas the parsed abstract syntax would be the slightly different: (program (+ 10 32)).
The next example demonstrates that expressions may be nested within each other, in this case nesting several additions and negations.
(+ 10 (- (+ 12 20)))
1.5. INTERPRETERS 13
What is the result of the above program?
As mentioned previously, the R0 language does not support arbitrarily-
large integers, but only 63-bit integers, so we interpret the arithmetic op- erations of R0 using fixnum arithmetic. What happens when we run the following program?
(define large 999999999999999999)
(interp-R0 ‘(program (+ (+ (+ ,large ,large) (+ ,large ,large))
(+ (+ ,large ,large) (+ ,large ,large)))))
It produces an error:
fx+: result is not a fixnum
We shall use the convention that if the interpreter for a language produces an error when run on a program, then the meaning of the program is un- specified. The compiler for the language is under no obligation for such a program; it can produce an executable that does anything.
Moving on, the read operation prompts the user of the program for an integer. If we interpret the AST (1.1) and give it the input 50
(interp-R0 ast1.1)
we get the answer to life, the universe, and everything:
42
We include the read operation in R0 so a clever student cannot implement a compiler for R0 simply by running the interpreter at compilation time to obtain the output and then generating the trivial code to return the output. (A clever student did this in a previous version of the course.)
The job of a compiler is to translate a program in one language into a program in another language so that the output program behaves the same way as the input program. This idea is depicted in the following diagram. Suppose we have two languages, L1 and L2, and an interpreter for each language. Suppose that the compiler translates program P1 in language L1 into program P2 in language L2. Then interpreting P1 and P2 on their respective interpreters with input i should yield the same output o.
compile
P1 P2
interp-L1 (i)
interp-L2 (i)
(1.7)
o
14 1. PRELIMINARIES In the next section we see our first example of a compiler, which is another
example of structural recursion.
1.6 Example Compiler: a Partial Evaluator
In this section we consider a compiler that translates R0 programs into R0 programs that are more efficient, that is, this compiler is an optimizer. Our optimizer will accomplish this by trying to eagerly compute the parts of the program that do not depend on any inputs. For example, given the following program
(+ (read) (- (+ 5 3)))
our compiler will translate it into the program
(+ (read) -8)
Figure 1.3 gives the code for a simple partial evaluator for the R0 lan- guage. The output of the partial evaluator is an R0 program, which we build up using a combination of quasiquotes and commas. (Though no quasiquote is necessary for integers.) In Figure 1.3, the normal structural recursion is captured in the main pe-arith function whereas the code for partially eval- uating negation and addition is factored into two separate helper functions: pe-neg and pe-add. The input to these helper functions is the output of partially evaluating the children nodes.
Our code for pe-neg and pe-add implements the simple idea of checking whether their arguments are integers and if they are, to go ahead and per- form the arithmetic. Otherwise, we use quasiquote to create an AST node for the appropriate operation (either negation or addition) and use comma to splice in the child nodes.
To gain some confidence that the partial evaluator is correct, we can test whether it produces programs that get the same result as the input program. That is, we can test whether it satisfies Diagram (1.7). The following code runs the partial evaluator on several examples and tests the output program. The assert function is defined in Appendix 12.2.
(define (test-pe p)
(assert “testing␣pe-arith”
(equal? (interp-R0 p) (interp-R0 (pe-arith p)))))
(test-pe ‘(+ (read) (- (+ 5 3)))) (test-pe ‘(+ 1 (+ (read) 1))) (test-pe ‘(- (+ (read) (- 5))))
1.6. EXAMPLE COMPILER: A PARTIAL EVALUATOR 15
(define (pe-neg r)
(cond [(fixnum? r) (fx- 0 r)]
[else ‘(- ,r)]))
(define (pe-add r1 r2)
(cond [(and (fixnum? r1) (fixnum? r2)) (fx+ r1 r2)]
[else ‘(+ ,r1 ,r2)]))
(define (pe-arith e)
(match e
[(? fixnum?) e] [‘(read) ‘(read)] [‘(- ,e1)
(pe-neg (pe-arith e1))]
[‘(+ ,e1 ,e2)
(pe-add (pe-arith e1) (pe-arith e2))]))
Figure 1.3: A partial evaluator for R0 expressions.
Exercise 1. We challenge the reader to improve on the simple partial eval- uator in Figure 1.3 by replacing the pe-neg and pe-add helper functions with functions that know more about arithmetic. For example, your partial evaluator should translate
(+ 1 (+ (read) 1)) into
(+ 2 (read))
To accomplish this, we recommend that your partial evaluator produce out- put that takes the form of the residual non-terminal in the following gram- mar.
exp ::= (read) | (- (read)) | (+ exp exp) residual ::= int | (+ int exp) | exp
16 1. PRELIMINARIES
2
Integers and Variables
This chapter concerns the challenge of compiling a subset of Racket that includes integer arithmetic and local variable binding, which we name R1, to x86-64 assembly code [Intel, 2015]. Henceforth we shall refer to x86-64 simply as x86. The chapter begins with a description of the R1 language (Section 2.1) followed by a description of x86 (Section 2.2). The x86 assem- bly language is quite large, so we only discuss what is needed for compiling R1. We introduce more of x86 in later chapters. Once we have introduced R1 and x86, we reflect on their differences and come up with a plan to break down the translation from R1 to x86 into a handful of steps (Section 2.3). The rest of the sections in this Chapter give detailed hints regarding each step (Sections 2.4 through 2.10). We hope to give enough hints that the well-prepared reader can implement a compiler from R1 to x86 while at the same time leaving room for some fun and creativity.
2.1 The R1 Language
The R1 language extends the R0 language (Figure 1.1) with variable defi- nitions. The syntax of the R1 language is defined by the grammar in Fig- ure 2.1. The non-terminal var may be any Racket identifier. As in R0, read is a nullary operator, – is a unary operator, and + is a binary opera- tor. Similar to R0, the R1 language includes the program construct to mark the top of the program, which is helpful in parts of the compiler. The info field of the program construct contain an association list that is used to communicating auxiliary data from one step of the compiler to the next.
The R1 language is rich enough to exhibit several compilation techniques but simple enough so that the reader, together with couple friends, can
17
18 2. INTEGERS AND VARIABLES
Figure 2.1: The syntax of R1, a language of integers and variables.
implement a compiler for it in a week or two of part-time work. To give the reader a feeling for the scale of this first compiler, the instructor solution for the R1 compiler is less than 500 lines of code.
Let us dive into the description of the R1 language. The let construct defines a variable for use within its body and initializes the variable with the value of an expression. So the following program initializes x to 32 and then evaluates the body (+ 10 x), producing 42.
(program ()
(let ([x (+ 12 20)]) (+ 10 x)))
When there are multiple let’s for the same variable, the closest enclosing let is used. That is, variable definitions overshadow prior definitions. Con- sider the following program with two let’s that define variables named x. Can you figure out the result?
(program ()
(let ([x 32]) (+ (let ([x 10]) x) x)))
For the purposes of showing which variable uses correspond to which defini- tions, the following shows the x’s annotated with subscripts to distinguish them. Double check that your answer for the above is the same as your answer for this annotated version of the program.
(program ()
(let ([x1 32]) (+ (let ([x2 10]) x2 ) x1 )))
The initializing expression is always evaluated before the body of the let, so in the following, the read for x is performed before the read for y. Given the input 52 then 10, the following produces 42 (and not -42).
(program ()
(let ([x (read)]) (let ([y (read)]) (+ x (- y)))))
Figure 2.2 shows the interpreter for the R1 language. It extends the interpreter for R0 with two new match clauses for variables and for let. For let, we will need a way to communicate the initializing value of a variable to all the uses of a variable. To accomplish this, we maintain a mapping from variables to values, which is traditionally called an environment. For
exp ::= int | (read) | (- exp) | (+ exp exp) | var | (let ([var exp]) exp)
R1 ::= (program info exp)
2.1. THE R1 LANGUAGE 19
(define (interp-exp env)
(lambda (e)
(match e
[(? fixnum?) e] [‘(read)
(define r (read)) (cond [(fixnum? r) r]
[else (error ’interp-R1 “expected␣an␣integer” r)])] [‘(- ,e)
(define v ((interp-exp env) e))
(fx- 0 v)]
[‘(+ ,e1 ,e2)
(define v1 ((interp-exp env) e1))
(define v2 ((interp-exp env) e2))
(fx+ v1 v2)]
[(? symbol?) (lookup e env)] [‘(let ([,x ,e]) ,body)
(define new-env (cons (cons x ((interp-exp env) e)) env))
((interp-exp new-env) body)]
)))
(define (interp-R1 env)
(lambda (p)
(match p
[‘(program ,info ,e) ((interp-exp ’()) e)])))
Figure 2.2: Interpreter for the R1 language.
simplicity, here we use an association list to represent the environment. The interp-R1 function takes the current environment, env, as an extra param- eter. When the interpreter encounters a variable, it finds the corresponding value using the lookup function (Appendix 12.2). When the interpreter encounters a let, it evaluates the initializing expression, extends the envi- ronment with the result bound to the variable, then evaluates the body of the let.
The goal for this chapter is to implement a compiler that translates any program P1 in the R1 language into an x86 assembly program P2 such that P2 exhibits the same behavior on an x86 computer as the R1 program running in a Racket implementation. That is, they both output the same
20 integer n.
2.
INTEGERS AND VARIABLES
P2 interp-x86
n
compile
P1
interp-R1
In the next section we introduce enough of the x86 assembly language to compile R1.
2.2 The x86 Assembly Language
An x86 program is a sequence of instructions. The program is stored in the computer’s memory and the program counter points to the address of the next instruction to be executed. For most instructions, once the instruction is executed, the program counter is incremented to point to the immedi- ately following instruction in memory. Each instruction may refer to integer constants (called immediate values), variables called registers, and instruc- tions may load and store values into memory. For our purposes, we can think of the computer’s memory as a mapping of 64-bit addresses to 64-bit values1. Figure 2.3 defines the syntax for the subset of the x86 assembly language needed for this chapter. We use the AT&T syntax expected by the GNU assembler, which comes with the gcc compiler that we use for compiling assembly code to machine code. Also, Appendix 12.3 includes a quick-reference of all the x86 instructions used in this book and a short explanation of what they do.
An immediate value is written using the notation $n where n is an in- teger. A register is written with a % followed by the register name, such as %rax. An access to memory is specified using the syntax n(%r), which obtains the address stored in register r and then offsets the address by n bytes (8 bits). The address is then used to either load or store to memory depending on whether it occurs as a source or destination argument of an instruction.
An arithmetic instruction, such as addq s, d, reads from the source s and destination d, applies the arithmetic operation, then writes the result in d.
1This simple story suffices for describing how sequential programs access memory but is not sufficient for multi-threaded programs. However, multi-threaded execution is beyond the scope of this book.
2.2. THE X86 ASSEMBLY LANGUAGE 21
reg ::= arg ::=
instr ::= prog ::=
rsp|rbp|rax|rbx|rcx|rdx|rsi|rdi|
r8 | r9 | r10 | r11 | r12 | r13 | r14 | r15
$int | %reg | int(%reg)
addq arg, arg | subq arg, arg | negq arg | movq arg, arg | callq label | pushq arg | popq arg | retq | label: instr .globl main
main: instr+
main:
.globl main
movq $10, %rax addq $32, %rax retq
Figure 2.4: An x86 program equivalent to (+ 10 32).
Figure 2.3: A subset of the x86 assembly language (AT&T syntax).
The move instruction, movq s d reads from s and stores the result in d. The callq label instruction executes the procedure specified by the label.
Figure 2.4 depicts an x86 program that is equivalent to (+ 10 32). The globl directive says that the main procedure is externally visible, which is necessary so that the operating system can call it. The label main: indicates the beginning of the main procedure which is where the operating system starts executing this program. The instruction movq $10, %rax puts 10 into register rax. The following instruction addq $32, %rax adds 32 to the 10 in rax and puts the result, 42, back into rax.
The last instruction, retq, finishes the main function by returning the integer in rax to the operating system. The operating system interprets this integer as the program’s exit code. By convention, an exit code of 0 indicates the program was successful, and all other exit codes indicate various errors. Nevertheless, we return the result of the program as the exit code.
Unfortunately, x86 varies in a couple ways depending on what operating system it is assembled in. The code examples shown here are correct on Linux and most Unix-like platforms, but when assembled on Mac OS X, labels like main must be prefixed with an underscore, as in _main.
We exhibit the use of memory for storing intermediate results in the next example. Figure 2.5 lists an x86 program that is equivalent to (+ 52 (- 10)).
22 2. start:
INTEGERS AND VARIABLES
main:
movq $10, -8(%rbp) negq -8(%rbp)
movq -8(%rbp), %rax addq $52, %rax
jmp conclusion
.globl main
pushq %rbp
movq %rsp, %rbp subq $16, %rsp jmp start
conclusion:
addq $16, %rsp
popq %rbp retq
Figure 2.5: An x86 program equivalent to (+ 52 (- 10)).
This program uses a region of memory called the procedure call stack (or stack for short). The stack consists of a separate frame for each procedure call. The memory layout for an individual frame is shown in Figure 2.6. The register rsp is called the stack pointer and points to the item at the top of the stack. The stack grows downward in memory, so we increase the size of the stack by subtracting from the stack pointer. The frame size is required to be a multiple of 16 bytes. In the context of a procedure call, the return address is the next instruction on the caller side that comes after the call instruction. During a function call, the return address is pushed onto the stack. The register rbp is the base pointer which serves two purposes: 1) it saves the location of the stack pointer for the calling procedure and 2) it is used to access variables associated with the current procedure. The base pointer of the calling procedure is pushed onto the stack after the return address. We number the variables from 1 to n. Variable 1 is stored at address −8(%rbp), variable 2 at −16(%rbp), etc.
Getting back to the program in Figure 2.5, the first three instructions are the typical prelude for a procedure. The instruction pushq %rbp saves the base pointer for the procedure that called the current one onto the stack and subtracts 8 from the stack pointer. The second instruction movq %rsp, %rbp changes the base pointer to the top of the stack. The instruction subq
2.2. THE X86 ASSEMBLY LANGUAGE 23
Position
Contents
8(%rbp)
0(%rbp) -8(%rbp) -16(%rbp)
… 0(%rsp)
return address old rbp variable 1 variable 2
… variable n
Figure 2.6: Memory layout of a frame.
$16, %rsp moves the stack pointer down to make enough room for storing variables. This program just needs one variable (8 bytes) but because the frame size is required to be a multiple of 16 bytes, it rounds to 16 bytes.
The four instructions under the label start carry out the work of com- puting (+ 52 (- 10)). The first instruction movq $10, -8(%rbp) stores 10 in variable 1. The instruction negq -8(%rbp) changes variable 1 to −10. The movq $52, %rax places 52 in the register rax and addq -8(%rbp), %rax adds the contents of variable 1 to rax, at which point rax contains 42.
The three instructions under the label conclusion are the typical finale of a procedure. The first two are necessary to get the state of the machine back to where it was at the beginning of the procedure. The addq $16, %rsp instruction moves the stack pointer back to point at the old base pointer. The amount added here needs to match the amount that was subtracted in the prelude of the procedure. Then popq %rbp returns the old base pointer to rbp and adds 8 to the stack pointer. The last instruction, retq, jumps back to the procedure that called this one and adds 8 to the stack pointer, which returns the stack pointer to where it was prior to the procedure call.
The compiler will need a convenient representation for manipulating x86 programs, so we define an abstract syntax for x86 in Figure 2.7. We re- fer to this language as x860 with a subscript 0 because later we introduce extended versions of this assembly language. The main difference com- pared to the concrete syntax of x86 (Figure 2.3) is that it does nto allow labelled instructions to appear anywhere, but instead organizes instructions into groups called blocks and a label is associated with every block, which is why the program form includes an association list mapping labels to blocks. The reason for this organization becomes apparent in Chapter 4.
24 2. INTEGERS AND VARIABLES
register
arg instr
block
x860
::= rsp|rbp|rax|rbx|rcx|rdx|rsi|rdi| r8 | r9 | r10 | r11 | r12 | r13 | r14 | r15
::= (int int ) | (reg register ) | (deref register int )
::= (addq arg arg) | (subq arg arg) | (movq arg arg) | (retq)
| (negq arg) | (callq label) | (pushq arg) | (popq arg) ::= (block info instr + )
::= (program info ((label . block )+ ))
Figure 2.7: Abstract syntax for x860 assembly.
2.3 Planning the trip to x86 via the C0 language
To compile one language to another it helps to focus on the differences between the two languages because the compiler will need to bridge them. What are the differences between R1 and x86 assembly? Here we list some of the most important ones.
(a) x86 arithmetic instructions typically have two arguments and update the second argument in place. In contrast, R1 arithmetic operations take two arguments and produce a new value. An x86 instruction may have at most one memory-accessing argument. Furthermore, some instructions place special restrictions on their arguments.
(b) An argument to an R1 operator can be any expression, whereas x86 instructions restrict their arguments to be simple expressions like inte- gers, registers, and memory locations. (All the other kinds are called complex expressions.)
(c) The order of execution in x86 is explicit in the syntax: a sequence of instructions and jumps to labeled positions, whereas in R1 it is a left-to-right depth-first traversal of the abstract syntax tree.
(d) An R1 program can have any number of variables whereas x86 has 16 registers and the procedure calls stack.
(e) Variables in R1 can overshadow other variables with the same name. The registers and memory locations of x86 all have unique names or addresses.
We ease the challenge of compiling from R1 to x86 by breaking down the problem into several steps, dealing with the above differences one at a time.
2.3. PLANNING THE TRIP TO X86 VIA THE C0 LANGUAGE 25
Each of these steps is called a pass of the compiler, because step traverses (passes over) the AST of the program. We begin by giving a sketch about how we might implement each pass, and give them names. We shall then figure out an ordering of the passes and the input/output language for each pass. The very first pass has R1 as its input language and the last pass has x86 as its output language. In between we can choose whichever language is most convenient for expressing the output of each pass, whether that be R1, x86, or new intermediate languages of our own design. Finally, to implement the compiler, we shall write one function, typically a structural recursive function, per pass.
Pass
Pass
Pass
Pass
Pass
select-instructions To handle the difference between R1 opera- tions and x86 instructions we shall convert each R1 operation to a short sequence of instructions that accomplishes the same task.
remove-complex-opera* To ensure that each subexpression (i.e. op- erator and operand, and hence opera*) is a simple expression, we shall introduce temporary variables to hold the results of subexpressions.
explicate-control To make the execution order of the program ex- plicit, we shall convert from the abstract syntax tree representation into a graph representation in which each node contains a sequence of actions and the edges say where to go after the sequence is complete.
assign-homes To handle the difference between the variables in R1 versus the registers and stack location in x86, we shall come up with an assignment of each variable to its home, that is, to a register or stack location.
uniquify This pass deals with the shadowing of variables by renaming every variable to a unique name, so that shadowing no longer occurs.
The next question is: in what order should we apply these passes? This question can be a challenging one to answer because it is difficult to know ahead of time which orders will be better (easier to implement, produce more efficient code, etc.) so often some trial-and-error is involved. Nevertheless, we can try to plan ahead and make educated choices regarding the orderings.
Let us consider the ordering of uniquify and remove-complex-opera*. The assignment of subexpressions to temporary variables involves introduc- ing new variables and moving subexpressions, which might change the shad- owing of variables and inadvertently change the behavior of the program. But if we apply uniquify first, this will not be an issue. Of course, this
26 2. INTEGERS AND VARIABLES
means that in remove-complex-opera*, we need to ensure that the tempo- rary variables that it creates are unique.
Next we shall consider the ordering of the explicate-control pass and select-instructions. It is clear that explicate-control must come first because the control-flow graph that it generates is needed when determin- ing where to place the x86 label and jump instructions. Regarding the ordering of explicate-control with respect to uniquify, it is important to apply uniquify first because in explicate-control we change all the let-bound variables to become local variables whose scope is the entire pro- gram. With respect to remove-complex-opera*, it perhaps does not matter very much, but it works well to place explicate-control after removing complex subexpressions.
The assign-homes pass should come after remove-complex-opera* and explicate-control. The remove-complex-opera* pass generates tempo- rary variables, which also need to be assigned homes. The explicate-control pass deletes branches that will never be executed, which can remove vari- ables. Thus it is good to place explicate-control prior to assign-homes so that there are fewer variables that need to be assigned homes. This is important because the assign-homes pass has the highest time complexity.
Last, we need to decide on the ordering of select-instructions and assign-homes. These two issues are intertwined, creating a bit of a Gordian Knot. To do a good job of assigning homes, it is helpful to have already determined which instructions will be used, because x86 instructions have restrictions about which of their arguments can be registers versus stack lo- cations. For example, one can give preferential treatment to variables that occur in register-argument positions. On the other hand, it may turn out to be impossible to make sure that all such variables are assigned to registers, and then one must redo the selection of instructions. Some compilers handle this problem by iteratively repeating these two passes until a good solution is found. We shall use a simpler approach in which select-instructions comes first, followed by the assign-homes, followed by a third pass, named patch-instructions, that uses a reserved register (rax) to patch-up out- standing problems regarding instructions with too many memory accesses.
