Model Checking Algorithms

Transcription

1 Model Checking Algorithms Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan November 14, 2018 Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

2 Outline 1 Model-checking algorithms The CTL model-checking algorithm CTL model checking with fairness The LTL model-checking algorithm 2 The fixed-point characterisation of CTL Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

3 The Model Checking Problem Let M = (S,, L) be a model, s S a state, and φ a temporal logic formula. The model checking problem is to decide whether M, s φ holds. We will discuss two model checking algorithms: one for LTL, the other for CTL. These algorithms help us understand basic principles of various verification tools (such as NuSMV). NuSMV does not implement the algorithms we discuss here. Yet the basic ideas are not very different. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

4 Outline 1 Model-checking algorithms The CTL model-checking algorithm CTL model checking with fairness The LTL model-checking algorithm 2 The fixed-point characterisation of CTL Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

5 The CTL Model Checking Algorithm I Let us first consider deciding whether M, s 0 φ where M is a finite transition system and φ a CTL formula. That is, M = (S,, L) with S a finite set of states. There are algorithms that solve the model checking problem for certain infinite transition systems. Our algorithm in fact computes all states that satisfy the top CTL formula. That is, it computes the set {s S M, s φ}. After computing all states satisfying the given CTL formula, the model checking problem is solved easily. We simply check if s0 {s S M, s φ}. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

6 The CTL Model Checking Algorithm II In our algorithm, we only consider temporal connectives {EX, AF, EU}. {EX, AF, EU} is adequate. For each state, we label it with subformulae of the given CTL formula. A state satisfies all subformulae in its label. Start from smallest subformulae; the algorithm works on each subformula until the given CTL formula. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

7 The CTL Model Checking Algorithm III Input: M = (S,, L) : a mode; φ : a CTL formula Output: {s S M, s φ} foreach subformula ψ of φ do switch ψ do case : do continue; case p: do label all s with p if p L(s); case ψ 1 ψ 2 : do label all s with ψ if it is lablled with ψ 1, ψ 2 ; case ψ 1 : do label all s with ψ if it is not labelled with ψ 1 ; case EXψ 1 : do label all s with ψ if one of its successors is labelled with ψ 1 ; case AFψ 1 : do label all s with ψ if it is labelled with ψ 1, or all successors of s are labeleed with ψ until no change ; case E[ψ 1 U ψ 2 ]: do label all s with ψ if it is labelled with ψ 2, or s is labelled with ψ 1 and one of its successors is labelled with ψ until no change ; Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

8 The CTL Model Checking Algorithm IV Let φ be the number of connectives in φ. Let M = (S,, L) be a transition system. The complexity of the algorithm is O( φ S ( S + )). There are O( φ ) subformulae. For AFψ 1 and E[ψ 1 U ψ 2 ], each iteration takes O( S + ) steps; there are at most O( S ) iterations. We now present the pseudo algorithm SAT(M, φ). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

9 SAT(M, φ) switch φ do case : do return S ; case : do return ; case p: do return {s S φ L(s)} ; case φ 1 : do return S SAT(M, φ 1 ) ; case φ 1 φ 2 : do return SAT(M, φ 1 ) SAT(M, φ 2 ) ; case EXφ 1 : do return SAT EX (M, φ 1 ) ; case E[φ 1 U φ 2 ]: do return SAT EU (M, φ 1, φ 2 ) ; case AFφ 1 : do return SAT AF (M, φ 1 ) ; Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

10 pre (M, Y ) and pre (M, Y ) In order to give the pseudo algorithm for SAT EX (M, φ), SAT EU (M, φ, ψ), and SAT AF (M, φ), we need two functions: pre (M, Y ) pre (M, Y ) = {s S there exists s (s s and s Y )} = {s S for all s (s s implies s Y )} pre (M, Y ) consists of states whose successors intersect Y is not empty. pre (M, Y ) consists of states whose successors are contained in Y. Observe that pre (M, Y ) = S pre (M, S Y ) Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

11 SAT EX (M, φ) X SAT(M, φ); Y pre (M, X ); return Y ; Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

12 SAT AF (M, φ) Y SAT(M, φ); repeat X Y ; Y Y pre (M, Y ); until X = Y ; return Y ; Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

13 SAT EU (M, φ, ψ) W SAT(M, φ); Y SAT(M, ψ); repeat X Y ; Y Y (W pre (M, Y )); until X = Y ; return Y ; Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

