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Precise Inter-procedural Analysis

Precise Inter-procedural Analysis. using Random Interpretation. Sumit Gulwani George C. Necula. UC Berkeley. presented by Kian Win Ong. Quick Overview. true. false. *. a := 0 b := i. a := i – 2 b := 2. false. true. *. c := b – a. c := 2a + b. assert (a + b = i) assert (c = a + i).

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Precise Inter-procedural Analysis

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  1. Precise Inter-procedural Analysis using Random Interpretation Sumit GulwaniGeorge C. Necula UC Berkeley presented by Kian Win Ong

  2. Quick Overview true false * a := 0b := i a := i – 2b := 2 false true * c := b – a c := 2a + b assert (a + b = i)assert (c = a + i)

  3. Quick Overview true false * a := 0b := i a := i – 2b := 2 Random testing needs to execute all 4 paths to verify assertions false true * c := b – a c := 2a + b assert (a + b = i)assert (c = a + i) û

  4. Quick Overview i = 3 true false * a := 0b := i a := i – 2b := 2 w1 = 5 i = 3, a = 0, b = 3 i = 3, a = 1, b = 2 ajoin = w1Î afalse + ( 1 – w1 ) Î atrue i = 3, a = -4, b = 7 false true * c := b – a c := 2a + b assert (a + b = i)assert (c = a + i)

  5. Quick Overview i = 3 true false * a := 0b := i a := i – 2b := 2 w1 = 5 i = 3, a = 0, b = 3 i = 3, a = 1, b = 2 i = 3, a = -4, b = 7 false true * c := b – a c := 2a + b i = 3, a = -4, b = 7, c = -1 w2 = 2 i = 3, a = -4, b = 7, c = 11 i = 3, a = -4, b = 7, c = 23 assert (a + b = i)assert (c = a + i) û

  6. Random Interpretation • Random Testingdynamically testing the program using randomly generated input • Pros: Simple implementation • Cons: Limited code coverage • Abstract Interpretationstatically analyzing selected properties of the program using symbolic execution • Pros: Static analysis • Cons: Conservative / Complicated

  7. Random Interpretation • Random Interpretation • statically analyzing selected properties of the program using symbolic random states • Pros: Static analysis, Simple implementation • Cons: Probabilistically sound Small number of runs guarantee a high probability of soundness

  8. Intra-procedural Framework • Program Model: • State captured as polynomials, which are linear in program variables • Goal: • To detect equivalences between polynomials c := b – a c := 2a + b i = 3, a = -4, b = 7, c = -1 w = 2 i = 3, a = -4, b = 7, c = 11 i = 3, a = -4, b = 7, c = 23

  9. Intra-procedural Framework • Algorithm • Choose random values for input variables • Execute assignments • Use property-specific Eval() to abstract program state as polynomials • Execute both branches of conditionals • Use Affine Join to combine both program states at join points • Compare polynomials to decide equality

  10. Intra-procedural Framework • Design ofEval()s • Property (abstraction) specific • Linear arithmetice := x | e1§e2 | cÎeP(e) := e • Un-interpreted functionse := x | F(e)P(x) := xP(F(e)) := c1ÎP(e) + c2 • Completeness and Soundness • P(e1) = P(e2) iff e1 = e2 • Linearity • P(e) is linear in program variables

  11. Intra-procedural Framework • Affine Join • To combine (branched) program states at join points=w(1,2) true false * a := 0b := i a := i – 2b := 2 1 2 w = 5 i = 3, a = 0, b = 3 i = 3, a = 1, b = 2  i = 3, a = -4, b = 7 (x) := wÎ1(x) + (1-w)Î2(x)

  12. Intra-procedural Framework • Affine Join • CompletenessIf polynomials P1 and P2 are equivalent in states 1 and 2, Then they are also equivalent in state  • SoundnessIf polynomials P1 and P2 are not equivalent in either state 1 and 2, Then it is unlikely that they are equivalent in state  Generate a small number tof runs

  13. Inter-Procedural Extensions • Maintain symbolic state summaries • Generate multiple fresh runs

  14. Inter-Procedural Extensions i = 3 true false * a := 0b := i a := i – 2b := 2 w1 = 5 i = 3, a = 0, b = 3 i = 3, a = 1, b = 2 i = 3, a = -4, b = 7 false true * c := b – a c := 2a + b i = 3, a = -4, b = 7, c = -1 w2 = 2 i = 3, a = -4, b = 7, c = 11 i = 3, a = -4, b = 7, c = 23 assert (a + b = i)assert (c = a + i) û

  15. Inter-Procedural Extensions i = 3 i = 2 true false * a := 0b := i a := i – 2b := 2 w1 = 5 i = 2, a = 0, b = 2 i = 2, a = 0, b = 2 i = 2, a = 0, b = 2 false true * c := b – a c := 2a + b i = 2, a = 0, b = 2, c = 2 w2 = 2 i = 2, a = 0, b = 2, c = 2 i = 2, a = 0, b = 2, c = 2 assert (a + b = i)assert (c = a + i) ü û

  16. Inter-Procedural Extensions 1. Maintain symbolic state summaries i true false * a := 0b := i a := i – 2b := 2 w1 = 5 a = 0, b = i a = i - 2, b = 2 a = 8 – 4i, b = 5i - 8 false true * c := b – a c := 2a + b a = 8 – 4i, b = 5i – 8, c = 8 – 3i w2 = 2 a = 8 – 4i, b = 5i – 8,c = 9i - 16 a = 8 – 4i, b = 5i – 8, c = 21i - 40 assert (a + b = i)assert (c = a + i)

  17. Inter-Procedural Extensions Unsound way of summarizing multiple calls i true false x := A(2)y := A(1)z := A(1) * u := i + 1 u := 3 u = 3 u = i + 1 x = 3y = -2z = -2 w = 5 u = 5i - 7 assert (x = 3)assert (y = z) return u Procedure B Procedure A

  18. Inter-Procedural Extensions 2. Generate multiple fresh runs x := A(2)y := A(1)z := A(1) i x = 7(5i – 7,7 – 2i)y = 3(5i – 7,7 – 2i)z = 5(5i – 7,7 – 2i) true false * u := i + 1 u := 3 w1 = 5 u = 3 u = i + 1 x = 6(5i – 7,7 – 2i)y = 0(5i – 7,7 – 2i)z = 1(5i – 7,7 – 2i) w2 = -2 u = i + 1 u = 3 u = 5i - 7 u = 7 – 2i assert (x = 3)assert (y = z) return u Procedure A Procedure B

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