CS293S SSA & Dead Code Elimination Yufei Ding Review of Last Class Two other flow analysis (DFA) Constant Propagation Reaching definitions (def-use chain) Static Single Assignment(SSA) Maximal SSA (all variables in every joint block) Minimal SSA (a def in block n results in an insertion in each of its DF(n)) Semi-pruned SSA (similar as minimal SSA, but only focus on global variable definitions) Maximal SSA Dominance Frontiers Dominance Frontiers B0 • DF(n ) is fringe just beyond the region n dominates B1 • m ÎDF(n) : iff n Ï(Dom(m) - {m}) but n Î DOM(p) for some p Î preds(m). B2 B3 i.e., n dominates m’s i.e., n doesn’t strictly some parent dominate m B4 B5 B6 B7 A node can be the dominance frontier of itself. Dominance frontier * e.g., for node n = 1. It dominates its own parent, of 3={7} node 7, but does not directly dominate itself. * it often indicates that there is a back edge. 4 Computing Dominance Frontiers • Only join points are in DF(n) for some n • Leads to a simple, intuitive algorithm for computing B0 B0 dominance frontiers For each join point x B1 B1 For each CFG predecessor of x (from the CFG) Walk up to IDOM(x ) in the dominator tree, adding x to DF(n) for each n in the walk except IDOM(x). B2 B3 B2 B3 B4 B5 B4 B5 B6 B6 • What if there is another back edge from B7 to B0? B7 B7 Control flow Dominance graph (CFG) Tree 5 Minimal SSA: Ø-functions insertion with DF for each variable x in the CFG work list = get all nodes (basic blocks) in which x is defined B0 for each node n in work list x¬B1Ø(...) for each node m in DF(n) if (there is no ϕ-function for x in m) insert a ϕ-function for x in m B2 B3 work list = work list ∪ { m } x¬ ... B4 B5 x¬B Ø(...) 6 • DF(4) is {6}, so ¬ in 4 forces Ø-function in 6 x¬B7Ø(...) • ¬ in 6 forces Ø-function in DF(6) = {7} • ¬ in 7 forces Ø-function in DF(7) = {1} • ¬ in 1 forces Ø-function in DF(1) = Ø (halt ) 6 Focus of This Class Static Single Assignment(SSA) Semi-pruned SSA (similar as minimal SSA, but on only global variable) Pruned SSA (similar as semi-pruned SSA, but dead are removed) Techniques for Removing ϕ-functions Dead code elimiation Semi-pruned SSA Observation: a variable that is only live in a single node can never have a live Ø-function. Therefore, the minimal technique can be further refined by first computing the set of global names – defined as the names that are live across more than one node – and producing Ø-functions for these names only. Step 1: Global Variables a ¬ ••• b ¬ ••• c ¬ ••• • Globals = ⋃"## $ %$ &'( UEVar(n) d ¬ ••• UEVar(n): Upper exposed variables in block n, i.e., i ¬ ••• i > 100 B0 variables used before it is redefined in block n. (We have learned this in liveness analysis. ) B 1 a ¬ ••• c ¬ ••• • Example B B3 UEVAR (B7) = {a, b, c, d, i}; 2 b ¬ ••• a ¬ ••• c ¬ ••• d ¬ ••• all others are empty set; d ¬ ••• Globals = {a, b, c, d, i}, (y, z are local names) B4 B5 d ¬ ••• c ¬ ••• • Get blocks where each of these Globals get b ¬ ••• defined B6 Names a b c d i B7 y ¬ a+b blocks 0,1,3 0,2,6 0,1,2,5 0,2,3,4 0,7 z ¬ c+d i ¬ i+1 i > 100 Phase 2: inserting ϕ-functions a ¬ ••• b ¬ ••• for each of the global name x c ¬ ••• d ¬ ••• work list = get all nodes in which x is defined i ¬ ••• B0 i > 100 for each node n in work list Excluding local for each node m in DF(n) B 1 a ¬ ••• names avoids if (there is no ϕ-function for x in m) c ¬ ••• Ø’s for y & z insert a ϕ-function for x to m work list = work list ∪ { m } B B3 2 b ¬ ••• a ¬ ••• c ¬ ••• d ¬ ••• d ¬ ••• B4 B5 d ¬ ••• c ¬ ••• b ¬ ••• B6 Names a b c d i blocks 0,1,3 0,2,6 0,1,2,5 0,2,3,4 0,7 B7 y ¬ a+b z ¬ c+d i ¬ i+1 Block 0 1 2 3 4 5 6 7 i > 100 DF - 1 7 7 6 6 7 1 Phase 3: renaming variables Renaming is done by a pre-order traversal of the dominator tree, as follows: for each node b in the dominator tree 1. rename definitions and uses of variables in b 2. rename ϕ-functions parameters corresponding to b in all successors of n in the CFG. Rename(b) for each Ø-function in b, x ¬ Ø(…) rename x as NewName(x) One possible Implementation via a for each operation “x ¬ y op z” in b set of stacks and counters. rewrite y as top(stack[y]) rewrite z as top(stack[z]) 1. Get the root node n0 of the rewrite x as NewName(x) CFG for each successor of b in the CFG 2. Call Rename(n0) rewrite appropriate Ø parameters for each successor s of b in dom. tree Rename(s) for each operation “x ¬ y op z” in b pop(stack[x]) a ¬ ••• Example b ¬ ••• c ¬ ••• d ¬ ••• B0 i ¬ ••• i > 100 B0 B 1 a ¬ Ø(a,a) B1 b ¬ Ø(b,b) c ¬ Ø(c,c) d ¬ Ø(d,d) i ¬ Ø(i,i) B2 B3 a ¬ ••• c ¬ ••• B4 B5 B B3 2 b ¬ ••• a ¬ ••• c ¬ ••• d ¬ ••• d ¬ ••• B4 B5 B d ¬ ••• c ¬ ••• 6 d ¬ Ø(d,d) c ¬ Ø(c,c) B7 B6 b ¬ ••• Dominance B7 a ¬ Ø(a,a) Tree b ¬ Ø(b,b) c ¬ Ø(c,c) d ¬ Ø(d,d) • pre-order traversal of the dominator y ¬ a+b z ¬ c+d tree: {0,1,2,3,4,5,6,7} i ¬ i+1 i > 100 Example a ¬ ••• Dominance Tree b ¬ ••• c ¬ ••• B0 d ¬ ••• B0 i ¬ ••• i > 100 B1 B1 a ¬ Ø(a,a) b ¬ Ø(b,b) c ¬ Ø(c,c) B2 B3 d ¬ Ø(d,d) i ¬ Ø(i,i) a ¬ ••• c ¬ ••• B4 B5 B6 B B3 2 b ¬ ••• a ¬ ••• c ¬ ••• d ¬ ••• d ¬ ••• B B4 B5 7 d ¬ ••• c ¬ ••• d ¬ Ø(d,d) Before processing B c ¬ Ø(c,c) 0 b ¬ ••• B6 a b c d i Counters 0 0 0 0 0 B7 a ¬ Ø(a,a) i ≤ 100 b ¬ Ø(b,b) c ¬ Ø(c,c) Stacks d ¬ Ø(d,d) y ¬ a+b z ¬ c+d • stacks for uses of variables i ¬ i+1 • counters for new definitions i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c ¬ ••• 0 B0 B0 d0 ¬ ••• i > 100 i0 ¬ ••• B1 B1 a ¬ Ø(a0,a) b ¬Ø(b0,b) c ¬ Ø(c0,c) d ¬ Ø(d0,d) B2 B3 i ¬ Ø(i0,i) a ¬ ••• c ¬ ••• B4 B5 B2 B3 a ¬ ••• b ¬ ••• B6 c ¬ ••• d ¬ ••• d ¬ ••• B4 B5 d ¬ ••• c ¬ ••• B7 d ¬ Ø(d,d) End of B0 c ¬ Ø(c,c) b ¬ ••• B6 a b c d i Counters 1 1 1 1 1 B7 a ¬ Ø(a,a) i ≤ 100 Stacks a b c d i b ¬ Ø(b,b) 0 0 0 0 0 c ¬ Ø(c,c) d ¬ Ø(d,d) y ¬ a+b z ¬ c+d i ¬ i+1 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• B0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a2 ¬ ••• B4 B5 c2 ¬ ••• B6 B B3 2 b ¬ ••• a ¬ ••• c ¬ ••• d ¬ ••• d ¬ ••• B7 B4 B5 d ¬ ••• c ¬ ••• d ¬ Ø(d,d) End of B1 c ¬ Ø(c,c) b ¬ ••• B6 a b c d i Counters 3 2 3 2 2 B7 a ¬ Ø(a,a) i ≤ 100 Stacks a b c d i b ¬ Ø(b,b) 0 0 0 0 0 c ¬ Ø(c,c) a b c d i d ¬ Ø(d,d) 1 1 1 1 1 y ¬ a+b a c z ¬ c+d 2 2 i ¬ i+1 