1. 3-Valued Abstraction and 3-Valued Model-Checking

2. 2 Abstraction Ü Abstraction:  an effective technique to combat state explosion problem  approximate sets of concrete states by an abstract state  approximate sets of concrete transitions by an abstract transition Ü Using 2-valued logic (over-approximation)  False variables represent “unknown” value  True transitions represent possible behaviour r (s 0,1 )=T ⇒ r (s 0 )=T p (s 0,1 )=F ⇒ ? p¬qrp¬qr pq¬rpq¬r ¬p¬qr¬p¬qr s0s0 s1s1 s2s2 r pq¬rpq¬r s 0,1 s2s2

3. 3 Abstraction, Cont’d Ü Using 2-valued logic  False variables represent “unknown” value  True transitions represent possible behaviour AX (r  p)(s 0,1 )=T ⇒ AX (r  p)(s 0 )=T AX r (s 0,1 )=F ⇒ ? EX (  r   p)(s 0,1 )=F ⇒ EX (  r   p)(s 0 )=F EX r(s 0,1 )=T ⇒ ? p¬qrp¬qr pq¬rpq¬r ¬p¬qr¬p¬qr s0s0 s1s1 s2s2 r pq¬rpq¬r s 0,1 s2s2

4. 4 Abstraction, Cont’d Ü Soundness:  Only with respect to True universal properties  For existential properties – use under-approximation  For False properties:  play counter-example to determine whether spurious  Use counter-example-based abstraction refinement

5. 5 3-valued abstraction Ü Goals:  Reason about mixed properties  Not have to tell which counterexamples are spurious  Not have an increase in statespace, when compared to 2- valued  Use counterexample for abstraction refinement Ü Outline:  3-valued logic, properties, models, model-checking  3-valued abstractions  Abstraction refinement

6. 6 Logic: 3-valued Kleene logic Logic order Ü Properties:  F ⊑ M, M ⊑ T  A  B = min (A, B)  A  B = max (A, B)   T = F,  F = T,  M = M Ü Preserves:  Commutativity, associativity, idempotence, De Morgan laws Ü Does not preserve  Law of excluded middle: A  A= T (top)  Law of non-contradiction: A  A=  (bottom) T F M

7. 7 Note Ü 3-valued logic forms a lattice  Ordering ⊑ : less than or equal  Meet operation ⊓ : min  Join operation ⊔ : max  Negation : horizontal symmetry Ü This is an example of a quasi-boolean algebra Ü Equality and Identity are different!  a  b  a = b T F M

8. 8 Logic Ü Information order  M contains least amount of information  T, F – maximum amount of information  If one refines M – it can change to T or F or stay at M TF M

9. 9 Overview of MV-Model CheckingOverview of Model Checking Yes/No Answer Yes/No Answer SW/HW artifact SW/HW artifact Correctness properties Correctness properties Temporal logic Temporal logic Model of System Model of System Model Extraction Model Extraction Translation Model Checker Model Checker Correct? MV-Logic Answer MV-Logic Answer MV-Model Checker MV-Model Checker How correct?

10. 10 Multi-valued state machines: Xkripke structures Ü Extension of conventional state machines (Kripke structures)  variables take any value from the logic ( T, F, M )  transitions between states take any value from the logic  False transitions are not shown (by convention) Ü Example: pressed = T request = F pressed = T request = F pressed = M request = T T TM T

11. 11 Formally, Ü Kripke structures extended for MV case  M =  L is a quasi-boolean algebra (ℒ, ⊑, ⊓,⊔,  ), where ( ℒ, ⊑ ) is a lattice  S is a (finite) set of states, each with a unique name  A is a set of atomic propositions  s 0 is a unique initial state ( s 0  S )  I: S  A  ℒ is the interpretation function that assigns a logic value to each atomic proposition  R: S  S  ℒ is the function that assigns a logic value to each transition between states

