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Path-Sensitive Analysis for Linear Arithmetic and Uninterpreted Functions

Path-Sensitive Analysis for Linear Arithmetic and Uninterpreted Functions. SAS 2004 Sumit Gulwani George Necula EECS Department University of California, Berkeley. Example. All 3 asserts are true. False. True. a=2?. y := a; z := 2;. y := 2; z := a;. True. False. a=2?.

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Path-Sensitive Analysis for Linear Arithmetic and Uninterpreted Functions

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  1. Path-Sensitive Analysis for Linear Arithmetic and Uninterpreted Functions SAS 2004 Sumit Gulwani George Necula EECS Department University of California, Berkeley

  2. Example All 3 asserts are true False True a=2? y := a; z := 2; y := 2; z := a; True False a=2? u := 1; v := 1+a; u := a-1; v := 3; t1 := y-u; t2 := v-z; Assert(t1=t2 Æ t1=1 Æ z=2);

  3. Path-Insensitive Analysis • Most PTIME analyses treat conditionals as non-deterministic. • They will verify only t1=t2 False True * y := a; z := 2; y := 2; z := a; True False * u := 1; v := 1+a; u := a-1; v := 3; t1 := y-u; t2 := v-z; Assert(t1=t2 Æ t1=1 Æ z=2);

  4. Path-Sensitive Analysis • We can do better by doing a boolean abstraction of conditionals. • Each atomic predicate is abstracted to a boolean variable • This will also verify t1=1 • This is still abstract though! • z=2 not verified • undecidable to reason completely False True c1 y := a; z := 2; y := 2; z := a; True False c1 u := 1; v := 1+a; u := a-1; v := 3; t1 := y-u; t2 := v-z; Assert(t1=t2 Æ t1=1 Æ z=2);

  5. Outline • Existing approach (MVR) vs. our approach (FCED) • FCEDs for linear arithmetic • FCEDs for uninterpreted function terms

  6. c1 t1 = c2 y= c1 u= c2 c2 2 a 1 a-1 1 -a+3 a-1 1 Multi-Valued ROBDDs (MVRs) True False c1 y := a; z := 2; y := 2; z := a; True False c2 • |MVR(t1)| = |MVR(y)| £ |MVR(u)| • MVR(t1) does not share nodes with MVR(y) and MVR(u) • Need a normal form for leaves u := 1; v := 1+a; u := a-1; v := 3; t1 := y-u; t2 := v-z; Assert(t1=t2); Assert(t1=1);

  7. Free Conditional Expression Diagrams (FCEDs) t1 = - True False c1 y= c1 u= c2 y := a; z := 2; y := 2; z := a; 2 a 1 a-1 True False c2 • |FCED(t1)| = |FCED(y)| + |FCED(u)| • FCED(t1) shares nodes with FCED(y) and FCED(u) • No need for normal form u := 1; v := 1+a; u := a-1; v := 3; t1 := y-u; t2 := v-z; Assert(t1=t2); Assert(t1=1);

  8. Outline • Existing approach (MVR) vs. our approach (FCEDs) • FCEDs for linear arithmetic • FCEDs for uninterpreted function terms

  9. Problem Definition e = q | y | e1§ e2| q £ e | if b then e1 else e2 b = c | b1Æ b2| b1Ç b2 e: conditional linear arithmetic expression b: boolean formula y: rational variable c: boolean variable q: rational constant • Construct FCED for an expression e, given FCEDs for its subexpressions. • Check 2 FCEDs for equivalence

  10. FCED An FCED f is a DAG with the following kind of nodes. f := y | q | Plus(f1,f2) | Minus(f1,f2) | Times(q,f) | Choose(f1,f2) | Guard(g,f) Choose(f1,f2) means f1 or f2 Guard(g,f) means if g then f Boolean expressions g are represented using ROBDDs g := true | false | c | If(c,g1,g2)

  11. plus + choose choose c1 c2 guard guard guard guard 2 a 1 a-1 a-1 R(:c2) R(c2) 1 a R(c1) 2 R(:c1) Example Formalization

  12. plus + choose choose c1 c2 guard guard guard guard 2 a 1 a-1 a-1 R(:c2) R(c2) 1 a R(c1) 2 R(:c1) Example Formalization

  13. FCED Construction • FCED(y) = Leaf(y) • FCED(q) = Leaf(q) • FCED(e1+e2) = Plus (FCED(e1), FCED(e2)) • FCED(q £ e) = Times(q,FCED(e)) • FCED(if b then e1 else e2) = Choose(Guard(R(b),e1), Guard(R(NOT(b)),e2)

