1 / 20

Cover Algorithms and Their Combination

Cover Algorithms and Their Combination. Sumit Gulwani, Madan Musuvathi Microsoft Research, Redmond. Cover Definition. Cover operation is useful for simplifying a formula by discarding facts related to a set of variables Given A quantifier-free formula  in theory T A set of symbols V

uriah-cruz
Download Presentation

Cover Algorithms and Their Combination

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Cover Algorithms and Their Combination Sumit Gulwani, Madan Musuvathi Microsoft Research, Redmond

  2. Cover Definition • Cover operation is useful for simplifying a formula by discarding facts related to a set of variables • Given • A quantifier-free formula  in theory T • A set of symbols V • Cover(, V) is • The most-precise quantifier-free formula implied by  that does not involve V • e.g. Cover(y=f(a+v)–f(b+v), {v}) : (a=b) ) y=0

  3. Cover vs. Quantifier Elimination • Quantifier Elimination: Given a quantified formula, output a logically equivalent quantifier-free formula • 9V ´CoverT(,V) if T admits quantifier elimination • Some theories do not: theory of uninterpreted functions • Example: f(y) = 0 • Cannot say “0 is in the domain of y” without using quantifiers • Cover(,V) is the most-precise quantifier-free approximation to 9V 

  4. Applications • Strongest post-condition • Useful for abstract interpretation on logical formulas • Existential quantification of dead variables • SP(, x := e) = 9 x’ ([x’/x] Æ x = e[x’/x]) • Image computation • Useful for reachability analysis in symbolic model checking • Existential quantification of old state variables • Ri+1(S) = 9S’(Ri[S’/S] Æ T(S’,S)) Ç Ri(S)

  5. Applications • Procedure summaries • Existential quantification of local variables • Useful for interprocedural analysis • Interpolants • Suppose A ) B. Then I is the Interpolant(A,B) if • A ) I ) B • I only contains variables common to A and B • Cover(A, VA) is most precise Interpolant(A,B) • :Cover(:B, VB) is least precise Interpolant(A,B)

  6. Outline • Symbolic model checking using Cover • Cover algorithm for uninterpreted functions • Cover algorithm for the combination of uninterpreted functions and linear arithmetic

  7. Symbolic Model Checking Algorithm • I(S) : initial states, E(S) : error states • T(S’,S) : transition from old state S’ to new state S • R(S): reachable states • R0(S) = I(S) • Ri+1(S) = 9S’(Ri[S’/S] Æ T(S’,S)) ÇRi(S) • Error found if Rn+1(S) Æ E(S) is satisfiable

  8. Symbolic Model Checking Using Cover • I(S) : initial states, E(S) : error states • T(S’,S) : transition from old state S’ to new state S • R(S): reachable states • R0(S) = I(S) • Ri+1(S) = Cover(Ri[S’/S] Æ T(S’,S), S’) ÇRi(S)

  9. Symbolic Model Checking Using Cover • I(S) : initial states, E(S) : error states • T(S’,S) : transition from old state S’ to new state S • R(S): reachable states • R0(S) = I(S) • Ri+1(S) = Cover(Ri[S’/S] Æ T(S’,S), S’) ÇRi(S) • This algorithm can find false errors • As Cover over-approximates the set of reachable states

  10. Symbolic Model Checking Using Cover • I(S) : initial states, E(S) : error states • T(S’,S) : transition from old state S’ to new state S • R(S): reachable states • R0(S) = I(S) • Ri+1(S) = Cover(Ri[S’/S] Æ T(S’,S), S’) ÇRi(S) • Theorem: If the transition system is described using quantifier-free formulas, symbolic model checking using cover is sound and precise

  11. Outline • Symbolic model checking using Cover • Cover algorithm for uninterpreted functions • Cover algorithm for the combination of uninterpreted functions and linear arithmetic

  12. Cover Algorithm for Unary Uninterpreted Functions • Cover(, V) = Erase V from congruence closure of  • Example: Let  be x=f(v1) Æ y=f(v2) Æ v1 = v2 Cover(, {v1,v2}) is x=y f x f y v2 v1

  13. Cover Algorithm for Binary Uninterpreted Functions • The erasure technique does not work • Let  be x=f(a,v) Æ y=f(b,v) • Erasure(, {v}) is true • Cover(, {v}) is a=b ) x=y • Cover(, V) is: For all partitions E of congruence classes in  E ) Erasure(Æ E, V)

  14. a1 = b1Æ a2 = b1 ) y f a1 = b1Æ a2 = b2 ) y f x1 x1 y f x1 x2 y a1 = b2Æ a2 = b1 ) f f f x1 f x2 x1 a2 v a1 v y b1 a1 = b2Æ a2 = b2 ) f v x2 x2 x2 f b2 v Example Cover(, {v}) can be exponential in   Cover(,{v})

  15. Outline • Cover algorithm for linear arithmetic • Cover algorithm for uninterpreted functions • Cover algorithm for combination of theories

  16. Combining Cover Algorithms: Idea 1 CoverT1[ T2(1Æ2, V): Return CoverT1(1,V) Æ CoverT2(2,V) Fails on x=v1+1 Æ y=v2+1Æv1=f(z) Æ v2=f(z) Algorithm returns true Cover is x=y Solution: Share variable equalities

  17. Combining Cover Algorithms: Idea 2 CoverT1[ T2(1Æ2, V): E Ã Saturate(1,2) Return CoverT1(1ÆE,V) Æ CoverT2(2ÆE,V) Fails on v=x+1Æy=f(v) Algorithm returns true Cover is y=f(x+1) Solution: Share equalities between variables and “simple”terms

  18. Combining Cover Algorithms: Idea 3 CoverT1[ T2(1Æ2, V): E Ã Saturate(1,2) Return CoverT1(1ÆE,V) Æ CoverT2(2ÆE,V) Fails on x·v Æ v·yÆv=f(z,v) Algorithm returns x·y Cover is x·y Æ (x=y ) x=f(z,x)) Solution: Share conditional equalities

  19. Example Cover(y=f(a+v)–f(b+v), {v}) a=b ) v1=v2 v3 = f(v1) v4 = f(v2) v1 = a+v v2 = b+v y = v3-v4 a=b ) v3=v4 true a=b ) y=0

  20. Conclusion • Cover is the most-precise quantifier-free approximation to quantifier elimination • Cover algorithm for uninterpreted functions • Cover algorithm for combination of theories • Exchange equalities between variables and good terms • Exchange conditional equalities

More Related