Figure 2.8 presents the ordering of the compiler passes in the form of a graph. Each pass is an edge and the input/output language of each pass is a node in the graph. The output of uniquify and remove-complex-opera* are programs that are still in the R1 language, but the output of the pass explicate-control is in a different language that is designed to make the order of evaluation explicit in its syntax, which we introduce in the next section. Also, there are two passes of lesser importance in Figure 2.8 that
2.3. PLANNING THE TRIP TO X86 VIA THE C0 LANGUAGE 27 uniquify remove-complex.
R1 R1 R1 uncover-locals
explicate-control
patch-instr.
select-instr.
C0
C0
assign-homes
print-x86
x86∗0
Figure 2.8: Overview of the passes for compiling R1.
x86∗0
x860 x86†0 we have not yet talked about, uncover-locals and print-x86. We shall
discuss them later in this Chapter.
2.3.1 The C0 Intermediate Language
It so happens that the output of explicate-control is vaguely similar to the C language [Kernighan and Ritchie, 1988], so we name it C0. The syntax for C0 is defined in Figure 2.9. The C0 language supports the same operators as R1 but the arguments of operators are now restricted to just variables and integers, thanks to the remove-complex-opera* pass. In the literature this style of intermediate language is called administrative normal form, or ANF for short [Danvy, 1991, Flanagan et al., 1993]. Instead of let expressions, C0 has assignment statements which can be executed in sequence using the seq construct. A sequence of statements always ends with return, a guarantee that is baked into the grammar rules for the tail non-terminal. The naming of this non-terminal comes from the term tail position, which refers to an expression that is the last one to execute within a function. (A expression in tail position may contain subexpressions, and those may or may not be in tail position depending on the kind of expression.)
A C0 program consists of an association list mapping labels to tails. This is overkill for the present Chapter, as we do not yet need to introduce goto for jumping to labels, but it saves us from having to change the syntax of the program construct in Chapter 4. For now there will be just one label, start, and the whole program will be it’s tail. The info field of the program construt, after the uncover-locals pass, will contain a mapping from the symbol locals to a list of variables, that is, a list of all the variables used in
28 2. INTEGERS AND VARIABLES
Figure 2.9: The C0 intermediate language.
the program. At the start of the program, these variables are uninitialized (they contain garbage) and each variable becomes initialized on its first assignment.
2.3.2 The dialects of x86
The x86∗0 language, pronounced “pseudo-x86”, is the output of the pass select-instructions. It extends x860 with variables and looser rules re- garding instruction arguments. The x86† language, the output of print-x86, is the concrete syntax for x86.
2.4 Uniquify Variables
The purpose of this pass is to make sure that each let uses a unique variable name. For example, the uniquify pass should translate the program on the left into the program on the right.
arg ::= exp ::= stmt ::= tail ::= C0 ::=
int | var arg|(read)|(-arg)|(+argarg) (assign var exp)
(return exp) | (seq stmt tail) (program info ((label . tail )+ ))
(program ()
(let ([x 32]) ⇒
(+ (let ([x 10]) x) x)))
(program ()
(let ([x.1 32])
(+ (let ([x.2 10]) x.2) x.1)))
The following is another example translation, this time of a program with a let nested inside the initializing expression of another let.
(program ()
(let ([x (let ([x 4])
(+ x 1))])
(+ x 2)))
(program ()
⇒ (let ([x.2 (let ([x.1 4])
(+ x.1 1))])
(+ x.2 2)))
We recommend implementing uniquify as a structurally recursive func- tion that mostly copies the input program. However, when encountering a let, it should generate a unique name for the variable (the Racket function gensym is handy for this) and associate the old name with the new unique name in an association list. The uniquify function will need to access this
2.4. UNIQUIFY VARIABLES 29
(define (uniquify-exp alist)
(lambda (e)
(match e
[(? symbol?) ___]
[(? integer?) e]
[‘(let ([,x ,e]) ,body) ___] [‘(,op ,es …)
‘(,op ,@(for/list ([e es]) ((uniquify-exp alist) e)))] )))
(define (uniquify alist)
(lambda (e)
(match e
[‘(program ,info ,e)
‘(program ,info ,((uniquify-exp alist) e))]
)))
Figure 2.10: Skeleton for the uniquify pass.
association list when it gets to a variable reference, so we add another param- eter to uniquify for the association list. It is quite common for a compiler pass to need a map to store extra information about variables. Such maps are often called symbol tables.
The skeleton of the uniquify function is shown in Figure 2.10. The function is curried so that it is convenient to partially apply it to an asso- ciation list and then apply it to different expressions, as in the last clause for primitive operations in Figure 2.10. In the last match clause for the primitive operators, note the use of the comma-@ operator to splice a list of S-expressions into an enclosing S-expression.
Exercise 2. Complete the uniquify pass by filling in the blanks, that is, implement the clauses for variables and for the let construct.
Exercise 3. Test your uniquify pass by creating five example R1 programs and checking whether the output programs produce the same result as the input programs. The R1 programs should be designed to test the most interesting parts of the uniquify pass, that is, the programs should include let constructs, variables, and variables that overshadow each other. The five programs should be in a subdirectory named tests and they should have the same file name except for a different integer at the end of the name, followed
30 2. INTEGERS AND VARIABLES by the ending .rkt. Use the interp-tests function (Appendix 12.2) from
utilities.rkt to test your uniquify pass on the example programs. 2.5 Remove Complex Operators and Operands
The remove-complex-opera* pass will transform R1 programs so that the arguments of operations are simple expressions. Put another way, this pass removes complex subexpressions, such as the expression (- 10) in the pro- gram below. This is accomplished by introducing a new let-bound variable, binding the complex subexpression to the new variable, and then using the new variable in place of the complex expression, as shown in the output of remove-complex-opera* on the right.
(program ()
(+ 52 (- 10)))
⇒
(program ()
(let ([tmp.1 (- 10)])
(+ 52 tmp.1)))
We recommend implementing this pass with two mutually recursive func- tions, rco-arg and rco-exp. The idea is to apply rco-arg to subexpressions that need to become simple and to apply rco-exp to subexpressions can stay complex. Both functions take an expression in R1 as input. The rco-exp function returns an expression. The rco-arg function returns two things: a simple expression and association list mapping temporary variables to com- plex subexpressions. You can return multiple things from a function using Racket’s values form and you can receive multiple things from a function call using the define-values form. If you are not familiar with these con- structs, the Racket documentation will be of help. Also, the for/lists construct is useful for applying a function to each element of a list, in the case where the function returns multiple values.
(rco-arg ‘(- 10)) ⇒
(values ‘tmp.1
‘((tmp.1 . (- 10))))
Take special care of programs such as the following that let-bind vari- ables with integers or other variables. It should leave them unchanged, as shown in to the program on the right
(program () (let ([a 42])
(let ([b a]) b)))
⇒
(program () (let ([a 42])
(let ([b a]) b)))
and not translate them to the following, which might result from a careless implementation of rco-exp and rco-arg.
2.6. EXPLICATE CONTROL 31
(program ()
(let ([tmp.1 42])
(let ([a tmp.1]) (let ([tmp.2 a])
(let ([b tmp.2]) b)))))
Exercise 4. Implement the remove-complex-opera* pass and test it on all of the example programs that you created to test the uniquify pass and create three new example programs that are designed to exercise all of the in- teresting code in the remove-complex-opera* pass. Use the interp-tests function (Appendix 12.2) from utilities.rkt to test your passes on the example programs.
2.6 Explicate Control
The explicate-control pass makes the order of execution explicit in the syntax of the program. For R1, this amounts to flattening let constructs into a sequence of assignment statements. For example, consider the follow- ing R1 program.
(program ()
(let ([y (let ([x 20])
(+ x (let ([x 22]) x)))])
y))
The output of remove-complex-opera* is shown below, on the left. The right-hand-side of a let executes before its body, so the order of evaluation for this program is to assign 20 to x.1, assign 22 to x.2, assign (+ x.1 x.2) to y, then return y. Indeed, the result of explicate-control produces code in the C0 language that makes this explicit.
(program ()
((start .
(seq (assign x.1 20)
(seq (assign x.2 22)
(seq (assign y (+ x.1 x.2)) (return y)))))))
(program ()
(let ([y (let ([x.1 20])
(let ([x.2 22]) ⇒ (+ x.1 x.2)))])
y))
We recommend implementing explicate-control using two mutually recursive functions: explicate-control-tail and explicate-control-assign. The explicate-control-tail function should be applied to expressions
in tail position, whereas explicate-control-assign should be applied
32 2. INTEGERS AND VARIABLES
to expressions that occur on the right-hand-side of a let. The function explicate-control-tail takes an R1 expression as input and produces a C0 tail (see the grammar in Figure 2.9). The explicate-control-assign function takes an R1 expression, the variable that it is to be assigned to, and C0 code (a tail) that should come after the assignment (e.g., the code generated for the body of the let).
2.7 Uncover Locals
The pass uncover-locals simply collects all of the variables in the program and places then in the info of the program construct. Here is the output for the example program of the last section.
(program ((locals . (x.1 x.2 y)))
((start .
2.8
(seq (assign x.1 20)
(seq (assign x.2 22)
(seq (assign y (+ x.1 x.2)) (return y)))))))
Select Instructions
In the select-instructions pass we begin the work of translating from C0 to x86. The target language of this pass is a pseudo-x86 language that still uses variables, so we add an AST node of the form (var var) to the x86 abstract syntax. We recommend implementing the select-instructions in terms of three auxilliary functions, one for each of the non-terminals of C0: arg, stmt, and tail.
The cases for arg are straightforward, simply putting variables and in- teger literals into the s-expression format expected of pseudo-x86, (var x) and (int n), respectively.
Next we discuss some of the cases for stmt, starting with arithmetic op- erations. For example, in C0 an addition operation can take the form below. To translate to x86, we need to use the addq instruction which does an in- place update. So we must first move 10 to x.
(assign x (+ 10 32)) ⇒
(movq (int 10) (var x))
(addq (int 32) (var x))
There are some cases that require special care to avoid generating needlessly complicated code. If one of the arguments is the same as the left-hand side of the assignment, then there is no need for the extra move instruction. For
2.9. ASSIGN HOMES 33 example, the following assignment statement can be translated into a single
addq instruction.
(assign x (+ 10 x)) ⇒ (addq (int 10) (var x))
The read operation does not have a direct counterpart in x86 assembly, so we have instead implemented this functionality in the C language, with the function read_int in the file runtime.c. In general, we refer to all of the functionality in this file as the runtime system, or simply the runtime for short. When compiling your generated x86 assembly code, you will need to compile runtime.c to runtime.o (an “object file”, using gcc option -c) and link it into the final executable. For our purposes of code generation, all you need to do is translate an assignment of read to some variable lhs (for left-hand side) into a call to the read_int function followed by a move from rax to the left-hand side. The move from rax is needed because the return value from read_int goes into rax, as is the case in general.
(assign lhs (read)) ⇒
(callq read_int)
(movq (reg rax) (var lhs))
There are two cases for the tail non-terminal: return and seq. Re- garding (return e), we recommend treating it as an assignment to the rax register followed by a jump to the conclusion of the program (so the conclu- sion needs to be labeled). For (seq s t), we simply process the statement s and tail t recursively and append the resulting instructions.
Exercise 5. Implement the select-instructions pass and test it on all of the example programs that you created for the previous passes and create three new example programs that are designed to exercise all of the inter- esting code in this pass. Use the interp-tests function (Appendix 12.2) from utilities.rkt to test your passes on the example programs.
2.9 Assign Homes
As discussed in Section 2.3, the assign-homes pass places all of the variables on the stack. Consider again the example R1 program (+ 52 (- 10)), which after select-instructions looks like the following.
(movq (int 10) (var tmp.1))
(negq (var tmp.1))
(movq (var tmp.1) (var tmp.2))
(addq (int 52) (var tmp.2))
(movq (var tmp.2) (reg rax)))
34 2. INTEGERS AND VARIABLES The variable tmp.1 is assigned to stack location -8(%rbp), and tmp.2 is
assign to -16(%rbp), so the assign-homes pass translates the above to
(movq (int 10) (deref rbp -8))
(negq (deref rbp -8))
(movq (deref rbp -8) (deref rbp -16))
(addq (int 52) (deref rbp -16))
(movq (deref rbp -16) (reg rax)))
In the process of assigning stack locations to variables, it is convenient to compute and store the size of the frame (in bytes) in the info field of the program node, with the key stack-space, which will be needed later to generate the procedure conclusion. Some operating systems place restric- tions on the frame size. For example, Mac OS X requires the frame size to be a multiple of 16 bytes.
Exercise 6. Implement the assign-homes pass and test it on all of the example programs that you created for the previous passes pass. We rec- ommend that assign-homes take an extra parameter that is a mapping of variable names to homes (stack locations for now). Use the interp-tests function (Appendix 12.2) from utilities.rkt to test your passes on the example programs.
2.10 Patch Instructions
The purpose of this pass is to make sure that each instruction adheres to the restrictions regarding which arguments can be memory references. For most instructions, the rule is that at most one argument may be a memory reference.
Consider again the following example.
(let ([a 42]) (let ([b a])
b))
After assign-homes pass, the above has been translated to
(movq (int 42) (deref rbp -8))
(movq (deref rbp -8) (deref rbp -16))
(movq (deref rbp -16) (reg rax))
(jmp conclusion)
The second movq instruction is problematic because both arguments are stack locations. We suggest fixing this problem by moving from the source to the register rax and then from rax to the destination, as follows.
2.11. PRINT X86 35
(movq (int 42) (deref rbp -8))
(movq (deref rbp -8) (reg rax))
(movq (reg rax) (deref rbp -16))
(movq (deref rbp -16) (reg rax))
Exercise 7. Implement the patch-instructions pass and test it on all of the example programs that you created for the previous passes and create three new example programs that are designed to exercise all of the inter- esting code in this pass. Use the interp-tests function (Appendix 12.2) from utilities.rkt to test your passes on the example programs.
2.11 Print x86
The last step of the compiler from R1 to x86 is to convert the x86 AST (defined in Figure 2.7) to the string representation (defined in Figure 2.3). The Racket format and string-append functions are useful in this regard. The main work that this step needs to perform is to create the main function and the standard instructions for its prelude and conclusion, as shown in Figure 2.5 of Section 2.2. You need to know the number of stack-allocated variables, so we suggest computing it in the assign-homes pass (Section 2.9) and storing it in the info field of the program node.
If you want your program to run on Mac OS X, your code needs to determine whether or not it is running on a Mac, and prefix underscores to labels like main. You can determine the platform with the Racket call (system-type ’os), which returns ’macosx, ’unix, or ’windows.
Exercise 8. Implement the print-x86 pass and test it on all of the example programs that you created for the previous passes. Use the compiler-tests function (Appendix 12.2) from utilities.rkt to test your complete com- piler on the example programs.
36 2. INTEGERS AND VARIABLES
3
Register Allocation
In Chapter 2 we simplified the generation of x86 assembly by placing all variables on the stack. We can improve the performance of the generated code considerably if we instead place as many variables as possible into registers. The CPU can access a register in a single cycle, whereas accessing the stack takes many cycles to go to cache or many more to access main memory. Figure 3.1 shows a program with four variables that serves as a running example. We show the source program and also the output of instruction selection. At that point the program is almost x86 assembly but not quite; it still contains variables instead of stack locations or registers.
The goal of register allocation is to fit as many variables into registers as possible. It is often the case that we have more variables than registers, so we cannot map each variable to a different register. Fortunately, it is common for different variables to be needed during different periods of time, and in such cases several variables can be mapped to the same register. Consider variables x and y in Figure 3.1. After the variable x is moved to z it is no longer needed. Variable y, on the other hand, is used only after this point, so x and y could share the same register. The topic of Section 3.2 is how we compute where a variable is needed. Once we have that information, we compute which variables are needed at the same time, i.e., which ones interfere, and represent this relation as graph whose vertices are variables and edges indicate when two variables interfere with eachother (Section 3.3). We then model register allocation as a graph coloring problem, which we discuss in Section 3.4.
In the event that we run out of registers despite these efforts, we place the remaining variables on the stack, similar to what we did in Chapter 2. It is common to say that when a variable that is assigned to a stack location,
37
38
R1 program:
(program ()
(let ([v 1])
(let ([w 46])
(let ([x (+ v 7)]) (let ([y (+ 4 x)]) (let ([z (+ x w)])
(+ z (- y))))))))
3. REGISTER ALLOCATION
After instruction selection:
(program
((locals . (v w x y z t.1)))
((start .
(block ()
(movq (int 1) (var v))
(movq (int 46) (var w))
(movq (var v) (var x))
(addq (int 7) (var x))
(movq (var x) (var y))
(addq (int 4) (var y))
(movq (var x) (var z))
(addq (var w) (var z))
(movq (var y) (var t.1))
(negq (var t.1))
(movq (var z) (reg rax))
(addq (var t.1) (reg rax))
(jmp conclusion)))))
Figure 3.1: An example program for register allocation.
it has been spilled. The process of spilling variables is handled as part of the graph coloring process described in 3.4.
3.1 Registers and Calling Conventions
As we perform register allocation, we will need to be aware of the conven- tions that govern the way in which registers interact with function calls. The convention for x86 is that the caller is responsible for freeing up some registers, the caller-saved registers, prior to the function call, and the callee is responsible for saving and restoring some other registers, the callee-saved registers, before and after using them. The caller-saved registers are
rax rdx rcx rsi rdi r8 r9 r10 r11
while the callee-saved registers are
rsp rbp rbx r12 r13 r14 r15
Another way to think about this caller/callee convention is the following. The caller should assume that all the caller-saved registers get overwritten with arbitrary values by the callee. On the other hand, the caller can safely assume that all the callee-saved registers contain the same values after the
3.2. LIVENESS ANALYSIS 39
call that they did before the call. The callee can freely use any of the caller- saved registers. However, if the callee wants to use a callee-saved register, the callee must arrange to put the original value back in the register prior to returning to the caller, which is usually accomplished by saving and restoring the value from the stack.
3.2 Liveness Analysis
A variable is live if the variable is used at some later point in the program and there is not an intervening assignment to the variable. To understand the latter condition, consider the following code fragment in which there are two writes to b. Are a and b both live at the same time?
1 (movq (int 5) (var a))
2 (movq (int 30) (var b))
3 (movq (var a) (var c))
4 (movq (int 10) (var b))
5 (addq (var b) (var c))
The answer is no because the value 30 written to b on line 2 is never used. The variable b is read on line 5 and there is an intervening write to b on line 4, so the read on line 5 receives the value written on line 4, not line 2.
The live variables can be computed by traversing the instruction se- quence back to front (i.e., backwards in execution order). Let I1, . . . , In be the instruction sequence. We write Lafter(k) for the set of live variables after instruction Ik and Lbefore(k) for the set of live variables before instruction Ik. The live variables after an instruction are always the same as the live variables before the next instruction.
Lafter(k) = Lbefore(k + 1)
To start things off, there are no live variables after the last instruction, so
Lafter(n) = ∅
We then apply the following rule repeatedly, traversing the instruction se-
quence back to front.
Lbefore(k) = (Lafter(k) − W (k)) ∪ R(k),
where W(k) are the variables written to by instruction Ik and R(k) are the variables read by instruction Ik. Figure 3.2 shows the results of live variables analysis for the running example, with each instruction aligned with its Lafter set to make the figure easy to read.
40
1 (block ()
2 (movq (int 1) (var v))
3 (movq (int 46) (var w))
4 (movq (var v) (var x))
5 (addq (int 7) (var x))
6 (movq (var x) (var y))
7 (addq (int 4) (var y))
8 (movq (var x) (var z))
9 (addq (var w) (var z))
10 (movq (var y) (var t.1))
11 (negq (var t.1))
12 (movq (var z) (reg rax))
13 (addq (var t.1) (reg rax))
14 (jmp conclusion))
3. REGISTER ALLOCATION
{}
{v} {v,w} {w, x} {w, x} {w, x, y} {w, x, y} {w, y, z} {y,z} {z, t.1} {z, t.1} {t.1}
{}
{}
Figure 3.2: An example block annotated with live-after sets.
Exercise 9. Implement the compiler pass named uncover-live that com- putes the live-after sets. We recommend storing the live-after sets (a list of lists of variables) in the info field of the block construct. We recommend organizing your code to use a helper function that takes a list of instruc- tions and an initial live-after set (typically empty) and returns the list of live-after sets. We recommend creating helper functions to 1) compute the set of variables that appear in an argument (of an instruction), 2) compute the variables read by an instruction which corresponds to the R function discussed above, and 3) the variables written by an instruction which corre- sponds to W.
3.3 Building the Interference Graph
Based on the liveness analysis, we know where each variable is needed. How- ever, during register allocation, we need to answer questions of the specific form: are variables u and v live at the same time? (And therefore cannot be assigned to the same register.) To make this question easier to answer, we create an explicit data structure, an interference graph. An interference graph is an undirected graph that has an edge between two variables if they are live at the same time, that is, if they interfere with each other.
