Browne M. C., Clarke E. M. and Grumberg O. [1988], `Characterizing nite Kripke structures in propositional temporal logic', Theoretical Computer Science 59(1-2), 115-131.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by those in a subset N (P ) N (P ) according to a notion of simulation that preserves properties. Once this is achieved, we can prove that P satis es by showing that the networks of N (P ) satisfy . We now de ne a stuttering version of simulation, similar to stuttering bisimulation [BCG88] Since in the simulation we consider all actions of the simulated network T will have a corresponding sequence of actions in the simulating network T , we do not have to consider stuttering in T , which simpli es the de nition. Since we only require fair paths to satisfy a property, we ....

Browne, M. C., Clarke, E. and Grumberg, O. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59: 115-131, 1988.

....under . Hence, if t i = t i , for 1 i r, M = M(P k ) t r . Clearly there is a bisimulation between M and M and so we have the following result: M; s j= M ; s j= The proof follows from the result that CTL is adequate with respect to bisimulation, established in [7]) This is similar to the result in section 3.1 but in this case, because we consider (unquotiented) structures, the formula itself can be transformed under a permutation as well as the model (often desirable) This approach can be applied also when the processes p i are not all isomorphic. ....

M. Browne, E. Clarke, and O. Grumberg. Characterizing nite kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115-131, 1989.

....context of model checking with abstraction, however, there remains a missing link: how to lift a deductive proof generated automatically for the abstract program back to the concrete program domain. While it is well known that property satisfaction can be lifted back through abstraction [HM85,BCG88] these results rely on the set based semantics of properties, and do not indicate how to lift deductive proofs of satisfaction. In this paper, we tackle this question for temporal properties in the Mucalculus [Koz82] and several types of abstractions. The Mu calculus is a very expressive ....

....an LTS M , we show how a proof of a property f on N can be lifted to a proof of the same property on M . We consider two common notions of abstraction: simulation [Mil71] which preserves only universal properties, and bisimulation [Par81] which preserves properties of the full mu calculus (cf. BCG88,Sti95] Let M and N be LTS s, with M = N and M = N . A relation SM SN is a simulation from M to N if, and only if: The initial states of M and N are related, i.e. s M s N , and For every s in SM and t in SN such that s t holds: LM (s) LN (t) and for every a 2 M , and ....

M. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59, 1988.

....adjusted to such designs. Then, they induce more accurate coverage metrics, with no additional computational cost. 5. 2 Properties of coverage metrics Sensitivity to abstraction and reduction One of the ways to cope with the state explosion problem is to consider an abstraction of the system [BCG88]. Essentially, in the abstraction we merge states that are indistinguishable by the speci cation to a single state. The covered sets de ned in Section 2.2 are sensitive to abstraction, in the sense that there is no correlation between 6 Note that this is di erent from the case where x can be xed ....

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115-131, 1988.

....nite domains. An unde ned array component represents nondeterminism which is still to be resolved. We describe a translation of such a program to a bisimulation equivalent data independent program without arrays. It follows that we have a procedure for calculus model checking in this case [BCG88, NK00] The calculus is a branching time logic, more expressive than CTL or CTL [AH98] For a program P , any transition system generated by P with nite instances of X and Y is simulated by the transition system generated by P with in nite instances of X and Y . It follows that there is a ....

....system M which has the same observables as, and is bisimulationequivalent to, the transition system hhP ] ii A ;B using the algorithm in [NK00] with as the initial condition 2 . Also note that states related by some bisimulation have exactly the same true calculus formulas [BCG88] Using these facts we proceed as follows: hhPii A ;B ; b 0 j= 8t 2 db 0 e hhPii A ;B ; t j= f Proposition 4.11 and De nition 4.4 g 8s 2 db 0 e ] s) hhP ] ii A ;B ; s j= f [NK00] g 8u 2 db 0 e M;u j= M; b 0 j= Hence the ....

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59, 1988.

....the color of the circles denotes their label, and the transition relation is denoted by a dashed line. States related by the equivalence relation B are joined by a solid line. To check that B is a WEB, let rank(u; v) tag of v, and use the well founded witness hrank ; hIN; ii. Theorem 1 (cf. [2, 18]) If B is a WEB on TS M and sBw, then s and w have the same fullpaths up to stuttering and for any CTL nX formula f , M; s j= f i M;w j= f . A consequence of Theorem 1 is that states related by a WEB satisfy the same next time free formulae of LTL (Linear Temporal Logic) and the same safety ....

M. Browne, E. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59, 1988.

....Temporal logic is alive and well, and: it may still have most of its future ahead of it References [1] G. d Agostino. Modal Logic and Set Theory: Translation, Bisimulation, and Interpolation. Forthcoming Dissertation, Institute for Logic, Language and Computation, University of Amsterdam, 1998. [2] A. Aliseda Lera and R. van Glabbeek, eds. Computing Natural Language. CSLI Publications, Stanford, 1998. 3] H. Andr eka, J. van Benthem and I. N emeti. Modal Logics and Bounded Fragments of Predicate Logic . Journal of Philosophical Logic, 27:3, 217 274, 1998. 4] F. Anger, H. Guesgen, R. ....

....Blackburn would like to thank Aravind Joshi and the sta of the IRCS, University of Pennsylvania, for their hospitality while the nal version of this paper was being prepared. References [1] J. Allen. Towards a general theory of knowledge and action. Arti cial Intelligence, 23:123 154, 1984. [2] J. van Benthem. The Logic of Time. Reidel, 1983. 3] P. Blackburn Nominal Tense Logic and Other Sorted Intensional Frameworks. PhD Thesis, Centre for Cognitive Science, University of Edinburgh, 1990. 4] P. Blackburn. Fine grained theories of time. In M. Aurnague, A. Borillo, M. Borillo and M. ....