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• B0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a2 ¬ ••• B4 B5 c2 ¬ ••• B6 B2 B3 a ¬ ••• b2 ¬ ••• d ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d ¬ ••• c ¬ ••• d ¬ Ø(d,d) c ¬ Ø(c,c) End of B2 b ¬ ••• B6 a b c d i Counters 3 3 4 3 2 B7 a ¬ Ø(a2,a) i ≤ 100 Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a b c d z ¬ c+d 2 2 2 2 i ¬ i+1 c3 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• B 0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a2 ¬ ••• B B c2 ¬ ••• 4 5 B6 B2 B3 a ¬ ••• b2 ¬ ••• d ¬ ••• c3 ¬ ••• d ¬ ••• 2 B7 B4 B5 d ¬ ••• c ¬ ••• d ¬ Ø(d,d) Before starting B c ¬ Ø(c,c) 3 b ¬ ••• B6 a b c d i Counters 3 3 4 3 2 B7 a ¬ Ø(a2,a) i ≤ 100 Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c z ¬ c+d 2 2 i ¬ i+1 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• B 0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a2 ¬ ••• B4 B5 c2 ¬ ••• B6 B B3 2 a3 ¬ ••• b2 ¬ ••• d3 ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d ¬ ••• c ¬ ••• d ¬ Ø(d,d) c ¬ Ø(c,c) End of B3 b ¬ ••• B6 a b c d i Counters 4 3 4 4 2 B7 a ¬ Ø(a2,a) Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c d z ¬ c+d 2 2 3 i ¬ i+1 a3 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• B0 c0 ¬ ••• d0 ¬ ••• B0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a ¬ ••• 2 B4 B5 c2 ¬ ••• B6 B B3 2 a3 ¬ ••• b2 ¬ ••• d3 ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d4 ¬ ••• c ¬ ••• End of B d ¬ Ø(d4,d) 4 c ¬ Ø(c2,c) b ¬ ••• B6 a b c d i Counters 4 3 4 5 2 B7 a ¬ Ø(a2,a) Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c d z ¬ c+d 2 2 3 i ¬ i+1 a3 d4 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• B0 c0 ¬ ••• d0 ¬ ••• B0 i0 ¬ ••• i > 100 B1 B1 a1 ¬ Ø(a0,a) b1 ¬ Ø(b0,b) c1 ¬ Ø(c0,c) B2 B3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a ¬ ••• 2 B4 B5 c2 ¬ ••• B6 B B3 2 a3 ¬ ••• b2 ¬ ••• d3 ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d4 ¬ ••• c ¬ ••• Before processing B d ¬ Ø(d4,d) 5 c ¬ Ø(c2,c) b ¬ ••• B6 a b c d i Counters 4 3 4 5 2 B7 a ¬ Ø(a2,a) Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c d z ¬ c+d 2 2 3 i ¬ i+1 a3 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• i0 ¬ ••• B0 i > 100 B1 B1 a1 ¬ Ø(a0,a) b ¬ Ø(b ,b) 1 0 B B c1 ¬ Ø(c0,c) 2 3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a ¬ ••• 2 B4 B5 c2 ¬ ••• B6 B B3 2 a3 ¬ ••• b2 ¬ ••• d3 ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d4 ¬ ••• c4 ¬ ••• d ¬ Ø(d4,d3) End of B5 c ¬ Ø(c2,c4) b ¬ ••• B6 a b c d i Counters 4 3 5 5 2 B7 a ¬ Ø(a2,a) Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c d z ¬ c+d 2 2 3 i ¬ i+1 a3 c4 i > 100 Example a0 ¬ ••• Dominance Tree b0 ¬ ••• c0 ¬ ••• B0 d0 ¬ ••• i0 ¬ ••• B0 i > 100 B1 B1 a1 ¬ Ø(a0,a) b ¬ Ø(b ,b) 1 0 B B c1 ¬ Ø(c0,c) 2 3 d1 ¬ Ø(d0,d) i1 ¬ Ø(i0,i) a ¬ ••• 2 B4 B5 c2 ¬ ••• B6 B B3 2 a3 ¬ ••• b2 ¬ ••• d3 ¬ ••• c3 ¬ ••• d2 ¬ ••• B7 B4 B5 d4 ¬ ••• c4 ¬ ••• Before B d ¬ Ø(d4,d3) 6 c ¬ Ø(c2,c4) b ¬ ••• B6 a b c d i Counters 4 3 5 5 2 B7 a ¬ Ø(a2,a) Stacks a0 b0 c0 d0 i0 b ¬ Ø(b2,b) c ¬ Ø(c3,c) a1 b1 c1 d1 i1 d ¬ Ø(d2,d) y ¬ a+b a c d z ¬ c+d 2 2 3 i ¬ i+1 a3 i > 100 Example a0 ¬ •••