# A Classification of Symbolic Transition Systems

## Abstract

We define five increasingly comprehensive classes of infinite-state systems, called STS1–5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.

STS1 These are the systems with finite *bisimilarity* quotients. They can be analyzed symbolically by (1) iterating the predecessor and boolean operations starting from a finite set of observable state sets, and (2) terminating when no new state sets are generated. This enables model checking of the *μ*-calculus.

STS2 These are the systems with finite *similarity* quotients. They can be analyzed symbolically by iterating the predecessor and positive boolean operations. This enables model checking of the existential and universal fragments of the *μ*-calculus.

STS3 These are the systems with finite *trace-equivalence* quotients. They can be analyzed symbolically by iterating the predecessor operation and a restricted form of positive boolean operations (intersection is restricted to intersection with observables). This enables model checking of linear temporal logic.

STS4 These are the systems with finite *distance-equivalence* quotients (two states are equivalent if for every distance *d*, the same observables can be reached in *d* transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new state sets are generated. This enables model checking of the existential conjunction-free and universal disjunction-free fragments of the *μ*-calculus.

STS5 These are the systems with finite *bounded-reachability* quotients (two states are equivalent if for every distance *d*, the same observables can be reached in *d* or fewer transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new states are encountered. This enables model checking of reachability properties.

## Preview

Unable to display preview. Download preview PDF.

## References

- [ACH+95]R. Alur, C. Courcoubetis, N. Halbwachs, T.A. Henzinger, P.-H. Ho, X. Nicollin, A. Olivero, J. Sifakis, and S. Yovine. The algorithmic analysis of hybrid systems.
*Theoretical Computer Science*, 138:3–34, 1995.MATHCrossRefMathSciNetGoogle Scholar - [AČJT96]P. A. Abdulla, K. Čerāns, B. Jonsson, and Y.-K. Tsay. General decidability theorems for infinite-state systems. In
*Proceedings of the 11th Annual Symposium on Logic in Computer Science*, pages 313–321. IEEE Computer Society Press, 1996.Google Scholar - [AD94]R. Alur and D.L. Dill. A theory of timed automata.
*Theoretical Computer Science*, 126:183–235, 1994.MATHCrossRefMathSciNetGoogle Scholar - [AH98]R. Alur and T.A. Henzinger.
*Computer-aided Verification: An Introduction to Model Building and Model Checking for Concurrent Systems*. Draft, 1998.Google Scholar - [AHH96]R. Alur, T.A. Henzinger, and P.-H. Ho. Automatic symbolic verification of embedded systems.
*IEEE Transactions on Software Engineering*, 22:181–201, 1996.CrossRefGoogle Scholar - [AJ98]P. Abdulla and B. Jonsson. Verifying networks of timed automata. In
*TACAS 98: Tools and Algorithms for Construction and Analysis of Systems*, Lecture Notes in Computer Science 1384, pages 298–312. Springer-Verlag, 1998.CrossRefGoogle Scholar - [BFH90]A. Bouajjani, J.-C. Fernandez, and N. Halbwachs. Minimal model generation. In
*CAV 90: Computer-aided Verification*, Lecture Notes in Computer Science 531, pages 197–203. Springer-Verlag, 1990.CrossRefGoogle Scholar - [Dam94]M. Dam. CTL* and ECTL* as fragments of the modal
*μ*-calculus.*Theoretical Computer Science*, 126:77–96, 1994.MATHCrossRefMathSciNetGoogle Scholar - [EJS93]E.A. Emerson, C.S. Jutla, and A.P. Sistla. On model checking for fragments of
*μ*-calculus. In*CAV 93: Computer-aided Verification*, Lecture Notes in Computer Science 697, pages 385–396. Springer-Verlag, 1993.Google Scholar - [FS98]A. Finkel and Ph. Schnoebelen.
*Well-structured Transition Systems Everywhere*. Technical Report LSV-98-4, Laboratoire Spécification et Vérification, ENS Cachan, 1998.Google Scholar - [Hen95]T.A. Henzinger. Hybrid automata with finite bisimulations. In
*ICALP 95: Automata, Languages, and Programming*, Lecture Notes in Computer Science 944, pages 324–335. Springer-Verlag, 1995.Google Scholar - [Hen96]T.A. Henzinger. The theory of hybrid automata. In
*Proceedings of the 11th Annual Symposium on Logic in Computer Science*, pages 278–292. IEEE Computer Society Press, 1996.Google Scholar - [HHK95]M.R. Henzinger, T.A. Henzinger, and P.W. Kopke. Computing simulations on finite and infinite graphs. In
*Proceedings of the 36rd Annual Symposium on Foundations of Computer Science*, pages 453–462. IEEE Computer Society Press, 1995.Google Scholar - [HHWT95]T.A. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: the next generation. In
*Proceedings of the 16th Annual Real-time Systems Symposium*, pages 56–65. IEEE Computer Society Press, 1995.Google Scholar - [HK96]T.A. Henzinger and P.W. Kopke. State equivalences for rectangular hybrid automata. In
*CONCUR 96: Concurrency Theory*, Lecture Notes in Computer Science 1119, pages 530–545. Springer-Verlag, 1996.Google Scholar - [HKPV98]T.A. Henzinger, P.W. Kopke, A. Puri, and P. Varaiya. What’s decidable about hybrid automata?
*Journal of Computer and System Sciences*, 57:94–124, 1998.MATHCrossRefMathSciNetGoogle Scholar - [HM99]T.A. Henzinger and R. Majumdar. Symbolic model checking for rectangular hybrid systems. Submitted for publication, 1999.Google Scholar
- [KS90]P.C. Kanellakis and S.A. Smolka. CCS expressions, finite-state processes, and three problems of equivalence.
*Information and Computation*, 86:43–68, 1990.MATHCrossRefMathSciNetGoogle Scholar - [vG90]R.J. van Glabbeek.
*Comparative Concurrency Semantics and Refinement of Actions*. PhD thesis, Vrije Universiteit te Amsterdam, The Netherlands, 1990.Google Scholar