12. 12 3-valued CTL Ü multi-valued extension of CTL  same syntax as CTL  plus constants from the logic ( T, M, F ) Ü semantics:  replace existential quantification by disjunction, universal quantification by conjunction, so (EX  ) (s) =  t  S s.t. ( R(s,t)   (t) )  t  S ( R(s,t)   (t) ) For all states s, (AX  ) (s) = (  EX(   )) (s) (EG  ) (s) =  (s)  (EX EG  ) (s) (AG  ) (s) = (  EF(   )) (s) Examples: AG (request -> AX pressed) ≟ AG (pressed \/ request) ≟  other operators are defined as in CTL: T F M pressed = T request = F pressed = T request = F pressed = M request = T T TM T

13. 13 Model-Checking Cont’d Ü Can a True property evaluate to M? Ü Answer:  Yes  AG (pressed \/  pressed) = M  Comes from law of excluded middle Ü Some terminology:  Compositional semantics  Evaluate each CTL operator, compose according to lattice rules  Thorough semantics [Bruns&Godefroid 00]  Property evaluates to M iff exists a refinement where it evaluates to T and a refinement where it evaluates to F.  $$ to evaluate

14. 14 Symbolic mv model-checking Ü Similar idea to classical model-checking  recursively go through the structure of XCTL property  encode sets of states symbolically  encode transition relation symbolically Ü Data structures  direct approach: MDDs  the number of terminal nodes and branching factor equal to number of values in logic  Example: x  y in 3-valued logic x yy FMT F F F M M M T TT  can use BDD vector …  or mixed approaches (MBTDDs, MTBDDs)

15. 15 Reduction to Classical Ü [Bruns&Godefroid’99]. Assumption: transition relation is classical  Move negation to level of atomic propositions  Create a positive and negative version of every atomic proposition  Let x = M.  Positive cut:  Set x and and  x to True  PosAnswer = check property  Negative cut:  Set x and and  x to False  NegAnswer = check property Ü If NegAnswer = PosAnswer (True or False)  Return this as answer Ü Else  Return Maybe

16. 16 Example p = T q = F z- =T z+=T p = T q = F z = F p = T q+ = T q- = T z = F Model Positive Cut [[p ∧¬q ∨z ]](s 0 ) = T p = T q = F z = M p = T q = F z = F p = T q = M z = F

17. 17 Example p = T q = F z- = F z = F p = T q = F z = F p = T q+ = F q- = F z = F Model Negative Cut p = T q = F z = M p = T q = F z = F p = T q = M z = F [[p ∧¬q ∨z ]](s 0 ) = F Therefore, the answer is M

18. 18 Example p = T q = F z- =T z+= T p = T q = F z = F p = T q+ = T q- = T z = F Model Positive Cut p = T q = F z = M p = T q = F z = F p = T q = M z = F [[EX (p ∧¬q ∨z )]](s 0 ) = T

19. 19 Example p = T q = F z- = F z = F p = T q = F z = F p = T q+ = F q- = F z = F Model Negative Cut p = T q = F z = M p = T q = F z = F p = T q = M z = F [[EX (p ∧¬q ∨z )]](s 0 ) = T Therefore, the answer is T

20. 20 Reduction to Classical (Take Two) Ü [Gurfinkel&Chechik 2003] Ü Assumptions:  States can be 3-valued, transition relation can be three- valued Ü Reduction steps  for True and Maybe, construct a cut formula equivalent to [[ ϕ ]](s)⊒j  logic: from mv CTL to restricted mv-logic with two-valued answers  model: unchanged  transform each cut to a classical model-checking problem  logic: from restricted mv-logic to classical CTL  model: from ΧKripke structure to classical Kripke structure

21. 21 Propositional Logic [[p ∧¬q ∨z ]](s 0 ) = M [[T ∧¬M ∨F ]](s 0 ) [[T ∧M ∨F ]](s 0 ) [[M ∨F ]](s 0 ) [[M ]](s 0 ) p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T

22. 22 Propositional Logic – the cut [[T ∧F ∨F ]](s 0 ) [[F ∨ F ]](s 0 ) F ‖p ∧¬q ∨z ‖(s 0 ) ⊒ M [[p ∧¬q ∨z ]](s 0 ) ⊒ T [[ ]](s 0 )(p ⊒ T)(¬q ⊒ T)(z ⊒ T)∧∨ T p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T

