Abstract
In this work we focus on model checking of probabilistic models. Probabilistic models are widely used to describe randomized protocols. A Markov chain induces a probability measure on sets of computations. The notion of correctness now becomes probabilistic. We solve here the general problem of linear-time probabilistic model checking with respect to ω-regular specifications. As specification formalism, we use alternating Büchi infinite-word automata, which have emerged recently as a generic specification formalism for developing model checking algorithms. Thus, the problem we solve is: given a Markov chain \({\cal {M}}\) and automaton \({\cal {A}}\), check whether the probability induced by \({\cal {M}}\) of \(L({\cal {A}})\) is one (or compute the probability precisely). We show that these problem can be solved within the same complexity bounds as model checking of Markov chains with respect to LTL formulas. Thus, the additional expressive power comes at no penalty.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Armoni, R., Bustan, D., Kupferman, O., Vardi, M.Y.: Resets vs. aborts in linear temporal logic. In: Int’l Conf. on Tools and Algorithms for Construction and Analysis of Systems, pp. 65–80 (2003)
Armoni, R., Fix, L., Gerth, R., Ginsburg, B., Kanza, T., Landver, A., Mador-Haim, S., Tiemeyer, A., Singerman, E., Vardi, M.Y., Zbar, Y.: The ForSpec temporal language: A new temporal property-specification language. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 296–311. Springer, Heidelberg (2002)
Baier, C., Clarke, E.M., Hartonas-Garmhausen, V., Kwiatkowska, M.Z., Ryan, M.: Symbolic model checking for probabilistic processes. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 430–440. Springer, Heidelberg (1997)
Beer, S., Ben-David, C., Eisner, D., Fisman, A.: The temporal logic sugar. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 363–367. Springer, Heidelberg (2001)
Berman, P., Garay, J.A.: Randomized distributed agreement revisited. In: Proceedings of the 23rd Inte’l Symp. on Fault-Tolerant Computing (FTCS 1993), pp. 412–421 (1993)
Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Trans. on Programming Languages and Systems 8(2), 244–263 (1986)
Clarke, E.M., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (1999)
Cole, R., Maggs, B.M.,, F.M.: Randomized protocols for low-congestion circuit routing in multistage interconnection networks. In: 30th ACM Symp. on Theo. of Comp (STOC), pp. 378–388 (1998)
Courcoubetis, C., Yannakakis, M.: Markov decision processes and regular events. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 336–349. Springer, Heidelberg (1990)
Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)
Couvreur, J.M., Saheb, N., Sutre, G.: An optimal automata approach to LTL model checking of probabilistic systems. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, Springer, Heidelberg (2003)
Fritz, C., Wilke, T.: State space reductions for alternating Büchi automata: Quotienting by simulation equivalences. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 157–168. Springer, Heidelberg (2002)
Gastin, P., Oddoux, D.: Fast LTL to Büchi automata translation. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 53–65. Springer, Heidelberg (2001)
Holzmann, G.J.: The model checker SPIN. IEEE Trans. on Software Engineering, 23 23(5), 279–295 (1997) ,Special issue on Formal Methods in Software Practice
Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov Chains. Springer, Heidelberg (1976)
Kupferman, O., Vardi, M.Y.: Weak alternating automata are not that weak. In: Proc. 5th Israeli Symp. on Theory of Computing and Systems, pp. 147–158. IEEE Computer Society Press, Los Alamitos (1997)
Kurshan, R.P.: FormalCheck User’s Manual. Cadence Design, Inc. (1998)
Lichtenstein, O., Pnueli, A.: Checking that finite state concurrent programs satisfy their linear specification. In: Proc. 12th ACM Symp. on Principles of Programming Languages, pp. 97–107 (1985)
Lichtenstein, O., Pnueli, A., Zuck, L.: The glory of the past. In: Parikh, R. (ed.) Logic of Programs 1985. LNCS, vol. 193, pp. 196–218. Springer, Heidelberg (1985)
Merz, S.: Weak alternating automata in Isabelle/HOL. In: Aagaard, M.D., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 423–440. Springer, Heidelberg (2000)
M.J. Morley. Semantics of temporal e. In T. F. Melham and F.G. Moller, editors, Banff’99 Higher Order Workshop (Formal Methods in Computation). University of Glasgow, Department of Computing Science Technic al Report, 1999.
Muller, D.E., Saoudi, A., Schupp, P.E.: Alternating automata, the weak monadic theory of the tree and its complexity. In: Kott, L. (ed.) ICALP 1986. LNCS, vol. 226, Springer, Heidelberg (1986)
Queille, J.P., Sifakis, J.: Specification and verification of concurrent systems in Cesar. In: Dezani-Ciancaglini, M., Montanari, U. (eds.) Programming 1982. LNCS, vol. 137, pp. 337–351. Springer, Heidelberg (1982)
Safra, S.: On the complexity of ω-automata. In: Proc. 29th IEEE Symp. on Foundations of Computer Science, White Plains, October 1988, pp. 319–327 (1988)
Savitch, W.J.: Relationship between nondeterministic and deterministic tape complexities. Journal on Computer and System Sciences 4, 177–192 (1970)
Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logic. J. ACM 32, 733–749 (1985)
Thrun, S.: Probabilistic algorithms in robotics. AI Magazine 21(4), 93–109 (2000)
Vardi, M.Y.: Probabilistic linear-time model checking: An overview of the automata-theoretic approach. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, pp. 265–276. Springer, Heidelberg (1999)
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. 26th IEEE Symp. on Foundations of Computer Science, Portland, October 1985, pp. 327–338 (1985)
Vardi, M.Y.: A temporal fixpoint calculus. In: Proc. 15th ACM Symp. on Principles of Programming Languages, San Diego, January 1988, pp. 250–259 (1988)
Vardi, M.Y.: Nontraditional applications of automata theory. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 575–597. Springer, Heidelberg (1994)
Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency. LNCS, vol. 1043, pp. 238–266. Springer, Heidelberg (1996)
Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Proc. 1st Symp. on Logic in Computer Science, Cambridge, June 1986, pp. 332–344 (1986)
Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)
Wolper, P.: Temporal logic can be more expressive. Information and Control 56(1–2), 72–99 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bustan, D., Rubin, S., Vardi, M.Y. (2004). Verifying ω-Regular Properties of Markov Chains. In: Alur, R., Peled, D.A. (eds) Computer Aided Verification. CAV 2004. Lecture Notes in Computer Science, vol 3114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27813-9_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-27813-9_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22342-9
Online ISBN: 978-3-540-27813-9
eBook Packages: Springer Book Archive