Bisimulation Minimisation Mostly Speeds Up Probabilistic Model Checking

  • Joost-Pieter Katoen
  • Tim Kemna
  • Ivan Zapreev
  • David N. Jansen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4424)

Abstract

This paper studies the effect of bisimulation minimisation in model checking of monolithic discrete-time and continuous-time Markov chains as well as variants thereof with rewards. Our results show that—as for traditional model checking—enormous state space reductions (up to logarithmic savings) may be obtained. In contrast to traditional model checking, in many cases, the verification time of the original Markov chain exceeds the quotienting time plus the verification time of the quotient. We consider probabilistic bisimulation as well as versions thereof that are tailored to the property to be checked.

Download to read the full conference paper text

References

  1. 1.
    Andova, S., Hermanns, H., Katoen, J.-P.: Discrete-time rewards model-checked. In: Larsen, K.G., Niebert, P. (eds.) FORMATS 2003. LNCS, vol. 2791, pp. 88–104. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Aziz, A., et al.: Model-checking continuous time Markov chains. ACM TOCL 1, 162–170 (2000)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Aziz, A., et al.: It usually works: the temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Baier, C., Ciesinski, F., Größer, M.: ProbMela and verification of Markov decision processes. Performance Evaluation Review 32, 22–27 (2005)CrossRefGoogle Scholar
  5. 5.
    Baier, C., et al.: On the logical characterisation of performability properties. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 780–792. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Baier, C., et al.: Model-checking algorithms for continuous-time Markov chains. IEEE TSE 29, 524–541 (2003)Google Scholar
  7. 7.
    Baier, C., et al.: Comparative branching-time semantics for Markov chains. Information and Computation 200, 149–214 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ben Mamoun, M., Pekergin, N., Younès, S.: Model checking of continuous-time Markov chains by closed-form bounding distributions. In: QEST, pp. 189–198. IEEE CS, Los Alamitos (2006)Google Scholar
  9. 9.
    Böde, E., et al.: Compositional performability evaluation for Statemate. In: QEST, pp. 167–178. IEEE CS, Los Alamitos (2006)Google Scholar
  10. 10.
    Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. Journal of Applied Probability 31, 59–75 (1994)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    D’Aprile, D., Donatelli, S., Sproston, J.: CSL model checking for the GreatSPN tool. In: Aykanat, C., Dayar, T., Körpeoğlu, İ. (eds.) ISCIS 2004. LNCS, vol. 3280, pp. 543–553. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    D’Argenio, P.R., et al.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Derisavi, S.: Solution of Large Markov Models using Lumping Techniques and Symbolic Data Structures. PhD thesis, Univ. of Illinois at Urbana-Champaign (2005)Google Scholar
  14. 14.
    Derisavi, S., Hermanns, H., Sanders, W.H.: Optimal state-space lumping in Markov chains. IPL 87, 309–315 (2003)CrossRefMathSciNetGoogle Scholar
  15. 15.
    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
  16. 16.
    Fisler, K., Vardi, M.Y.: Bisimulation minimization in an automata-theoretic verification framework. In: Gopalakrishnan, G.C., Windley, P. (eds.) FMCAD 1998. LNCS, vol. 1522, pp. 115–132. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    Fisler, K., Vardi, M.Y.: Bisimulation and model checking. In: Pierre, L., Kropf, T. (eds.) CHARME 1999. LNCS, vol. 1703, pp. 338–342. Springer, Heidelberg (1999)Google Scholar
  18. 18.
    Fisler, K., Vardi, M.Y.: Bisimulation minimization and symbolic model checking. Formal Methods in System Design 21, 39–78 (2002)MATHCrossRefGoogle Scholar
  19. 19.
    Groesser, M., Baier, C.: Partial order reduction for Markov decision processes: a survey. In: de Boer, F.S., et al. (eds.) FMCO 2005. LNCS, vol. 4111, pp. 408–427. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 512–535 (1994)MATHCrossRefGoogle Scholar
  21. 21.
    Haverkort, B., et al.: Model checking performability properties. In: DSN, pp. 103–112. IEEE CS, Los Alamitos (2002)Google Scholar
  22. 22.
    Haverkort, B.R., Hermanns, H., Katoen, J.-P.: On the use of model checking techniques for quantitative dependability evaluation. In: 19th IEEE Symposium on Reliable Distributed Systems, pp. 228–237. IEEE CS, Los Alamitos (2000)Google Scholar
  23. 23.
    Hermanns, H., et al.: On the use of MTBDDs for performability analysis and verification of stochastic systems. J. of Logic and Alg. Progr. 56, 23–67 (2003)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996)Google Scholar
  25. 25.
    Huth, M.: An abstraction framework for mixed non-deterministic and probabilistic systems. In: Baier, C., et al. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 419–444. Springer, Heidelberg (2004)Google Scholar
  26. 26.
    Huth, M.: On finite-state approximants for probabilistic computation tree logic. TCS 346, 113–134 (2005)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ibe, O.C., Trivedi, K.S.: Stochastic Petri net models of polling systems. IEEE J. on Selected Areas in Communications 8, 1649–1657 (1990)CrossRefGoogle Scholar
  28. 28.
    Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. Information and Computation 88, 60–87 (1990)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Katoen, J.-P., Khattri, M., Zapreev, I.S.: A Markov reward model checker. In: QEST, pp. 243–244. IEEE CS, Los Alamitos (2005)Google Scholar
  30. 30.
    Katoen, J.-P., Zapreev, I.S.: Safe on-the-fly steady-state detection for time-bounded reachability. In: QEST, pp. 301–310. IEEE CS, Los Alamitos (2006)Google Scholar
  31. 31.
    Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: a hybrid approach. Int. J. on STTT 6, 128–142 (2004)Google Scholar
  32. 32.
    Kwiatkowska, M., Norman, G., Parker, D.: Game-based abstraction for Markov decision processes. In: QEST, pp. 157–166. IEEE CS, Los Alamitos (2006)Google Scholar
  33. 33.
    Kwiatkowska, M., Norman, G., Parker, D.: Symmetry reduction for probabilistic model checking. In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 234–248. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  34. 34.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Massink, M., Katoen, J.-P., Latella, D.: Model checking dependability attributes of wireless group communication. In: DSN, pp. 711–720. IEEE CS, Los Alamitos (2004)Google Scholar
  36. 36.
    Monniaux, D.: Abstract interpretation of programs as Markov decision processes. Science of Computer Programming 58, 179–205 (2005)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Pnueli, A., Zuck, L.: Verification of multiprocess probabilistic protocols. Distributed Computing 1, 53–72 (1986)MATHCrossRefGoogle Scholar
  38. 38.
    Reiter, M.K., Rubin, A.D.: Crowds: anonymity for web transactions. ACM Transactions on Information and System Security 1, 66–92 (1998)CrossRefGoogle Scholar
  39. 39.
    Sproston, J., Donatelli, S.: Backward bisimulation in Markov chain model checking. IEEE TSE 32, 531–546 (2006)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Joost-Pieter Katoen
    • 1
    • 2
  • Tim Kemna
    • 2
  • Ivan Zapreev
    • 1
    • 2
  • David N. Jansen
    • 1
    • 2
  1. 1.Software Modeling and Verification Group, RWTH AachenGermany
  2. 2.Formal Methods and Tools, University of TwenteThe Netherlands

Personalised recommendations