# Verifying systems with infinite but regular state spaces

## Abstract

Thanks to the development of a number of efficiency enhancing techniques, state-space exploration based verification, and in particular model checking, has been quite successful for finite-state systems. This has prompted efforts to apply a similar approach to systems with infinite state spaces. Doing so amounts to developing algorithms for computing a symbolic representation of the infinite state space, as opposed to requiring the user to characterize the state space by assertions. Of course, in most cases, this can only be done at the cost of forgoing any general guarantee of success. The goal of this paper is to survey a number of results in this area and to show that a surprisingly common characteristic of the systems that can be analyzed with this approach is that their state space can be represented as a regular language.

## Keywords

Model Check Regular Language Finite Automaton Reachable State Integer Vector## References

- [AD94]R. Alur and D. Dill. A theory of timed automata.
*Theoretical Computer Science*, 126(2):183–236, 1994.CrossRefGoogle Scholar - [BBR97]B. Boigelot, L. Bronne, and S. Rassart. An improved reachability analysis method for strongly linear hybrid systems. In
*Proc. 9th Int. Conf on Computer Aided Verification*, volume 1254 of*Lecture Notes in Computer Science*, pages 167–178, Haifa, June 1997. Springer-Verlag.Google Scholar - [BC96]A. Boudet and H. Comon. Diophantine equations, Presburger arithmetic and finite automata. In
*Proceedings of CAAP'96*, number 1059 in Lecture Notes in Computer Science, pages 30–43. Springer-Verlag, 1996.Google Scholar - [BCM+92]J.R. Burch, E.M. Clarke, K.L. McMillan, D.L. Dill, and L.J. Hwang. Symbolic model checking: 102° states and beyond.
*Information and Computation*, 98(2):142–170, June 1992.CrossRefGoogle Scholar - [BC96]B. Boigelot and P. Godefroid. Symbolic verification of communication protocols with infinite state spaces using QDDs. In
*Proceedings of Computer-Aided Verification*, volume 1102 of*Lecture Notes in Computer Science*, pages 1–12, New-Brunswick, NJ, USA, July 1996. Springer-Verlag.Google Scholar - [BGWW97]B. Boigelot, P. Godefroid, B. Willems, and P. Wolper. The power of QDD's. In
*Proc. of Int. Static Analysis Symposium*, volume 1302 of*Lecture Notes in Computer Science*, pages 172–186, Paris, September 1997. Springer-Verlag.Google Scholar - [Boi98]B. Boigelot.
*Symbolic Methods for Exploring Infinite State Spaces*. PhD thesis, Université de Liege, 1998.Google Scholar - [BRW98]B. Boigelot, S. Rassart, and P. Wolper. On the expressiveness of real and integer arithmetic automata. to appear in Proc. ICALP'98, 1998.Google Scholar
- [Bry92]R.E. Bryant. Symbolic boolean manipulation with ordered binary-decision diagrams.
*ACM Computing Surveys*, 24(3):293–318, 1992.CrossRefGoogle Scholar - [BS95]O. Burkart and B. Steffen. Composition, decomposition and model checking of pushdown processes.
*Nordic Journal of Computing*, 2(2):89–125, 1995.Google Scholar - [Büc60]J. R. Büchi. Weak second-order arithmetic and finite automata.
*Zeitschrift Math. Logik and Grundlagen der Mathematik*, 6:66–92, 1960.Google Scholar - [BVW94]O. Bernholtz, M.Y. Vardi, and P. Wolper. An automata-theoretic approach to branching-time model checking. In
*Computer Aided Verification, Proc. 6th Int. Workshop*, volume 818 of*Lecture Notes in Computer Science*, pages 142–155, Stanford, California, June 1994. Springer-Verlag. full version available from authors.Google Scholar - [BW94]B. Boigelot and P. Wolper. Symbolic verification with periodic sets. In
*Computer Aided Verification, Proc. 6th Int. Conference*, volume 818 of*Lecture Notes in Computer Science*, pages 55–67, Stanford, California, June 1994. Springer-Verlag.Google Scholar - [Cau92]D. Caucal. On the regular structure of prefix rewriting.
*Theoretical Computer Science*, 106:61–86, 1992.CrossRefGoogle Scholar - [Cob69]A. Cobham. On the base-dependence of sets of numbers recognizable by finite automata.
*Mathematical Systems Theory*, 3:186–192, 1969.CrossRefGoogle Scholar - [EN94]J. Esparza and M. Nielsen. Decidability issues for Petri nets — a survey.
*Bulletin of the EATCS*, 52:245–262, 1994.Google Scholar - [FWW97]A. Finkel, B. Willems, and P. Wolper. A direct symbolic approach to model checking pushdown systems (extended abstract). Presented at Infinity'97 (Bologna), Electronic notes in theoretical computer science, August 1997.Google Scholar
- [Sem77]A. L. Semenov. Presburgerness of predicates regular in two number systems.
*Siberian Mathematical Journal*, 18:289–299, 1977.CrossRefGoogle Scholar - [Val92]A. Valmari. A stubborn attack on state explosion.
*Formal Methods in System Design*, 1:297–322, 1992.CrossRefGoogle Scholar - [VW86]M.Y. Vardi and P. Wolper. An automata-theoretic approach to automatic program verification. In
*Proceedings of the First Symposium on Logic in Computer Science*, pages 322–331, Cambridge, June 1986.Google Scholar - [WB95]P. Wolper and B. Boigelot. An automata-theoretic approach to presburger arithmetic constraints. In
*Proc. Static Analysis Symposium*, volume 983 of*Lecture Notes in Computer Science*, pages 21–32, Glasgow, September 1995. Springer-Verlag.Google Scholar - [WG93]P. Wolper and P. Godefroid. Partial-order methods for temporal verification. In
*Proc. CONCUR '93*, volume 715 of*Lecture Notes in Computer Science*, pages 233–246, Hildesheim, August 1993. Springer-Verlag.Google Scholar