# Symbolic Algorithms for Infinite-State Games

## Abstract

*symbolic*if it manipulates not individual states, but sets of states that are represented by constraints. Such a procedure can be used for the analysis of

*infinite*state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of

*infinite-state games*, which occur in the control and modular verification of infinite-state systems. To characterize the termination of symbolic procedures for solving infinite-state games, we classify these game structures into four increasingly restrictive categories:

- 1
Class 1 consists of infinite-state structures for which all safety and reachability games can be solved.

- 2
Class 2 consists of infinite-state structures for which all ω-regular games can be solved.

- 3
Class 3 consists of infinite-state structures for which all nested positive boolean combinations of ω-regular games can be solved.

- 4
Class 4 consists of infinite-state structures for which all nested boolean combinations of ω-regular games can be solved.

We give a structural characterization for each class, using *equivalence relations* on the state spaces of games which range from game versions of trace equivalence to a game version of bisimilarity. We provide infinite-state examples for all four classes of games from control problems for *hybrid systems.* We conclude by presenting symbolic algorithms for the *synthesis* of winning strategies (“controller synthesis”) for infinitestate games with arbitrary ω-regular objectives, and prove termination over all class-2 structures. This settles, in particular, the symbolic controller synthesis problem for rectangular hybrid systems.

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## References

- 1.P. Abdulla and B. Jonsson. Verifying networks of timed automata. In
*TACAS 98*, LNCS 1384, pp. 298–312. Springer-Verlag, 1998.Google Scholar - 2.R. Alur and D. Dill. A theory of timed automata.
*Theoretical Computer Science*, 126:183–235, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 3.R. Alur, T. Henzinger, and O. Kupferman. Alternating-time temporal logic. In
*FOCS 97*, pp. 100–109. IEEE Computer Society Press, 1997.Google Scholar - 4.R. Alur, T. Henzinger, O. Kupferman, and M. Vardi. Alternating refinement relations. In
*CONCUR 97*, LNCS 1466, pp. 163–178. Springer-Verlag, 1998.Google Scholar - 5.A. Bouajjani, J.-C. Fernandez, and N. Halbwachs. Minimal model generation. In
*CAV 90*, LNCS 531, pp. 197–203. Springer-Verlag, 1990.Google Scholar - 6.J. Büchi and L. Landweber. Solving sequential conditions by finite-state strategies.
*Transactions of the AMS*, 138:295–311, 1969.CrossRefGoogle Scholar - 7.E. Emerson and C. Jutla. Tree automata, mu-calculus, and determinacy. In
*FOCS 91*, pp. 368–377. IEEE Computer Society Press, 1991.Google Scholar - 8.E. Emerson, C. Jutla, and A. Sistla. On model checking for fragments of μ-calculus. In
*CAV 93*, LNCS 697, pp. 385–396. Springer-Verlag, 1993.Google Scholar - 9.T. Henzinger, P.-H. Ho, and H. Wong-Toi. HyTech: a model checker for hybrid systems.
*Software Tools for Technology Transfer*, 1:110–122, 1997.zbMATHCrossRefGoogle Scholar - 10.T. Henzinger, B. Horowitz, and R. Majumdar. Rectangular hybrid games. In
*CONCUR 99*, LNCS 1664, pp. 320–335. Springer-Verlag, 1999.Google Scholar - 11.T. Henzinger and R. Majumdar. A classification of symbolic transition systems. In
*STACS 2000*, LNCS 1770, pp. 13–35. Springer-Verlag, 2000.CrossRefGoogle Scholar - 12.T. Henzinger, X. Nicollin, J. Sifakis, and S. Yovine. Symbolic model checking for real-time systems.
*Information and Computation*, 111:193–244, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 13.P. Kanellakis and S. Smolka. CCS expressions, finite-state processes, and three problems of equivalence.
*Information and Computation*, 86:43–68, 1990.zbMATHCrossRefMathSciNetGoogle Scholar - 14.D. Kozen. Results on the propositional μ-calculus.
*Theoretical Computer Science*, 27:333–354, 1983.zbMATHCrossRefMathSciNetGoogle Scholar - 15.O. Maler, A. Pnueli, and J. Sifakis. On the synthesis of discrete controllers for timed systems. In
*STACS 95*, LNCS 900, pp. 229–242. Springer-Verlag, 1995.Google Scholar - 16.R. McNaughton. Infinite games played on finite graphs.
*Annals of Pure and Applied Logic*, 65:149–184, 1993.zbMATHCrossRefMathSciNetGoogle Scholar - 17.A. Mostowski. Regular expressions for infinite trees and a standard form of automata. In
*Symp. Comp. Theory*, LNCS 208, pp. 157–168. Springer-Verlag, 1984.Google Scholar - 18.P. Ramadge and W. Wonham. Supervisory control of a class of discrete-event processes.
*SIAM J. Control and Optimization*, 25:206–230, 1987.zbMATHCrossRefMathSciNetGoogle Scholar - 19.W. Thomas. Automata on infinite objects. In J. van Leeuwen, ed.,
*Handbook of Theoretical Computer Science*, volume B, pp. 133–191. Elsevier, 1990.Google Scholar - 20.W. Thomas. On the synthesis of strategies in infinite games. In
*STACS 95*, LNCS 900, pp. 1–13. Springer-Verlag, 1995.Google Scholar