M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1 - 37, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....or safety property is decidable and is a PSPACEcomplete problem. Proof The characterizations given by Lemma 4.3 and Lemma 4. 4 reduce the problem to questions decidable in PSPACE [24, 10] notice that for PLTL formulas one can build in PSPACE an automaton for the formula and for its complement [28]) Hardness can be established by a reduction from regular language inclusion [10] 2 Note that Lemma 4.3 provides the link between relative liveness and machine closure. Indeed, recall the following de nition [1, 2, 4] De nition 4.6 Let L , for an alphabet . L ; is called a ....

Vardi, M. Y., and Wolper, P. Reasoning about in nite computations. Information and Computation 115, 1 (November 1994), 1-37.

....of project Metodi Formali per la Sicurezza (MEFISTO) Automata on in nite words and trees turned out to be very useful for those areas of computer science where nonterminating computations are studied. They give a unifying paradigm to specify, verify, and synthesize nonterminating systems [7, 15, 16]. A system speci cation can be translated to an automaton, and thus, questions about systems and their speci cations are reduced to decision problems in the automata theory. For example, the satis ability of a speci cation and the correctness of a system with respect to its speci cation can ....

M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1 - 37, 1994.

....tree automaton, information logic, hybrid logic 1 Introduction From logic to automata. After the works of B uchi and Rabin [B uc62,Rab69] various classes of automata turned out to be well suited to solve decision procedures for logical problems, including some for temporal logics (see e.g. VW94,Var97,KVW00] for the calculus and its fragments (see e.g. EJ99,SE89,VW86,EJS01,Var98] and for description logics (see e.g. CDGL99,CGL02] to quote three families of logics. For instance, translating formulae in temporal logics to automata is a standard approach for implementing model ....

M. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1-37, 1994.

....checking works. The system under inspection is modelled as a Kripke model M and the speci cation is given as an LTL formula . The Kripke model M can be see as an automaton accepting the language L(M ) It is also possible to create an automaton on in nite words A which exactly accepts L( [48]. Clearly, a system M has the property if the languages have no common words. This is equivalent to that L(M) L( The following procedure can thus be applied to model check M against [34, 10] 1. Construct a B uchi automaton A : with the language L( 2. Construct the Kripke ....

....the alternating automaton resulting from the construction. Another possibility is to de ne the translation directly to nite automata. Kupferman and Vardi [32] have presented one direct translation from an LTL formula. The translation is based on the reverse deterministic automaton de ned in [48]. The automaton associates states to subsets S in cl( The set of formulas associated with a state represent the unful lled conditions of the formula. Thus, a successful run will start in a initial state which contains : and will nish in the single accepting state which maps to ; ....

Vardi M.Y. and P. Wolper. Reasoning about in nite computations. Information and Computation, 115(1):1-37, November 1994.

....problem of determining the smallest number k such that the language de ned by a given FTL formula belongs to L(Ut k ) can be solved e ectively. Proof. It suces to note that given an FTL formula it is easy to compute a nite automaton that recognizes the language de ned by , see, for instance, [25] for the case of words. Obviously, analogues of the previous two results hold for the since hierarchy. 5. Extension to Words. In this chapter we extend the algebraic characterization of the until hierarchy and its decidability to the situation where temporal logic formulas are interpreted in ....

M. Y. Vardi and P. Wolper, Reasoning about in nite computations, Information and Computation, 115 (1994), pp. 1-37.

....are based on tableaux techniques as in [Bal98, BGM98] see e.g. the proofs of Theorem 6 and Lemma 27) Related work. Formal language theory and automata theory are already used for logics such as modal calculus, Propositional Dynamic Logic PDL, Propositional Temporal Logic PTL, CTL (see e.g. [VW86, VW94, EJ99]) but from a di erent perspective than here. We can also mention the Extended Temporal Logic (ETL) that can express properties of a sequence de nable by a right linear grammar [Wol83, VW94] In our work, we are only dealing with automata on nite words. Furthermore, our work continues the line of ....

....as modal calculus, Propositional Dynamic Logic PDL, Propositional Temporal Logic PTL, CTL (see e.g. VW86, VW94, EJ99] but from a di erent perspective than here. We can also mention the Extended Temporal Logic (ETL) that can express properties of a sequence de nable by a right linear grammar [Wol83, VW94]. In our work, we are only dealing with automata on nite words. Furthermore, our work continues the line of research relating regular expressions and Propositional Dynamic Logic (see e.g. BM75, Pra79, FL79, Pra81, HPS83, HKT00] Plan of the paper. The rest of the paper is structured as ....

M. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1-37, 1994. Journal version of the FOCS'83 paper.

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M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115(1):1-37, November 1994.

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M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115(1):1-37, November 1994.

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Vardi, M. and P. Wolper, Reasoning about in nite computations, Information and Computation 115 (1994), pp. 1-37. 13

....in particular program synthesis [MW84,EC82] triggered a second look at the problem. Indeed, it was rather obvious that a nonelementary construction was not necessary to build an automaton from a temporal logic formula; it could be done within a single exponential by a direct construction [WVS83,VW94]. As originally presented, this This work was partially funded by a grant of the Communaut e fran caise de Belgique Direction de la recherche scienti que Actions de recherche concert ees . construction was worst and best case exponential. Though it was fairly clear that it could be modi ....

....B uchi in nite word automata from linear temporal logic formulas. After an introduction to temporal logic and a presentation of in nite word automata that stresses their kinship to logic, a rst simple, but always exponential, construction is presented. This construction is similar to the one of [WVS83,VW94], but is more streamlined since it does not deal with the extended temporal logic considered in the earlier work. Thereafter, it is shown how this construction can be adapted to obtain a more e ective construction that only builds the needed states of the automaton, as described in [GPVW95] and ....

Moshe Y. Vardi and Pierre Wolper. Reasoning about in nite computations. Information and Computation, 115(1):1-37, November 1994.

.... that do not t into the restricted syntax that is present in the problems above (e.g. formulas of the form A d for some CTL formula ) VW86b] and let A 0 be nondeterministic B uchi word automaton that accepts exactly all words (i.e. trees of branching degree 1) that satisfy [VW94]. We expand A 0 to a B uchi tree automaton A 00 that accepts a tree i the tree has a path accepted by A 0 (in each state, A 00 guesses a direction in which it follows A 0 ) We prove that is linearly witnessable i L(A ) L(A 00 ) Since the containment problem L(A) ....

....in M . When is an LTL formula, so is :witness( and the generation of can be done by generating a path in the intersection of M and a B uchi word automaton for :witness( Membership in PSPACE then follows from the fact that the automaton for an LTL formula is of size exponential in [VW94], and the generation of a path in the intersection of the automaton with M can be done on the y and nondeterministically in space that is logarithmic in the sizes of M and the automaton. When is a CTL formula, we know, as discussed in the proof of Lemma 3, that a counterexample in M for ....

M.Y. Vardi and P. Wolper. Reasoning about in- nite computations. Information and Computation, 115(1):1-37, November 1994.

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M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1 - 37, 1994.

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M.Y. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1 - 37, 1994.

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M. Vardi and P. Wolper. Reasoning about in nite computations. Information and Computation, 115:1 - 37, 1994.

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