Decision Problems for Interval Markov Chains

  • Benoît Delahaye
  • Kim G. Larsen
  • Axel Legay
  • Mikkel L. Pedersen
  • Andrzej Wąsowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6638)


Interval Markov Chains (IMCs) are the base of a classic probabilistic specification theory by Larsen and Jonsson in 1991. They are also a popular abstraction for probabilistic systems.

In this paper we study complexity of several problems for this abstraction, that stem from compositional modeling methodologies. In particular we close the complexity gap for thorough refinement of two IMCs and for deciding the existence of a common implementation for an unbounded number of IMCs, showing that these problems are EXPTIME-complete. We also prove that deciding consistency of an IMC is polynomial and discuss suitable notions of determinism for such specifications.


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  1. 1.
    Andova, S.: Process algebra with probabilistic choice. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, pp. 111–129. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Antonik, A., Huth, M., Larsen, K.G., Nyman, U., Wąsowski, A.: Modal and mixed specifications: key decision problems and their complexities. MSC 20(01), 75–103 (2010)MathSciNetMATHGoogle Scholar
  3. 3.
    Beneš, N., Křetínský, J., Larsen, K.G., Srba, J.: Checking thorough refinement on modal transition systems is exptime-complete. In: Leucker, M., Morgan, C. (eds.) ICTAC 2009. LNCS, vol. 5684, pp. 112–126. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Caillaud, B., Delahaye, B., Larsen, K.G., Legay, A., Pedersen, M.L., Wąsowski, A.: Compositional design methodology with constraint markov chains. In: QEST. IEEE Computer Society, Los Alamitos (2010)Google Scholar
  5. 5.
    Dams, D.: Abstract Interpretation and Partition Refinement for Model Checking. PhD thesis, Eindhoven University of Technology (July 1996)Google Scholar
  6. 6.
    Delahaye, B., Larsen, K.G., Legay, A., Pedersen, M.L., Wąsowski, A.: Decision problems for interval markov chains (2011),
  7. 7.
    Fecher, H., Leucker, M., Wolf, V.: Don’t Know in probabilistic systems. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 71–88. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Asp. Comput. 6(5), 512–535 (1994)CrossRefMATHGoogle Scholar
  9. 9.
    Henzinger, M.R., Henzinger, T.A., Kopke, P.W.: Computing simulations on finite and infinite graphs. In: Proc. FOCS 1995, pp. 453–462 (1995)Google Scholar
  10. 10.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: LICS, pp. 266–277. IEEE Computer, Los Alamitos (1991)Google Scholar
  11. 11.
    Jonsson, B., Larsen, K.G., Yi, W.: Probabilistic extensions of process algebras. In: Handbook of Process Algebra, pp. 685–710. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  12. 12.
    Katoen, J., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for continuous-time Markov chains. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 311–324. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Katoen, J., Klink, D., Neuhäußer, M.R.: Compositional abstraction for stochastic systems. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 195–211. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Khachiyan, L.G.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244(5), 1093–1096 (1979)MathSciNetMATHGoogle Scholar
  15. 15.
    Larsen, K.G.: Modal specifications. In: Sifakis, J. (ed.) AVMS 1989. LNCS, vol. 407, pp. 232–246. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  16. 16.
    López, N., Núñez, M.: An overview of probabilistic process algebras and their equivalences. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 89–123. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Benoît Delahaye
    • 1
  • Kim G. Larsen
    • 2
  • Axel Legay
    • 3
  • Mikkel L. Pedersen
    • 2
  • Andrzej Wąsowski
    • 4
  1. 1.Université de Rennes 1/IRISAFrance
  2. 2.Aalborg UniversityDenmark
  3. 3.INRIA/IRISAFrance
  4. 4.IT University of CopenhagenDenmark

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