The most obvious way to compute the interference graph is to look at the set of live variables between each statement in the program, and add an edge to the graph for every pair of variables in the same set. This approach
3.3. BUILDING THE INTERFERENCE GRAPH 41
is less than ideal for two reasons. First, it can be rather expensive because it takes O(n2) time to look at every pair in a set of n live variables. Second, there is a special case in which two variables that are live at the same time do not actually interfere with each other: when they both contain the same value because we have assigned one to the other.
A better way to compute the interference graph is to focus on the writes. That is, for each instruction, create an edge between the variable being written to and all the other live variables. (One should not create self edges.) For a callq instruction, think of all caller-saved registers as being written to, so and edge must be added between every live variable and every caller-saved register. For movq, we deal with the above-mentioned special case by not adding an edge between a live variable v and destination d if v matches the source of the move. So we have the following three rules.
1. If instruction Ik is an arithmetic instruction such as (addq s d), then add the edge (d, v) for every v ∈ Lafter(k) unless v = d.
2. If instruction Ik is of the form (callq label), then add an edge (r,v) for every caller-saved register r and every variable v ∈ Lafter(k).
3. If instruction Ik is a move: (movq s d), then add the edge (d,v) for every v ∈ Lafter(k) unless v = d or v = s.
Working from the top to bottom of Figure 3.2, we obtain the following interference for the instruction at the specified line number.
Line 2: no interference,
Line 3: w interferes with v,
Line 4: x interferes with w,
Line 5: x interferes with w,
Line 6: y interferes with w,
Line 7: y interferes with w and x, Line 8: z interferes with w and y, Line 9: z interferes with y,
Line 10: t.1 interferes with z, Line 11: t.1 interferes with z, Line 12: no interference,
Line 13: no interference.
Line 14: no interference.
The resulting interference graph is shown in Figure 3.3.
42 3. REGISTER ALLOCATION vwx
Exercise 10. Implement the compiler pass named build-interference ac- cording to the algorithm suggested above. We recommend using the Racket graph package to create and inspect the interference graph. The output graph of this pass should be stored in the info field of the program, under the key conflicts.
3.4 Graph Coloring via Sudoku
We now come to the main event, mapping variables to registers (or to stack locations in the event that we run out of registers). We need to make sure not to map two variables to the same register if the two variables interfere with each other. In terms of the interference graph, this means that adjacent vertices must be mapped to different registers. If we think of registers as colors, the register allocation problem becomes the widely-studied graph coloring problem [Balakrishnan, 1996, Rosen, 2002].
The reader may be more familiar with the graph coloring problem than he or she realizes; the popular game of Sudoku is an instance of the graph coloring problem. The following describes how to build a graph out of an initial Sudoku board.
• There is one vertex in the graph for each Sudoku square.
• There is an edge between two vertices if the corresponding squares are in the same row, in the same column, or if the squares are in the same 3 × 3 region.
• Choose nine colors to correspond to the numbers 1 to 9.
• Based on the initial assignment of numbers to squares in the Sudoku board, assign the corresponding colors to the corresponding vertices in the graph.
yz
t.1
Figure 3.3: The interference graph of the example program.
3.4. GRAPH COLORING VIA SUDOKU 43
1
2
3
2
3
3
1
2
2
3
1
Figure 3.4: A Sudoku game board and the corresponding colored graph.
If you can color the remaining vertices in the graph with the nine colors, then you have also solved the corresponding game of Sudoku. Figure 3.4 shows an initial Sudoku game board and the corresponding graph with colored vertices. We map the Sudoku number 1 to blue, 2 to yellow, and 3 to red. We only show edges for a sampling of the vertices (those that are colored) because showing edges for all of the vertices would make the graph unreadable.
Given that Sudoku is an instance of graph coloring, one can use Sudoku strategies to come up with an algorithm for allocating registers. For ex- ample, one of the basic techniques for Sudoku is called Pencil Marks. The idea is that you use a process of elimination to determine what numbers no longer make sense for a square, and write down those numbers in the square (writing very small). For example, if the number 1 is assigned to a square, then by process of elimination, you can write the pencil mark 1 in all the squares in the same row, column, and region. Many Sudoku com- puter games provide automatic support for Pencil Marks. The Pencil Marks technique corresponds to the notion of color saturation due to Brélaz [1979]. The saturation of a vertex, in Sudoku terms, is the set of colors that are no longer available. In graph terminology, we have the following definition:
saturation(u) = {c | ∃v.v ∈ neighbors(u) and color(v) = c}
where neighbors(u) is the set of vertices that share an edge with u.
Using the Pencil Marks technique leads to a simple strategy for filling in numbers: if there is a square with only one possible number left, then
44 3. REGISTER ALLOCATION
Algorithm: DSATUR
Input: a graph G
Output: an assignment color[v] for each vertex v ∈ G
W ← vertices(G) while W ̸= ∅ do
pick a vertex u from W with the highest saturation, breaking ties randomly
find the lowest color c that is not in {color[v] : v ∈ adjacent(u)} color[u] ← c
W ← W − {u}
Figure 3.5: The saturation-based greedy graph coloring algorithm.
write down that number! But what if there are no squares with only one possibility left? One brute-force approach is to just make a guess. If that guess ultimately leads to a solution, great. If not, backtrack to the guess and make a different guess. One good thing about Pencil Marks is that it reduces the degree of branching in the search tree. Nevertheless, backtracking can be horribly time consuming. One way to reduce the amount of backtrack- ing is to use the most-constrained-first heuristic. That is, when making a guess, always choose a square with the fewest possibilities left (the vertex with the highest saturation). The idea is that choosing highly constrained squares earlier rather than later is better because later there may not be any possibilities.
In some sense, register allocation is easier than Sudoku because we can always cheat and add more numbers by mapping variables to the stack. We say that a variable is spilled when we decide to map it to a stack location. We would like to minimize the time needed to color the graph, and backtracking is expensive. Thus, it makes sense to keep the most-constrained-first heuris- tic but drop the backtracking in favor of greedy search (guess and just keep going). Figure 3.5 gives the pseudo-code for this simple greedy algorithm for register allocation based on saturation and the most-constrained-first heuristic, which is roughly equivalent to the DSATUR algorithm of Brélaz [1979] (also known as saturation degree ordering [Gebremedhin, 1999, Al- Omari and Sabri, 2006]). Just as in Sudoku, the algorithm represents colors with integers, with the first k colors corresponding to the k registers in a given machine and the rest of the integers corresponding to stack locations.
With this algorithm in hand, let us return to the running example and
3.4. GRAPH COLORING VIA SUDOKU 45
consider how to color the interference graph in Figure 3.3. We shall not use register rax for register allocation because we use it to patch instructions, so we remove that vertex from the graph. Initially, all of the vertices are not yet colored and they are unsaturated, so we annotate each of them with a dash for their color and an empty set for the saturation.
v : −, {} w : −, {} x : −, {}
y : −, {} z : −, {} t.1 : −, {}
We select a maximally saturated vertex and color it 0. In this case we have a 7-way tie, so we arbitrarily pick t.1. The then mark color 0 as no longer available for z because it interferes with t.1.
v : −, {} w : −, {} x : −, {}
y : −, {} z : −, {0} t.1 : 0, {}
Now we repeat the process, selecting another maximally saturated vertex,
which in this case is z. We color z with 1.
v : −, {} w : −, {1} x : −, {}
y : −, {1} z : 1, {0} t.1 : 0, {1}
The most saturated vertices are now w and y. We color y with the first
available color, which is 0.
v : −, {} w : −, {0, 1} x : −, {0, }
y : 0, {1} z : 1, {0} t.1 : 0, {1} Vertex w is now the most highly saturated, so we color w with 2.
v : −,{2} w : 2,{0,1} x : −,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{}
46 3. REGISTER ALLOCATION Now x has the highest saturation, so we color it 1.
v : −,{2} w : 2,{0,1} x : 1,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{} In the last step of the algorithm, we color v with 0.
v : 0,{2} w : 2,{0,1} x : 1,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{}
With the coloring complete, we can finalize the assignment of variables to registers and stack locations. Recall that if we have k registers, we map the first k colors to registers and the rest to stack locations. Suppose for the moment that we have just one register to use for register allocation, rcx. Then the following is the mapping of colors to registers and stack allocations.
{0 → %rcx, 1 → -8(%rbp), 2 → -16(%rbp), . . .}
Putting this mapping together with the above coloring of the variables, we
arrive at the assignment:
{v → %rcx, w → -16(%rbp), x → -8(%rbp), y → %rcx, z → -8(%rbp), t.1 → %rcx}
Applying this assignment to our running example, on the left, yields the program on the right.
(block ()
(movq (int 1) (var v))
(movq (int 46) (var w)) (movq (var v) (var x))
(addq (int 7) (var x))
(movq (var x) (var y))
(addq (int 4) (var y)) ⇒ (movq (var x) (var z))
(addq (var w) (var z))
(movq (var y) (var t.1)) (negq (var t.1))
(movq (var z) (reg rax)) (addq (var t.1) (reg rax)) (jmp conclusion))
(block ()
(movq (int 1) (reg rcx))
(movq (int 46) (deref rbp -16))
(movq (reg rcx) (deref rbp -8))
(addq (int 7) (deref rbp -8))
(movq (deref rbp -8) (reg rcx))
(addq (int 4) (reg rcx))
(movq (deref rbp -8) (deref rbp -8))
(addq (deref rbp -16) (deref rbp -8))
(movq (reg rcx) (reg rcx))
(negq (reg rcx))
(movq (deref rbp -8) (reg rax))
(addq (reg rcx) (reg rax))
(jmp conclusion))
3.4. GRAPH COLORING VIA SUDOKU 47 uniquify remove-complex.
R1 R1 R1
explicate-control
patch-instr. print-x86
x86∗ x86 x86† allocate-reg.
x86∗
Figure 3.6: Diagram of the passes for R1 with register allocation.
The resulting program is almost an x86 program. The remaining step is to apply the patch instructions pass. In this example, the trivial move of -8(%rbp) to itself is deleted and the addition of -16(%rbp) to -8(%rbp) is fixed by going through rax as follows.
(movq (deref rbp -16) (reg rax)
(addq (reg rax) (deref rbp -8))
An overview of all of the passes involved in register allocation is shown in Figure 3.6.
Exercise 11. Implement the pass allocate-registers, which should come after the build-interference pass. The three new passes, uncover-live, build-interference, and allocate-registers replace the assign-homes pass of Section 2.9.
We recommend that you create a helper function named color-graph that takes an interference graph and a list of all the variables in the pro- gram. This function should return a mapping of variables to their colors (represented as natural numbers). By creating this helper function, you will be able to reuse it in Chapter 6 when you add support for functions.
Once you have obtained the coloring from color-graph, you can as- sign the variables to registers or stack locations and then reuse code from the assign-homes pass from Section 2.9 to replace the variables with their
uncover-locals
C0 C0
select-instr.
x86∗ uncover-live
x86∗
build-inter.
48 3. REGISTER ALLOCATION assigned location.
Test your updated compiler by creating new example programs that exercise all of the register allocation algorithm, such as forcing variables to be spilled to the stack.
3.5 Print x86 and Conventions for Registers
Recall the print-x86 pass generates the prelude and conclusion instructions for the main function. The prelude saved the values in rbp and rsp and the conclusion returned those values to rbp and rsp. The reason for this is that our main function must adhere to the x86 calling conventions that we described in Section 3.1. In addition, the main function needs to restore (in the conclusion) any callee-saved registers that get used during register allocation. The simplest approach is to save and restore all of the callee- saved registers. The more efficient approach is to keep track of which callee- saved registers were used and only save and restore them. Either way, make sure to take this use of stack space into account when you are calculating the size of the frame. Also, don’t forget that the size of the frame needs to be a multiple of 16 bytes.
3.6 Challenge: Move Biasing∗
This section describes an optional enhancement to register allocation for those students who are looking for an extra challenge or who have a deeper interest in register allocation.
We return to the running example, but we remove the supposition that we only have one register to use. So we have the following mapping of color numbers to registers.
{0 → %rbx, 1 → %rcx, 2 → %rdx,…}
Using the same assignment that was produced by register allocator described in the last section, we get the following program.
3.6. CHALLENGE: MOVE BIASING∗ 49
(block ()
(movq (int 1) (var v))
(movq (int 46) (var w))
(movq (var v) (var x))
(addq (int 7) (var x))
(movq (var x) (var y))
(addq (int 4) (var y))
(movq (var x) (var z))
(addq (var w) (var z))
(movq (var y) (var t.1))
(negq (var t.1))
(movq (var z) (reg rax))
(addq (var t.1) (reg rax))
(jmp conclusion))
(block ()
(movq (int 1) (reg rbx))
(movq (int 46) (reg rdx))
(movq (reg rbx) (reg rcx))
(addq (int 7) (reg rcx))
(movq (reg rcx) (reg rbx))
⇒ (addq (int 4) (reg rbx)) (movq (reg rcx) (reg rcx)) (addq (reg rdx) (reg rcx)) (movq (reg rbx) (reg rbx)) (negq (reg rbx))
(movq (reg rcx) (reg rax))
(addq (reg rbx) (reg rax))
(jmp conclusion))
While this allocation is quite good, we could do better. For example, the variables v and x ended up in different registers, but if they had been placed in the same register, then the move from v to x could be removed.
We say that two variables p and q are move related if they participate together in a movq instruction, that is, movq p, q or movq q, p. When the register allocator chooses a color for a variable, it should prefer a color that has already been used for a move-related variable (assuming that they do not interfere). Of course, this preference should not override the preference for registers over stack locations, but should only be used as a tie breaker when choosing between registers or when choosing between stack locations.
We recommend that you represent the move relationships in a graph, similar to how we represented interference. The following is the move graph for our running example.
vwx
v. So we have the following coloring and the most saturated vertex is x. v : −,{2} w : 2,{0,1} x : −,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{}
yz
t.1
Now we replay the graph coloring, pausing to see the coloring of x and
50 3. REGISTER ALLOCATION
Last time we chose to color x with 1, which so happens to be the color of z, and x is move related to z. This was rather lucky, and if the program had been a little different, and say z had been already assigned to 2, then x would still get 1 and our luck would have run out. With move biasing, we use the fact that x and z are move related to influence the choice of color for x, in this case choosing 1 because that’s the color of z.
v : −,{2} w : 2,{0,1} x : 1,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{}
Next we consider coloring the variable v, and we just need to avoid choosing 2 because of the interference with w. Last time we choose the color 0, simply because it was the lowest, but this time we know that v is move related to x, so we choose the color 1.
v : 1,{2} w : 2,{0,1} x : 1,{0,2}
y : 0,{1,2} z : 1,{0,2} t.1 : 0,{}
We apply this register assignment to the running example, on the left,
to obtain the code on right.
(block ()
(movq (int 1) (var v))
(movq (int 46) (var w))
(movq (var v) (var x))
(addq (int 7) (var x))
(movq (var x) (var y))
(addq (int 4) (var y))
(movq (var x) (var z))
(addq (var w) (var z))
(movq (var y) (var t.1))
(negq (var t.1))
(movq (var z) (reg rax))
(addq (var t.1) (reg rax))
(jmp conclusion))
(block ()
(movq (int 1) (reg rcx))
(movq (int 46) (reg rbx))
(movq (reg rcx) (reg rcx))
(addq (int 7) (reg rcx))
(movq (reg rcx) (reg rdx))
⇒ (addq (int 4) (reg rdx)) (movq (reg rcx) (reg rcx)) (addq (reg rbx) (reg rcx)) (movq (reg rdx) (reg rbx)) (negq (reg rbx))
(movq (reg rcx) (reg rax))
(addq (reg rbx) (reg rax))
(jmp conclusion))
The patch-instructions then removes the trivial moves from v to x and from x to z to obtain the following result.
3.6. CHALLENGE: MOVE BIASING∗ 51
(block ()
(movq (int 1) (reg rcx))
(movq (int 46) (reg rbx))
(addq (int 7) (reg rcx))
(movq (reg rcx) (reg rdx))
(addq (int 4) (reg rdx))
(addq (reg rbx) (reg rcx))
(movq (reg rdx) (reg rbx))
(negq (reg rbx))
(movq (reg rcx) (reg rax))
(addq (reg rbx) (reg rax))
(jmp conclusion))
Exercise 12. Change your implementation of allocate-registers to take move biasing into account. Make sure that your compiler still passes all of the previous tests. Create two new tests that include at least one opportu- nity for move biasing and visually inspect the output x86 programs to make sure that your move biasing is working properly.
52 3. REGISTER ALLOCATION
4
Booleans and Control Flow
The R0 and R1 languages only had a single kind of value, the integers. In this Chapter we add a second kind of value, the Booleans, to create the R2 language. The Boolean values true and false are written #t and #f respectively in Racket. We also introduce several operations that involve Booleans (and, not, eq?, <, etc.) and the conditional if expression. With the addition of if expressions, programs can have non-trivial control flow which has an impact on several parts of the compiler. Also, because we now have two kinds of values, we need to worry about programs that apply an operation to the wrong kind of value, such as (not 1).
There are two language design options for such situations. One option is to signal an error and the other is to provide a wider interpretation of the operation. The Racket language uses a mixture of these two options, depending on the operation and the kind of value. For example, the result of (not 1) in Racket is #f because Racket treats non-zero integers like #t. On the other hand, (car 1) results in a run-time error in Racket stating that car expects a pair.
The Typed Racket language makes similar design choices as Racket, except much of the error detection happens at compile time instead of run time. Like Racket, Typed Racket accepts and runs (not 1), producing #f. But in the case of (car 1), Typed Racket reports a compile-time error because Typed Racket expects the type of the argument to be of the form (Listof T) or (Pairof T1 T2).
For the R2 language we choose to be more like Typed Racket in that we shall perform type checking during compilation. In Chapter 8 we study the alternative choice, that is, how to compile a dynamically typed language like Racket. The R2 language is a subset of Typed Racket but by no means
53
54 4. BOOLEANS AND CONTROL FLOW
cmp ::= eq?|<|<=|>|>=
exp ::= |
|
|
R2 ::=
int|(read)|(-exp)|(+expexp)|(-expexp) var | (let ([var exp]) exp)
#t | #f | (and exp exp) | (or exp exp) | (not exp) (cmp exp exp) | (if exp exp exp)
(program info exp)
Figure 4.1: The syntax of R2, extending R1 (Figure 2.1) with Booleans and conditionals.
includes all of Typed Racket. Furthermore, for many of the operations we shall take a narrower interpretation than Typed Racket, for example, rejecting (not 1).
This chapter is organized as follows. We begin by defining the syntax and interpreter for the R2 language (Section 4.1). We then introduce the idea of type checking and build a type checker for R2 (Section 4.2). To compile R2 we need to enlarge the intermediate language C0 into C1, which we do in Section 4.5. The remaining sections of this Chapter discuss how our compiler passes need to change to accommodate Booleans and conditional control flow.
4.1 The R2 Language
The syntax of the R2 language is defined in Figure 4.1. It includes all of R1 (shown in gray), the Boolean literals #t and #f, and the conditional if expression. Also, we expand the operators to include subtraction, and, or and not, the eq? operations for comparing two integers or two Booleans, and the <, <=, >, and >= operations for comparing integers.
Figure 4.2 defines the interpreter for R2, omitting the parts that are the same as the interpreter for R1 (Figure 2.2). The literals #t and #f simply evaluate to themselves. The conditional expression (if cnd thn els) evaluates the Boolean expression cnd and then either evaluates thn or els depending on whether cnd produced #t or #f. The logical operations not and and behave as you might expect, but note that the and operation is short-circuiting. That is, given the expression (and e1 e2), the expression e2 is not evaluated if e1 evaluates to #f.
With the addition of the comparison operations, there are quite a few primitive operations and the interpreter code for them is somewhat repet- itive. In Figure 4.2 we factor out the different parts into the interp-op
4.2. TYPE CHECKING R2 PROGRAMS 55
function and the similar parts into the one match clause shown in Fig- ure 4.2. We do not use interp-op for the and operation because of the short-circuiting behavior in the order of evaluation of its arguments.
4.2 Type Checking R2 Programs
It is helpful to think about type checking in two complementary ways. A type checker predicts the type of value that will be produced by each expres- sion in the program. For R2, we have just two types, Integer and Boolean. So a type checker should predict that
(+ 10 (- (+ 12 20))) produces an Integer while
(and (not #f) #t)
produces a Boolean.
As mentioned at the beginning of this chapter, a type checker also rejects
programs that apply operators to the wrong type of value. Our type checker for R2 will signal an error for the following expression because, as we have seen above, the expression (+ 10 …) has type Integer, and we require the argument of a not to have type Boolean.
(not (+ 10 (- (+ 12 20))))
The type checker for R2 is best implemented as a structurally recur- sive function over the AST. Figure 4.3 shows many of the clauses for the type-check-exp function. Given an input expression e, the type checker ei- ther returns the type (Integer or Boolean) or it signals an error. Of course, the type of an integer literal is Integer and the type of a Boolean literal is Boolean. To handle variables, the type checker, like the interpreter, uses an association list. However, in this case the association list maps variables to types instead of values. Consider the clause for let. We type check the initializing expression to obtain its type T and then associate type T with the variable x. When the type checker encounters the use of a variable, it can find its type in the association list.
