Finite-Horizon Bisimulation Minimisation for Probabilistic Systems

  • Nishanthan Kamaleson
  • David Parker
  • Jonathan E. Rowe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9641)

Abstract

We present model reduction techniques to improve the efficiency and scalability of verifying probabilistic systems over a finite time horizon. We propose a finite-horizon variant of probabilistic bisimulation for discrete-time Markov chains, which preserves a bounded fragment of the temporal logic PCTL. In addition to a standard partition-refinement based minimisation algorithm, we present on-the-fly finite-horizon minimisation techniques, which are based on a backwards traversal of the Markov chain, directly from a high-level model description. We investigate both symbolic and explicit-state implementations, using SMT solvers and hash functions, respectively, and implement them in the PRISM model checker. We show that finite-horizon reduction can provide significant reductions in model size, in some cases outperforming PRISM’s existing efficient implementations of probabilistic verification.

References

  1. 1.
    Aljazzar, H., Fischer, M., Grunske, L., Kuntz, M., Leitner, F., Leue, S.: Safety analysis of an airbag system using probabilistic FMEA and probabilistic counterexamples. In: Proceedings of the QEST 2009 (2009)Google Scholar
  2. 2.
    Angluin, D., Aspnes, J., Eisenstat, D.: A simple population protocol for fast robust approximate majority. Distrib. Comput. 21(2), 87–102 (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Aziz, A., Singhal, V., Balarin, F., Brayton, R.K., Sangiovanni-Vincentelli, A.L.: It usually works: the temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. 4.
    de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    De Moura, L., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69–77 (2011)CrossRefGoogle Scholar
  6. 6.
    Dehnert, C., Katoen, J.-P., Parker, D.: SMT-based bisimulation minimisation of Markov models. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 28–47. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Della Penna, G., Intrigila, B., Melatti, I., Tronci, E., Zilli, M.V.: Finite horizon analysis of Markov chains with the mur\(\phi \) verifier. STTT 8(4–5), 397–409 (2006)CrossRefMATHGoogle Scholar
  8. 8.
    Derisavi, S.: Signature-based symbolic algorithm for optimal Markov chain lumping. In: Proceedings of the QEST 2007, pp. 141–150. IEEE Computer Society (2007)Google Scholar
  9. 9.
    Derisavi, S., Hermanns, H., Sanders, W.H.: Optimal state-space lumping in Markov chains. Inf. Process. Lett. 87(6), 309–315 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. FAC 6(5), 512–535 (1994)MATHGoogle Scholar
  11. 11.
    Heath, J., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic model checking of complex biological pathways. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 32–47. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Katoen, J.-P., Kemna, T., Zapreev, I., Jansen, D.N.: Bisimulation minimisation mostly speeds up probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 87–101. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Katoen, J.P., Zapreev, I.S., Hahn, E.M., Hermanns, H., Jansen, D.N.: The ins and outs of the probabilistic model checker MRMC. Perform. Eval. 68(2), 90–104 (2011)CrossRefGoogle Scholar
  14. 14.
    Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains, 2nd edn. Springer, Heidelberg (1976)CrossRefMATHGoogle Scholar
  15. 15.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Kwiatkowska, M., Norman, G., Sproston, J., Wang, F.: Symbolic model checking for probabilistic timed automata. Inf. Comput. 205(7), 1027–1077 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kwiatkowska, M., Norman, G., Parker, D.: The PRISM benchmark suite. In: Proceedings of the QEST 2012, pp. 203–204 (2012)Google Scholar
  18. 18.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Valmari, A., Franceschinis, G.: Simple O(m logn) time Markov chain lumping. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 38–52. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Vose, M.: The Simple Genetic Algorithm: Foundations and Theory. MIT Press, Cambridge (1999)MATHGoogle Scholar
  22. 22.
    Wimmer, R., Becker, B.: Correctness issues of symbolic bisimulation computationfor Markov chains. In: MüllerClostermann, B., Echtle, K., Rathgeb, E.P. (eds.) MMB & DFT 2010. LNCS, vol. 5987, pp. 287–301. Springer, Heidelberg (2010)Google Scholar
  23. 23.
  24. 24.

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Nishanthan Kamaleson
    • 1
  • David Parker
    • 1
  • Jonathan E. Rowe
    • 1
  1. 1.School of Computer ScienceUniversity of BirminghamBirminghamUK

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