SSA: Static Single-Assignment Form

1. SSA: Static Single-Assignment Form Compiler Baojian Hua bjhua@ustc.edu.cn

2. Middle End translation AST IR1 translation IR2 other IR and translation asm

3. Optimizations opt opt translation AST IR1 opt translation IR2 opt opt other IR and translation asm

4. SSA in General

5. SSA History • Optimization is IR-dependent: • We’ve discussed optimizations on the AST, TAC or CFG • Key problem: is there a best IR for optimization? • In late 1980’, researchers at IBM watson research center developed a new IR: SSA • with many fancy properties • great for optimizations • Later, SSA is used in many optimizing compilers: • GCC, MS VC, Jikes, MLton, LLVM, hotspot, … • As a result, SSA becomes a de-factor standard IR for optimizing compilers

6. Values ≠ Locations int i, sum1=0, sum2=0; for (i=0; i<10; i++) sum1 += i; for (i=0; i<100; i++) sum2 += i;

7. Use-Def Chains & Def-Use Chains int i, sum1=0, sum2=0; for (i=0; i<10; i++) sum1 += i; for (i=0; i<100; i++) sum2 += i;

8. UD Chains & DU Chains can be very Expensive int i, j, x, y; switch (i){ case 1: x = 1; break; case 2: x = 2; break; case 3: x = 3; break; case 4: x = 4; break; default: x = 5; break; } switch (j){ case 1: y = x+1; break; case 2: y = x-2; break; case 3: y = x*8; break; default: y = x/3; break; } For M uses and N defs: O(M *N) space and time.

9. UD Chains & DU Chains can be Expensive int i, j, x, y, z; switch (i){ case 1: x = 1; break; case 2: x = 2; break; case 3: x = 3; break; case 4: x = 4; break; default: x = 5; break; } z = x; switch (j){ case 1: y = z+1; break; case 2: y = z-2; break; case 3: y = z*8; break; default: y = z/3; break; } A solution to this is to limit each variable to have just one definition site.

10. SSA • SSA is an IR in which every variable is assigned at most once (in text) x = 2 x = 1 x = 2 x = 1 y = x

11. Advantages of SSA • Make DU and UD chains explicit • each definition knows its use • each use knows its unique definition • Make optimizations: • easier • faster • For most optimizations, also reduce the time/space requirements

12. Example: Constant propagation on SSA x = 3 … Each variable has just one definition, so we need not do reaching definition analysis. … a = x Can we replace this “x” with constant “3”?

13. Example: Copy propagation on SSA x = y … Each variable has just one definition, so we need not do reaching definition analysis. … a = x Can we replace this “x” with the variable “y”?

14. Example: Dead code elimination (DCE) on SSA x = v … Can we eliminate this assignment? … If “x” is not used anywhere, then one can eliminate this. (No liveness analysis!) …

15. Example: Common Sub-expression Elimination (CSE) on SSA x = a+b … Both a and b should have been assigned before this definition! SSA: each variable assigned at most once! If x dominates y, then one can do this substitution! (Why?) … y = a+b Can we replace this “a+b” with the variable “x”? No available expression analysis and reaching expression analysis!

16. Construction of SSA

17. Converting to SSA • Easy for a basic block • rename each definition • and rewrite each use to the recent def. a = x+y b = a+x a = b+2 c = y+1 a = c+a a1 = x+y b1 = a1+x a2 = b1+2 c1 = y+1 a3 = c1+a2

18. CFG c = 1 c1 = 1 a = x + y b = a + x a = b + 2 c = y + 1 a1 = x + y b1 = a1 +x a2 = b +2 c2 = y +1 a = c + a a3 = c? + a? So how to rewrite such kind of uses? Use a fiction of Φfunction.

19. Merging at joins with Φ function c = 1 c1 = 1 a = x + y b = a + x a = b + 2 c = y + 1 a1 = x + y b1 = a1 +x a2 = b +2 c2 = y +1 a = c + a a3 = Φ(a1, a2) b2 = Φ(b1, ?) c3 = Φ(c1, c2) a4 = c3 + a3 Φfunction indicates which definition to use.

20. The Φ function • Φ merges multiple definitions along multiple control paths into a single definition • For a basic block b with p predessors: • x = Φ (x1, x2, …, xp) • How does Φ know which xi to choose? • don’t care for analysis • can eliminate Φ before execution

21. Eliminating Φ function c1 = 1 c1 = 1 a1 = x + y b1 = a1 +x a2 = b +2 c2 = y +1 a1 = x + y b1 = a1 +x a3 = a1 c3 = c1 a2 = b +2 c2 = y +1 a3 = a2 c3 = c2 a3 = Φ(a1, a2) c3 = Φ(c1, c2) a4 = c3 + a3 a4 = c3 + a3

22. Trivial SSA Construction • At each join point, insert Φ for every live variable x = 1 x1 = 1 y = 2 y = 3 y1 = 2 y2 = 3 z = x + y x2 =Φ(x1, x1) y3 =Φ(y1, y2) z1 = x2 + y3 This Φ function is unnecessary!

