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Abstraction for Falsification

Abstraction for Falsification. Thomas Ball Orna Kupferman Greta Yorsh. Microsoft Research, Redmond, US Hebrew University, Jerusalem, Israel Tel Aviv University, Israel. CAV’05. Abstraction for Verification. Goal: prove properties Sound abstraction for verification

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Abstraction for Falsification

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  1. Abstraction for Falsification Thomas Ball Orna Kupferman Greta Yorsh Microsoft Research, Redmond, US Hebrew University, Jerusalem, Israel Tel Aviv University, Israel CAV’05

  2. Abstraction for Verification • Goal: prove properties • Sound abstraction for verification • properties of abstract system hold for corresponding concrete system •  : C  A • if abstract state a satisfies property P then all concrete states represented by a satisfy P

  3. Abstraction for Verification • Goal: prove properties • Sound abstraction for verification • properties of abstract system hold for corresponding concrete system •  : C  A • a  A if a P then  c  C . (c)=a  c  P

  4. Falsification detect errors Abstraction for Verification • Goal: prove properties • Sound abstraction for verification • properties of abstract system hold for corresponding concrete system •  : C  A • a  A if a P then  c  C . (c)=a  c  P

  5. Falsification detect errors falsification Abstraction for Verification • Goal: prove properties • Sound abstraction for verification • errors of the abstract system exist in corresponding concrete system •  : C  A • a  A if a P then  c  C . (c)=a  c  P

  6. Falsification detect errors falsification Abstraction for Verification • Goal: prove properties • Sound abstraction for verification • errors of the abstract system exist in corresponding concrete system •  : C  A • a  A if a P then  c  C . (c)=a  c  P  c  C . (c)=a  c  P

  7. Motivation • An abstraction that is sound for falsification need not be sound for verification. • Existing frameworks for abstraction for verification • Modal Transition System (MTS) • MTS, PKS,KMTS - equivalent in expressive power [ Godefroid,Jagadessan – VMCAI’03 ] • can be too restrictive for falsification

  8. Main Results • New framework for abstraction • Ternary Modal Transition System (TMTS) • TMTS is stronger than MTS • Semantics of -calculus for TMTS • Weak reachability • TMTS with parameterized transitions gives tighter underapproximation • TMTS with assume-guarantee transitions for complete reasoning

  9. Modal Transition Systems Concrete Abstract a  (existential abstraction) MAY(a,a’)  may c, c’ . c  c’  (c) = a  (c’) = a’ a’   a MUST+(a,a’)  c. (c) = a  c’ . (c’) = a’  c  c’ must a’  [ T. Ball - FMCO’04 ] a MUST–(a,a’)  c’. (c’) = a’  c. (c) = a  c  c’ must onto underapproximation a’  must  may overapproximation  total underapproximation  must  may   must+ and must– are incomparable

  10. TMTS strictly more expressive than MTS MTS • may and must+ transitions • precision preorder is logically characterized by PML  ::= p | AX  |   |    TMTS • may, must+ and must– transitions • precision preorder is logically characterized by full-PML  ::= p | AX  | AY  |  |    • full-PML is strictly more expressive than PML [Pinter,Wolper - PODC’84] [Kupferman,Pnueli - LICS’95]

  11. TMTS: what does it buy us? • Verifying specifications with past operators • Reasoning about specifications in falsification setting • must+ for verification and must- for falsification • Tighter weak reachability in abstract system • combine must+ and must- along the path

  12. Semantics of -calculus for TMTS •  : C  A • (C, c1)   • [ (A, a1)   ] - the value of the -calculus formula  in state a1 of TMTS A

  13. Semantics of -calculus for TMTS • [ (A, a)   ] = T • for all concrete state c with (c) = a, (C, c)   • [ (A, a)   ] = T • there exists a concrete state c with (c) = a and (C, c)   • [ (A, a)   ] = F • for all concrete state c with (c) = a, (C, c)   • [ (A, a)   ] = F • there exists a concrete state c with (c) = a and (C, c)   • [ (A, a)   ] = M • there exist concrete states c and c’ such that (c) = (c’) = a and (C, c)   and (C, c’)   • [ (A, a)   ] = 