23. 23 Combining Results [[p ∧¬q ∨z ]](s 0 ) ⋣ T [[p ∧¬q ∨z ]](s 0 ) ⊒ M Therefore, [[p ∧¬q ∨z ]](s 0 ) = M T M F  p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T

24. 24 Propositional Logic – final step [[p + ∧q - ∨z + ]](s 0 ) [[T∧F ∨F]](s 0 ) [[F]](s 0 ) [[(p ⊒ T) ∧(¬q ⊒ T) ∨(z ⊒ T)]](s 0 ) Legend p + represents p ⊒ j p - represents ¬p ⊒ j p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T p+z-p+z- p+q-z-p+q-z- p+q-p+q-

25. 25 Existential Temporal Logic – the cut [[ EX ( p ∧¬q ∨z )]](s 0 ) ⊒ T ∨ t∈S R(s 0,t) ∧[[p ∧¬q ∨z ]](t) ⊒ T ∨ t∈S (R(s 0,t) ⊒ T) ∧([[p ∧¬q ∨z ]](t) ⊒ T) ∨ t∈S (R(s 0,t) ⊒ T) ∧[[(p ⊒T)∧(¬q ⊒T)∨(z ⊒T)]](t) [[ EX ⊒ T (( p ⊒T)∧(¬q ⊒T)∨(z ⊒T))]](s 0 ) p = T q = F z = M p = T q = F z = F p = T q = M z = F

26. 26 Existential Temporal Logic – final step [[EX ⊒T ( p + ∧q - ∨z + )]](s 0 ) [[ EX( p + ∧q - ∨z + )]](s 0 ) [[EX ⊒T ((p ⊒T)∧(¬q ⊒T)∨(z ⊒T))]](s 0 ) p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T p + z - p + q - z - p + q -

27. 27 Universal Temporal Logic – the cut [[AX(p ∧¬q ∨z )]](s 0 ) ⊒ T ∧ t∈S R(s 0,t) ⇒[[p ∧¬q ∨z ]](t) ⊒ T ∧ t∈S ¬R(s 0,t) ∨ [[p ∧¬q ∨z ]](t) ⊒ T ∧ t∈S ¬R(s 0,t) ⊒ T ∨ [[p ∧¬q ∨z ]](t) ⊒ T ∧ t∈S ¬(R(s 0,t) ⊒ M) ∨ [[p ∧¬q ∨z ]](t) ⊒ T ∧ t∈S (R(s 0,t) ⊒ M) ⇒ [[p ∧¬q ∨z ]](t) ⊒ T [[AX ⊒ M (( p ⊒T) ∧(¬q ⊒T) ∨(z ⊒T))]](s 0 ) p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T Dealing with negation  In 3-valued logic  ¬b ⊒ T iff ¬(b ⊒ M)  since ¬b ⊒ T iff b = F T M F

28. 28 Universal Temporal Logic – final step [[AX ⊒ M (( p ⊒T)∧(¬q ⊒T)∨(z ⊒T))]](s 0 ) [[AX ⊒ M (p + ∧q - ∨z + )]](s 0 ) [[AX(p + ∧q - ∨z + )]](s 0 ) p = T q = F z = M p = T q = F z = F p = T q = M z = F M M T T T p + z - p + q - z - p + q -

29. 29 Handling Mixed Modalities Ü The first reduction step does not change  [[AX EXp]](s 0 )⊒T is transformed into [[AX ⊒M EX ⊒T (p⊒T)]](s 0 ) Ü Problem with the second step  need a Kripke structure with two types of transitions  ⊒M for universal modality  ⊒T for existential modality Ü Solution  treat transitions labels as actions  convert the resulting Labeled Transition System into a Kripke structure Ü Disadvantage  introduces a new variable  size of the statespace doubles

30. 30 Summary of the Reduction Ü Multi-valued model-checking problem is reduced to several classical problems  one classical problem for True and one for Maybe  size of the formula does not change  atomic literals are changed to “plus” and “minus” versions  other parts remain unchanged  for universal and existential fragments  statespace of resulting Kripke structure is similar to the original  for formulas with both universal and existential modalities  statespace of the resulting Kripke structure is double of the original  formulas with fixpoint operators are handled similarly  (see Gurfinkel, Chechik, CONCUR’03)