14 A More Efficient Model Checking Algorithm Using backward breadth-first search, we can in fact do better. We use the adequate set {EX, EG, EU} instead. Observe that EXψ 1 and E[ψ 1 U ψ 2 ] can be labelled in time O( S + ) if we perform backward breadth-first search. For the case EGψ 1, Consider the subgraph with states satisfying ψ1 ; Find maximal strongly connected components (SCC s) in the subgraph; Use backward breadth-first search on the subgraph to find states that can reach an SCC. The new algorithm takes time O( φ ( S + )). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

15 The State Explosion Problem Although the complexity of CTL model checking algorithm is linear in the number of states, the number of states can be exponential in the number of variables and the number of components of the system. Adding a Boolean variable can double the number of states. This is called the state explosion problem. Lots of researches try to overcome the state explosion problem. Efficient data structures. Abstraction. Partial order reduction. Induction. Compositional reasoning. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

16 Outline 1 Model-checking algorithms The CTL model-checking algorithm CTL model checking with fairness The LTL model-checking algorithm 2 The fixed-point characterisation of CTL Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

17 Fairness I We often simplify the system when we build a model for it. This is called abstraction. Abstraction sometimes introduces unrealistic model behaviors. For instance, a process may stay in its critical section forever. Such unrealistic behaviors may disprove intended properties. For instance, it is possible that the other process won t get to its critical section forever. In order to consider realistic model behaviors, we impose fairness constraints. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

18 Fairness II We consider only fair computation paths specified by fair constraints. Instead of standard path quantifiers A (for all) and E (for some), we use their variants A C (for all fair paths) and E C (for some fair paths). As an example, we can ask a process will eventually be permitted to enter its critical section if it requests so along all fair paths. Clearly, we have to slightly modify the CTL model checking algorithm. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

19 Fair Computation Paths Definition Let C = {ψ 1, ψ 2,..., ψ n } be fairness constraints. A computation path s 0 s 1 is fair with respect to C if for every i there are infinitely s j s such that s j ψ i. We write A C and E C for the path quantifers A and E restricted to fair paths. For example, M, s 0 A C Gφ if φ holds in every state along all fair paths. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

20 CTL Model Checking Algorithm with Fairness Constraints Let C = {ψ 1, ψ 2,..., ψ n } be a set of fairness constraints. We consider the adequate set {E C U, E C G, E C X}. Observe that E C [φ U ψ] E[φ U (ψ E C G )] E C Xφ EX(φ E C G ). Note that a computation path is fair iff all its suffixes are fair. M, s E C G ensures that s is on a fair path. Since E C U and E C X can be reduced to E C G, it remains to compute {s S M, s E C Gφ}. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

21 Computing {s S M, s E C Gφ} It turns out that the algorithm for computing E C G is similar for computing EG. Let C = {ψ 1, ψ 2,..., ψ n } be a set of fairness constraints. To compute {s S M, s E C Gφ}, do the following: Consider the subgraph with states satisfying φ; Find maximal strongly connected components (SCC s) in the subgraph; Remove an SCC if it does not contain a state satisfying ψi for some i. The remaining SCC s are fair SCC s; Use backward breadth-first search on the subgraph to find states that can reach a fair SCC. The complexity of the algorithm with fairness constraints is O( C φ ( S + )). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

22 Fairness in NuSMV In NuSMV two different types of fair paths can be specified. Justice Let C = {p1, p 2,..., p n } be a set of atomic formulae. A path s0 s 1 satisfies the justice constraint C if for every i, there are infinitely s j s such that s j p i. NuSMV uses the keywords FAIRNESS or JUSTICE. Compassion Let C = {(p1, q 1 ), (p 2, q 2 ),..., (p n, q n )} be a set of pairs of atomic formulae. A path s0 s 1 satisfies the compassion constraint C if for every i, there are infinitely s j s such that s j p i then there are infinitely s j s such that s j q i. NuSMV uses the keyword COMPASSION. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

23 Outline 1 Model-checking algorithms The CTL model-checking algorithm CTL model checking with fairness The LTL model-checking algorithm 2 The fixed-point characterisation of CTL Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

24 LTL Model Checking Algorithm The CTL model checking algorithm is rather straightforward. It labels each state by satisfied subformulae. It works because CTL formulae specify state properties. LTL formulae, on the other hand, specify path properties. Labelling states no longer work. We need a formal model for paths. Automata theory is required! Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