  14. Information Lattice Truth Lattice T T F   F

  15. Information Lattice Truth Lattice T T F M T  M T F F  F

  16. Semantics of -calculus for TMTS • [ (A, a)  1  2 ] • [ (A, a)  EX ] • [ (A, a)    ]

  17. [ (A, a)  1  2 ] = [ (A, a)  1 ] # [ (A, a)  2 ] 6-valued Semantics of 1 2

  18. 6-valued Semantics of 1 2

  19. 6-valued Semantics of 1 2

  20. 6-valued Semantics of 1 2

  21. Information Lattice Truth Lattice T T F M T  M T F F  F

  22. 6-valued Semantics of 1 2

  23. 6-valued Semantics of 1 2

  24. [ (A, a)  EX ] = Semantics of EX F if for all a’, if may(a,a’) then [(A, a’)  ] = F T if exists a’ s.t. must+(a,a’) and [(A,a’)  ] = T Tif exists a’ s.t. must–(a,a’) and [(A,a’)  ]  T  otherwise

  25. EX = T a  must– a’   = T c’ c  if [ (A, a)  EX ] = Tthen there exists c with (c) = a and c  EX • [ (A, a)  EX ] = T • exists a’ s.t. must–(a,a’) and [(A,a’)  ] = T • exists c’ such that (c’)=a’ and c’   • for all c’ with (c’)=a’ there is c with (c)=a such that cc’  EX  

  26. Semantics of  • The semantics of PML operators is monotonic • Least fixpoint operator can be computed by iterations from F is the usual way: • [(A,a)  Z . (Z) ] = [ (A, a)  *(F) ]

  27. x > 6 ... 7 8 9  x:=x–3 x > 6 ... 7 8 9  Semantics of -calculus for TMTS • The 6-valued semantics is at least as precise as the standard 3-valued semantics of -calculus for MTS • [(A,a)  ] =  • 3-valued abstraction refinement of must+ transitions [Shoham,Grumberg – CAV’03] adapt for must- • Hypermust transitions • [Larsen,Xinxin-LICS‘90] [Shoham,Grumberg – CAV’04] • adapt for must– • MTS with hypermust+ is incomparable with TMTS x = 7 x = 10  EX(x>6) = ?  EX(x>6) =T  EX(x>6) F  EX(x>6) T must – may

  28. Semantics of -calculus for TMTS • The 6-valued semantics is at least as precise as the standard 3-valued semantics of -calculus for MTS • [(A,a)  ] =  • 3-valued abstraction refinement of must+ transitions [Shoham,Grumberg – CAV’03] adapt for must- • Hypermust transitions • [Larsen,Xinxin-LICS‘90] [Shoham,Grumberg – CAV’04] • adapt for must– • MTS with hypermust+ is incomparable with TMTS

  29. c a  a’  c’ Weak Reachability • a’ is weakly-reachable from a • c, c’ . (c)=a  (c’)=a’  c * c’ initial state error trace error state Related to testing

  30. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– may L2: TF L3: FT L2: FF may must– must– L4: TF L4: FT L4: FF

  31. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– may L2: TF L3: FT L2: FF may must– must– L4: TF L4: FT L4: FF x = 5

  32. Underapproximation of Weak Reachability • if [must+]*(a,a’) then a’ is weakly reachable from a • Arbitrary combinations of must+ and must– transitions do not preserve weak reachability • Find a tighter underapproximation of weak-reachability

  33. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: x = 2  x = 6  x = 9  x = 5  Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– may must – ? must + ? L2: TF L3: FT L2: FF may must– must– L4: TF L4: FT L4: FF

  34. Underapproximation of Weak Reachability • if [must+]*(a,a’) then a’ is weakly reachable from a • Arbitrary combinations of must+ and must– transitions do not preserve weak reachability • Find a tighter underapproximation of weak-reachability

  35. Observations • a3 is weakly reachable from a1 if there exists a2 such that must–(a1,a2) and must+(a2,a3) • Onto nature of must– is preserved by [must-]* • Total nature of must+ is preserved by [must+]* a1  must– a2  must+ a3  [T.Ball – FMCO’04]