  14. FCED Construction • FCED(y) = Leaf(y) • FCED(q) = Leaf(q) • FCED(e1+e2) = Plus (FCED(e1), FCED(e2)) • FCED(q £ e) = Times(q,FCED(e)) • FCED(if b then e1 else e2) = Choose(||R(b),FCED(e1)||, ||NOT R(b), FCED(e2)||)

  15. Normalize Guard Operator • Inputs: guard g, FCED f • Output: FCED f’ s.t. • f ´ f’ • 8 guard nodes Guard(g,f’’) in f’, BV(g) < BV(f’’) • ||g,f|| = Guard(g,f), if BV(g) < BV(f) • ||g, Plus(f1,f2) = Plus(||g,f1||, ||g, f2||) • ||g, Choose(f1,f2) = Choose(||g,f1||, ||g, f2||) • ||g1, Guard(g2,f )|| = Guard(|| INTERSECT(g1,g2),f ||) • …

  16. guard guard guard R(c1) R(c1) R(c1) choose choose guard guard guard guard guard z 6 R(:c2) R(c2) 2 R(c1) R(:c1) 3 3 R(:c1Æc1) choose guard guard guard 2 R(c1Æc1) 2 R(c1) R(:c1) 3 Example: Normalize Guard Operator Given f, construct ||R(c1),f|| plus choose

  17. Randomized Equivalence Testing for FCEDs Assign hash values to nodes of FCEDs in bottom-up manner V: FCED Node ! Integer • V(Leaf(q)) = q • V(Leaf(y)) = ry • V(Plus(f1,f2)) = V(f1) + V(f2) • V(Choose(f1,f2)) = V(f1) + V(f2) • V(Guard(g,f)) = H(g) £ V(f) H: Guard ! Integer • H(true) = 1, H(false) = 0 • H(c) = rc • H(If(c,g1,g2)) = rc£ H(g1) + (1-rc) £ H(g2)

  18. Randomized Equivalence Testing for FCEDs Completeness f1´ f2) V(f1) = V(f2) Soundness f1´ f2) Pr[V(f1) = V(f2)] · s/t s: maximum # of nodes in a FCED t: size of set from which random values are chosen Proof: 9 1-1 Poly: FCED ! Polynomials such that V(f) is the value of Poly(f)

  19. Outline • Existing approach (MVR) vs. our approach (FCEDs) • FCEDs for linear arithmetic • FCEDs for uninterpreted function terms

  20. Problem Definition e = y | F(e1,e2) | if b then e1 else e2 b = c | b1Æ b2| b1Ç b2 e: conditional uninterpreted function term b: boolean formula y: variable c: boolean variable • Construct FCED for an expression e, given FCEDs for its subexpressions. • Check 2 FCEDs for equivalence

  21. FCED An FCED f is a DAG with the following kind of nodes. f := y | F(f1,f2) | Choose(f1,f2) | Guard(g,f) Choose(f1,f2) means f1 or f2 Guard(g,f) means if g then f Boolean expressions g are represented using ROBDDs g := true | false | c | If(c,g1,g2)

  22. FCED Construction FCED(y) = Leaf(y) FCED(F(e1,e2)) = F(FCED(e1), FCED(e2)) FCED(if b then e1 else e2) = Choose(||R(b),FCED(e1)||, ||NOT R(b), FCED(e2)||)

  23. Randomized Equivalence Testing of FCEDs Assign hash values to nodes of FCEDs in bottom-up manner V: FCED Node ! Tuple of k integers K ¸ depth of any FCED • V(y) = [ry,…ry] • V(Choose(f1,f2)) = V(f1) + V(f2) • V(Guard(g,f)) = H(g) £ V(f) • V(F(f1,f2)) = V(f1) £ M + V(f2) £ N M, N: random k £ k matrices

  24. Randomized Equivalence Testing for FCEDs Completeness f1´ f2) V(f1) = V(f2) Soundness f1´ f2) Pr[V(f1) = V(f2)] · s: maximum # of nodes in a FCED t: size of set from which random values are chosen Proof: more involved

  25. Conclusion and Future Work • Randomization can help achieve simplicity and efficiency at the expense of making soundness probabilistic. • Integrate randomized techniques with symbolic algorithms • Few interesting possible extensions: • Combination of uninterpreted functions with arithmetic • Partially interpreted functions like commutative and/or associative functions • Model memory

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