Exercise 13. Complete the implementation of type-check-R2 and test it on 10 new example programs in R2 that you choose based on how thoroughly they test the type checking algorithm. Half of the example programs should have a type error, to make sure that your type checker properly rejects them. The other half of the example programs should not have type errors. Your testing should check that the result of the type checker agrees with
56
4. BOOLEANS AND CONTROL FLOW
(define primitives (set ’+ ’- ’eq? ’< ’<= ’> ’>= ’not ’read))
(define (interp-op op)
(match op
…
[’not (lambda (v) (match v [#t #f] [#f #t]))] [’eq? (lambda (v1 v2)
(cond [(or (and (fixnum? v1) (fixnum? v2)) (and (boolean? v1) (boolean? v2)))
(eq? v1 v2)]))] [’< (lambda (v1 v2)
(cond [(and (fixnum? v1) (fixnum? v2)) (< v1 v2)]))] [’<= (lambda (v1 v2)
(cond [(and (fixnum? v1) (fixnum? v2)) (<= v1 v2)]))] [’> (lambda (v1 v2)
(cond [(and (fixnum? v1) (fixnum? v2)) (> v1 v2)]))] [’>= (lambda (v1 v2)
(cond [(and (fixnum? v1) (fixnum? v2)) (>= v1 v2)]))] [else (error ’interp-op “unknown␣operator”)]))
(define (interp-exp env)
(lambda (e)
(define recur (interp-exp env))
(match e
…
[(? boolean?) e]
[‘(if ,cnd ,thn ,els)
(define b (recur cnd))
(match b
[#t (recur thn)]
[#f (recur els)])] [‘(and ,e1 ,e2)
(define v1 (recur e1))
(match v1
[#t (match (recur e2) [#t #t] [#f #f])]
[#f #f])]
[‘(,op ,args …)
#:when (set-member? primitives op)
(apply (interp-op op) (for/list ([e args]) (recur e)))] )))
(define (interp-R2 env)
(lambda (p)
(match p
[‘(program ,info ,e)
((interp-exp ’()) e)])))
Figure 4.2: Interpreter for the R2 language.
4.2. TYPE CHECKING R2 PROGRAMS 57
(define (type-check-exp env)
(lambda (e)
(define recur (type-check-exp env))
(match e
[(? fixnum?) ’Integer]
[(? boolean?) ’Boolean]
[(? symbol? x) (dict-ref env x)] [‘(read) ’Integer]
[‘(let ([,x ,e]) ,body)
(define T (recur e))
(define new-env (cons (cons x T) env)) (type-check-exp new-env body)]
…
[‘(not ,e)
(match (recur e)
[’Boolean ’Boolean]
[else (error ’type-check-exp “’not’␣expects␣a␣Boolean” e)])]
… )))
(define (type-check-R2 env)
(lambda (e)
(match e
[‘(program ,info ,body)
(define ty ((type-check-exp ’()) body))
‘(program ,info ,body)]
)))
Figure 4.3: Skeleton of a type checker for the R2 language.
58 4. BOOLEANS AND CONTROL FLOW
the value returned by the interpreter, that is, if the type checker returns Integer, then the interpreter should return an integer. Likewise, if the type checker returns Boolean, then the interpreter should return #t or #f. Note that if your type checker does not signal an error for a program, then interpreting that program should not encounter an error. If it does, there is something wrong with your type checker.
4.3 Shrink the R2 Language
The R2 language includes several operators that are easily expressible in terms of other operators. For example, subtraction is expressible in terms of addition and negation.
(-e1 e2) ⇒ (+e1 (-e2))
Several of the comparison operations are expressible in terms of less-than
and logical negation.
(<= e1 e2) ⇒ (not (< e2 e1))
By performing these translations near the front-end of the compiler, the later passes of the compiler will not need to deal with these constructs, making those passes shorter. On the other hand, sometimes these translations make it more difficult to generate the most efficient code with respect to the number of instructions. However, these differences typically do not affect the number of accesses to memory, which is the primary factor that determines execution time on modern computer architectures.
Exercise 14. Implement the pass shrink that removes subtraction, and, or, <=, >, and >= from the language by translating them to other constructs in R2. Create tests to make sure that the behavior of all of these constructs stays the same after translation.
4.4 XOR, Comparisons, and Control Flow in x86
To implement the new logical operations, the comparison operations, and the if expression, we need to delve further into the x86 language. Fig- ure 4.4 defines the abstract syntax for a larger subset of x86 that includes instructions for logical operations, comparisons, and jumps.
One small challenge is that x86 does not provide an instruction that directly implements logical negation (not in R2 and C1). However, the
4.4. XOR, COMPARISONS, AND CONTROL FLOW IN X86 59
arg ::= (int int) | (reg register) | (derefregister int) | (byte-reg register)
cc ::= e|l|le|g|ge
instr ::= (addq arg arg) | (subq arg arg) | (negq arg) | (movq arg arg)
| (callq label) | (pushq arg) | (popq arg) | (retq) | (xorq arg arg) | (cmpq arg arg) | (set cc arg)
| (movzbq arg arg) | (jmp label) | (jmp-if cc label) | (label label )
x861 ::= (program info (type type) instr+)
Figure 4.4: The x861 language (extends x860 of Figure 2.7).
xorq instruction can be used to encode not. The xorq instruction takes two arguments, performs a pairwise exclusive-or operation on each bit of its arguments, and writes the results into its second argument. Recall the truth table for exclusive-or:
01 001 110
For example, 0011 XOR 0101 = 0110. Notice that in row of the table for the bit 1, the result is the opposite of the second bit. Thus, the not operation can be implemented by xorq with 1 as the first argument: 0001 XOR 0000 = 0001 and 0001 XOR 0001 = 0000.
Next we consider the x86 instructions that are relevant for compiling the comparison operations. The cmpq instruction compares its two arguments to determine whether one argument is less than, equal, or greater than the other argument. The cmpq instruction is unusual regarding the order of its arguments and where the result is placed. The argument order is backwards: if you want to test whether x < y, then write cmpq y, x. The result of cmpq is placed in the special EFLAGS register. This register cannot be accessed directly but it can be queried by a number of instructions, including the set instruction. The set instruction puts a 1 or 0 into its destination depending on whether the comparison came out according to the condition code cc (e for equal, l for less, le for less-or-equal, g for greater, ge for greater-or- equal). The set instruction has an annoying quirk in that its destination argument must be single byte register, such as al, which is part of the rax register. Thankfully, the movzbq instruction can then be used to move from a single byte register to a normal 64-bit register.
60 4. BOOLEANS AND CONTROL FLOW
arg ::= cmp ::= exp ::= stmt ::= tail ::=
int |var |#t|#f
eq?|< arg|(read)|(-arg)|(+argarg)|(notarg)|(cmpargarg) (assign var exp)
(return exp) | (seq stmt tail)
C1 ::=
(program info ((label . tail )+ ))
| (goto label) | (if (cmp arg arg) (goto label) (goto label))
Figure 4.5: The C1 language, extending C0 with Booleans and conditionals.
For compiling the if expression, the x86 instructions for jumping are relevant. The jmp instruction updates the program counter to point to the instruction after the indicated label. The jmp-if instruction updates the program counter to point to the instruction after the indicated label depending on whether the result in the EFLAGS register matches the con- dition code cc, otherwise the jmp-if instruction falls through to the next instruction. Because the jmp-if instruction relies on the EFLAGS register, it is quite common for the jmp-if to be immediately preceeded by a cmpq instruction, to set the EFLAGS regsiter. Our abstract syntax for jmp-if differs from the concrete syntax for x86 to separate the instruction name from the condition code. For example, (jmp-if le foo) corresponds to jle foo.
4.5 The C1 Intermediate Language
As with R1, we shall compile R2 to a C-like intermediate language, but we need to grow that intermediate language to handle the new features in R2: Booleans and conditional expressions. Figure 4.5 shows the new features of C1; we add logic and comparison operators to the exp non-terminal, the literals #t and #f to the arg non-terminal. Regarding control flow, C1 dif- fers considerably from R2. Instead of if expressions, C1 has goto’s and conditional goto’s in the grammar for tail. This means that a sequence of statements may now end with a goto or a conditional goto, which jumps to one of two labeled pieces of code depending on the outcome of the com- parison. In Section 4.6 we discuss how to translate from R2 to C1, bridging this gap between if expressions and goto’s.
4.6. EXPLICATE CONTROL 61 4.6 Explicate Control
Recall that the purpose of explicate-control is to make the order of evaluation explicit in the syntax of the program. With the addition of if in R2, things get more interesting.
As a motivating example, consider the following program that has an if expression nested in the predicate of another if.
(program ()
(if (if (eq? (read) 1)
(eq? (read) 0)
(eq? (read) 2)) (+ 10 32)
(+ 700 77)))
The naive way to compile if and eq? would be to handle each of them in isolation, regardless of their context. Each eq? would be translated into a cmpq instruction followed by a couple instructions to move the result from the EFLAGS register into a general purpose register or stack location. Each if would be translated into the combination of a cmpq and jmp-if. However, if we take context into account we can do better and reduce the use of cmpq and EFLAG-accessing instructions.
One idea is to try and reorganize the code at the level of R2, pushing the outer if inside the inner one. This would yield the following code.
(if (eq? (read) 1)
(if (eq? (read) 0)
(+ 10 32)
(+ 700 77))
(if (eq? (read) 2))
(+ 10 32)
(+ 700 77))
Unfortunately, this approach duplicates the two branches, and a compiler must never duplicate code!
We need a way to perform the above transformation, but without dupli- cating code. The solution is straightforward if we think at the level of x86 assembly: we can label the code for each of the branches and insert goto’s in all the places that need to execute the branches. Put another way, we need to move away from abstract syntax trees and instead use graphs. In particular, we shall use a standard program representation called a control flow graph (CFG), due to Frances Elizabeth Allen [1970]. Each vertex is a labeled sequence of code, called a basic block, and each edge represents a jump to another block. The program construct of C0 and C1 represents
62
(program ()
(if (if (eq? (read) 1)
(eq? (read) 0)
(eq? (read) 2)) (+ 10 32)
(+ 700 77)))
⇓
4.
BOOLEANS AND CONTROL FLOW
(program ()
((block62 .
(seq (assign tmp54 (read)) (if (eq? tmp54 2)
(goto block59)
(goto block60))))
(block61 .
(seq (assign tmp53 (read)) (if (eq? tmp53 0)
(goto block57)
(goto block58))))
(block60 . (goto block56))
(block59 . (goto block55))
(block58 . (goto block56))
(block57 . (goto block55))
(block56 . (return (+ 700 77))) (block55 . (return (+ 10 32))) (start .
(seq (assign tmp52 (read)) (if (eq? tmp52 1)
(goto block61)
(goto block62))))))
(program () ⇒ (if (if (let ([tmp52 (read)])
(eq? tmp52 1))
(let ([tmp53 (read)])
(eq? tmp53 0))
(let ([tmp54 (read)])
(eq? tmp54 2))) (+ 10 32)
(+ 700 77)))
Figure 4.6: Example translation from R2 to C1 via the explicate-control. a control flow graph as an association list mapping labels to basic blocks.
Each block is represented by the tail non-terminal.
Figure 4.6 shows the output of the remove-complex-opera* pass and then the explicate-control pass on the example program. We shall walk through the output program and then discuss the algorithm. Following the order of evaluation in the output of remove-complex-opera*, we first have the (read) and comparison to 1 from the predicate of the inner if. In the output of explicate-control, in the start block, this becomes a (read) followed by a conditional goto to either block61 or block62. Each of these contains the translations of the code (eq? (read) 0) and (eq? (read) 1), respectively. Regarding block61, we start with the (read) and comparison to 0 and then have a conditional goto, either to block59 or block60, which indirectly take us to block55 and block56, the two branches of the outer if, i.e., (+ 10 32) and (+ 700 77). The story for block62 is similar.
The nice thing about the output of explicate-control is that there
4.6. EXPLICATE CONTROL 63
are no unnecessary uses of eq? and every use of eq? is part of a conditional jump. The down-side of this output is that it includes trivial blocks, such as block57 through block60, that only jump to another block. We discuss a solution to this problem in Section 4.11.
Recall that in Section 2.6 we implement the explicate-control pass for R1 using two mutually recursive functions, explicate-control-tail and explicate-control-assign. The former function translated expressions in tail position whereas the later function translated expressions on the right- hand-side of a let. With the addition of if expression in R2 we have a new kind of context to deal with: the predicate position of the if. So we shall need another function, explicate-control-pred, that takes an R2 expression and two pieces of C1 code (two tail’s) for the then-branch and else-branch. The output of explicate-control-pred is a C1 tail. However, these three functions also need to contruct the control-flow graph, which we recommend they do via updates to a global variable. Next we consider the specific additions to the tail and assign functions, and some of cases for the pred function.
The explicate-control-tail function needs an additional case for if. The branches of the if inherit the current context, so they are in tail posi- tion. Let B1 be the result of explicate-control-tail on the thn branch and B2 be the result of apply explicate-control-tail to the else branch. Then the if translates to the block B3 which is the result of applying explicate-control-pred to the predicate cnd and the blocks B1 and B2.
(if cnd thn els) ⇒ B3
Next we consider the case for if in the explicate-control-assign function. So the context of the if is an assignment to some variable x and then the control continues to some block B1. The code that we generate for both the thn and els branches shall both need to continue to B1, so we add B1 to the control flow graph with a fresh label l1. Again, the branches of the if inherit the current context, so that are in assignment positions. Let B2 be the result of applying explicate-control-assign to the thn branch, variable x, and the block (goto l1). Let B3 be the result of applying explicate-control-assign to the else branch, variable x, and the block (goto l1). The if translates to the block B4 which is the result of applying explicate-control-pred to the predicate cnd and the blocks B2 and B3.
(if cnd thn els) ⇒ B4
The function explicate-control-pred will need a case for every ex- pression that can have type Boolean. We detail a few cases here and leave
64 4. BOOLEANS AND CONTROL FLOW
the rest for the reader. The input to this function is an expression and two blocks, B1 and B2, for the branches of the enclosing if. One of the base cases of this function is when the expression is a less-than comparision. We translate it to a conditional goto. We need labels for the two branches B1 and B2, so we add them to the control flow graph and obtain some labels l1 and l2. The translation of the less-than comparison is as follows.
(< e1 e2) ⇒ (if (< e1 e2) (goto l1) (goto l2))
The case for if in explicate-control-pred is particularly illuminating, as it deals with the challenges that we discussed above regarding the example of the nested if expressions. Again, we add the two input branches B1 and B2 to the control flow graph and obtain the labels l1 and l2. The branches thn and els of the current if inherit their context from the current one, i.e., predicate context. So we apply explicate-control-pred to thn with the two blocks (goto l1) and (goto l2), to obtain B3. Similarly for the els branch, to obtain B4. Finally, we apply explicate-control-pred to the predicate cnd and the blocks B3 and B4 to obtain the result B5.
(if cnd thn els) ⇒ B5
Exercise 15. Implement the pass explicate-code by adding the cases for if to the functions for tail and assignment contexts, and implement the function for predicate contexts. Create test cases that exercise all of the new cases in the code for this pass.
4.7 Select Instructions
Recall that the select-instructions pass lowers from our C-like inter- mediate representation to the pseudo-x86 language, which is suitable for conducting register allocation. The pass is implemented using three auxil- liary functions, one for each of the non-terminals arg, stmt, and tail.
For arg, we have new cases for the Booleans. We take the usual approach of encoding them as integers, with true as 1 and false as 0.
#t ⇒ 1 #f ⇒ 0
For stmt, we discuss a couple cases. The not operation can be imple- mented in terms of xorq as we discussed at the beginning of this section. Given an assignment (assign lhs (not arg)), if the left-hand side lhs is the same as arg, then just the xorq suffices:
(assign x (not x)) ⇒ ((xorq (int 1) x′))
4.8. REGISTER ALLOCATION 65 Otherwise, a movq is needed to adapt to the update-in-place semantics of
x86. Let arg′ be the result of recursively processing arg. Then we have (assign lhs (not arg)) ⇒ ((movq arg′ lhs′) (xorq (int 1) lhs′))
Next consider the cases for eq? and less-than comparison. Translating these operations to x86 is slightly involved due to the unusual nature of the cmpq instruction discussed above. We recommend translating an assignment from eq? into the following sequence of three instructions.
(cmpq arg′2 arg′1) (assign lhs (eq? arg1 arg2)) ⇒ (set e (byte-reg al))
(movzbq (byte-reg al) lhs′)
Regarding the tail non-terminal, we have two new cases, for goto and conditional goto. Both are straightforward to handle. A goto becomes a jump instruction.
(goto l) ⇒ ((jmp l))
A conditional goto becomes a compare instruction followed by a conditional jump (for “then”) and the fall-through is to a regular jump (for “else”).
(if (eq? arg1 arg2) (goto l1)
(goto l2))
((cmpq arg′2 arg′1) ⇒ (jmp-if e l1)
(jmp l2))
Exercise 16. Expand your select-instructions pass to handle the new features of the R2 language. Test the pass on all the examples you have created and make sure that you have some test programs that use the eq? and < operators, creating some if necessary. Test the output using the interp-x86 interpreter (Appendix 12.1).
4.8 Register Allocation
The changes required for R2 affect the liveness analysis, building the inter- ference graph, and assigning homes, but the graph coloring algorithm itself does not need to change.
4.8.1 Liveness Analysis
Recall that for R1 we implemented liveness analysis for a single basic block (Section 3.2). With the addition of if expressions to R2, explicate-control
66 4. BOOLEANS AND CONTROL FLOW
now produces many basic blocks arranged in a control-flow graph. The first question we need to consider is in what order should we process the basic blocks? Recall that to perform liveness analysis, we need to know the live- after set. If a basic block has no successor blocks, then it has an empty live-after set and we can immediately apply liveness analysis to it. If a basic block has some successors, then we need to complete liveness analysis on those blocks first. Furthermore, we know that the control flow graph does not contain any cycles (it is a DAG, that is, a directed acyclic graph)1. What all this amounts to is that we need to process the basic blocks in reverse topological order. We recommend using the tsort and transpose functions of the Racket graph package to obtain this ordering.
The next question is how to compute the live-after set of a block given the live-before sets of all its successor blocks. During compilation we do not know which way the branch will go, so we do not know which of the successor’s live-before set to use. The solution comes from the observation that there is no harm in identifying more variables as live than absolutely necessary. Thus, we can take the union of the live-before sets from all the successors to be the live-after set for the block. Once we have computed the live-after set, we can proceed to perform liveness analysis on the block just as we did in Section 3.2.
The helper functions for computing the variables in an instruction’s ar- gument and for computing the variables read-from (R) or written-to (W) by an instruction need to be updated to handle the new kinds of arguments and instructions in x861.
4.8.2 Build Interference
Many of the new instructions in x861 can be handled in the same way as the instructions in x860. Thus, if your code was already quite general, it will not need to be changed to handle the new instructions. If not, I recommend that you change your code to be more general. The movzbq instruction should be handled like the movq instruction.
Exercise 17. Update the register-allocation pass so that it works for R2 and test your compiler using your previously created programs on the interp-x86 interpreter (Appendix 12.1).
1If we were to add loops to the language, then the CFG could contain cycles and we would instead need to use the classic worklist algorithm for computing the fixed point of the liveness analysis [Aho et al., 1986].
4.9. PATCH INSTRUCTIONS 67 4.9 Patch Instructions
The second argument of the cmpq instruction must not be an immediate value (such as a literal integer). So if you are comparing two immediates, we recommend inserting a movq instruction to put the second argument in rax. The second argument of the movzbq must be a register. There are no special restrictions on the x86 instructions jmp-if, jmp, and label.
Exercise 18. Update patch-instructions to handle the new x86 instruc- tions. Test your compiler using your previously created programs on the interp-x86 interpreter (Appendix 12.1).
4.10 An Example Translation
Figure 4.7 shows a simple example program in R2 translated to x86, showing the results of explicate-control, select-instructions, and the final x86 assembly code.
Figure 4.8 lists all the passes needed for the compilation of R2.
4.11 Challenge: Optimize Jumps∗ UNDER CONSTRUCTION
68 4.
(program ()
(if (eq? (read) 1) 42 0))
⇓
(program ()
((block32 . (return 0))
(block31 . (return 42)) (start .
(seq (assign tmp30 (read)) (if (eq? tmp30 1)
(goto block31)
(goto block32))))))
⇓
(program ((locals . (tmp30)))
((block32 .
(block ()
(movq (int 0) (reg rax))
(jmp conclusion)))
(block31 .
(block ()
(movq (int 42) (reg rax))
(jmp conclusion)))
(start .
(block ()
(callq read_int)
(movq (reg rax) (var tmp30))
(cmpq (int 1) (var tmp30))
(jmp-if e block31)
(jmp block32)))))
BOOLEANS AND CONTROL FLOW
_block31:
movq $42, %rax
jmp _conclusion
_block32:
movq $0, %rax
jmp _conclusion
_start:
callq _read_int movq %rax, %rcx cmpq $1, %rcx je _block31
jmp _block32
.globl _main
_main:
Figure 4.7: Example compilation of an if expression to x86.
