Abstract
In this paper we study time-bounded verification of a finite continuous-time Markov chain (CTMC) \(\mathcal{C}\) against a real-time specification, provided either as a metric temporal logic (MTL) property ϕ, or as a timed automaton (TA) \(\mathcal{A}\). The key question is: what is the probability of the set of timed paths of \(\mathcal{C}\) that satisfy ϕ (or are accepted by \(\mathcal{A}\)) over a time interval of fixed, bounded length? We provide approximation algorithms to solve these problems. We first derive a bound N such that timed paths of \(\mathcal{C}\) with at most N discrete jumps are sufficient to approximate the desired probability up to ε. Then, for each discrete (untimed) path σ of length at most N, we generate timed constraints over variables determining the residence time of each state along σ, depending on the real-time specification under consideration. The probability of the set of timed paths, determined by the discrete path and the associated timed constraints, can thus be formulated as a multidimensional integral. Summing up all such probabilities yields the result. For MTL, we consider both the continuous and the pointwise semantics. The approximation algorithms differ mainly in constraints generation for the two types of specifications.
This work is supported by the ERC Advanced Grant VERIWARE.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)
Alur, R., Feder, T., Henzinger, T.A.: The benefits of relaxing punctuality. J. ACM 43(1), 116–146 (1996)
Alur, R., Henzinger, T.A.: A Really Temporal Logic. J. ACM 41(1), 181–204 (1994)
Alur, R., Henzinger, T.A.: Real-time logics: Complexity and expressiveness. In: LICS, pp. 390–401 (1990)
Alur, R., Kurshan, R.P., Viswanathan, M.: Membership questions for timed and hybrid automata. In: IEEE Real-Time Systems Symposium, pp. 254–263 (1998)
Baier, C., Cloth, L., Haverkort, B.R., Kuntz, M., Siegle, M.: Model checking Markov chains with actions and state labels. IEEE Trans. Software Eng. 33(4), 209–224 (2007)
Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Software Eng. 29(6), 524–541 (2003)
Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Performance evaluation and model checking join forces. Commun. ACM 53(9), 76–85 (2010)
Baier, C., Hermanns, H., Katoen, J.-P., Haverkort, B.R.: Efficient computation of time-bounded reachability probabilities in uniform continuous-time Markov decision processes. Theor. Comput. Sci. 345(1), 2–26 (2005)
Bouyer, P., Chevalier, F., Markey, N.: On the expressiveness of TPTL and MTL. Inf. Comput. 208(2), 97–116 (2010)
Barbot, B., Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Efficient CTMC model checking of linear real-time objectives. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 128–142. Springer, Heidelberg (2011)
Bemporad, A., Fukuda, K., Torrisi, F.D.: Convexity recognition of the union of polyhedra. Comput. Geom. 18(3), 141–154 (2001)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer, Heidelberg (1998)
Bouyer, P.: From Qualitative to Quantitative Analysis of Timed Systems. Mémoire d’habilitation, Université Paris 7, Paris, France (January 2009)
Chen, T., Diciolla, M., Kwiatkowska, M., Mereacre, A.: Time-bounded verification of CTMCs against real-time specifications. Tech. Rep. RR-11-06, Department of Computer Science, University of Oxford (2011)
Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Quantitative model checking of continuous-time Markov chains against timed automata specifications. In: LICS, pp. 309–318 (2009)
Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Model checking of continuous-time Markov chains against timed automata specifications. Logical Methods in Computer Science 7(1–2), 1–34 (2011)
Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)
Donatelli, S., Haddad, S., Sproston, J.: Model checking timed and stochastic properties with CSL\(^{\textrm{\uppercase{ta}}}\). IEEE Trans. Software Eng. 35(2), 224–240 (2009)
Hahn, E.M., Hermanns, H., Wachter, B., Zhang, L.: Time-bounded model checking of infinite-state continuous-time Markov chains. Fundam. Inform. 95(1), 129–155 (2009)
Hiriart-Urruty, J., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I.: Fundamentals. Springer, Heidelberg (1994)
Jenkins, M., Ouaknine, J., Rabinovich, A., Worrell, J.: Alternating timed automata over bounded time. In: LICS, pp. 60–69. IEEE Computer Society, Los Alamitos (2010)
Katoen, J.-P., Zapreev, I.S.: Safe on-the-fly steady-state detection for time-bounded reachability. In: QEST, pp. 301–310 (2006)
Koymans, R.: Specifying real-time properties with metric temporal logic. Real-Time Systems 2(4), 255–299 (1990)
Lasserre, J.B., Zeron, E.S.: A Laplace transform algorithm for the volume of a convex polytope. J. ACM 48(6), 1126–1140 (2001)
Nickovic, D., Piterman, N.: From MTL to deterministic timed automata. In: Chatterjee, K., Henzinger, T.A. (eds.) FORMATS 2010. LNCS, vol. 6246, pp. 152–167. Springer, Heidelberg (2010)
Ouaknine, J., Rabinovich, A., Worrell, J.: Time-bounded verification. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 496–510. Springer, Heidelberg (2009)
Ouaknine, J., Worrell, J.: On the decidability and complexity of metric temporal logic over finite words. Logical Methods in Computer Science 3(1) (2007)
Ouaknine, J., Worrell, J.: Towards a theory of time-bounded verification. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010 Part II. LNCS, vol. 6199, pp. 22–37. Springer, Heidelberg (2010)
Roux, O., Rusu, V.: Verifying time-bounded properties for ELECTRE reactive programs with stopwatch automata. In: Antsaklis, P.J., Kohn, W., Nerode, A., Sastry, S.S. (eds.) HS 1994 Part II. LNCS, vol. 999, pp. 405–416. Springer, Heidelberg (1995)
Schrijver, A.: Theory of linear and integer programming. Wiley-Interscience series in discrete mathematics and optimization. Wiley, Chichester (1999)
Sharma, A., Katoen, J.-P.: Weighted lumpability on Markov chains. In: 8th Ershov Informatics Conference. LNCS (2011)
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: FOCS, pp. 327–338 (1985)
Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: LICS, pp. 332–344 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chen, T., Diciolla, M., Kwiatkowska, M., Mereacre, A. (2011). Time-Bounded Verification of CTMCs against Real-Time Specifications. In: Fahrenberg, U., Tripakis, S. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2011. Lecture Notes in Computer Science, vol 6919. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24310-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-24310-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24309-7
Online ISBN: 978-3-642-24310-3
eBook Packages: Computer ScienceComputer Science (R0)