23. Minimal SSA Construction • At each join point, insert Φ for every variable with different outstanding definitions x = 1 x1 = 1 y = 2 y = 3 y1 = 2 y2 = 3 z = x + y y3 =Φ(y1, y2) z1 = x2 + y3

24. Where to insert Φ? 1 Suppose there is a definition of “x” in block 4, which node needs a Φ? 2 3 4 Alternatively, which node does not need a Φ? 5 6 8 7 9 11 10 12

25. Where to insert Φ? 1 1 2 2 3 4 4 3 5 5 6 6 7 12 8 7 8 11 9 11 9 10 12 10

26. Dominance Frontier (DF) 1 1 2 2 3 4 4 3 5 5 6 6 7 12 8 7 8 11 9 11 9 10 12 10

27. Example: DF for node 5 1 1 2 2 3 4 4 3 5 5 6 6 7 12 8 7 8 11 9 11 9 10 12 10

28. Using DF to Place Φ • If there is a definition in node n, then there should be a Φ in DF(n) • for the definition “x=v”, insert • x=Φ(x, …, x) • where the number of arguments equal with the number of predecessors of n • One can use a worklist algorithm

29. Placing Φ place_phi () compute defsites(x) for every variable x; for (each variable x) for (each defsite d in defsites(x)) for (each node n in DF(d)) if we haven’t place Φ, then put one for x of the form x = Φ(x, …, x); if (n not in defsites(x)) defsites(x) \/= {n};

30. Example 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} i = 1 j = 1 k = 0 1 1 2 3 4 k < 100? 2 3 4 5 6 7 j < 20? return j 5 6 j = i k = k+1 j = k k = k+2 7

31. Placing Φ 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} Var i: w = defsites(i) = {1} i = 1 j = 1 k = 0 1 k < 100? 2 3 4 j < 20? return j 5 6 j = i k = k+1 j = k k = k+2 7

32. Placing Φ 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} Var j: w = defsites(j) = {1,5,6} i = 1 j = 1 k = 0 1 n = 1, w={5,6} n = 5, w={6} k < 100? 2 3 4 j < 20? return j 5 6 j = i k = k+1 j = k k = k+2 7 {j}

33. Placing Φ i = 1 j = 1 k = 0 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} Var j: w = defsites(j) = {1,5,6} 1 n = 1, w={5,6} {j} n = 5, w={6} 2 k < 100? n = 5, w={6, 7} n = 6, w={7} 3 j < 20? 4 return j n = 7, w={} 5 j = i k = k+1 j = k k = k+2 6 7 {j} j = Φ(j, j)

34. Placing Φ i = 1 j = 1 k = 0 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} Var j: w = defsites(j) = {1,5,6} 1 n = 1, w={5,6} n = 5, w={6} {j} 2 j = Φ(j, j)k < 100? n = 5, w={6, 7} n = 6, w={7} n = 7, w={} 3 j < 20? 4 return j n = 7, w={2} n = 2, w={} 5 j = i k = k+1 j = k k = k+2 6 {j} 7 j = Φ(j, j)

35. Placing Φ i = 1 j = 1 k = 0 1: {} 2: {2} 3: {2} 4: {} 5: {7} 6: {7} 7: {2} Similar for k. Leave it to you. 1 {j, k} 2 j = Φ(j, j)k = Φ(k, k) k < 100? 3 4 j < 20? return j 5 6 j = i k = k+1 j = k k = k+2 {j, k} j = Φ(j, j) k = Φ(k, k) 7

36. Renaming variables • Walk the dominator tree, renaming variable as you go • Replace each variable use with most recent renamed definition • easy for a basic block • for branches, use the most recent definition above the dominator tree nodes • that is, a pre-order tree walk

37. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 1 3 4 1 3 4 5 6 7 j < 20? return j 5 6 j = i k = k + 1 j = k k = k + 2 1 Walk the dominator tree (in pre-order). 3 2 j = Φ(j , j ) k = Φ(k , k ) 7

38. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 3 4 2 1 2 3 4 5 6 7 j < 20? return j 5 6 j = i k = k + 1 j = k k = k + 2 Walk the dominator tree (in pre-order). j = Φ(j , j ) k = Φ(k , k ) 7

39. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 3 4 2 1 2 3 4 5 6 7 j < 20? return j 2 5 6 j = i k = k + 1 j = k k = k + 2 Walk the dominator tree (in pre-order). j = Φ(j , j ) k = Φ(k , k ) 7

40. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 3 4 2 1 2 3 4 5 6 7 j < 20? return j 2 5 6 j = i k = k + 1 j = k k = k + 2 3 1 Walk the dominator tree (in pre-order). 3 2 j = Φ(j , j ) k = Φ(k , k ) 7 3 3

41. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 3 4 2 1 2 3 4 5 6 7 j < 20? return j 2 5 6 j = i k = k + 1 j = k k = k + 2 3 1 2 Walk the dominator tree (in pre-order). 4 3 2 4 2 j = Φ(j , j ) k = Φ(k , k ) 7 3 4 3 4

42. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 5 3 4 2 1 5 2 3 4 5 6 7 j < 20? return j 2 5 6 j = i k = k + 1 j = k k = k + 2 3 1 2 Walk the dominator tree (in pre-order). 4 3 2 4 2 j = Φ(j , j ) k = Φ(k , k ) 7 5 3 4 5 3 4

43. Rename Variables i = 1 j = 1 k = 0 1 1 1 1 1 2 2 j = Φ(j , j )k = Φ(k , k ) k < 100? 2 1 5 3 4 2 1 5 2 3 4 5 6 7 j < 20? return j 2 2 5 6 j = i k = k + 1 j = k k = k + 2 3 1 2 Walk the dominator tree (in pre-order). 4 3 2 4 2 j = Φ(j , j ) k = Φ(k , k ) 7 5 3 4 5 3 4

44. Summary • SSA is an intermediate representation • static single definition for every variable • invariant maintained by fake Φ functions • Conversion to SSA can be very fast • linear-time algorithm exists • SSA is very compact • linear size of the original program • Optimizations on SSA is very easy • no need to do data-flow analysis • more on this next time