25 Paths and Traces Let M = (S,, L) be a model. Recall that a path s 0 s 1 is a sequence of states such that s i s i+1 for every i 0. A trace is a sequence of valuations of propositional atoms. The trace of a path s 0 s 1 is L(s 0 )L(s 1 ). Recall that L S 2 Atoms. L(si ) is the set of propositional atoms that hold in s i. L(si ) hence is a valuation of propositional atoms. Note that different paths may have the same trace. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

26 Basic Ideas Let M = (S,, L) be a model, s S, and φ an LTL formula. We check whether M, s φ holds as follows. 1 Construct an automaton A φ. A φ accepts the traces satisfying φ; 2 Construct the combination of the automaton A φ and M. Each path of the combination is a path of A φ and also a path of M. 3 Check if there is an accepting path starting from a combined state including s. Such a path is a path of M from s. Moreover its trace satisfying φ. If an accepting path is found, we report M, s / φ ; Otherwise, M, s φ. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

27 Illustration I init (a) := 1; init (b) := 0; next (a) := case!a : 0; b : 1; 1 : { 0, 1 }; esac; next (b) := case a & next (a) :!a : 1; 1 : { 0, 1 }; esac;!b; s 1 s 3 ab ab ab ab s 2 s 4 M = (S,, L) with Atoms = {a, b} and φ = (a U b) Does M, s 3 φ hold? Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

28 Illustration II q 1 q 2 abψ abψ A trace t is accepting if there is a path π whose trace is t such that an accepting state occurs infinitely often. q 3 abψ abψ abψ q 4 {a}{a}{a, b}{a}{a} is accepting. {a}{a} is not. q 3 φ = (a U b), ψ = (a U b) and A ψ Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

29 Illustration III (s 1, q 1 ) abψ abψ (s 3, q 3 ) abψ (s 3, q 3 ) (s 2, q 2 ) abψ abψ (s 4, q 4 ) (s 3, q 3 )(s 2, q 2 ) is accepting. Hence M, s 3 / (a U b) because of the path s 3 s 2. In fact, s 3 s 4 s 3 s 4 is another counterexample. Combination of M and A ψ Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

30 Construct A φ I Let φ be an LTL formula. We want to construct an automaton A φ such that A φ accepts precisely traces on which φ holds. We assume φ contains only the temporal connectives U and X. Recall that {U, X} is adequate. Define the closure C(φ) of an LTL formula φ by C(φ) = {ψ, ψ ψ is a subformula of φ} where we identify ψ and ψ. Example: C(a U b) = {a, b, a, b, a U b, (a U b)}. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

31 Construct A φ II Let φ be an LTL formula. A maximal subset q of C(φ) satisfies the following: for all (non-negated) ψ C(φ), either ψ q or ψ q; ψ1 ψ 2 q iff ψ 1 q or ψ 2 q; Conditions for other Boolean connectives are similar; If ψ1 U ψ 2 q, then ψ 2 q or ψ 1 q; and If (ψ1 U ψ 2 ) q, then ψ 2 q. The states of A φ are the maximal subsets of C(φ). That is, {q C(φ) q is maximal}. Informally, ψ q means ψ C is true at state q. The initial states of A φ are those containing φ. Formally, {q C(φ) q is maximal and φ q}. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

32 Construct A φ III The transition relation δ of A φ is defined as follows. (q, q ) δ if Xψ q implies ψ q ; Xψ q implies ψ q ; ψ 1 U ψ 2 q and ψ 2 / q imply ψ 1 U ψ 2 q ; and (ψ1 U ψ 2 ) q and ψ 1 q imply (ψ 1 U ψ 2 ) q. Informally, ψ q enforces certain ψ q. Recall the definition of maximal subsets of C(φ). Observe that ψ1 U ψ2 ψ2 (ψ1 X(ψ1 U ψ2) (ψ 1 U ψ 2 ) ψ 2 ( ψ 1 X (ψ 1 U ψ 2 )). Consider C(a U b) = {a, a, b, b, a U b, (a U b)}. Let q = {a, b, a U b} be a maximal subset of C(a U b). We have (q, q) δ, the transition relation of A aub. Informally, a U b q means a U b holds at all traces from q. Apparently, qq does not satisfy a U b. We define acceptance conditions to disallow such traces. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

