1. Introduction to Compilers Ben Livshits Based in part of Stanford class slides from http://infolab.stanford.edu/~ullman/dragon/w06/w06.html

2. Organization Really basic stuff Flow Graphs Constant Folding Global Common Subexpressions Induction Variables/Reduction in Strength Data-flow analysis Proving Little Theorems Data-Flow Equations Major Examples Pointer analysis

3. Compiler Organization

4. Dataflow Analysis Basics L2: Compiler Organization Dataflow analysis basics L3: Dataflow lattices Integrative dataflow solution Gen/kill frameworks

5. Pointer Analysis L10: Pointer analysis L11 Pointer analysis and bddbddb

7. Constant Folding

8. Global Common Subexpressions

10. 8 Intermediate Code for (i=0; i<n; i++) A[i] = 1; Intermediate code exposes optimizable constructs we cannot see at source-code level. Make flow explicit by breaking into basic blocks = sequences of steps with entry at beginning, exit at end.

11. 9 i = 0 if i>=n goto … t1 = 8*i A[t1] = 1 i = i+1 Basic Blocks for (i=0; i<n; i++) A[i] = 1;

12. 10 Induction Variables x is an induction variable in a loop if it takes on a linear sequence of values each time through the loop. Common case: loop index like i and computed array index like t1. Eliminate “superfluous” induction variables. Replace multiplication by addition (reduction in strength ).

13. 11 Example i = 0 if i>=n goto … t1 = 8*i A[t1] = 1 i = i+1 t1 = 0 n1 = 8*n if t1>=n1 goto … A[t1] = 1 t1 = t1+8

14. 12 Loop-Invariant Code Motion Sometimes, a computation is done each time around a loop. Move it before the loop to save n-1 computations. Be careful: could n=0? I.e., the loop is typically executed 0 times.

15. 13 Example i = 0 i = 0 t1 = y+z if i>=n goto … if i>=n goto … t1 = y+z x = x+t1 i = i+1 x = x+t1 i = i+1

16. 14 Constant Folding Sometimes a variable has a known constant value at a point. If so, replacing the variable by the constant simplifies and speeds-up the code. Easy within a basic block; harder across blocks.

17. 15 Example i = 0 n = 100 if i>=n goto … t1 = 8*i A[t1] = 1 i = i+1 t1 = 0 if t1>=800 goto … A[t1] = 1 t1 = t1+8

18. 16 Global Common Subexpressions Suppose block B has a computation of x+y. Suppose we are sure that when we reach this computation, we are sure to have: Computed x+y, and Not subsequently reassigned x or y. Then we can hold the value of x+y and use it in B.

19. 17 Example a = x+y t = x+y a = t b = x+y t = x+y b = t c = x+y c = t

20. 18 Example --- Even Better t = x+y a = t t = x+y b = t c = t t = x+y a = t b = t t = x+y b = t c = t

22. Data-Flow Equations

24. 21 As a Flow Graph x = true if x == true “body”

25. 22 Formulation: Reaching Definitions Each place some variable x is assigned is a definition. Ask: for this use of x, where could x last have been defined. In our example: only at x=true.

26. 23 d1 d2 Example: Reaching Definitions d1: x = true d1 if x == true d2 d1 d2: a = 10

27. 24 Clincher Since at x == true, d1 is the only definition of x that reaches, it must be that x is true at that point. The conditional is not really a conditional and can be replaced by a branch.

28. 25 Not Always That Easy int i = 2; int j = 3; while (i != j) { if (i < j) i += 2; else j += 2; } We’ll develop techniques for this problem, but later …

29. 26 d1 d2 d3 d4 d2, d3, d4 d1, d3, d4 d1, d2, d3, d4 d1, d2, d3, d4 The Flow Graph d1: i = 2 d2: j = 3 if i != j d1, d2, d3, d4 if i < j d4: j = j+2 d3: i = i+2

30. 27 DFA Is Sometimes Insufficient In this example, i can be defined in two places, and j in two places. No obvious way to discover that i!=j is always true. But OK, because reaching definitions is sufficient to catch most opportunities for constant folding (replacement of a variable by its only possible value).

31. 28 Be Conservative! (Code optimization only) It’s OK to discover a subset of the opportunities to make some code-improving transformation. It’s notOK to think you have an opportunity that you don’t really have.

32. 29 Example: Be Conservative boolean x = true; while (x) { . . . *p = false; . . . } Is it possible that p points to x?

33. 30 Another def of x d2 As a Flow Graph d1: x = true d1 if x == true d2: *p = false

34. 31 Possible Resolution Just as data-flow analysis of “reaching definitions” can tell what definitions of x might reach a point, another DFA can eliminate cases where p definitely does not point to x. Example: the only definition of p is p = &y and there is no possibility that y is an alias of x.