  36. a1  [must–]* a2 [must+]* a3  Underapproximation If there exists a1, a2, a3 such that [must–]*(a1,a2) and [must+]*(a2,a3) then a3 is weakly-reachable from a1 [T.Ball – FMCO’04]

  37. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– may L2: TF L3: FT L2: FF may must– must– L4: TF L4: FT L4: FF

  38. MUST+ ? ( total from a? )  MUST– ? ( onto a’ ?) Parameterized Transitions NO a  NO a’  MAY

  39.  MUST+() c. (c) = a  c    c’ . (c’) = a’  c  c’ total from   a MUST–()   c’. (c’) = a’  c’    c. (c) = a  c  c’ must–() a’ onto     Parameterized Transitions a  must+() a’  if  is TRUE then must+() is must+ and must–() is must–

  40. Observation a1 • a3 is weakly reachable from a1 if there exists a2 such that • must–(1)(a1,a2) • must+(2) (a2,a3) • 1 2 a2 is satisfiable  must–(1) a2 1 2  must+(2) a3 

  41. Observation a1 • a3 is weakly reachable from a1 if there exists a2 such that • must–(1)(a1,a2) • must+(2) (a2,a3) • 1 2 a2 is satisfiable • Strongest parameters 1 and2  must–(1) a2 1 2  must+(2) a3 

  42. MUST– ( SP (s,a) ) a  s c’. (c’) = a’  c’    c. (c) = a  c  c’ a’ if must–() then a  (  SP(s,a))   Strongest Parameters MUST+ ( WP(s,a’) )  a  s c. (c) = a  c    c’ . (c’) = a’  c  c’ if must+() then a  (  WP(s,a’)) a’  Generated automatically as part of the construction of TMTS

  43. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– may L2: TF L3: FT L2: FF SP(x:=x+3, x<6) = x < 9 WP(x:=x-3, x<6) = x < 9 may must– must– L4: TF L4: FT L4: FF

  44. L0: if x<6 then L1: x:= x + 3 L2: if x > 7 then L3: x :=x – 3 L4: Example Predicates: (x < 6) (x > 7) x<6 x>7 (x=6)(x=7) L0: FT L0: FF L1: TF must– must– must–(x<9) L2: TF L3: FT L2: FF must+(x<9) SP(x:=x+3, x<6) = x < 9 WP(x:=x-3, x<6) = x < 9 must– must–  must– (x < 9) L4: TF L4: FT L4: FF  must+ (x < 9)

  45. [must–]* must–(1) must+(2) [must+]* Tighter Underapproximation a1  If there exists a1,...,a5 s.t. [must–]*(a1,a2) must–(1)(a2,a3) must+(2) (a3,a4) [must+]*(a4,a5) 1 2 a3 is satisfiable then a5 is weakly-reachable from a1 a2  a3 1 2  a4  a5 

  46. Complete Reasoning • a’ is reachable by a certain sequence of abstract transitions from a • a’ is weakly-reachable from a • Assume-guarantee transitions • another type of parameterized transitions: <> must+ <’>

  47. < > MUST–<  ’ > a   c’. (c’) = a’  c’   ’  c . (c) = a  c    c  c’ <>must–<‘ > a’   ’ Assume-Guarantee Transitions   < > MUST+ <  ’> a  c. (c) = a  c    c’ . (c’) = a’  c’   ’  c  c’ <>must+<‘ > a’   ’ Which  and ’ predicates do we need?

  48. a1  s1 a2  s2 a3 3 3  s3 a4  s4 a5  The idea... 1 = a1 2 = SP(s1, 1)  a2 3 = SP(s2, 2)  a3 <1>must– <2> <2>must– <3> 3 = WP(s3,4)  a3 4 = WP(s4,5)  a4 5 = a5 <3>must+ < 4> 3 3 is satisfiable <4>must+ < 5>

  49. Assume-guarantee transitions • Complete Reasoning about Weak Reachability • a’ is reachable by a certain sequence of assume-guarantee transitions from a • a’ is weakly-reachable from a • Finding right parameters ~ computing loop invariants

  50. [must–] * [must+]* [must–] * must–(1) must+(2) [must+]* Weak Reachability: Summary • Previous work [T.Ball – FMCO’04]: • Parameterized transitions • Assume-guarantee transitions • complete reasoning

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