33 Construct A φ IV Let χ 1 U ψ 1,..., χ k U ψ k be all formulae of this form in C(φ). The acceptance condition of A φ is as follows. A state q of A φ is i-accepting if { (χ i U ψ i ), ψ i } q. A path π is accepted if for every 1 i k, π has infinitely many i-accepting states. A state can be i-accepting for every 1 i k. To see why it works, consider a path π = q 0 q 1 with only finitely many i-accepting states. Hence for some h, we have π h = q h q h+1 with q j { (χ i U ψ i ), ψ i } = for every j h. By the maximality of q j, we have {χ i U ψ i, ψ i } q j for every j h. The path π is precisely what we wan to eliminate. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

34 Construct A φ V q 1 abψ abψ q 2 C(a U b) = {a, a, b, b, aub, (aub)}. Maximal subsets of C(a U b): q 3 abψ q 3 abψ abψ q 4 q 1 = { a, b, (a U b)} q 2 = { a, b, a U b} q 3 = {a, b, a U b} q 3 = {a, b, (a U b)} q 4 = {a, b, a U b} Accepting states are {q i (a U b) q i or b q i }. ψ = a U b and A ψ Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

35 Construct A φ VI q 1 q 2 (a U b), ( a U b), a, b, ψ a U b, ( a U b), a, b, ψ q 5 a U b, a U b, a, b, ψ a U b, a U b, a, b, ψ q 6 For a U b, the accepting states are {q 1, q 3, q 4, q 5, q 6 }. For a U b, the accepting states are {q 1, q 2, q 3, q 5, q 6 }. q 3 (a U b), ( a U b), a, b, ψ q 4 (a U b), a U b, a, b, ψ ψ = (a U b) ( a U b) and A ψ Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

36 LTL Model Checking with Fairness Constraints Our CTL model checking algorithm is slightly modified to check CTL properties with fairness constraints. However, it is not necessary for LTL model checking. Consider, for example, checking an LTL formula φ with a justice constraint ψ. Recall that a justice constraint ψ considers only paths that ψ occurs infinitely often. We would like to check if φ holds on all fair paths. This is equivalent to checking GFψ φ. because LTL can specify justice constraints. Hence the LTL model checking algorithm works even if there are fairness constraints. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

37 NuSMV LTL Model Checking Algorithm We can in fact implement the LTL model checking algorithm by the CTL model checking algorithm with justice constraints. Let M be a NuSMV model and φ an LTL formula. Here is how it works: Construct A φ as a NuSMV model. Construct M A φ. Check EG with justice constraint (χ U ψ) ψ for each χ U ψ in φ. The NuSMV LTL model checking algorithm uses justice constraints to search paths accepted by M A φ. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

38 Outline 1 Model-checking algorithms 2 The fixed-point characterisation of CTL Monotone functions The correctness of SAT EG The correctness of SAT EU Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

39 The CTL Model Checking Algorithm Revisited I We have presented a CTL model checking algorithm. Let M = (S,, L) be a transition system. The algorithm computes the set [φ] for any CTL formula φ, where We use the following equations: [φ] = {s S M, s φ}. [p ] = {s S p L(s)} [ ] = [ φ] = S [φ] [φ ψ ] = [φ] [ψ ] [EXφ] = pre (M, [φ]) [AFφ] = [φ] pre (M, [AFφ]) [E[φ U ψ]] = [ψ ] ([φ] pre (M, [E[φ U ψ]])) The last two recursive equations are puzzling. Circular definitions are always problematic! Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

40 The CTL Model Checking Algorithm Revisited II To simplify our presentation, we will use a different adequate set of temporal connectives {EX, EG, EU}. To get rid of pre (, ). Hence we use the following puzzling equations: [EGφ] = [φ] pre (M, [EGφ]) [E[φ U ψ]] = [ψ ] ([φ] pre (M, [E[φ U ψ]])) We will explain them by fixed-point theory. The theory will also establish the correctness of the algorithm. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

41 Outline 1 Model-checking algorithms 2 The fixed-point characterisation of CTL Monotone functions The correctness of SAT EG The correctness of SAT EU Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

42 Monotone Functions Definition Let S be a set of states and F 2 S 2 S. 1 F is monotone if X Y implies F (X ) F (Y ) for every X, Y S; 2 X S is a fixed point of F if F (X ) = X. Examples: let S = {s 0, s 1 }. F (Y ) = Y {s0 }. F (Y ) is monotone. {s 0 } and {s 0, s 1 } are fixed points of F (Y ). G(Y ) = { {s 1} if Y = {s 0 }. G(Y ) is not monotone. G(Y ) has no {s 0 } otherwise fixed points. Let F 2 S 2 S. We write F i (X ) for the expression i F (F F (X ) ). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