35. 32 Reaching Definitions Formalized A definition d of a variable x is said to reach a point p in a flow graph if: Every path from the entry of the flow graph to p has d on the path, and After the last occurrence of d there is no possibility that x is redefined.

36. 33 Data-Flow Equations --- (1) A basic block can generate a definition. A basic block can either Kill a definition of x if it surely redefines x. Transmit a definition if it may not redefine the same variable(s) as that definition.

37. 34 Data-Flow Equations --- (2) Variables: IN(B) = set of definitions reaching the beginning of block B. OUT(B) = set of definitions reaching the end of B.

38. 35 Data-Flow Equations --- (3) Two kinds of equations: Confluence equations : IN(B) in terms of outs of predecessors of B. Transfer equations : OUT(B) in terms of of IN(B) and what goes on in block B.

39. 36 Confluence Equations IN(B) = ∪predecessors P of B OUT(P) {d2, d3} {d1, d2} P2 P1 {d1, d2, d3} B

40. 37 Transfer Equations Generate a definition in the block if its variable is not definitely rewritten later in the basic block. Kill a definition if its variable is definitely rewritten in the block. An internal definition may be both killed and generated.

41. 38 Example: Gen and Kill IN = {d2(x), d3(y), d3(z), d5(y), d6(y), d7(z)} d1: y = 3 d2: x = y+z d3: *p = 10 d4: y = 5 Kill includes {d1(x), d2(x), d3(y), d5(y), d6(y),…} Gen = {d2(x), d3(x), d3(z),…, d4(y)} OUT = {d2(x), d3(x), d3(z),…, d4(y), d7(z)}

42. 39 Transfer Function for a Block For any block B: OUT(B) = (IN(B) – Kill(B)) ∪Gen(B)

43. 40 Iterative Solution to Equations For an n-block flow graph, there are 2n equations in 2n unknowns. Alas, the solution is not unique. Use iterative solution to get the least fixed-point. Identifies any def that might reach a point.

44. 41 Iterative Solution --- (2) IN(entry) = ∅; for each block B do OUT(B)= ∅; while (changes occur) do for each block B do { IN(B) = ∪predecessors P of B OUT(P); OUT(B) = (IN(B) – Kill(B)) ∪Gen(B); }

45. 42 IN(B1) = {} OUT(B1) = { IN(B2) = { d1, OUT(B2) = { IN(B3) = { d1, OUT(B3) = { Example: Reaching Definitions d1: x = 5 B1 d1} d2} if x == 10 B2 d1, d2} d2} d2: x = 15 B3 d2}

46. 43 Aside: Notice the Conservatism Not only the most conservative assumption about when a def is killed or gen’d. Also the conservative assumption that any path in the flow graph can actually be taken.

48. Pointer Analysis

50. 46 Flow/Context Insensitivity Not so bad when program units are small (few assignments to any variable). Example: Java code often consists of many small methods. Remember: you can distinguish variables by their full name, e.g., class.method.block.identifier.

51. 47 Context Sensitivity Can distinguish paths to a given point. Example: If we remembered paths, we would not have the problem in the constant-propagation framework where x+y = 5 but neither x nor y is constant over all paths.

52. 48 The Example Again x = 3 y = 2 x = 2 y = 3 z = x+y

53. 49 An Interprocedural Example int id(int x) {return x;} void p() {a=2; b=id(a);…} void q() {c=3; d=id(c);…} If we distinguish p calling id from q calling id, then we can discover b=2 and d=3. Otherwise, we think b, d = {2, 3}.

54. 50 Context-Sensitivity --- (2) Loops and recursive calls lead to an infinite number of contexts. Generally used only for interprocedural analysis, so forget about loops. Need to collapse strong components of the calling graph to a single group. “Context” becomes the sequence of groups on the calling stack.

55. 51 Example: Calling Graph t Contexts: Green Green, pink Green, yellow Green, pink, yellow s r p q main

56. 52 Comparative Complexity Insensitive: proportional to size of program (number of variables). Flow-Sensitive: size of program, squared (points times variables). Context-Sensitive: worst-case exponential in program size (acyclic paths through the code).

57. 53 Logical Representation We have used a set-theoretic formulation of DFA. IN = set of definitions, e.g. There has been recent success with a logical formulation, involving predicates. Example: Reach(d,x,i) = “definition d of variable x can reach point i.”

58. 54 Comparison: Sets Vs. Logic Both have an efficiency enhancement. Sets: bit vectors and boolean ops. Logic: BDD’s, incremental evaluation. Logic allows integration of different aspects of a flow problem. Think of PRE as an example. We needed 6 stages to compute what we wanted.

