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Qualitative Reachability for Open Interval Markov Chains

  • Jeremy SprostonEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11123)

Abstract

Interval Markov chains extend classical Markov chains with the possibility to describe transition probabilities using intervals, rather than exact values. While the standard formulation of interval Markov chains features closed intervals, previous work has considered also open interval Markov chains, in which the intervals can also be open or half-open. In this paper we focus on qualitative reachability problems for open interval Markov chains, which consider whether the optimal (maximum or minimum) probability with which a certain set of states can be reached is equal to 0 or 1. We present polynomial-time algorithms for these problems for both of the standard semantics of interval Markov chains. Our methods do not rely on the closure of open intervals, in contrast to previous approaches for open interval Markov chains, and can characterise situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely.

References

  1. 1.
    Baier, C., de Alfaro, L., Forejt, V., Kwiatkowska, M.: Model checking probabilistic systems. Handbook of Model Checking, pp. 963–999. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-10575-8_28CrossRefzbMATHGoogle Scholar
  2. 2.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  3. 3.
    Caillaud, B., Delahaye, B., Larsen, K.G., Legay, A., Pedersen, M.L., Wasowski, A.: Constraint Markov chains. Theor. Comput. Sci. 412(34), 4373–4404 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chakraborty, S., Katoen, J.-P.: Model checking of open interval Markov chains. In: Gribaudo, M., Manini, D., Remke, A. (eds.) ASMTA 2015. LNCS, vol. 9081, pp. 30–42. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-18579-8_3CrossRefGoogle Scholar
  5. 5.
    Chatterjee, K., Sen, K., Henzinger, T.A.: Model-checking \(\omega \)-regular properties of interval Markov chains. In: Amadio, R. (ed.) FoSSaCS 2008. LNCS, vol. 4962, pp. 302–317. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78499-9_22CrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, T., Han, T., Kwiatkowska, M.: On the complexity of model checking interval-valued discrete time Markov chains. Inf. Process. Lett. 113(7), 210–216 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Daws, C.: Symbolic and parametric model checking of discrete-time Markov chains. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 280–294. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-31862-0_21CrossRefzbMATHGoogle Scholar
  9. 9.
    de Alfaro, L.: Formal verification of probabilistic systems. Ph.D. thesis, Stanford University, Department of Computer Science (1997)Google Scholar
  10. 10.
    Alfaro, L.: Computing minimum and maximum reachability times in probabilistic systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 66–81. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48320-9_7CrossRefGoogle Scholar
  11. 11.
    Forejt, V., Kwiatkowska, M., Norman, G., Parker, D.: Automated verification techniques for probabilistic systems. In: Bernardo, M., Issarny, V. (eds.) SFM 2011. LNCS, vol. 6659, pp. 53–113. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-21455-4_3CrossRefGoogle Scholar
  12. 12.
    Haddad, S., Monmege, B.: Interval iteration algorithm for MDPs and IMDPs. Theor. Comput. Sci. 735, 111–131 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects Comput. 6(5), 512–535 (1994)CrossRefGoogle Scholar
  14. 14.
    Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: 1991 Proceedings of LICS, pp. 266–277. IEEE Computer Society (1991)Google Scholar
  15. 15.
    Kozine, I.O., Utkin, L.V.: Interval-valued finite Markov chains. Reliable Comput. 8(2), 97–113 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lanotte, R., Maggiolo-Schettini, A., Troina, A.: Parametric probabilistic transition systems for system design and analysis. Formal Aspects Comput. 19(1), 93–109 (2007)CrossRefGoogle Scholar
  17. 17.
    Puggelli, A., Li, W., Sangiovanni-Vincentelli, A.L., Seshia, S.A.: Polynomial-time verification of PCTL properties of MDPs with convex uncertainties. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 527–542. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_35CrossRefGoogle Scholar
  18. 18.
    Sen, K., Viswanathan, M., Agha, G.: Model-checking Markov chains in the presence of uncertainties. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 394–410. Springer, Heidelberg (2006).  https://doi.org/10.1007/11691372_26CrossRefzbMATHGoogle Scholar
  19. 19.
    Sproston, J.: Probabilistic timed automata with clock-dependent probabilities. In: Hague, M., Potapov, I. (eds.) RP 2017. LNCS, vol. 10506, pp. 144–159. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-67089-8_11CrossRefGoogle Scholar
  20. 20.
    Sproston, J.: Qualitative reachability for open interval Markov chains. CoRR (2018)Google Scholar
  21. 21.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: 1985 Proceedings of FOCS, pp. 327–338. IEEE Computer Society (1985)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversity of TurinTurinItaly

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