⇒ pushq movq
%rbp
%rsp, %rbp
%r12
%rbx
%r13
%r14
$0, %rsp
pushq
pushq
pushq
pushq
subq
jmp _start
_conclusion:
addq $0, %rsp
popq %r14 popq %r13 popq %rbx popq %r12 popq %rbp retq
4.11. CHALLENGE: OPTIMIZE JUMPS∗ 69
typecheck shrink uniquify remove-complex. R2 R2 R2 R2 R2
uncover-locals
C1 C1
explicate-control
print-x86
x86†
select-instr.
patch-instr.
x86∗ x86∗ x86∗ uncover-live allocate-reg.
x86∗
Figure 4.8: Diagram of the passes for R2, a language with conditionals.
x86∗
build-inter.
70 4. BOOLEANS AND CONTROL FLOW
5
Tuples and Garbage Collection
In this chapter we study the implementation of mutable tuples (called “vec- tors” in Racket). This language feature is the first to use the computer’s heap because the lifetime of a Racket tuple is indefinite, that is, a tuple lives forever from the programmer’s viewpoint. Of course, from an implemen- tor’s viewpoint, it is important to reclaim the space associated with a tuple when it is no longer needed, which is why we also study garbage collection techniques in this chapter.
Section 5.1 introduces the R3 language including its interpreter and type checker. The R3 language extends the R2 language of Chapter 4 with vectors and Racket’s “void” value. The reason for including the later is that the vector-set! operation returns a value of type Void1.
Section 5.2 describes a garbage collection algorithm based on copying live objects back and forth between two halves of the heap. The garbage collec- tor requires coordination with the compiler so that it can see all of the root pointers, that is, pointers in registers or on the procedure call stack. Sec- tions 5.3 through 5.8 discuss all the necessary changes and additions to the compiler passes, including a new compiler pass named expose-allocation.
1This may sound contradictory, but Racket’s Void type corresponds to what is more commonly called the Unit type. This type is inhabited by a single value that is usually written unit or ()[Pierce, 2002].
71
72
5. TUPLES AND GARBAGE COLLECTION
(let ([t (vector 40 #t (vector 2))]) (if (vector-ref t 1)
(+ (vector-ref t 0)
(vector-ref (vector-ref t 2) 0))
44))
Figure 5.1: Example program that creates tuples and reads from them.
type ::= Integer | Boolean | (Vector type+) | Void cmp ::= eq?|<|<=|>|>=
exp ::= |
| | | | |
R3 ::=
int|(read)|(-exp)|(+expexp)|(-expexp) var | (let ([var exp]) exp) #t|#f|(andexpexp)|(orexpexp)|(notexp) (cmp exp exp) | (if exp exp exp)
(vector exp+) | (vector-ref exp int) (vector-set! exp int exp)
(void)
(program exp)
Figure 5.2: The syntax of R3, extending R2 (Figure 4.1) with tuples. 5.1 The R3 Language
Figure 5.2 defines the syntax for R3, which includes three new forms for creating a tuple, reading an element of a tuple, and writing to an element of a tuple. The program in Figure 5.1 shows the usage of tuples in Racket. We create a 3-tuple t and a 1-tuple. The 1-tuple is stored at index 2 of the 3-tuple, demonstrating that tuples are first-class values. The element at index 1 of t is #t, so the “then” branch is taken. The element at index 0 of t is 40, to which we add the 2, the element at index 0 of the 1-tuple.
Tuples are our first encounter with heap-allocated data, which raises several interesting issues. First, variable binding performs a shallow-copy when dealing with tuples, which means that different variables can refer to the same tuple, i.e., different variables can be aliases for the same thing. Consider the following example in which both t1 and t2 refer to the same tuple. Thus, the mutation through t2 is visible when referencing the tuple from t1, so the result of this program is 42.
(let ([t1 (vector 3 7)]) (let ([t2 t1])
5.2. GARBAGE COLLECTION 73 (let ([_ (vector-set! t2 0 42)])
(vector-ref t1 0))))
The next issue concerns the lifetime of tuples. Of course, they are created by the vector form, but when does their lifetime end? Notice that the grammar in Figure 5.2 does not include an operation for deleting tuples. Furthermore, the lifetime of a tuple is not tied to any notion of static scoping. For example, the following program returns 3 even though the variable t goes out of scope prior to accessing the vector.
(vector-ref
(let ([t (vector 3 7)])
t) 0)
From the perspective of programmer-observable behavior, tuples live forever. Of course, if they really lived forever, then many programs would run out of memory.2 A Racket implementation must therefore perform automatic garbage collection.
Figure 5.3 shows the definitional interpreter for the R3 language and Figure 5.4 shows the type checker. The additions to the interpreter are straightforward but the updates to the type checker deserve some explana- tion. As we shall see in Section 5.2, we need to know which variables are pointers into the heap, that is, which variables are vectors. Also, when al- locating a vector, we shall need to know which elements of the vector are pointers. We can obtain this information during type checking and when we uncover local variables. The type checker in Figure 5.4 not only computes the type of an expression, it also wraps every sub-expression e with the form (has-type e T ), where T is e’s type. Subsequently, in the uncover-locals pass (Section 5.5) this type information is propagated to all variables (in- cluding the temporaries generated by remove-complex-opera*).
5.2 Garbage Collection
Here we study a relatively simple algorithm for garbage collection that is the basis of state-of-the-art garbage collectors [Lieberman and Hewitt, 1983, Ungar, 1984, Jones and Lins, 1996, Detlefs et al., 2004, Dybvig, 2006, Tene et al., 2011]. In particular, we describe a two-space copying collector [Wilson, 1992] that uses Cheney’s algorithm to perform the copy [Cheney, 1970].
2The R3 language does not have looping or recursive function, so it is nigh impossible to write a program in R3 that will run out of memory. However, we add recursive functions in the next Chapter!
74
5. TUPLES AND GARBAGE COLLECTION
(define primitives (set … ’vector ’vector-ref ’vector-set!))
(define (interp-op op)
(match op
…
[’vector vector]
[’vector-ref vector-ref]
[’vector-set! vector-set!]
[else (error ’interp-op “unknown␣operator”)]))
(define (interp-R3 env)
(lambda (e)
(match e …
[else (error ’interp-R3 “unrecognized␣expression”)] )))
Figure 5.3: Interpreter for the R3 language.
Figure 5.5 gives a coarse-grained depiction of what happens in a two-space collector, showing two time steps, prior to garbage collection on the top and after garbage collection on the bottom. In a two-space collector, the heap is divided into two parts, the FromSpace and the ToSpace. Initially, all allocations go to the FromSpace until there is not enough room for the next allocation request. At that point, the garbage collector goes to work to make more room.
The garbage collector must be careful not to reclaim tuples that will be used by the program in the future. Of course, it is impossible in general to predict what a program will do, but we can overapproximate the will-be- used tuples by preserving all tuples that could be accessed by any program given the current computer state. A program could access any tuple whose address is in a register or on the procedure call stack. These addresses are called the root set. In addition, a program could access any tuple that is transitively reachable from the root set. Thus, it is safe for the garbage collector to reclaim the tuples that are not reachable in this way.
So the goal of the garbage collector is twofold:
1. preserve all tuple that are reachable from the root set via a path of
pointers, that is, the live tuples, and
2. reclaim the memory of everything else, that is, the garbage.
5.2. GARBAGE COLLECTION 75
(define (type-check-exp env)
(lambda (e)
(define recur (type-check-exp env))
(match e
…
[’(void) (values ’(has-type (void) Void) ’Void)] [‘(vector ,es …)
(define-values (e* t*) (for/lists (e* t*) ([e es])
(recur e)))
(let ([t ‘(Vector ,@t*)])
(debug “vector/type-check-exp␣finished␣vector” t) (values ‘(has-type (vector ,@e*) ,t) t))]
[‘(vector-ref ,e ,i)
(define-values (e^ t) (recur e))
(match t
[‘(Vector ,ts …)
(unless (and (exact-nonnegative-integer? i) (< i (length ts)))
(error ’type-check-exp "invalid␣index␣~a" i)) (let ([t (list-ref ts i)])
(values ‘(has-type (vector-ref ,e^ (has-type ,i Integer)) ,t) t))]
[else (error "expected␣a␣vector␣in␣vector-ref,␣not" t)])] [‘(eq? ,arg1 ,arg2)
(define-values (e1 t1) (recur arg1))
(define-values (e2 t2) (recur arg2))
(match* (t1 t2)
[(‘(Vector ,ts1 ...) ‘(Vector ,ts2 ...))
(values ‘(has-type (eq? ,e1 ,e2) Boolean) ’Boolean)]
[(other wise) ((super type-check-exp env) e)])]
...
)))
Figure 5.4: Type checker for the R3 language.
76 5. TUPLES AND GARBAGE COLLECTION
A copying collector accomplishes this by copying all of the live objects from the FromSpace into the ToSpace and then performs a slight of hand, treating the ToSpace as the new FromSpace and the old FromSpace as the new ToSpace. In the example of Figure 5.5, there are three pointers in the root set, one in a register and two on the stack. All of the live objects have been copied to the ToSpace (the right-hand side of Figure 5.5) in a way that preserves the pointer relationships. For example, the pointer in the register still points to a 2-tuple whose first element is a 3-tuple and second element is a 2-tuple. There are four tuples that are not reachable from the root set and therefore do not get copied into the ToSpace. (The sitation in Figure 5.5, with a cycle, cannot be created by a well-typed program in R3. However, creating cycles will be possible once we get to R6. We design the garbage collector to deal with cycles to begin with, so we will not need to revisit this issue.)
There are many alternatives to copying collectors (and their older sib- lings, the generational collectors) when its comes to garbage collection, such as mark-and-sweep and reference counting. The strengths of copying col- lectors are that allocation is fast (just a test and pointer increment), there is no fragmentation, cyclic garbage is collected, and the time complexity of collection only depends on the amount of live data, and not on the amount of garbage [Wilson, 1992]. The main disadvantage of two-space copying col- lectors is that they use a lot of space, though that problem is ameliorated in generational collectors. Racket and Scheme programs tend to allocate many small objects and generate a lot of garbage, so copying and genera- tional collectors are a good fit. Of course, garbage collection is an active research topic, especially concurrent garbage collection [Tene et al., 2011]. Researchers are continuously developing new techniques and revisiting old trade-offs [Blackburn et al., 2004, Jones et al., 2011, Shahriyar et al., 2013, Cutler and Morris, 2015, Shidal et al., 2015].
5.2.1 Graph Copying via Cheney’s Algorithm
Let us take a closer look at how the copy works. The allocated objects and pointers can be viewed as a graph and we need to copy the part of the graph that is reachable from the root set. To make sure we copy all of the reachable vertices in the graph, we need an exhaustive graph traversal algorithm, such as depth-first search or breadth-first search [Moore, 1959, Cormen et al., 2001]. Recall that such algorithms take into account the possibility of cycles by marking which vertices have already been visited, so as to ensure termination of the algorithm. These search algorithms also use
5.2. GARBAGE COLLECTION 77
Registers Heap
1
#f
...
Stack
...
FromSpace
ToSpace
#t
42
9
7
5
3
6
#t
0
8
4
5
2
Registers Heap
1
#f
...
Stack
...
FromSpace
ToSpace
#t
42
#t
42
9
#t
0
7
5
3
7
5
3
6
8
8
4
4
2
5
Figure 5.5: A copying collector in action.
78 5. TUPLES AND GARBAGE COLLECTION
a data structure such as a stack or queue as a to-do list to keep track of the vertices that need to be visited. We shall use breadth-first search and a trick due to Cheney [1970] for simultaneously representing the queue and copying tuples into the ToSpace.
Figure 5.6 shows several snapshots of the ToSpace as the copy progresses. The queue is represented by a chunk of contiguous memory at the beginning of the ToSpace, using two pointers to track the front and the back of the queue. The algorithm starts by copying all tuples that are immediately reachable from the root set into the ToSpace to form the initial queue. When we copy a tuple, we mark the old tuple to indicate that it has been visited. (We discuss the marking in Section 5.2.2.) Note that any pointers inside the copied tuples in the queue still point back to the FromSpace. Once the initial queue has been created, the algorithm enters a loop in which it repeatedly processes the tuple at the front of the queue and pops it off the queue. To process a tuple, the algorithm copies all the tuple that are directly reachable from it to the ToSpace, placing them at the back of the queue. The algorithm then updates the pointers in the popped tuple so they point to the newly copied tuples. Getting back to Figure 5.6, in the first step we copy the tuple whose second element is 42 to the back of the queue. The other pointer goes to a tuple that has already been copied, so we do not need to copy it again, but we do need to update the pointer to the new location. This can be accomplished by storing a forwarding pointer to the new location in the old tuple, back when we initially copied the tuple into the ToSpace. This completes one step of the algorithm. The algorithm continues in this way until the front of the queue is empty, that is, until the front catches up with the back.
5.2.2 Data Representation
The garbage collector places some requirements on the data representations used by our compiler. First, the garbage collector needs to distinguish be- tween pointers and other kinds of data. There are several ways to accomplish this.
1. Attached a tag to each object that identifies what type of object it is [McCarthy, 1960].
2. Store different types of objects in different regions [Steele, 1977].
3. Use type information from the program to either generate type-specific code for collecting or to generate tables that can guide the collec- tor [Appel, 1989, Goldberg, 1991, Diwan et al., 1992].
5.2. GARBAGE COLLECTION 79
7
5
4
scan pointer
free pointer
7
5
4
#t
42
scan pointer
free pointer
7
5
4
#t
42
3
scan pointer
free pointer
free pointer
scan free pointer pointer
7
5
4
#t
42
3
8
scan pointer
7
5
4
#t
42
3
8
Figure 5.6: Depiction of the Cheney algorithm copying the live tuples.
80 5. TUPLES AND GARBAGE COLLECTION
Registers Heap ...
Stack Root Stack
...
1
#f
7
5
9
#t
0
4
Figure 5.7: Maintaining a root stack to facilitate garbage collection.
Dynamically typed languages, such as Lisp, need to tag objects anyways, so option 1 is a natural choice for those languages. However, R3 is a statically typed language, so it would be unfortunate to require tags on every object, especially small and pervasive objects like integers and Booleans. Option 3 is the best-performing choice for statically typed languages, but comes with a relatively high implementation complexity. To keep this chapter to a 2-week time budget, we recommend a combination of options 1 and 2, with separate strategies used for the stack and the heap.
Regarding the stack, we recommend using a separate stack for point- ers [Siebert, 2001, Henderson, 2002, Baker et al., 2009], which we call a root stack (a.k.a. “shadow stack”). That is, when a local variable needs to be spilled and is of type (Vector type1 . . . typen), then we put it on the root stack instead of the normal procedure call stack. Furthermore, we always spill vector-typed variables if they are live during a call to the collector, thereby ensuring that no pointers are in registers during a collection. Fig- ure 5.7 reproduces the example from Figure 5.5 and contrasts it with the data layout using a root stack. The root stack contains the two pointers from the regular stack and also the pointer in the second register.
The problem of distinguishing between pointers and other kinds of data also arises inside of each tuple. We solve this problem by attaching a tag, an extra 64-bits, to each tuple. Figure 5.8 zooms in on the tags for two of the tuples in the example from Figure 5.5. Note that we have drawn the bits in a big-endian way, from right-to-left, with bit location 0 (the least
5.2. GARBAGE COLLECTION 81
...
1
1
0
0
0
0
1
0
1
unused pointer mask vector length forwarding
7
5
...
1
0
0
0
0
0
0
1
1
1
Figure 5.8: Representation for tuples in the heap.
significant bit) on the far right, which corresponds to the directionality of the x86 shifting instructions salq (shift left) and sarq (shift right). Part of each tag is dedicated to specifying which elements of the tuple are pointers, the part labeled “pointer mask”. Within the pointer mask, a 1 bit indicates there is a pointer and a 0 bit indicates some other kind of data. The pointer mask starts at bit location 7. We have limited tuples to a maximum size of 50 elements, so we just need 50 bits for the pointer mask. The tag also contains two other pieces of information. The length of the tuple (number of elements) is stored in bits location 1 through 6. Finally, the bit at location 0 indicates whether the tuple has yet to be copied to the ToSpace. If the bit has value 1, then this tuple has not yet been copied. If the bit has value 0 then the entire tag is in fact a forwarding pointer. (The lower 3 bits of an pointer are always zero anyways because our tuples are 8-byte aligned.)
5.2.3 Implementation of the Garbage Collector
The implementation of the garbage collector needs to do a lot of bit-level data manipulation and we need to link it with our compiler-generated x86 code. Thus, we recommend implementing the garbage collector in C [Kernighan and Ritchie, 1988] and putting the code in the runtime.c file. Figure 5.9 shows the interface to the garbage collector. The initialize function cre- ates the FromSpace, ToSpace, and root stack. The initialize function is meant to be called near the beginning of main, before the rest of the program executes. The initialize function puts the address of the beginning of the FromSpace into the global variable free_ptr. The global fromspace_end points to the address that is 1-past the last element of the FromSpace. (We use half-open intervals to represent chunks of memory [Dijkstra, 1982].) The
82
5. TUPLES AND GARBAGE COLLECTION
void initialize(uint64_t rootstack_size, uint64_t heap_size); void collect(int64_t** rootstack_ptr, uint64_t bytes_requested); int64_t* free_ptr;
int64_t* fromspace_begin;
int64_t* fromspace_end;
int64_t** rootstack_begin;
Figure 5.9: The compiler’s interface to the garbage collector.
rootstack_begin global points to the first element of the root stack.
As long as there is room left in the FromSpace, your generated code can allocate tuples simply by moving the free_ptr forward. The amount of room left in FromSpace is the difference between the fromspace_end and the free_ptr. The collect function should be called when there is not enough room left in the FromSpace for the next allocation. The collect function takes a pointer to the current top of the root stack (one past the last item that was pushed) and the number of bytes that need to be allocated. The collect function performs the copying collection and leaves the heap
in a state such that the next allocation will succeed.
Exercise 19. In the file runtime.c you will find the implementation of initialize and a partial implementation of collect. The collect func- tion calls another function, cheney, to perform the actual copy, and that function is left to the reader to implement. The following is the prototype for cheney.
static void cheney(int64_t** rootstack_ptr);
The parameter rootstack_ptr is a pointer to the top of the rootstack (which is an array of pointers). The cheney function also communicates with collect through the global variables fromspace_begin and fromspace_end mentioned in Figure 5.9 as well as the pointers for the ToSpace:
static int64_t* tospace_begin;
static int64_t* tospace_end;
The job of the cheney function is to copy all the live objects (reachable from the root stack) into the ToSpace, update free_ptr to point to the next unused spot in the ToSpace, update the root stack so that it points to the objects in the ToSpace, and finally to swap the global pointers for the FromSpace and ToSpace.
The introduction of garbage collection has a non-trivial impact on our compiler passes. We introduce one new compiler pass called expose-allocation
5.3. EXPOSE ALLOCATION 83
and make non-trivial changes to type-check, flatten, select-instructions, allocate-registers, and print-x86. The following program will serve as our running example. It creates two tuples, one nested inside the other. Both tuples have length one. The example then accesses the element in the inner tuple tuple via two vector references.
(vector-ref (vector-ref (vector (vector 42)) 0) 0))
Next we proceed to discuss the new expose-allocation pass.
5.3 Expose Allocation
The pass expose-allocation lowers the vector creation form into a con- ditional call to the collector followed by the allocation. We choose to place the expose-allocation pass before flatten because expose-allocation introduces new variables, which can be done locally with let, but let is gone after flatten. In the following, we show the transformation for the vector form into let-bindings for the intializing expressions, by a conditional collect, an allocate, and the initialization of the vector. (The len is the length of the vector and bytes is how many total bytes need to be allocated for the vector, which is 8 for the tag plus len times 8.)
(has-type (vector e0...en−1) type) =⇒
(let ([x0 e0 ]) ... (let ([xn−1 en−1 ])
(let ([_ (if (< (+ (global-value free_ptr) bytes)
(global-value fromspace_end))
(void)
(collect bytes))]) (let ([v (allocate len type)])
(let ([_ (vector-set! v 0 x0)]) ... (let ([_ (vector-set! v n − 1 xn−1 )])
v) ... )))) ...)
(In the above, we suppressed all of the has-type forms in the output for the sake of readability.) The placement of the initializing expressions e0, . . . , en−1 prior to the allocate and the sequence of vector-set!’s is important, as those expressions may trigger garbage collection and we do not want an allocated but uninitialized tuple to be present during a garbage collection.
The output of expose-allocation is a language that extends R3 with the three new forms that we use above in the translation of vector.
exp ::= · · · | (collect int) | (allocate int type) | (global-value name)
84 5. TUPLES AND GARBAGE COLLECTION
(program ()
(vector-ref
(vector-ref
(let ((vecinit48
(let ((vecinit44 42)) (let ((collectret46
(if (<
(+ (global-value free_ptr) 16)
(global-value fromspace_end))
(void)
(collect 16))))
(let ((alloc43 (allocate 1 (Vector Integer))))
(let ((initret45 (vector-set! alloc43 0 vecinit44))) alloc43))))))
(let ((collectret50
(if (< (+ (global-value free_ptr) 16)
(global-value fromspace_end))
(void)
(collect 16))))
(let ((alloc47 (allocate 1 (Vector (Vector Integer)))))
(let ((initret49 (vector-set! alloc47 0 vecinit48))) alloc47))))
0) 0))
Figure 5.10: Output of the expose-allocation pass, minus all of the has-type forms.