59. 55 Datalog --- (1) Predicate Arguments: variables or constants The body : For each assignment of values to variables that makes all these true … Make this atom true (the head ). Atom = Reach(d,x,i) Literal = Atom or NOT Atom Rule = Atom :- Literal & … & Literal

60. 56 Example: Datalog Rules Reach(d,x,j) :- Reach(d,x,i) & StatementAt(i,s) & NOT Assign(s,x) & Follows(i,j) Reach(s,x,j) :- StatementAt(i,s) & Assign(s,x) & Follows(i,j)

61. 57 Datalog --- (2) Intuition: subgoals in the body are combined by “and” (strictly speaking: “join”). Intuition: Multiple rules for a predicate (head) are combined by “or.”

62. 58 Datalog --- (3) Predicates can be implemented by relations (as in a database). Each tuple, or assignment of values to the arguments, also represents a propositional (boolean) variable.

63. 59 Iterative Algorithm for Datalog Start with the EDB predicates = “whatever the code dictates,” and with all IDB predicates empty. Repeatedly examine the bodies of the rules, and see what new IDB facts can be discovered from the EDB and existing IDB facts.

64. 60 Example: Seminaive Path(x,y) :- Arc(x,y) Path(x,y) :- Path(x,z) & Path(z,y) NewPath(x,y) = Arc(x,y); Path(x,y) = ∅; while (NewPath != ∅) do { NewPath(x,y) = {(x,y) | NewPath(x,z) && Path(z,y) || Path(x,z) && NewPath(z,y)} – Path(x,y); Path(x,y) = Path(x,y) ∪ NewPath(x,y); }

65. Pointer analysis 61

66. 62 New Topic: Pointer Analysis We shall consider Andersen’s formulation of Java object references. Flow/context insensitive analysis. Cast of characters: Local variables, which point to: Heap objects, which may have fields that are references to other heap objects.

67. 63 Representing Heap Objects A heap object is named by the statement in which it is created. Note many run-time objects may have the same name. Example: h: T v = new T;says variable v can point to (one of) the heap object(s) created by statement h. v h

68. 64 Other Relevant Statements v.f = w makes the f field of the heap object h pointed to by v point to what variable w points to. f v w f h g i

69. 65 Other Statements --- (2) v = w.f makes v point to what the f field of the heap object h pointed to by w points to. v w i f h g

70. 66 Other Statements --- (3) v = w makes v point to whatever w points to. Interprocedural Analysis : Also models copying an actual parameter to the corresponding formal or return value to a variable. v w h

71. 67 Datalog Rules Pts(V,H) :- “H: V = new T” Pts(V,H) :- “V=W” & Pts(W,H) Pts(V,H) :- “V=W.F” & Pts(W,G) & Hpts(G,F,H) Hpts(H,F,G) :- “V.F=W” & Pts(V,H) & Pts(W,G)

72. 68 Example T p(T x) { h: T a = new T; a.f = x; return a; } void main() { g: T b = new T; b = p(b); b = b.f; }

73. 69 Apply Rules Recursively --- Round 1 Pts(a,h) Pts(b,g) T p(T x) {h: T a = new T; a.f = x; return a;} void main() {g: T b = new T; b = p(b); b = b.f;}

74. 70 Apply Rules Recursively --- Round 2 Pts(x,g) Pts(b,h) T p(T x) {h: T a = new T; a.f = x; return a;} void main() {g: T b = new T; b = p(b); b = b.f;} Pts(a,h) Pts(b,g)

75. 71 Apply Rules Recursively --- Round 3 Hpts(h,f,g) Pts(x,h) T p(T x) {h: T a = new T; a.f = x; return a;} void main() {g: T b = new T; b = p(b); b = b.f;} Pts(a,h) Pts(b,g) Pts(x,g) Pts(b,h)

76. 72 Apply Rules Recursively --- Round 4 Hpts(h,f,h) T p(T x) {h: T a = new T; a.f = x; return a;} void main() {g: T b = new T; b = p(b); b = b.f;} Pts(a,h) Pts(b,g) Pts(x,g) Pts(b,h) Pts(x,h) Hpts(h,f,g)

77. 73 Adding Context Sensitivity Include a component C = context. C doesn’t change within a function. Call and return can extend the context if the called function is not mutually recursive with the caller.

78. 74 Example of Rules: Context Sensitive Pts(V,H,B,I+1,C) :- “B,I: V=W” & Pts(W,H,B,I,C) Pts(X,H,B0,0,D) :- Pts(V,H,B,I,C) & “B,I: call P(…,V,…)” & “X is the corresponding actual to V in P” & “B0 is the entry of P” & “context D is C extended by P”