43 Fixpoint Theorem I Theorem Let S be a set with S = n. If F 2 S 2 S is a monotone function, F n ( ) is the least fixed point of F and F n (S) is the greatest fixed point of F (by set inclusion order). Proof. Since F ( ) and F is monotone, F ( ) F 2 ( ). More generally, F ( ) F 2 ( ) F n ( ) F n+1 ( ). Since S = n and F is monotone, there is 1 k n that F k ( ) = F k+1 ( ). Thus F n ( ) is a fixed point of F. Suppose H = F (H) is a fixed point of F. Since H, F ( ) F (H) = H. F 2 ( ) F (H) = H. Hence F i ( ) H for every i. Particularly, F n ( ) H. The case for the greatest fixed point is similar. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

44 Fixpoint Theorem II Knaster-Tarski theorem in fact shows the least and greatest fixed points exist for any (not necessarily finite) set. Our fixpoint theorem is a special case of Knaster-Tarski theorem. The fixpoint theorem shows how to compute least and greatest fixed points for monotone functions over finite sets. We will show [EGφ] and [E[φ U ψ]] are in fact greatest and least fixed points of certain monotone functions respectively. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

45 Outline 1 Model-checking algorithms 2 The fixed-point characterisation of CTL Monotone functions The correctness of SAT EG The correctness of SAT EU Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

46 Computing [EGφ] Recall that EGφ φ EXEGφ. Hence [EGφ] = [φ] pre (M, [EGφ]). [EGφ] is a fixed point of F (X ) = [φ] pre (M, X ). We will show that [EGφ] is in fact a greatest fixed point of F. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

47 [EGφ] as a Greatest Fixed Point Theorem Let M = (S,, L) be a model with S = n and F (X ) = [φ] pre (M, X ). Then F is monotone, [EGφ] is the greatest fixed point of F, and [EGφ] = F n (S). Proof. Let X, Y S and X Y, and s F (X ). Then s [φ] and there is an s such that s s and s X Y. That is, s F (Y ) and hence F (X ) F (Y ). F is monotone. Since [EGφ] = F ([EGφ]), [EGφ] is a fixed point of F. It remains to show that [EGφ] is the greatest fixed point of F. Consider any X S with X = F (X ). Let s 0 X. Then s 0 F (X ) = [φ] pre (M, X ). s 0 [φ] and there is an s 1 with s 0 s 1 such that s 1 X. By induction, we have a path s 0 s 1 such that s i [φ] for s 0. Hence s 0 [EGφ]. Therefore X [EGφ] for every fixed point X of F. The greatest fixed point of F is F n (S). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

48 SAT EG (M, φ) I Y SAT(φ); X ; repeat X Y ; Y Y pre (M, Y ); until X = Y ; return (Y ); Note that SAT EG (M, φ) does not apply the previous theorem exactly. Recall that F (X ) = [φ] pre (M, X ). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

49 SAT EG (M, φ) II By the theorem, we would have Y 0 = F 0 (S) = S Y 1 = F (Y 0) = [φ] pre (M, Y 0) = [φ] Y 2 = F (Y 1) = Y 1 pre (M, Y 1) Y 3 = F (Y 2) = Y 1 pre (M, Y 2) where Y 0 Y 1 Y 2 SAT EG(M, φ) on the other hand computes Y 0 = [φ] = Y 1 Y 1 = Y 0 pre (M, Y 0) = Y 1 pre (M, Y 1) = Y 2 Y 2 = Y 1 pre (M, Y 1) = Y 1 pre (M, Y 1) pre (M, Y 2) = Y 1 pre (M, Y 2) = Y 3 Y 3 = Y 2 pre (M, Y 2) = Y 1 pre (M, Y 2) pre (M, Y 3) = Y 1 pre (M, Y 3) = Y 4 Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

50 Outline 1 Model-checking algorithms 2 The fixed-point characterisation of CTL Monotone functions The correctness of SAT EG The correctness of SAT EU Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

51 Computing [E[φ U ψ]] Recall that E[φ U ψ] ψ (φ EXE[φ U ψ]). Hence [E[φ U ψ]] = [ψ ] ([φ] pre (M, [E[φ U ψ]])). [E[φ U ψ]] is a fixed point of G(X ) = [ψ ] ([φ] pre (M, X )). We will show that [E[φ U ψ]] is in fact a least fixed point of G. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

