LTL Model Checking of Interval Markov Chains

  • Michael Benedikt
  • Rastislav Lenhardt
  • James Worrell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7795)


Interval Markov chains (IMCs) generalize ordinary Markov chains by having interval-valued transition probabilities. They are useful for modeling systems in which some transition probabilities depend on an unknown environment, are only approximately known, or are parameters that can be controlled. We consider the problem of computing values for the unknown probabilities in an IMC that maximize the probability of satisfying an ω-regular specification. We give new upper and lower bounds on the complexity of this problem. We then describe an approach based on an expectation maximization algorithm. We provide some analytical guarantees on the algorithm, and show how it can be combined with translation of logic to automata. We give experiments showing that the resulting system gives a practical approach to model checking IMCs.


Markov Chain Model Check Expectation Maximization Algorithm Product Graph Linear Temporal Logic 
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.


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  1. 1.
    Abe, N., Warmuth, M.K.: On the computational complexity of approximating distributions by probabilistic automata. Machine Learning 9, 205–260 (1992)zbMATHGoogle Scholar
  2. 2.
    Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. SIAM J. Comput. 38(5), 1987–2006 (2009)zbMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Or, M., Kozen, D., Reif, J.H.: The complexity of elementary algebra and geometry. JCSS 32(2), 251–264 (1986)zbMATHCrossRefGoogle Scholar
  4. 4.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer (1997)Google Scholar
  5. 5.
    Borodin, A., Cook, S.A., Pippenger, N.: Parallel computation for well-endowed rings and space-bounded probabilistic machines. Information and Control 58(1-3), 113–136 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Canny, J.F.: Some algebraic and geometric computations in pspace. In: STOC (1988)Google Scholar
  7. 7.
    Chatterjee, K., Sen, K., Henzinger, T.A.: Model-Checking ω-Regular Properties of Interval Markov Chains. In: Amadio, R. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 302–317. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Couvreur, J.-M., Saheb, N., Sutre, G.: An Optimal Automata Approach to LTL Model Checking of Probabilistic Systems. In: Vardi, M.Y., Voronkov, A. (eds.) LPAR 2003. LNCS, vol. 2850, pp. 361–375. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability Analysis of Probabilistic Systems by Successive Refinements. In: de Luca, L., Gilmore, S. (eds.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Delahaye, B., Larsen, K.G., Legay, A., Pedersen, M.L., Wąsowski, A.: Decision Problems for Interval Markov Chains. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 274–285. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Fehnker, A., Gao, P.: Formal Verification and Simulation for Performance Analysis for Probabilistic Broadcast Protocols. In: Kunz, T., Ravi, S.S. (eds.) ADHOC-NOW 2006. LNCS, vol. 4104, pp. 128–141. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Gerth, R., Peled, D., Vardi, M.Y., Wolper, P.: Simple on-the-fly automatic verification of linear temporal logic. In: Protocol Specification Testing and Verification (1995)Google Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, vol. 2. Springer (1993)Google Scholar
  14. 14.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: LICS (1991)Google Scholar
  15. 15.
    Klein, J., Baier, C.: Experiments with deterministic ω-automata for formulas of linear temporal logic. Theor. Comput. Sci. 363(2), 182–195 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Papadimitriou, C., Tsitsiklis, J.N.: The complexity of markov decision processes. Math. Oper. Res. 12(3), 441–450 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115, 1–37 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Benedikt
    • 1
  • Rastislav Lenhardt
    • 1
  • James Worrell
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUnited Kingdom

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