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M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Journal of Theoretical Computer Science, 59:115-131, 1988.

....s j= f denotes that f holds at state s of model M . A path is a sequence of states such that for adjacent states s; u; s 9 9 Ku. A path is a fullpath if it is in nite. fp: s denotes that is a fullpath starting at s. 2. 2 Stuttering Bisimulation We de ne a variant of stuttering bisimulation [2]. This notion is similar to the notion of a bisimulation [20, 18] but allows for nite stuttering, and therefore di ers from weak bisimulation [18] The idea is to partition the state space of a transition system into equivalence classes such that states in the same class have the same in nite ....

....We have the following theorems. Theorem 1 (cf. 19, 16] B is an ESTB on TS M i B is a WEB on M . Theorem 1 says that the notion of stuttering bisimulation is exactly captured by the notion of WEB. The signi cance is that any stuttering bisimulation can be proved using WEBs. Theorem 2 (cf. [2, 19]) If B is a WEB on TS M and sBw, then for any CTL nX formula f , M; s j= f i M;w j= f . Theorem 2 says that states related by a WEB satisfy the same next time free formulae of the branching time logic CTL . As a consequence, they satisfy the same LTL (Linear Temporal Logic) formulae and ....

M. Browne, E. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59, 1988.

....by the non at versions of the logics. Section 4 concludes. Some proofs in this article have been moved into the appendices. Comparative expressivity of CTL like temporal logics is studied in, among others, 9, 12, 10] Behavioural equivalences ( 6] induced by temporal logics are the subject of [13, 2, 19, 15, 7, 1, 11, 20, 5]. 2 Flat Linear time Temporal Logic: Expressivity Throughout this article, we assume given a nonempty set Prop of propositions. De nition 2.1 The logic LTL is the set of formulae de ned inductively by the following grammar, where p 2 Prop. p j : j j X j U( LTL(U) is the ....

....that L1 = L2 . Another example, more related to the topic of this article, is the comparison between CTL and CTL. On the one hand, the star does increase the expressive power: in [9] it is shown that the CTL formula 8F(p Xp) has no equivalent in CTL. On the other hand, as shown in [2], the two logics are equally distinguishing: for both of them, the induced logical equivalence coincides with bisimulation 3 ( 17] Below, these results will be extended for at versions of the logics. We start by de ning the syntax and semantics of the various Computation Tree Logics. De ....

[Article contains additional citation context not shown here]

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Journal of Theoretical Computer Science, 59:115-131, 1988.

....X, modality is not allowed. The importance of such fragments was stressed in [Lam83] The subscript operators of TCTL s nX, or the formula clocks of TCTL c nX, clearly add expressive power to CTLnX. Indeed no CTLnX formula can distinguish two Kripke structures which are stuttering equivalent [BCG88] while such structures can sometimes be distinguished by TCTL s nX formulae and their ability to count the number of tick. For example, the two structures of Figure 5 (where only grey states satisfy tick) are stuttering equivalent but q j= EF 0 tick and q 0 6j= EF 0 tick. Therefore TCTL s nX ....

M. C. Browne, E. M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59(1-2):115-131, 1988.

....a ects in M is NP complete. We rst prove membership in NP. Let M = hAP; W;R;w 0 ; Li. For a CTL formula , let [ W be the set of states that satisfy . The set [ respects bisimulation; thus, if two states are bisimilar in M , then either both are in [ or both are not in [ By [BCG88], for every set V W that respects bisimulation, there is a CTL formula V of length polynomial in M such that V holds at exactly all the states in V . It follows that for every formula , there is a formula 0 of length polynomial in M such that [ 0 ] Accordingly, a ects in M ....

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115-131, 1988.

....states of B only and does not involve the transitions of A. As a consequence, the method of [GS95] does not always apply to diverging processes. Norm functions very similar to ours were also studied by Namjoshi [Nam97] He uses them to obtain a characterization of the stuttering bisimulation of [BCG88] which is the equivalent of branching bisimulation in a setting where states rather than actions are labeled (see [DNV95] Both [GS95] and [Nam97] do not address e ectiveness issues. Although we present normed simulations in a setting of labeled transition systems, it should not be dicult to ....

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59(1,2):115-131, 1988.

....(lumping) equivalence In this section, we discuss some techniques to reduce the state space of a CTMC. These techniques are mainly based on the observation that (a slight variant of) ordinary lumping equivalence (i.e. bisimulation) preserves all CSL formulas. This result is in the spirit of [8] where bisimilar states of an ordinary transition system are shown to satisfy the same CTL formulas. Similar results have been established for many types of transition systems and branching time logics; e.g. in the probabilistic setting, 2] shows that probabilistic bisimulation on DTMCs ....

M. Brown, E. Clarke, O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Th. Comp. Sc., 59: 115-131, 1988.

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Browne M. C., Clarke E. M. and Grumberg O. [1988], `Characterizing nite Kripke structures in propositional temporal logic', Theoretical Computer Science 59(1-2), 115-131.

No context found.

M. Brown, E. Clarke, O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Th. Comp. Sc., 59: 115-131, 1988.

No context found.

M. Brown, E. Clarke and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Th. Comp. Sc., 59: 115-131, 1988.

No context found.

Browne, M. C., Clarke, E. and Grumberg, O. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59: 115-131, 1988.

No context found.

M.C. Browne, E.M. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59, 1988.

No context found.

M. Browne, E. Clarke, and O. Grumberg. Characterizing nite Kripke structures in propositional temporal logic. Theoretical Computer Science, 59:115-131, 1988.

No context found.

M. C. Browne, E. M. Clarke, and O. Grumberg, Characterizing nite Kripke structures in propositional temporal logic, Theoretical Computer Science 59(1/2) (1988), 115-131.

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