52 [E[φ U ψ]] as a Least Fixed Point Theorem Let M = (S,, L) be a model with S = n and G(X ) = [ψ ] ([φ] pre (M, X )). Then G is monotone, [E[φ U ψ]] is the least fixed point of G, and [E[φ U ψ]] = G n ( ). Proof. Since pre (M, X ) is monotone, G is monotone. Since G n ( ) is the least fixed point of G, it remains to show that [E[φ U ψ]] = G n ( ). Recall M, s E[φ U ψ] if for some path s 0(= s) s 1 there is i 0 that M, s i ψ and for every 0 j < i we have M, s j φ. Observe that G 1 ( ) = [ψ ] ([φ] pre (M, )) = [ψ ]. s G 1 ( ) iff M, s E[φ U ψ] by taking i = 0. Similarly, G 2 ( ) = G(G 1 ( )) = [ψ ] ([φ] pre (M, G 1 ( ))). That is, s G 2 ( ) iff M, s E[φ U ψ] by taking i = 0, 1. By induction, one can show s G k ( ) iff M, s E[φ U ψ] by taking i = 0,..., k 1. Hence [E[φ U ψ]] = G i ( ). i N Now recall that G 0 ( ) G 1 ( ) G 2 ( ) and G n ( ) is a fixed point. We have i N G i ( ) = G n ( ). That is, [E[φ U ψ]] = G n ( ). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

53 SAT EU (M, φ, ψ) Revisited I W SAT(M, φ); X S; Y SAT(M, ψ); repeat X Y ; Y Y (W pre (M, Y )); until X = Y ; return Y ; Note again that SAT EU (M, φ, ψ) does not exactly follow the theorem. Recall G(X ) = [ψ ] ([φ] pre (M, X )). Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

54 SAT EU (M, φ, ψ) Revisited II By the theorem, we would have Y 0 = G 0 ( ) = Y 1 = G(Y 0) = [ψ ] ([φ] pre (M, Y 0)) = [ψ ] Y 2 = G(Y 1) = [ψ ] ([φ] pre (M, Y 1)) = Y 1 ([φ] pre (M, Y 1)) Y 3 = G(Y 2) = Y 1 ([φ] pre (M, Y 2)) where Y 0 Y 1 Y 2 SAT EU(M, φ, ψ) on the other hand computes Y 0 = [ψ ] = Y 1 Y 1 = Y 0 ([φ] pre (M, Y 0)) = Y 1 ([φ] pre (M, Y 1)) = Y 2 Y 2 = Y 1 ([φ] pre (M, Y 1)) = Y 1 ([φ] pre (M, Y 1)) ([φ] pre (M, Y 2)) = Y 1 ([φ] pre (M, Y 2)) = Y 3 Y 3 = Y 2 ([φ] pre (M, Y 2)) = Y 1 ([φ] pre (M, Y 2)) ([φ] pre (M, Y 3)) = Y 1 ([φ] pre (M, Y 3)) = Y 4 Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

55 Example I s 0 q s 2 s 1 s 4 p q s 3 Compute [EFp ]. Since [EFp ] = [E[ U p]], consider G(X ) = [p ] ([ ] pre (M, X )) = {s 3 } pre (M, X ). G 1 ( ) = {s 3 }, G 2 ( ) = G({s 3 }) = {s 3, s 1 }, G 3 ( ) = G({s 1, s 3 }) = {s 3, s 0, s 2, s 1 }, G 4 ( ) = G({s 0, s 1, s 2, s 3 }) = {s 3, s 0, s 2, s 1 } = G 3 ( ). Hence [EFp ] = {s 0, s 1, s 2, s 3 }. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56

56 Example II s 0 q s 2 s 1 s 4 p q s 3 Compute [EGq ]. Consider F (X ) = [q ] pre (M, X ) = {s 0, s 4 } pre (M, X ). F 1 (S) = {s 0, s 4 }. F 2 (S) = F ({s 0, s 4 }) = {s 0, s 4 } {s 3, s 0, s 2, s 4 } = {s 0, s 4 } = F 1 (S). Hence [EGq ] = {s 0, s 4 }. Bow-Yaw Wang (Academia Sinica) Model Checking Algorithms November 14, / 56