Figure 5.10 shows the output of the expose-allocation pass on our running example.
5.4 Explicate Control and the C2 language
The output of explicate-control is a program in the intermediate lan- guage C2, whose syntax is defined in Figure 5.11. The new forms of C2 include the allocate, vector-ref, and vector-set!, and global-value expressions and the collect statement. The explicate-control pass can treat these new forms much like the other forms.
5.5. UNCOVER LOCALS 85
arg ::= cmp ::= exp ::=
int |var |#t|#f
eq?|< arg|(read)|(-arg)|(+argarg)|(notarg)|(cmpargarg)
stmt ::= tail ::=
C2 ::=
(assign var exp) | (return exp) | (collect int)
| (allocate int type) | (vector-ref arg int)
| (vector-set! arg int arg) | (global-value name) | (void)
(return exp) | (seq stmt tail)
| (goto label) | (if (cmp arg arg) (goto label) (goto label))
(program info ((label . tail )+ ))
Figure 5.11: The C2 language, extending C1 (Figure 4.5) with vectors. 5.5 Uncover Locals
Recall that the uncover-locals function collects all of the local variables so that it can store them in the info field of the program form. Also recall that we need to know the types of all the local variables for purposes of identifying the root set for the garbage collector. Thus, we change uncover-locals to collect not just the variables, but the variables and their types in the form of an association list. Thanks to the has-type forms, the types are readily available. Figure 5.12 lists the output of the uncover-locals pass on the running example.
86 5. TUPLES AND GARBAGE COLLECTION
(program
((locals . ((tmp54 . Integer) (tmp51 . Integer) (tmp53 . Integer)
(alloc43 . (Vector Integer)) (tmp55 . Integer)
(initret45 . Void) (alloc47 . (Vector (Vector Integer)))
(collectret46 . Void) (vecinit48 . (Vector Integer))
(tmp52 . Integer) (tmp57 . (Vector Integer))
(vecinit44 . Integer) (tmp56 . Integer) (initret49 . Void)
(collectret50 . Void))))
((block63 . (seq (collect 16) (goto block61)))
(block62 . (seq (assign collectret46 (void)) (goto block61))) (block61 . (seq (assign alloc43 (allocate 1 (Vector Integer)))
(seq (assign initret45 (vector-set! alloc43 0 vecinit44))
(seq (assign vecinit48 alloc43)
(seq (assign tmp54 (global-value free_ptr))
(seq (assign tmp55 (+ tmp54 16))
(seq (assign tmp56 (global-value fromspace_end))
(if (< tmp55 tmp56) (goto block59) (goto block60))))))))) (block60 . (seq (collect 16) (goto block58)))
(block59 . (seq (assign collectret50 (void)) (goto block58)))
(block58 . (seq (assign alloc47 (allocate 1 (Vector (Vector Integer))))
(seq (assign initret49 (vector-set! alloc47 0 vecinit48)) (seq (assign tmp57 (vector-ref alloc47 0))
(return (vector-ref tmp57 0))))))
(start . (seq (assign vecinit44 42)
(seq (assign tmp51 (global-value free_ptr))
(seq (assign tmp52 (+ tmp51 16))
(seq (assign tmp53 (global-value fromspace_end))
(if (< tmp52 tmp53) (goto block62) (goto block63)))))))))
Figure 5.12: Output of uncover-locals for the running example.
5.6. SELECT INSTRUCTIONS 87 5.6 Select Instructions
In this pass we generate x86 code for most of the new operations that were needed to compile tuples, including allocate, collect, vector-ref, vector-set!, and (void). We postpone global-value to print-x86.
The vector-ref and vector-set! forms translate into movq instruc- tions with the appropriate deref. (The plus one is to get past the tag at the beginning of the tuple representation.)
(assign lhs (vector-ref vec n)) =⇒
(movq vec′ (reg r11))
(movq (deref r11 8(n+1)) lhs)
(assign lhs (vector-set! vec n arg)) =⇒
(movq vec′ (reg r11))
(movq arg′ (deref r11 8(n + 1))) (movq (int 0) lhs)
The vec′ and arg′ are obtained by recursively processing vec and arg. The move of vec′ to register r11 ensures that offsets are only performed with register operands. This requires removing r11 from consideration by the register allocating.
We compile the allocate form to operations on the free_ptr, as shown below. The address in the free_ptr is the next free address in the FromSpace, so we move it into the lhs and then move it forward by enough space for the tuple being allocated, which is 8(len + 1) bytes because each element is 8 bytes (64 bits) and we use 8 bytes for the tag. Last but not least, we initialize the tag. Refer to Figure 5.8 to see how the tag is organized. We recommend using the Racket operations bitwise-ior and arithmetic-shift to com- pute the tag. The type annoation in the vector form is used to determine the pointer mask region of the tag.
(assign lhs (allocate len (Vector type...))) =⇒
(movq (global-value free_ptr) lhs′)
(addq (int 8(len + 1)) (global-value free_ptr)) (movq lhs′ (reg r11))
(movq (int tag) (deref r11 0))
The collect form is compiled to a call to the collect function in the runtime. The arguments to collect are the top of the root stack and the number of bytes that need to be allocated. We shall use a dedicated register,
88 5. TUPLES AND GARBAGE COLLECTION
arg ::= (int int) | (reg register) | (derefregister int) | (byte-reg register) | (global-value name)
cc ::= e|l|le|g|ge
instr ::= (addq arg arg) | (subq arg arg) | (negq arg) | (movq arg arg)
| (callq label) | (pushq arg) | (popq arg) | (retq) | (xorq arg arg) | (cmpq arg arg) | (setcc arg)
| (movzbq arg arg) | (jmp label) | (jmp-ifcc label) | (label label )
x862 ::= (program info (type type) instr+)
Figure 5.13: The x862 language (extends x861 of Figure 4.4).
r15, to store the pointer to the top of the root stack. So r15 is not available
for use by the register allocator.
(collect bytes)
=⇒
(movq (reg r15) (reg rdi)) (movq bytes (reg rsi)) (callq collect)
The syntax of the x862 language is defined in Figure 5.13. It differs from x861 just in the addition of the form for global variables. Figure 5.14 shows the output of the select-instructions pass on the running example.
5.6. SELECT INSTRUCTIONS 89
(program
((locals . ((tmp54 . Integer) (tmp51 . Integer) (tmp53 . Integer)
(alloc43 . (Vector Integer)) (tmp55 . Integer)
(initret45 . Void) (alloc47 . (Vector (Vector Integer)))
(collectret46 . Void) (vecinit48 . (Vector Integer))
(tmp52 . Integer) (tmp57 Vector Integer) (vecinit44 . Integer)
(tmp56 . Integer) (initret49 . Void) (collectret50 . Void))))
((block63 . (block ()
(movq (reg r15) (reg rdi))
(movq (int 16) (reg rsi)) (callq collect)
(jmp block61)))
(block62 . (block () (movq (int 0) (var collectret46)) (jmp block61)))
(block61 . (block ()
(movq (global-value free_ptr) (var alloc43))
(addq (int 16) (global-value free_ptr))
(movq (var alloc43) (reg r11))
(movq (int 3) (deref r11 0))
(movq (var alloc43) (reg r11))
(movq (var vecinit44) (deref r11 8))
(movq (int 0) (var initret45))
(movq (var alloc43) (var vecinit48))
(movq (global-value free_ptr) (var tmp54))
(movq (var tmp54) (var tmp55))
(addq (int 16) (var tmp55))
(movq (global-value fromspace_end) (var tmp56))
(cmpq (var tmp56) (var tmp55))
(jmp-if l block59)
(jmp block60)))
(block60 . (block ()
(movq (reg r15) (reg rdi))
(movq (int 16) (reg rsi)) (callq collect)
(jmp block58))
(block59 . (block ()
(movq (int 0) (var collectret50))
(jmp block58)))
(block58 . (block ()
(movq (global-value free_ptr) (var alloc47))
(addq (int 16) (global-value free_ptr))
(movq (var alloc47) (reg r11))
(movq (int 131) (deref r11 0))
(movq (var alloc47) (reg r11))
(movq (var vecinit48) (deref r11 8))
(movq (int 0) (var initret49))
(movq (var alloc47) (reg r11))
(movq (deref r11 8) (var tmp57))
(movq (var tmp57) (reg r11))
(movq (deref r11 8) (reg rax))
(jmp conclusion)))
(start . (block ()
(movq (int 42) (var vecinit44))
(movq (global-value free_ptr) (var tmp51))
(movq (var tmp51) (var tmp52))
(addq (int 16) (var tmp52))
(movq (global-value fromspace_end) (var tmp53))
(cmpq (var tmp53) (var tmp52))
(jmp-if l block62)
(jmp block63))))))
Figure 5.14: Output of the select-instructions pass.
90 5. TUPLES AND GARBAGE COLLECTION 5.7 Register Allocation
As discussed earlier in this chapter, the garbage collector needs to access all the pointers in the root set, that is, all variables that are vectors. It will be the responsibility of the register allocator to make sure that:
1. the root stack is used for spilling vector-typed variables, and
2. if a vector-typed variable is live during a call to the collector, it must be spilled to ensure it is visible to the collector.
The later responsibility can be handled during construction of the in- ference graph, by adding interference edges between the call-live vector- typed variables and all the callee-saved registers. (They already interfere with the caller-saved registers.) The type information for variables is in the program form, so we recommend adding another parameter to the build-interference function to communicate this association list.
The spilling of vector-typed variables to the root stack can be handled after graph coloring, when choosing how to assign the colors (integers) to registers and stack locations. The program output of this pass changes to also record the number of spills to the root stack.
5.8 Print x86
Figure 5.15 shows the output of the print-x86 pass on the running example. In the prelude and conclusion of the main function, we treat the root stack very much like the regular stack in that we move the root stack pointer (r15) to make room for all of the spills to the root stack, except that the root stack grows up instead of down. For the running example, there was just one spill so we increment r15 by 8 bytes. In the conclusion we decrement r15 by 8 bytes.
One issue that deserves special care is that there may be a call to collect prior to the initializing assignments for all the variables in the root stack. We do not want the garbage collector to accidentaly think that some unini- tialized variable is a pointer that needs to be followed. Thus, we zero-out all locations on the root stack in the prelude of main. In Figure 5.15, the instruction movq $0, (%r15) accomplishes this task. The garbage collector tests each root to see if it is null prior to dereferencing it.
Figure 5.16 gives an overview of all the passes needed for the compilation of R3.
5.8. PRINT X86
_block58:
movq _free_ptr(%rip), %rcx
addq $16, _free_ptr(%rip) movq %rcx, %r11
movq $131, 0(%r11)
movq %rcx, %r11
movq -8(%r15), %rax movq %rax, 8(%r11) movq $0, %rdx
movq %rcx, %r11 movq 8(%r11), %rcx movq %rcx, %r11 movq 8(%r11), %rax jmp _conclusion
_block59:
movq $0, %rcx
jmp _block58
_block62:
movq $0, %rcx
jmp _block61
_block60:
movq %r15, %rdi movq $16, %rsi callq _collect jmp _block58
_block63:
movq %r15, %rdi
91
_start:
movq $16, %rsi callq _collect jmp _block61
movq $42, %rbx
movq _free_ptr(%rip), %rdx
addq $16, %rdx
movq _fromspace_end(%rip), %rcx cmpq %rcx, %rdx
jl _block62
jmp _block63
Figure 5.15: Output of the print-x86 pass.
_block61:
movq _free_ptr(%rip), %rcx
addq $16, _free_ptr(%rip) movq %rcx, %r11
movq $3, 0(%r11)
movq %rcx, %r11
movq %rbx, 8(%r11)
movq $0, %rdx
movq %rcx, -8(%r15)
movq _free_ptr(%rip), %rcx addq $16, %rcx
movq _fromspace_end(%rip), %rdx cmpq %rdx, %rcx
jl _block59
jmp _block60
.globl _main
_main:
pushq %rbp
movq
pushq %r12 pushq %rbx pushq %r13 pushq %r14
%rsp, %rbp
subq
movq $16384, %rdi
movq $16, %rsi
callq _initialize
movq _rootstack_begin(%rip), %r15
movq $0, (%r15)
addq $8, %r15
jmp _start
_conclusion:
subq $8, %r15
addq $0, %rsp popq %r14 popq %r13 popq %rbx popq %r12 popq %rbp retq
$0, %rsp
92 5. TUPLES AND GARBAGE COLLECTION
typecheck uniquify expose-alloc. remove-complex. R3 R3 R3 R3 R3
uncover-locals
explicate-control
x86∗2 print-x86
C2
x86∗2 uncover-live
C2
select-instr.
patch-instr.
x86∗2 allocate-reg.
x86∗2
Figure 5.16: Diagram of the passes for R3, a language with tuples.
x86∗2
x86†2
build-inter.
6 Functions
This chapter studies the compilation of functions at the level of abstraction of the C language. This corresponds to a subset of Typed Racket in which only top-level function definitions are allowed. These kind of functions are an important stepping stone to implementing lexically-scoped functions in the form of lambda abstractions, which is the topic of Chapter 7.
6.1 The R4 Language
The syntax for function definitions and function application is shown in Figure 6.1, where we define the R4 language. Programs in R4 start with zero or more function definitions. The function names from these definitions are in-scope for the entire program, including all other function definitions (so the ordering of function definitions does not matter). The syntax for function application does not include an explicit keyword, which is error prone when using match. To alleviate this problem, we change the syntax from (exp exp∗) to (app exp exp∗) during type checking.
Functions are first-class in the sense that a function pointer is data and can be stored in memory or passed as a parameter to another function. Thus, we introduce a function type, written
(type1 · · · typen -> typer )
for a function whose n parameters have the types type1 through typen and whose return type is typer. The main limitation of these functions (with respect to Racket functions) is that they are not lexically scoped. That is, the only external entities that can be referenced from inside a function body are other globally-defined functions. The syntax of R4 prevents functions from being nested inside each other.
93
94 6. FUNCTIONS
type ::= cmp ::= exp ::=
Integer | Boolean | (Vector type+) | Void | (type∗ -> type) eq?|<|<=|>|>= int|(read)|(-exp)|(+expexp)|(-expexp)
def ::= R4 ::=
(define (var [var:type]∗):type exp) (program info def ∗ exp)
| var | (let ([var exp]) exp)
| #t|#f|(andexpexp)|(orexpexp)|(notexp)
| (cmp exp exp) | (if exp exp exp)
| (vector exp+) | (vector-ref exp int)
| (vector-set! exp int exp) | (void)
| (exp exp∗)
Figure 6.1: Syntax of R4, extending R3 (Figure 5.2) with functions.
(program ()
(define (map-vec [f : (Integer -> Integer)]
[v : (Vector Integer Integer)])
: (Vector Integer Integer)
(vector (f (vector-ref v 0)) (f (vector-ref v 1)))) (define (add1 [x : Integer]) : Integer
(+ x 1))
(vector-ref (map-vec add1 (vector 0 41)) 1) )
Figure 6.2: Example of using functions in R4.
The program in Figure 6.2 is a representative example of defining and using functions in R4. We define a function map-vec that applies some other function f to both elements of a vector (a 2-tuple) and returns a new vector containing the results. We also define a function add1 that does what its name suggests. The program then applies map-vec to add1 and (vector 0 41). The result is (vector 1 42), from which we return the 42.
The definitional interpreter for R4 is in Figure 6.3. The case for the program form is responsible for setting up the mutual recursion between the top-level function definitions. We use the classic backpatching approach that uses mutable variables and makes two passes over the function defini- tions [Kelsey et al., 1998]. In the first pass we set up the top-level environ- ment using a mutable cons cell for each function definition. Note that the lambda value for each function is incomplete; it does not yet include the en-
6.2. FUNCTIONS IN X86 95 vironment. Once the top-level environment is constructed, we then iterate
over it and update the lambda value’s to use the top-level environment. 6.2 Functions in x86
The x86 architecture provides a few features to support the implementation of functions. We have already seen that x86 provides labels so that one can refer to the location of an instruction, as is needed for jump instructions. Labels can also be used to mark the beginning of the instructions for a function. Going further, we can obtain the address of a label by using the leaq instruction and rip-relative addressing. For example, the following puts the address of the add1 label into the rbx register.
leaq add1(%rip), %rbx
In Section 2.2 we saw the use of the callq instruction for jumping to a function whose location is given by a label. Here we instead will be jumping to a function whose location is given by an address, that is, we need to make an indirect function call. The x86 syntax is to give the register name prefixed with an asterisk.
callq *%rbx
6.2.1 Calling Conventions
The callq instruction provides partial support for implementing functions, but it does not handle (1) parameter passing, (2) saving and restoring frames on the procedure call stack, or (3) determining how registers are shared by different functions. These issues require coordination between the caller and the callee, which is often assembly code written by different programmers or generated by different compilers. As a result, people have developed conventions that govern how functions calls are performed. Here we shall use the same conventions used by the gcc compiler [Matz et al., 2013].
Regarding (1) parameter passing, the convention is to use the following six registers: rdi, rsi, rdx, rcx, r8, and r9, in that order. If there are more than six arguments, then the convention is to use space on the frame of the caller for the rest of the arguments. However, to ease the implementation of efficient tail calls (Section 6.2.2), we shall arrange to never have more than six arguments. The register rax is for the return value of the function.
Regarding (2) frames and the procedure call stack, the convention is that the stack grows down, with each function call using a chunk of space called
96 6. FUNCTIONS
(define (interp-exp env)
(lambda (e)
(define recur (interp-exp env))
(match e
…
[‘(,fun ,args …)
(define arg-vals (for/list ([e args]) (recur e))) (define fun-val (recur fun))
(match fun-val
[‘(lambda (,xs …) ,body ,fun-env)
(define new-env (append (map cons xs arg-vals) fun-env)) ((interp-exp new-env) body)]
[else (error “interp-exp,␣expected␣function,␣not” fun-val)])] [else (error ’interp-exp “unrecognized␣expression”)]
)))
(define (interp-def d)
(match d
[‘(define (,f [,xs : ,ps] …) : ,rt ,body)
(mcons f ‘(lambda ,xs ,body ()))]
))
(define (interp-R4 p)
(match p
[‘(program ,ds … ,body)
(let ([top-level (for/list ([d ds]) (interp-def d))])
(for/list ([b top-level]) (set-mcdr! b (match (mcdr b)
[‘(lambda ,xs ,body ())
‘(lambda ,xs ,body ,top-level)])))
((interp-exp top-level) body))]
))
Figure 6.3: Interpreter for the R4 language.
6.2. FUNCTIONS IN X86
Caller View Frame 8(%rbp)
0(%rbp)
-8(%rbp) Caller
97
Callee View
Contents
return address old rbp callee-saved 1 … callee-saved j local 1
… local k
8(%rbp)
0(%rbp) -8(%rbp) … −8n(%rbp) −8(n + 1)(%rbp)
… −8(n + m)(%rsp)
return address old rbp callee-saved 1 … callee-saved n local 1
… local m
… −8j (%rbp) −8(j + 1)(%rbp)
… −8(j + k)(%rbp)
Callee
Figure 6.4: Memory layout of caller and callee frames.
a frame. The caller sets the stack pointer, register rsp, to the last data item in its frame. The callee must not change anything in the caller’s frame, that is, anything that is at or above the stack pointer. The callee is free to use locations that are below the stack pointer.
Regarding (3) the sharing of registers between different functions, recall from Section 3.1 that the registers are divided into two groups, the caller- saved registers and the callee-saved registers. The caller should assume that all the caller-saved registers get overwritten with arbitrary values by the callee. Thus, the caller should either 1) not put values that are live across a call in caller-saved registers, or 2) save and restore values that are live across calls. We shall recommend option 1). On the flip side, if the callee wants to use a callee-saved register, the callee must save the contents of those registers on their stack frame and then put them back prior to returning to the caller. The base pointer, register rbp, is used as a point-of-reference within a frame, so that each local variable can be accessed at a fixed offset from the base pointer. Figure 6.4 shows the layout of the caller and callee frames.
98 6. FUNCTIONS 6.2.2 Efficient Tail Calls
In general, the amount of stack space used by a program is determined by the longest chain of nested function calls. That is, if function f1 calls f2, f2 calls f3, . . ., and fn−1 calls fn, then the amount of stack space is bounded by O(n). The depth n can grow quite large in the case of recursive or mutually recursive functions. However, in some cases we can arrange to use only constant space, i.e. O(1), instead of O(n).
If a function call is the last action in a function body, then that call is said to be a tail call. In such situations, the frame of the caller is no longer needed, so we can pop the caller’s frame before making the tail call. With this approach, a recursive function that only makes tail calls will only use O(1) stack space. Functional languages like Racket typically rely heavily on recursive functions, so they typically guarantee that all tail calls will be optimized in this way.
