Symbolic Approximation of the Bounded Reachability Probability in Large Markov Chains

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8657)


We present a novel technique to analyze the bounded reachability probability problem for large Markov chains. The essential idea is to incrementally search for sets of paths that lead to the goal region and to choose the sets in a way that allows us to easily determine the probability mass they represent. To effectively analyze the system dynamics using an SMT solver, we employ a finite-precision abstraction on the Markov chain and a custom quantifier elimination strategy. Through experimental evaluation on PRISM benchmark models we demonstrate the feasibility of the approach on models that are out of reach for previous methods.


Markov Chain Model Check Probabilistic Choice Precision Parameter Goal Region 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kwiatkowska, M.Z., Norman, G., Parker, D.: Using probabilistic model checking in systems biology. SIGMETRICS Performance Evaluation Review 35(4) (2008)Google Scholar
  2. 2.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Katoen, J.P., Zapreev, I.S., Hahn, E.M., Hermanns, H., Jansen, D.N.: The ins and outs of the probabilistic model checker MRMC. In: Proc. of QEST, pp. 167–176. IEEE Computer Society (2009),
  5. 5.
    Hermanns, H., Wachter, B., Zhang, L.: Probabilistic CEGAR. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 162–175. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Fränzle, M., Teige, T., Eggers, A.: Engineering constraint solvers for automatic analysis of probabilistic hybrid automata. Journal of Logic and Algebraic Programming 79(7), 436–466 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Teige, T.: Stochastic Satisfiability Modulo Theories: A Symbolic Technique for the Analysis of Probabilistic Hybrid Systems. Doctoral dissertation, Carl von Ossietzky Universität Oldenburg, Department of Computing Science, Germany (2012)Google Scholar
  8. 8.
    Teige, T., Fränzle, M.: Generalized Craig interpolation for stochastic Boolean satisfiability problems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 158–172. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    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
  11. 11.
    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
  12. 12.
    Wimmer, R., Braitling, B., Becker, B.: Counterexample generation for discrete-time markov chains using bounded model checking. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 366–380. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Braitling, B., Wimmer, R., Becker, B., Jansen, N., Ábrahám, E.: Counterexample generation for Markov chains using SMT-based bounded model checking. In: Bruni, R., Dingel, J. (eds.) FMOODS/FORTE 2011. LNCS, vol. 6722, pp. 75–89. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Jansen, N., Ábrahám, E., Zajzon, B., Wimmer, R., Schuster, J., Katoen, J.-P., Becker, B.: Symbolic counterexample generation for discrete-time Markov chains. In: Păsăreanu, C.S., Salaün, G. (eds.) FACS 2012. LNCS, vol. 7684, pp. 134–151. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Ash, R., Doléans-Dade, C.: Probability and Measure Theory. Harcourt/Academic Press (2000)Google Scholar
  16. 16.
    Wintersteiger, C.M., Hamadi, Y., de Moura, L.M.: Efficiently solving quantified bit-vector formulas. Proc. of FMSD 42(1) (2013)Google Scholar
  17. 17.
    Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiabiliby modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Helmink, L., Sellink, M., Vaandrager, F.: Proof-checking a data link protocol. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, pp. 127–165. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  19. 19.
    Reiter, M., Rubin, A.: Crowds: Anonymity for web transactions. ACM Transactions on Information and System Security (TISSEC) 1(1), 66–92 (1998)CrossRefGoogle Scholar
  20. 20.
    Even, S., Goldreich, O., Lempel, A.: A randomized protocol for signing contracts. Communications of the ACM 28(6), 637–647 (1985)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Herman, T.: Probabilistic self-stabilization. Information Processing Letters 35(2), 63–67 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    von Neumann, J.: Probabilistic logics and synthesis of reliable organisms from unreliable components. In: Shannon, C., McCarthy, J. (eds.) Proc. of Automata Studies, pp. 43–98. Princeton University Press (1956)Google Scholar
  23. 23.
    Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. Information and Computation 88(1) (1990)Google Scholar
  24. 24.
    Rabe, M.N., Wintersteiger, C.M., Kugler, H., Yordanov, B., Hamadi, Y.: Symbolic approximation of the bounded reachability probability in large Markov chains. Technical Report MSR-TR-2014-74, Microsoft Research (2014)Google Scholar
  25. 25.
    Bellman, R.: Dynamic programming. Princeton University Press (1957)Google Scholar
  26. 26.
    Shapley, L.S.: Stochastic games. Proceedings of the National Academy of Sciences 39(10) (1953)Google Scholar
  27. 27.
    Hahn, E.M., Li, Y., Schewe, S., Turrini, A., Zhang, L.: iscasMc: A web-based probabilistic model checker. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 312–317. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  28. 28.
    Della Penna, G., Intrigila, B., Melatti, I., Tronci, E., Zilli, M.V.: Bounded probabilistic model checking with the Murϕ verifier. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 214–229. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  29. 29.
    de Alfaro, L., Kwiatkowska, M., Norman, G., Parker, D., Segala, R.: Symbolic model checking of probabilistic processes using MTBDDs and the Kronecker representation. In: Graf, S. (ed.) TACAS 2000. LNCS, vol. 1785, pp. 395–410. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  30. 30.
    Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proc. of POPL. ACM (1977)Google Scholar
  31. 31.
    Esparza, J., Gaiser, A.: Probabilistic abstractions with arbitrary domains. In: Yahav, E. (ed.) SAS 2011. LNCS, vol. 6887, pp. 334–350. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  32. 32.
    Kattenbelt, M., Kwiatkowska, M., Norman, G., Parker, D.: Abstraction refinement for probabilistic software. In: Jones, N.D., Müller-Olm, M. (eds.) VMCAI 2009. LNCS, vol. 5403, pp. 182–197. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  33. 33.
    Monniaux, D.: Abstract interpretation of probabilistic semantics. In: Palsberg, J. (ed.) SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  34. 34.
    Fränzle, M., Hermanns, H., Teige, T.: Stochastic satisfiability modulo theory: A novel technique for the analysis of probabilistic hybrid systems. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 172–186. Springer, Heidelberg (2008)Google Scholar
  35. 35.
    Andrés, M.E., D’Argenio, P., van Rossum, P.: Significant diagnostic counterexamples in probabilistic model checking. In: Chockler, H., Hu, A.J. (eds.) HVC 2008. LNCS, vol. 5394, pp. 129–148. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  36. 36.
    Han, T., Katoen, J.P., Berteun, D.: Counterexample generation in probabilistic model checking. IEEE Transactions on Software Engineering 35(2), 241–257 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Saarland UniversityGermany
  2. 2.Microsoft ResearchU.S.A.

Personalised recommendations