31. 31 Abstraction  Abstraction Function  : S ! S’ S S’

32. 32 Abstraction Ü Using 3-valued logic  introduce new special value Maybe to stand for “unknown”  Formally:  [[v]] (a) = T iff  s   (a) [[v]](s) = T  [[v]] (a) = F iff  s   (a) [[v]](s) = F  [[v]] (a) = M iff  s   (a) [[v]](s) = T and  t   (a) [[v]](t) = F  Examples:  r (s 0,1 )=T ⇒ r (s 0 )=Tp (s 0,1 )=M p¬qrp¬qr pq¬rpq¬r ¬p¬qr¬p¬qr s0s0 s1s1 s2s2 s 0,1 s2s2 p=M q=F r=T p=T q=T r=F T F M

33. 33 Refresher: Over- and Under-approximations Ü M’ is an over-approximation of M, or M’ simulates M if  R  [Dams’97]: (t, t 1 )  R’ iff  s   (t) s.t.  s 1   (t 1 ) and (s, s 1 )  R Ü M’ is an under-approximation of M, or M simulates M’ if  R  [Dams’97]: (t, t 1 )  R’ iff  s   (t) s.t.  s 1   (t’) and (s, s 1 )  R

34. 34 Existential Abstraction (Over-Approximation) I I

35. 35 Universal Abstraction (Under-Approximation) I I

36. 36 3-Val Transition Relation Ü Let R(s,t) = T if R(s,t)  R   R  [Dams’97]: (t, t 1 )  R’ iff  s   (t) s.t.  s 1   (t’) and (s, s 1 )  R  Let R(s,t) = F if R(s,t) ∉ R   R  [Dams’97]: (t, t 1 )  R’ iff  s   (t) s.t.  s 1   (t 1 ) and (s, s 1 )  R Ü Else R(s,t) = M

37. 37 3-valued abstraction I I M T T M M M M M M T

38. 38 T F M Abstraction Ü Using 3-valued logic  introduce new special value Maybe to stand for “unknown” AX r (s 0,1 )=M EX r (s 0,1 )=T p¬qrp¬qr pq¬rpq¬r ¬p¬qr¬p¬qr s0s0 s1s1 s2s2 r pq¬rpq¬r s 0,1 s2s2 T M T p=M q=F r=T p=T q=T r=F

39. 39 Model Checking 3-Val abstract Models Ü Preservation Theorem M’ ⊨ φ ⇔ M ⊨ φ False counterexample cannot be spurious No need for simulation! Maybe counterexample requires refinement No guarantee is given about a “Maybe” answer Let φ be an arbitrary property (i.e., expressed in LTL, CTL, mu-calculus) and M’ is a 3-val abstraction of M

40. 40 3-Val Abstraction-Refinement Loop Check Counterexample Refine Model CheckAbstract M’, φ M, φ,  No Bug Pass Fail Bug Spurious ’’

41. 41 No spurious counterexamples, but abstraction can be too coarse I I Deadend states Bad States Failure State f T MM

42. 42 Refinement ’’ ’’ ’’ ’’ ’’ Refinement :  ’ ’’ ’’

43. 43 Other use of 3-valued logic Ü Algebra:  use three-valued algebra (Kleene)  intermediate value represents incomplete information or uncertainty T F M  compact representation for all possible refinements of this model  if a property is True/False on the partial model, it is True/False on a refined one  initial theory developed by Bruns & Godefroid, CAV’99 p= T q= F r= T p= M q= M r= F p= T q= M r= T s0s0 s2s2 s1s1 T T M M T Application: Most models are incomplete! Allows verification before specification is completed

44. 44 Summary Ü Abstraction  Effective tool for combating state explosion  Over-approximation – sound for true universal properties, otherwise – check if counterexample is feasible and then refine  Under-approximation – same for existential properties Ü 3-Valued Abstraction  Specified in 3-val Kleene logic  Allows reasoning about mixed-quantifier properties  No need to check if counter-example is spurious  Counterexample used for refinement Ü 3-Val Model-Checking  Reduces to two runs of classical model-checker  Or can be done directly, say, using MDDs

45. 45 Next topic: Ü Software model-checking  (and software model-checking with 3-valued logic)