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A Polynomial-Time Algorithm for Global Value Numbering SAS 2004 Sumit Gulwani George C. Necula

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1 Global Value Numbering Goal: Discover equivalent expressions in procedures Applications: Compiler optimizations –Copy propagation, Constant propagation, Common sub- expression elimination, Induction variable elimination etc. Program verification –Discover loop invariants, verify program assertions Discover equivalent computations across programs –Plagiarism detection tools, Translation validation

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2 Global Value Numbering x := b £ a; y := a £ 3; c := a £ b; If (b == 3) z := a £ b; Equivalence problem is undecidable. Simplification Assumptions: Operators are uninterpreted (will not discover x = c) Conditionals are non-deterministic (will not discover y = c) Will discover z = c TrueFalse

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3 Non-trivial Example assert(x = y); assert(z = F(y)); * x := a; y := a; z := F(a); x := b; y := b; z := F(b);

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4 Existing Algorithms Algorithms that work on SSA form of the program –Alpern, Wegman, Zadecks (AWZ) algorithm: POPL 1988 Polynomial, Incomplete –Ruthing, Knoop, Steffens (RKS) Algorithm: SAS 1999 Polynomial, Incomplete, Improvement on AWZ Dataflow analysis or Abstract interpretation based –Kildalls Algorithm: POPL 1973 Exponential, Complete –Our Algorithm: POPL 2004 Polynomial, Complete, Randomized –Our Algorithm: this paper Polynomial, Complete

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5 Why SSA based algorithms are incomplete? assert(x = y); assert(z = F(y)); * x = (a,b) y = (a,b) z = (F(a),F(b)) F(y) = F( (a,b)) AWZ Algorithm: functions are uninterpreted –fails to discover second assertion RKS Algorithm: uses rewrite rules for normalization –Does not discover all assertions in little more involved examples. –Rewrite rules not applied exhaustively (exp applications o.w.) –Rules are pessimistic in handling loops x := a; y := a; z := F(a); x := b; y := b; z := F(b);

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6 Abstract Interpretation based algorithm G = SP(G 0,x := e) Assignment Node G0G0 x := e G 2 = G 0 Conditional Node G 1 = G 0 * G0G0 G = Join(G 1 0,G 2 0 ) G10G10 Join Node G20G20

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7 Outline Strong equivalence DAG (SED) The join operation: Idea #1 Pruning an SED: Idea #2 The strongest postcondition operation Fixed point computation

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8 Representing Equivalences a := 1; b := 2; x := F(1,2); { a,1 } { b,2 } { x, F(1,2) }

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9 Representing Equivalences a := 1; b := 2; x := F(1,2); { a,1 } { b,2 } { x, F(1,2), F(a,2), F(1,b), F(a,b) } Such an explicit representation can be exponential.

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10 Strong Equivalence DAG (SED) A data structure for representing equivalences. Nodes n: Type: c, ?, F(n 1,n 2 ) Terms(n): set of equivalent expressions –Terms( ) = V –Terms( ) = V [ { c } –Terms( ) = V [ { F(e 1,e 2 ) | e 1 2 Terms(n 1 ), e 2 2 Terms(n 2 ) } 8 variables x, 9 at most one node s.t. x 2 V – called Node(x)

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11 SED: Example This SED represents the following partition: Terms(n 1 ) = { a, 2 } Terms(n 2 ) = { b} Terms(n 3 ) = { c, d, F(a,b), F(2,b) } Terms(n 4 ) = { e, F(c,b), F(d,b), F(F(a,b),b), F(F(2,b),b) } a, 2 d,c, F b, ? e, F n1n1 n4n4 n3n3 n2n2

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12 Outline Strong equivalence DAG (SED) The join operation: Idea #1 Pruning an SED: Idea #2 The strongest postcondition operation Fixed point computation

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13 The Join Operation G = Join(G 1, G 2 ) G is obtained by product construction of G 1 and G 2 If n= 2 G 1 and m= 2 G 2, then [n,m]= 2 G Definition of t 1 t t 2 c t c = c F(l 1,r 1 ) t F(l 2,r 2 ) = F ([l 1,l 2 ],[r 1,r 2 ]) t 1 t t 2 = ?, otherwise Proof of Correctness Terms([n,m]) = Terms(n) Å Terms(m) (Thus product construction = partition intersection)

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14 Example: The Join Operation G1G1 G2G2 G F y 2, F y 1, F y 3,y 4 y 5, ? F y6,?y6,?y7,?y7,? F y 2, F y 1, F y 4,y 5 ? F y 6,y 7 ? y3,?y3,? G = Join(G 1,G 2 ) F y 2, F y 1, F y 4,y 5 ? F y6,?y6,? y3,?y3,? y7,?y7,?

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15 Outline Strong equivalence DAG (SED) The join operation: Idea #1 Pruning an SED: Idea #2 The strongest postcondition operation Fixed point computation

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16 Motivation: The Prune Operation Discovering equivalences among all expressions For the latter, it is sufficient to discover equivalences among all terms of size at most t at each program point (where t = #variables * size of program). Thus, SEDs can be pruned to have a small size. Discovering equivalences among program expressions vs. If G=Join(G 1,G 2 ), then Size(G) can be Size(G 1 ) £ Size(G 2 ) There are programs, where size of SEDs after n joins is exponential in n.

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17 The Prune Operation Prune(G,k) For each node, check if x 2 V is equal to some F-term of size less than k. If not, then delete all the nodes that are reachable from only

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18 Example: The Prune Operation G Prune(G,2) y 2, ? y 1, G y 4,y 5 ? G F y 2, F y 1, G y 4,y 5 ? F y6,?y6,? y3,?y3,? y7,?y7,?

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19 Outline Strong equivalence DAG (SED) The join operation: Idea #1 Pruning an SED: Idea #2 The strongest postcondition operation Fixed point computation

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20 The Strongest Postcondition Operation G = SP(G 0, x := e) To obtain G from G, do: Delete label x from Node(x) in G 0 Let n= be the node in G 0 s.t. e 2 Terms(n) (Add such a node to G 0 if it does not already exists) Add x to V.

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21 F Example: The Strongest Postcondition Operation G0G0 z, u, F x, ? G = SP(G 0, u := F(z,x)) z, F x, ? u, F

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22 Outline Strong equivalence DAG (SED) The join operation: Idea #1 Pruning an SED: Idea #2 The strongest postcondition operation Fixed point computation

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23 Fixed Point Computation and Complexity The lattice of sets of equivalences (among uninterpreted function terms) has height at most k. Complexity –Dominated by the cost of join operations –# of join operations: O(j £ k) –Each join operation: O(k 2 £ N) This requires doing pruning while computing join –Total cost: O(k 3 £ N £ j) k: # of variables N: size of program j: # of join points in program

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24 Example x := 1; y := 1; z := F(1,1); x := 2; y := 2; z := F(2,2); u := F(x,y); Assert(u = z); L1L1 L2L2 L3L3 L4L4 G1G1 z, F x,y, 1 G2G2 z, F x,y, 2 G 3 = Join(G 1,G 2 ) G3G3 z, F x,y,? G 4 = Assignment(G 3, u := F(x,y)) G4G4 u,z, F x,y, ?

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25 Conclusion Idea #1: Join of 2 SEDs = Product construction Idea #2: Prune SEDs (Discovering equivalences among program expressions does not require computing equivalences involving large terms) Future Work Inter-procedural value numbering Abstract interpretation for combined theory of linear arithmetic and uninterpreted functions

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