On (Omega-)Regular Model Checking

arXiv:0809.2214v1 [cs.LO] 12 Sep 2008 On (Omega-)Regular Model Checking Axel Legay Carnegie Mellon University Computer Science Department Pittsbugh, USA and Pierre Wolper Universi´e de Li`ege Institut Montefiore, B28 4000 Li`ege, Belgium Checking infinite-state systems is frequently done by encoding infinite sets of states as regular languages. Computing such a regular representation of, say, the set of reachable states of a system requires acceleration techniques that can finitely compute the effect of an unbounded number of transitions. Among the acceleration techniques that have been proposed, one finds both specific and generic techniques. Specific techniques exploit the particular type of system being analyzed, e.g. a system manipulating queues or integers, whereas generic techniques only assume that the transition relation is represented by a finite-state transducer, which has to be iterated. In this paper, we investigate the possibility of using generic techniques in cases where only specific techniques have been exploited so far. Finding that existing generic techniques are often not applicable in cases easily handled by specific techniques, we have developed a new approach to iterating transducers. This new approach builds on earlier work, but exploits a number of new conceptual and algorithmic ideas, often induced with the help of experiments, that give it a broad scope, as well as good performances. Categories and Subject Descriptors: D.2.4 [Formal Methods]: Model checking—Software/Program Verification; F.1.1 [Automata]: General Terms: Verification, Theory, Algorithms, Implementation Additional Key Words and Phrases: (Omega-)Regular Model Checking, Transducers, Extrapola- tion, Infinite-State System. Authors’ e-mail : {legay,pw}@montefiore.ulg.ac.be Authors’ website : http://www.montefiore.ulg.ac.be/∼{legay,pw}/ Axel Legay is supported by a B.A.E.F. grant. The present article is an improved version of [Boigelot et al. 2003], [Boigelot et al. 2004], and [Legay 2008]. Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee. c⃝20YY ACM 1529-3785/YY/00-0001 $5.00 ACM Transactions on Computational Logic, Vol. V, No. N, 20YY, Pages 1–0??. 2 · A. Legay, and P. Wolper 1. INTRODUCTION At the heart of all the techniques that have been proposed for exploring infinite state spaces, is a symbolic representation that can finitely represent infinite sets of states. In early work on the subject, this representation was domain specific, for example linear constraints for sets of real vectors. For several years now, the idea that a generic finite-automaton based representation could be used in many settings has gained ground, starting with systems manipulating queues and integers [Wolper and Boigelot 1995; Boigelot et al. 1997; Wolper and Boigelot 1998; 2000], then moving to parametric systems [Kesten et al. 1997], and, finally, reaching systems using real variables [Boigelot et al. 1998; Boigelot et al. 2001; 2005; Boigelot and Wolper 2002]. For exploring an infinite state space, one does not only need a finite representation of infinite sets, but also techniques for finitely computing the effect of an unbounded number of transitions. Such techniques can be domain specific or generic. Domain specific techniques exploit the specific properties and representations of the do- main being considered and were, for instance, obtained for queues in [Boigelot and Godefroid 1996; Bouajjani and Habermehl 1997], for integers and reals in [Boigelot 1999; Boigelot and Wolper 2002; Boigelot et al. 2003; Boigelot and Herbreteau 2006; Finkel and Leroux 2002; Bardin et al. 2004; Bardin et al. 2005], for push- down system in [Finkel et al. 1997; Bouajjani et al. 1997], and for lossy channels in [Abdulla and Jonsson 1996]. Generic techniques consider finite-automata rep- resentations and provide algorithms that operate directly on this representation, mostly disregarding the domain for which it is used. Generic techniques appeared first in the context of the verification of systems whose states can be encoded by finite words, such as parametric systems. The idea used there is that a configuration being a finite word, a transition relation is a relation on finite words, or equivalently a language of pairs of finite words. If this language is regular, it can be represented by a finite state automaton, more specifically a finite-state transducer, and the problem then becomes the one of iter- ating such a transducer. Finite state transducers are quite powerful (the transition relation of a Turing machine can be modeled by a finite-state transducer), the flip si

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