However, some care is needed with regards to argument passing in tail calls. As mentioned above, for arguments beyond the sixth, the convention is to use space in the caller’s frame for passing arguments. But here we’ve popped the caller’s frame and can no longer use it. Another alternative is to use space in the callee’s frame for passing arguments. However, this option is also problematic because the caller and callee’s frame overlap in memory. As we begin to copy the arguments from their sources in the caller’s frame, the target locations in the callee’s frame might overlap with the sources for later arguments! We solve this problem by not using the stack for paramter passing but instead use the heap, as we describe in the next section.
As briefly mentioned above, for a tail call we pop the caller’s frame prior to making the tail call. The instructions for popping a frame are the instructions that we usually place in the conclusion of a function. Thus, we also need to place such code immediately before each tail call. These instructions include restoring the callee-saved registers, so it is good that the argument passing registers are all caller-saved registers.
One last note regarding which instruction to use to make the tail call. When the callee is finished, it should not return to the current function, but it should return to the function that called the current one. Thus, the return address that is already on the stack is the right one, and we should not use callq to make the tail call, as that would unnecessarily overwrite the return address. Instead we can simply use the jmp instruction. Like the indirect function call, we write an indirect jump with a register prefixed with an asterisk. We recommend using rax to hold the jump target because the preceeding “conclusion” overwrites just about everything else.
6.3. SHRINK R4 99 jmp *%rax
6.3 Shrink R4
The shrink pass performs a couple minor modifications to the grammar to ease the later passes. This pass adds an empty info field to each function definition:
(define (f [x1 : type1 …) : typer exp)
⇒ (define (f [x1 : type1 …) : typer () exp)
and introduces an explicit main function.
(program info ds … exp) ⇒ (program info ds′ mainDef)
where mainDef is
(define (main) : Integer () exp′)
6.4 Reveal Functions
Going forward, the syntax of R4 is inconvenient for purposes of compilation because it conflates the use of function names and local variables. This is a problem because we need to compile the use of a function name differently than the use of a local variable; we need to use leaq to convert the function name (a label in x86) to an address in a register. Thus, it is a good idea to create a new pass that changes function references from just a symbol f to (fun-ref f). A good name for this pass is reveal-functions and the output language, F1, is defined in Figure 6.5.
Placing this pass after uniquify is a good idea, because it will make sure that there are no local variables and functions that share the same name. On the other hand, reveal-functions needs to come before the explicate-control pass because that pass will help us compile fun-ref into assignment statements.
6.5 Limit Functions
This pass transforms functions so that they have at most six parameters and transforms all function calls so that they pass at most six arguments. A simple strategy for imposing an argument limit of length n is to take all arguments i where i ≥ n and pack them into a vector, making that subsequent vector the nth argument.
100 6. FUNCTIONS
type ::= exp ::=
| | | | |
def ::= F1 ::=
Integer | Boolean | (Vector type+) | Void | (type∗ -> type) int | (read) | (- exp) | (+ exp exp)
var | (let ([var exp]) exp)
#t | #f | (not exp) | (cmp exp exp) | (ifexp exp exp) (vector exp+) | (vector-ref exp int)
(vector-set! exp int exp) | (void) | (app exp exp∗) (fun-ref label )
(define (label [var:type]∗):type exp)
(program info def ∗)
Figure 6.5: The F1 language, an extension of R4 (Figure 6.1). (f x1 … xn) ⇒ (f x1 … x5 (vector x6 … xn))
In the body of the function, all occurrances of the ith argument in which i > 5 must be replaced with a vector-ref.
6.6 Remove Complex Operators and Operands
The primary decisions to make for this pass is whether to classify fun-ref and app as either simple or complex expressions. Recall that a simple ex- pression will eventually end up as just an “immediate” argument of an x86 instruction. Function application will be translated to a sequence of instruc- tions, so app must be classified as complex expression. Regarding fun-ref, as discussed above, the function label needs to be converted to an address using the leaq instruction. Thus, even though fun-ref seems rather sim- ple, it needs to be classified as a complex expression so that we generate an assignment statement with a left-hand side that can serve as the target of the leaq.
6.7 Explicate Control and the C3 language
Figure 6.6 defines the syntax for C3, the output of explicate-control. The three mutually recursive functions for this pass, for assignment, tail, and predicate contexts, must all be updated with cases for fun-ref and app. In assignment and predicate contexts, app becomes call, whereas in tail position app becomes tailcall. We recommend defining a new function for processing function definitions. This code is similar to the case
6.8. UNCOVER LOCALS 101
arg ::= cmp ::= exp ::=
int |var |#t|#f
eq?|< arg|(read)|(-arg)|(+argarg)|(notarg)|(cmpargarg)
stmt ::= tail ::=
def ::= C3 ::=
(assign var exp) | (return exp) | (collect int)
| (allocate int type) | (vector-ref arg int)
| (vector-set! arg int arg) | (global-value name) | (void)
| (fun-ref label) | (call arg arg∗)
(return exp) | (seq stmt tail)
| (goto label) | (if (cmp arg arg) (goto label) (goto label))
| (tailcall arg arg∗)
(define (label [var:type]∗):type ((label . tail)+)) (program info def ∗ )
Figure 6.6: The C3 language, extending C2 (Figure 5.11) with functions.
for program in R3. The top-level explicate-control function that handles the program form of R4 can then apply this new function to all the function definitions.
6.8 Uncover Locals
The function for processing tail should be updated with a case for tailcall. We also recommend creating a new function for processing function defini- tions. Each function definition in C3 has its own set of local variables, so the code for function definitions should be similar to the case for the program form in C2.
6.9 Select Instructions
The output of select instructions is a program in the x863 language, whose syntax is defined in Figure 6.7.
An assignment of fun-ref becomes a leaq instruction as follows: (assign lhs (fun-ref f )) ⇒ (leaq (fun-ref f ) lhs)
Regarding function definitions, we need to remove their parameters and instead perform parameter passing in terms of the conventions discussed in Section 6.2. That is, the arguments will be in the argument passing
102 6. FUNCTIONS
arg ::= (int int ) | (reg register ) | (deref register int ) | (byte-reg register) | (global-value name)
| (fun-ref label )
cc ::= e|l|le|g|ge
instr ::= (addq arg arg) | (subq arg arg) | (negq arg) | (movq arg arg)
| (callq label) | (pushq arg) | (popq arg) | (retq)
| (xorq arg arg) | (cmpq arg arg) | (setcc arg)
| (movzbq arg arg) | (jmp label) | (jcc label) | (label label) | (indirect-callq arg) | (tail-jmp arg)
| (leaq arg arg)
block ::= (block info instr + )
def ::= (define (label ) info ((label . block )+ )) x863 ::= (program info def ∗ )
Figure 6.7: The x863 language (extends x862 of Figure 5.13).
registers, and inside the function we should generate a movq instruction for each parameter, to move the argument value from the appropriate register to a new local variable with the same name as the old parameter.
Next, consider the compilation of function calls, which have the following form upon input to select-instructions.
(assign lhs (call fun args ...))
In the mirror image of handling the parameters of function definitions, the arguments args need to be moved to the argument passing registers. Once the instructions for parameter passing have been generated, the function call itself can be performed with an indirect function call, for which I recommend creating the new instruction indirect-callq. Of course, the return value from the function is stored in rax, so it needs to be moved into the lhs.
(indirect-callq fun) (movq (reg rax) lhs)
Regarding tail calls, the parameter passing is the same as non-tail calls: generate instructions to move the arguments into to the argument passing registers. After that we need to pop the frame from the procedure call stack. However, we do not yet know how big the frame is; that gets determined during register allocation. So instead of generating those instructions here, we invent a new instruction that means “pop the frame and then do an indirect jump”, which we name tail-jmp.
6.10. UNCOVER LIVE 103
Recall that in Section 2.6 we recommended using the label start for the initial block of a program, and in Section 2.8 we recommended labelling the conclusion of the program with conclusion, so that (return arg) can be compiled to an assignment to rax followed by a jump to conclusion. With the addition of function definitions, we will have a starting block and con- clusion for each function, but their labels need to be unique. We recommend prepending the function’s name to start and conclusion, respectively, to obtain unique labels. (Alternatively, one could gensym labels for the start and conclusion and store them in the info field of the function definition.)
6.10 Uncover Live
Inside uncover-live, when computing the W set (written variables) for an indirect-callq instruction, we recommend including all the caller-saved registers, which will have the affect of making sure that no caller-saved register actually needs to be saved.
6.11 Build Interference Graph
With the addition of function definitions, we compute an interference graph for each function (not just one for the whole program).
Recall that in Section 5.7 we discussed the need to spill vector-typed variables that are live during a call to the collect. With the addition of functions to our language, we need to revisit this issue. Many functions will perform allocation and therefore have calls to the collector inside of them. Thus, we should not only spill a vector-typed variable when it is live during a call to collect, but we should spill the variable if it is live during any function call. Thus, in the build-interference pass, we recommend adding interference edges between call-live vector-typed variables and the callee-saved registers (in addition to the usual addition of edges between call-live variables and the caller-saved registers).
6.12 Patch Instructions
In patch-instructions, you should deal with the x86 idiosyncrasy that the destination argument of leaq must be a register. Additionally, you should ensure that the argument of tail-jmp is rax, our reserved register—this is to make code generation more convenient, because we will be trampling many registers before the tail call (as explained below).
104 6. FUNCTIONS 6.13 Print x86
For the print-x86 pass, we recommend the following translations: (fun-ref label) ⇒ label(%rip)
(indirect-callq arg) ⇒ callq *arg
Handling tail-jmp requires a bit more care. A straightforward translation of tail-jmp would be jmp *arg, which is what we will want to do, but before the jump we need to pop the current frame. So we need to restore the state of the registers to the point they were at when the current function was called. This sequence of instructions is the same as the code for the conclusion of a function.
Note that your print-x86 pass needs to add the code for saving and restoring callee-saved registers, if you have not already implemented that. This is necessary when generating code for function definitions.
6.14 An Example Translation
Figure 6.8 shows an example translation of a simple function in R4 to x86. The figure also includes the results of the explicate-control and select-instructions passes. We have ommited the has-type AST nodes for readability. Can you see any ways to improve the translation?
Exercise 20. Expand your compiler to handle R4 as outlined in this sec- tion. Create 5 new programs that use functions, including examples that pass functions and return functions from other functions and including re- cursive functions. Test your compiler on these new programs and all of your previously created test programs.
Figure 6.9 gives an overview of the passes needed for the compilation of R4.
6.14. AN EXAMPLE TRANSLATION
105
(program
(define (add [x : Integer]
[y : Integer])
: Integer (+ x y))
(add 40 2))
⇓
(program ()
(define (add86 [x87 : Integer]
[y88 : Integer]) : Integer () ((add86start . (return (+ x87 y88)))))
(define (main) : Integer ()
((mainstart .
(seq (assign tmp89 (fun-ref add86))
(tailcall tmp89 40 2))))))
⇒
((locals (x87 . Integer) (y88 . Integer))
(num-params . 2))
((add86start .
(block ()
(movq (reg rcx) (var x87))
(movq (reg rdx) (var y88))
(movq (var x87) (reg rax))
(addq (var y88) (reg rax))
(jmp add86conclusion)))))
(define (main)
((locals . ((tmp89 . (Integer Integer -> Integer))))
(num-params . 0))
((mainstart .
(block ()
(leaq (fun-ref add86) (var tmp89))
(movq (int 40) (reg rcx))
(movq (int 2) (reg rdx))
(tail-jmp (var tmp89))))))
⇓
_add90start:
movq %rcx, %rsi
movq %rdx, %rcx movq %rsi, %rax addq %rcx, %rax jmp _add90conclusion .globl _add90
.align 16
_add90:
_mainstart:
leaq
movq
movq
movq
addq
popq
popq
popq
popq
_add90(%rip), %rsi
$40, %rcx
$2, %rdx
%rsi, %rax
pushq
movq
pushq
pushq
pushq
pushq
subq
jmp _add90start
subq $0, %r15 popq %rbp jmp *%rax
.globl _main
.align 16
_mainconclusion:
addq $0, %rsp
popq %r14 popq %r13 popq %rbx popq %r12 subq $0, %r15 popq %rbp
%rbp
%rsp, %rbp
%r12
%rbx
%r13
%r14
$0, %rsp
_main:
pushq
%rbp
%rsp, %rbp retq
%r12
%rbx
%r13
%r14
$0, %rsp
Figure 6.8: Example compilation of a simple function to x86.
_add90conclusion:
addq $0, %rsp
popq %r14 popq %r13 popq %rbx popq %r12 subq $0, %r15 popq %rbp retq
movq
pushq
pushq
pushq
pushq
subq
movq $16384, %rdi
movq $16, %rsi
callq _initialize
movq _rootstack_begin(%rip), %r15
jmp _mainstart
(program ()
(define (add86)
$0, %rsp
%r14
%r13
%rbx
%r12
106 6. FUNCTIONS
typecheck uniquify
R4 R4 R4
reveal-functions
remove-complex. expose-alloc.
F1 F1 F1 F1
C3
C3
explicate-control
limit-functions
select-instr.
uncover-locals
patch-instr.
x86∗3 uncover-live
x86∗3 allocate-reg.
x86∗3 print-x86
x86∗3
Figure 6.9: Diagram of the passes for R4, a language with functions.
x86∗3
x86†3
build-inter.
7
Lexically Scoped Functions
This chapter studies lexically scoped functions as they appear in functional languages such as Racket. By lexical scoping we mean that a function’s body may refer to variables whose binding site is outside of the function, in an enclosing scope. Consider the example in Figure 7.1 featuring an anonymous function defined using the lambda form. The body of the lambda, refers to three variables: x, y, and z. The binding sites for x and y are outside of the lambda. Variable y is bound by the enclosing let and x is a parameter of f. The lambda is returned from the function f. Below the definition of f, we have two calls to f with different arguments for x, first 5 then 3. The functions returned from f are bound to variables g and h. Even though these two functions were created by the same lambda, they are really different functions because they use different values for x. Finally, we apply g to 11 (producing 20) and apply h to 15 (producing 22) so the result of this program is 42.
(define (f [x : Integer]) : (Integer -> Integer) (let ([y 4])
(lambda: ([z : Integer]) : Integer
(+ x (+ y z)))))
(let ([g (f 5)]) (let ([h (f 3)])
(+ (g 11) (h 15))))
Figure 7.1: Example of a lexically scoped function. 107
108 7. LEXICALLY SCOPED FUNCTIONS
type ::= exp ::=
| | | | | | |
def ::= R5 ::=
Integer | Boolean | (Vector type+) | Void | (type∗ -> type) int|(read)|(-exp)|(+expexp)|(-expexp)
var | (let ([var exp]) exp) #t|#f|(andexpexp)|(orexpexp)|(notexp)
(eq? exp exp) | (if exp exp exp) (vector exp+) | (vector-ref exp int) (vector-set! exp int exp) | (void) (exp exp∗)
(lambda: ([var:type]∗):type exp) (define (var [var:type]∗):type exp) (program def ∗ exp)
Figure 7.2: Syntax of R5, extending R4 (Figure 6.1) with lambda. 7.1 The R5 Language
The syntax for this language with anonymous functions and lexical scoping, R5, is defined in Figure 7.2. It adds the lambda form to the grammar for R4, which already has syntax for function application. In this chapter we shall descibe how to compile R5 back into R4, compiling lexically-scoped functions into a combination of functions (as in R4) and tuples (as in R3).
To compile lexically-scoped functions to top-level function definitions, the compiler will need to provide special treatment to variable occurences such as x and y in the body of the lambda of Figure 7.1, for the functions of R4 may not refer to variables defined outside the function. To identify such variable occurences, we review the standard notion of free variable.
Definition 21. A variable is free with respect to an expression e if the variable occurs inside e but does not have an enclosing binding in e.
For example, the variables x, y, and z are all free with respect to the expression (+ x (+ y z)). On the other hand, only x and y are free with respect to the following expression becuase z is bound by the lambda.
(lambda: ([z : Integer]) : Integer
(+ x (+ y z)))
Once we have identified the free variables of a lambda, we need to arrange for some way to transport, at runtime, the values of those variables from the point where the lambda was created to the point where the lambda is applied. Referring again to Figure 7.1, the binding of x to 5 needs to be
7.2. INTERPRETING R5 109
g
h
xy code
xy
5
4
3
4
Figure 7.3: Example closure representation for the lambda’s in Figure 7.1.
used in the application of g to 11, but the binding of x to 3 needs to be used in the application of h to 15. An efficient solution to the problem, due to Cardelli [1983], is to bundle into a vector the values of the free variables together with the function pointer for the lambda’s code, an arrangement called a flat closure (which we shorten to just “closure”) . Fortunately, we have all the ingredients to make closures, Chapter 5 gave us vectors and Chapter 6 gave us function pointers. The function pointer shall reside at index 0 and the values for free variables will fill in the rest of the vector. Figure 7.3 depicts the two closures created by the two calls to f in Figure 7.1. Because the two closures came from the same lambda, they share the same function pointer but differ in the values for the free variable x.
7.2 Interpreting R5
Figure 7.4 shows the definitional interpreter for R5. The clause for lambda saves the current environment inside the returned lambda. Then the clause for app uses the environment from the lambda, the lam-env, when inter- preting the body of the lambda. The lam-env environment is extended with the mapping of parameters to argument values.
7.3 Type Checking R5
Figure 7.5 shows how to type check the new lambda form. The body of the lambda is checked in an environment that includes the current environment (because it is lexically scoped) and also includes the lambda’s parameters. We require the body’s type to match the declared return type.
110 7. LEXICALLY SCOPED FUNCTIONS
(define (interp-exp env)
(lambda (e)
(define recur (interp-exp env))
(match e
…
[‘(lambda: ([,xs : ,Ts] …) : ,rT ,body)
‘(lambda ,xs ,body ,env)]
[‘(app ,fun ,args …)
(define fun-val ((interp-exp env) fun)) (define arg-vals (map (interp-exp env) args)) (match fun-val
[‘(lambda (,xs …) ,body ,lam-env)
(define new-env (append (map cons xs arg-vals) lam-env)) ((interp-exp new-env) body)]
[else (error “interp-exp,␣expected␣function,␣not” fun-val)])] [else (error ’interp-exp “unrecognized␣expression”)]
)))
Figure 7.4: Interpreter for R5.
(define (typecheck-R5 env)
(lambda (e)
(match e
[‘(lambda: ([,xs : ,Ts] …) : ,rT ,body)
(define new-env (append (map cons xs Ts) env)) (define bodyT ((typecheck-R5 new-env) body)) (cond [(equal? rT bodyT)
… )))
‘(,@Ts -> ,rT)]
[else
(error “mismatch␣in␣return␣type” bodyT rT)])]
Figure 7.5: Type checking the lambda’s in R5.
7.4. CLOSURE CONVERSION 111 7.4 Closure Conversion
The compiling of lexically-scoped functions into top-level function defini- tions is accomplished in the pass convert-to-closures that comes after reveal-functions and before limit-functions.
As usual, we shall implement the pass as a recursive function over the AST. All of the action is in the clauses for lambda and app. We transform a lambda expression into an expression that creates a closure, that is, creates a vector whose first element is a function pointer and the rest of the elements are the free variables of the lambda. The name is a unique symbol generated to identify the function.
(lambda: (ps …) : rt body) ⇒ (vector name fvs …)
In addition to transforming each lambda into a vector, we must create a
top-level function definition for each lambda, as shown below.
(define (name [clos : (Vector _ fvts …)] ps …) (let ([fvs1 (vector-ref clos 1)])
…
(let ([fvsn (vector-ref clos n)])
body′)…))
The clos parameter refers to the closure. The ps parameters are the normal parameters of the lambda. The types fvts are the types of the free variables in the lambda and the underscore is a dummy type because it is rather difficult to give a type to the function in the closure’s type, and it does not matter. The sequence of let forms bind the free variables to their values obtained from the closure.
We transform function application into code that retreives the function pointer from the closure and then calls the function, passing in the closure as the first argument. We bind e′ to a temporary variable to avoid code duplication.
(app e es …) ⇒
(let ([tmp e′])
(app (vector-ref tmp 0) tmp es′))
There is also the question of what to do with top-level function defini- tions. To maintain a uniform translation of function application, we turn function references into closures.
(fun-ref f ) ⇒ (vector (fun-ref f ))
The top-level function definitions need to be updated as well to take an extra closure parameter.
112 7. LEXICALLY SCOPED FUNCTIONS 7.5 An Example Translation
Figure 7.6 shows the result of closure conversion for the example program demonstrating lexical scoping that we discussed at the beginning of this chapter.
(define (f [x : Integer]) : (Integer -> Integer) (let ([y 4])
(lambda: ([z : Integer]) : Integer
(+ x (+ y z)))))
(let ([g (f 5)]) (let ([h (f 3)])
(+ (g 11) (h 15)))))
⇓
⇓
(define (f (x : Integer)) : (Integer -> Integer) (let ((y 4))
(lambda: ((z : Integer)) : Integer
(+ x (+ y z)))))
(let ((g (app (fun-ref f) 5))) (let ((h (app (fun-ref f) 3)))
(+ (app g 11) (app h 15)))))
(define (f (clos.1 : _) (x : Integer)) : (Integer -> Integer) (let ((y 4))
(vector (fun-ref lam.1) x y)))
(define (lam.1 (clos.2 : _) (z : Integer)) : Integer
(let ((x (vector-ref clos.2 1))) (let ((y (vector-ref clos.2 2)))
(+ x (+ y z)))))
(let ((g (let ((t.1 (vector (fun-ref f))))
(app (vector-ref t.1 0) t.1 5)))) (let ((h (let ((t.2 (vector (fun-ref f))))
(app (vector-ref t.2 0) t.2 3))))
(+ (let ((t.3 g)) (app (vector-ref t.3 0) t.3 11))
(let ((t.4 h)) (app (vector-ref t.4 0) t.4 15)))))) Figure 7.6: Example of closure conversion.
Figure 7.7 provides an overview of all the passes needed for the compi- lation of R5.
7.5. AN EXAMPLE TRANSLATION
113
typecheck uniquify
R4 R4 R4
remove-complex. expose-alloc.
reveal-functions
limit-functions
F1 F1 F1 F1 F1 convert-to-clos.
explicate-control
C3
select-instr.
x86∗3 uncover-live
x86∗3
C3
uncover-locals
x86∗3 allocate-reg.
x86∗3
patch-instr.
x86∗3 print-x86
x86†3
build-inter.
Figure 7.7: Diagram of the passes for R5, a language with lexically-scoped functions.
114 7. LEXICALLY SCOPED FUNCTIONS
8
Dynamic Typing
In this chapter we discuss the compilation of a dynamically typed language, named R7, that is a subset of the Racket language. (Recall that in the previous chapters we have studied subsets of the Typed Racket language.) In dynamically typed languages, an expression may produce values of differing type. Consider the following example with a conditional expression that may return a Boolean or an integer depending on the input to the program.
(not (if (eq? (read) 1) #f 0))
Languages that allow expressions to produce different kinds of values are called polymorphic. There are many kinds of polymorphism, such as subtype polymorphism and parametric polymorphism [Cardelli and Wegner, 1985]. The kind of polymorphism are talking about here does not have a special name, but it is the usual kind that arrises in dynamically typed languages.
Another characteristic of dynamically typed languages is that primitive operations, such as not, are often defined to operate on many different types of values. In fact, in Racket, the not operator produces a result for any kind of value: given #f it returns #t and given anything else it returns #f. Furthermore, even when primitive operations restrict their inputs to values of a certain type, this restriction is enforced at runtime instead of during compilation. For example, the following vector reference results in a run-time contract violation.
(vector-ref (vector 42) #t)
The syntax of R7, our subset of Racket, is defined in Figure 8.1. The definitional interpreter for R7 is given in Figure 8.2.
Let us consider how we might compile R7 to x86, thinking about the first example above. Our bit-level representation of the Boolean #f is zero and
115
116 8. DYNAMIC TYPING
cmp ::= eq?|<|<=|>|>=
exp ::= |
| | | | | | |
def ::= R7 ::=
int|(read)|(-exp)|(+expexp)|(-expexp) var | (let ([var exp]) exp) #t|#f|(andexpexp)|(orexpexp)|(notexp) (cmp exp exp) | (if exp exp exp)
(vector exp+) | (vector-ref exp exp) (vector-set! exp exp exp) | (void)
(exp exp∗) | (lambda (var∗) exp)
(boolean? exp) | (integer? exp)
(vector? exp) | (procedure? exp) | (void? exp) (define (var var∗) exp)
(program def ∗ exp)
Figure 8.1: Syntax of R7, an untyped language (a subset of Racket).
similarly for the integer 0. However, (not #f) should produce #t whereas (not 0) should produce #f. Furthermore, the behavior of not, in general, cannot be determined at compile time, but depends on the runtime type of its input, as in the example above that depends on the result of (read).
The way around this problem is to include information about a value’s runtime type in the value itself, so that this information can be inspected by operators such as not. In particular, we shall steal the 3 right-most bits from our 64-bit values to encode the runtime type. We shall use 001 to identify integers, 100 for Booleans, 010 for vectors, 011 for procedures, and 101 for the void value. We shall refer to these 3 bits as the tag and we define the following auxilliary function.
tagof (Integer) = 001
tagof (Boolean) = 100 tagof ((Vector . . .)) = 010 tagof ((Vectorof . . .)) = 010 tagof((…->…)) = 011 tagof (Void) = 101
(We shall say more about the new Vectorof type shortly.) This stealing of 3 bits comes at some price: our integers are reduced to ranging from −260 to 260. The stealing does not adversely affect vectors and procedures because those values are addresses, and our addresses are 8-byte aligned so the rightmost 3 bits are unused, they are always 000. Thus, we do not lose
(define (get-tagged-type v) (match v [‘(tagged ,v1 ,ty) ty]))
(define (valid-op? op) (member op ’(+ – and or not)))
(define (interp-r7 env)
(lambda (ast)
(define recur (interp-r7 env))
(match ast
[(? symbol?) (lookup ast env)]
[(? integer?) ‘(inject ,ast Integer)]
[#t ‘(inject #t Boolean)]
[#f ‘(inject #f Boolean)]
[‘(read) ‘(inject ,(read-fixnum) Integer)] [‘(lambda (,xs …) ,body)
‘(inject (lambda ,xs ,body ,env) (,@(map (lambda (x) ’Any) xs) -> Any))] [‘(define (,f ,xs …) ,body)
(mcons f ‘(lambda ,xs ,body))]
[‘(program ,ds … ,body)
(let ([top-level (for/list ([d ds]) ((interp-r7 ’()) d))]) (for/list ([b top-level])
(set-mcdr! b (match (mcdr b)
[‘(lambda ,xs ,body)
‘(inject (lambda ,xs ,body ,top-level)
(,@(map (lambda (x) ’Any) xs) -> Any))])))
((interp-r7 top-level) body))] [‘(vector ,(app recur elts) …)
(define tys (map get-tagged-type elts))
‘(inject ,(apply vector elts) (Vector ,@tys))] [‘(vector-set! ,(app recur v1) ,n ,(app recur v2))
(match v1
[‘(inject ,vec ,ty)
(vector-set! vec n v2)
‘(inject (void) Void)])]
[‘(vector-ref ,(app recur v) ,n)
(match v [‘(inject ,vec ,ty) (vector-ref vec n)])]
[‘(let ([,x ,(app recur v)]) ,body) ((interp-r7 (cons (cons x v) env)) body)]
[‘(,op ,es …) #:when (valid-op? op) (interp-r7-op op (for/list ([e es]) (recur e)))]
[‘(eq? ,(app recur l) ,(app recur r)) ‘(inject ,(equal? l r) Boolean)]
[‘(if ,(app recur q) ,t ,f)
(match q
[‘(inject #f Boolean) (recur f)]
[else (recur t)])]
[‘(,(app recur f-val) ,(app recur vs) …)
(match f-val
[‘(inject (lambda (,xs …) ,body ,lam-env) ,ty)
(define new-env (append (map cons xs vs) lam-env))
((interp-r7 new-env) body)]
[else (error “interp-r7,␣expected␣function,␣not” f-val)])])))
Figure 8.2: Interpreter for the R7 language. UPDATE ME -Jeremy
117
118 8. DYNAMIC TYPING
type ::= Integer | Boolean | (Vector type+) | (Vectorof type) | Void | (type∗ -> type) | Any
ftype ::= Integer | Boolean | Void | (Vectorof Any) | (Vector Any∗) | (Any∗ -> Any)
cmp ::= eq?|<|<=|>|>=
exp ::= int|(read)|(-exp)|(+expexp)|(-expexp)
| var | (let ([var exp]) exp)
| #t|#f|(andexpexp)|(orexpexp)|(notexp) | (cmp exp exp) | (if exp exp exp)
| (vector exp+) | (vector-ref exp int)
| (vector-set! exp int exp) | (void)
| (exp exp∗) | (lambda: ([var:type]∗):type exp)
| (inject exp ftype) | (project exp ftype)
| (boolean? exp) | (integer? exp)
| (vector? exp) | (procedure? exp) | (void? exp)
def ::= (define (var [var:type]∗):type exp) R6 ::= (program def ∗ exp)
Figure 8.3: Syntax of R6, extending R5 (Figure 7.2) with Any.
information by overwriting the rightmost 3 bits with the tag and we can simply zero-out the tag to recover the original address.
In some sense, these tagged values are a new kind of value. Indeed, we can extend our typed language with tagged values by adding a new type to classify them, called Any, and with operations for creating and using tagged values, yielding the R6 language that we define in Section 8.1. The R6 language provides the fundamental support for polymorphism and runtime types that we need to support dynamic typing.
We shall implement our untyped language R7 by compiling it to R6 (Section 8.4), but first we describe the how to extend our compiler to handle the new features of R6 (Sections 8.2 and 8.3).
8.1 The R6 Language: Typed Racket + Any
The syntax of R6 is defined in Figure 8.3. The (inject e T ) form converts the value produced by expression e of type T into a tagged value. The (project e T ) form converts the tagged value produced by expression e into a value of type T or else halts the program if the type tag is equivalent to
8.1. THE R6 LANGUAGE: TYPED RACKET + ANY 119
T . We treat (Vectorof Any) as equivalent to (Vector Any . . .).
Note that in both inject and project, the type T is restricted to the flat types ftype, which simplifies the implementation and corresponds with what is needed for compiling untyped Racket. The type predicates, (boolean? e) etc., expect a tagged value and return #t if the tag corresponds to the predicate, and return #t otherwise. Selections from the type checker for R6 are shown in Figure 8.4 and the definitional interpreter for R6 is in
Figure 8.5.
120 8.
(define (flat-ty? ty) …)
(define (typecheck-R6 env)
(lambda (e)
(define recur (typecheck-R6 env))
(match e
[‘(inject ,e ,ty) (unless (flat-ty? ty)
DYNAMIC TYPING
(error “may␣only␣inject␣a␣value␣of␣flat␣type,␣not␣~a” ty)) (define-values (new-e e-ty) (recur e))
(cond
[(equal? e-ty ty)
(values ‘(inject ,new-e ,ty) ’Any)]
[else
(error “inject␣expected␣~a␣to␣have␣type␣~a” e ty)])]
[‘(project ,e ,ty) (unless (flat-ty? ty)
(error “may␣only␣project␣to␣a␣flat␣type,␣not␣~a” ty)) (define-values (new-e e-ty) (recur e))
(cond
[(equal? e-ty ’Any)
(values ‘(project ,new-e ,ty) ty)]
[else
(error “project␣expected␣~a␣to␣have␣type␣Any” e)])]
[‘(vector-ref ,e ,i)
(define-values (new-e e-ty) (recur e))
(match e-ty
[‘(Vector ,ts …) …]
[‘(Vectorof ,ty)
(unless (exact-nonnegative-integer? i) (error ’type-check “invalid␣index␣~a” i))
(values ‘(vector-ref ,new-e ,i) ty)]
[else (error “expected␣a␣vector␣in␣vector-ref,␣not” e-ty)])]
… )))
Figure 8.4: Type checker for parts of the R6 language.
8.1. THE R6 LANGUAGE: TYPED RACKET + ANY 121
(define primitives (set ’boolean? …))
(define (interp-op op)
(match op
[’boolean? (lambda (v)
(match v
[‘(tagged ,v1 Boolean) #t]
[else #f]))]
…))
;; Equavalence of flat types
(define (tyeq? t1 t2)
(match ‘(,t1 ,t2)
[‘((Vectorof Any) (Vector ,t2s …)) (for/and ([t2 t2s]) (eq? t2 ’Any))] [‘((Vector ,t1s …) (Vectorof Any)) (for/and ([t1 t1s]) (eq? t1 ’Any))]
[else (equal? t1 t2)]))
(define (interp-R6 env)
(lambda (ast)
(match ast
[‘(inject ,e ,t)
‘(tagged ,((interp-R6 env) e) ,t)]
[‘(project ,e ,t2)
(define v ((interp-R6 env) e))
(match v
[‘(tagged ,v1 ,t1) (cond [(tyeq? t1 t2)
v1] [else
(error “in␣project,␣type␣mismatch” t1 t2)])]
[else
(error “in␣project,␣expected␣tagged␣value” v)])]
…)))
Figure 8.5: Interpreter for R6.
122 8. DYNAMIC TYPING 8.2 Shrinking R6
In the shrink pass we recommend compiling project into an explicit if expression that uses three new operations: tag-of-any, value-of-any, and exit. The tag-of-any operation retrieves the type tag from a tagged value of type Any. The value-of-any retrieves the underlying value from a tagged value. Finally, the exit operation ends the execution of the program by in- voking the operating system’s exit function. So the translation for project is as follows. (We have ommitted the has-type AST nodes to make this output more readable.)
(project e type) ⇒
(let ([tmp e′])
(if (eq? (tag-of-any tmp) tag)
(value-of-any tmp) (exit)))
Similarly, we recommend translating the type predicates (boolean?, etc.) into uses of tag-of-any and eq?.
8.3 Instruction Selection for R6
Inject We recommend compiling an inject as follows if the type is Integer or Boolean. The salq instruction shifts the destination to the left by the number of bits specified its source argument (in this case 3, the length of the tag) and it preserves the sign of the integer. We use the orq instruction to combine the tag and the value to form the tagged value.
(movq e′ lhs’) (assign lhs (inject e T)) ⇒ (salq (int 3) lhs’)
(orq (int tagof(T)) lhs’)
The instruction selection for vectors and procedures is different because their is no need to shift them to the left. The rightmost 3 bits are already zeros as described above. So we just combine the value and the tag using orq.
(assign lhs (inject e T)) ⇒
(movq e′ lhs’)
(orq (int tagof(T)) lhs’)
Tag of Any Recall that the tag-of-any operation extracts the type tag from a value of type Any. The type tag is the bottom three bits, so we obtain the tag by taking the bitwise-and of the value with 111 (7 in decimal).
8.4. COMPILING R7 TO R6 123
(assign lhs (tag-of-any e)) ⇒
(movq e′ lhs’)
(andq (int 7) lhs’)
Value of Any Like inject, the instructions for value-of-any are differ- ent depending on whether the type T is a pointer (vector or procedure) or not (Integer or Boolean). The following shows the instruction selection for Integer and Boolean. We produce an untagged value by shifting it to the right by 3 bits.
(assign lhs (project e T)) ⇒
(movq e′ lhs’)
(sarq (int 3) lhs’)
In the case for vectors and procedures, there is no need to shift. Instead we just need to zero-out the rightmost 3 bits. We accomplish this by creating the bit pattern …0111 (7 in decimal) and apply bitwise-not to obtain . . . 1000 which we movq into the destination lhs. We then generate andq with the tagged value to get the desired result.
(assign lhs (project e T)) ⇒
8.4 Compiling R7 to R6
(movq (int …1000) lhs’) (andq e′ lhs’)
Figure 8.6 shows the compilation of many of the R7 forms into R6. An important invariant of this pass is that given a subexpression e of R7, the pass will produce an expression e′ of R6 that has type Any. For example, the first row in Figure 8.6 shows the compilation of the Boolean #t, which must be injected to produce an expression of type Any. The second row of Figure 8.6, the compilation of addition, is representative of compilation for many operations: the arguments have type Any and must be projected to Integer before the addition can be performed.
The compilation of lambda (third row of Figure 8.6) shows what hap- pens when we need to produce type annotations: we simply use Any. The compilation of if and eq? demonstrate how this pass has to account for some differences in behavior between R7 and R6. The R7 language is more permissive than R6 regarding what kind of values can be used in various places. For example, the condition of an if does not have to be a Boolean. For eq?, the arguments need not be of the same type (but in that case, the result will be #f).
124 8. DYNAMIC TYPING
#t ⇒ (inject #t Boolean)
(inject
(+ e e ) ⇒ (+ (project e′1 Integer)
1 2 (project e′2 Integer)) Integer)
(lambda (x1 . . .) e) ⇒
(inject (lambda: ([x1:Any]. . .):Any e′) (Any…Any -> Any))
(app e e …e ) ⇒ (app (project e′0 (Any…Any -> Any)) 0 1 n e ′1 . . . e ′n )
(let ([tmp1 (project e′1 (Vectorof Any))]) (vector-ref e1 e2) ⇒ (let ([tmp2 (project e′2 Integer)])
(vector-ref tmp1 tmp2)))
(if (eq? e′1 (inject #f Boolean)) (if e1 e2 e3) ⇒ e′3
e ′2 )
(eq? e1 e2 ) ⇒ (inject (eq? e′1 e′2 ) Boolean)
Figure 8.6: Compiling R7 to R6.
9
Gradual Typing
This chapter will be based on the ideas of Siek and Taha [2006].
125
126 9. GRADUAL TYPING
10
Parametric Polymorphism
This chapter may be based on ideas from Cardelli [1984], Leroy [1992], Shao [1997], or Harper and Morrisett [1995].
127
128 10. PARAMETRIC POLYMORPHISM
11
High-level Optimization
This chapter will present a procedure inlining pass based on the algorithm of Waddell and Dybvig [1997].
129
130 11. HIGH-LEVEL OPTIMIZATION
12 Appendix
12.1 Interpreters
We provide several interpreters in the interp.rkt file. The interp-scheme function takes an AST in one of the Racket-like languages considered in this book (R1,R2,…) and interprets the program, returning the result value. The interp-C function interprets an AST for a program in one of the C-like languages (C0, C1, . . .), and the interp-x86 function interprets an AST for an x86 program.
12.2 Utility Functions
The utility function described in this section can be found in the utilities.rkt file.
The read-program function takes a file path and parses that file (it must be a Racket program) into an abstract syntax tree (as an S-expression) with a program AST at the top.
The assert function displays the error message msg if the Boolean bool is false.
(define (assert msg bool) …)
The lookup function takes a key and an association list (a list of key- value pairs), and returns the first value that is associated with the given key, if there is one. If not, an error is triggered. The association list may contain both immutable pairs (built with cons) and mutable mapirs (built with mcons).
The map2 function …
131
132 12. APPENDIX 12.2.1 Testing
The interp-tests function takes a compiler name (a string), a description of the passes, an interpreter for the source language, a test family name (a string), and a list of test numbers, and runs the compiler passes and the interpreters to check whether the passes correct. The description of the passes is a list with one entry per pass. An entry is a list with three things: a string giving the name of the pass, the function that implements the pass (a translator from AST to AST), and a function that implements the interpreter (a function from AST to result value) for the language of the output of the pass. The interpreters from Appendix 12.1 make a good choice. The interp-tests function assumes that the subdirectory tests has a bunch of Scheme programs whose names all start with the family name, followed by an underscore and then the test number, ending in .scm. Also, for each Scheme program there is a file with the same number except that it ends with .in that provides the input for the Scheme program.
(define (interp-tests name passes test-family test-nums) …
The compiler-tests function takes a compiler name (a string) a descrip- tion of the passes (see the comment for interp-tests) a test family name (a string), and a list of test numbers (see the comment for interp-tests), and runs the compiler to generate x86 (a .s file) and then runs gcc to generate machine code. It runs the machine code and checks that the output is 42.
(define (compiler-tests name passes test-family test-nums) …)
The compile-file function takes a description of the compiler passes (see the comment for interp-tests) and returns a function that, given a pro- gram file name (a string ending in .scm), applies all of the passes and writes the output to a file whose name is the same as the program file name but with .scm replaced with .s.
(define (compile-file passes)
(lambda (prog-file-name) …))
12.3 x86 Instruction Set Quick-Reference
Table 12.1 lists some x86 instructions and what they do. We write A → B to mean that the value of A is written into location B. Address offsets are given in bytes. The instruction arguments A,B,C can be immediate con- stants (such as $4), registers (such as %rax), or memory references (such as
12.3. X86 INSTRUCTION SET QUICK-REFERENCE 133
−4(%ebp)). Most x86 instructions only allow at most one memory reference per instruction. Other operands must be immediates or registers.
134 12. APPENDIX
Instruction Operation
addq A, B negq A subq A, B callq L callq *A retq
A+B→B
−A→A
B−A→B
Pushes the return address and jumps to label L Calls the function at the address A.
Pops the return address and jumps to it ∗rsp→A;rsp+8→rsp rsp−8→rsp;A→∗rsp
popq A
pushq A
leaq A,B A→B(Cmustbearegister)
cmpq A,B jeL
jlL
jle L
compare A and B and set the flag register
Jump to label L if the flag register matches the condition code of the instruction, otherwise go to the next instructions. The condition codes are e for “equal”, l for “less”, le for “less or equal”, g for “greater”, and ge for “greater or equal”.
Jump to label L
A→B
jg L
jge L
jmp L
movq A, B
movzbq A, B A → B, where A is a single-byte register (e.g., al or
cl), B is a 8-byte register, and the extra bytes of B are set to zero.
notq A orq A, B andq A, B salq A, B sarq A, B sete A setl A setle A setg A setge A
∼ A → A
A|B → B
A&B → B
B « A → B (arithmetic shift left, where A is a constant) B » A → B (arithmetic shift right, where A is a constant)
If the flag matches the condition code, then 1 → A, else 0 → A. Refer to je above for the description of the condition codes. A must be a single byte register (e.g., al or cl).
(bitwise complement) (bitwise-or)
(bitwise-and)
Table 12.1: Quick-reference for the x86 instructions used in this book.
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