Minimal Critical Subsystems for Discrete-Time Markov Models

  • Ralf Wimmer
  • Nils Jansen
  • Erika Ábrahám
  • Bernd Becker
  • Joost-Pieter Katoen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7214)


We propose a new approach to compute counterexamples for violated ω-regular properties of discrete-time Markov chains and Markov decision processes. Whereas most approaches compute a set of system paths as a counterexample, we determine a critical subsystem that already violates the given property. In earlier work we introduced methods to compute such subsystems based on a search for shortest paths. In this paper we use SMT solvers and mixed integer linear programming to determine minimal critical subsystems.


Model Check Target State Mixed Integer Linear Program Markov Decision Process Redundant Constraint 
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.


  1. 1.
    Bustan, D., Rubin, S., Vardi, M.Y.: Verifying ω-Regular Properties of Markov Chains. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 189–201. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. The MIT Press (2008)Google Scholar
  3. 3.
    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
  4. 4.
    Chadha, R., Viswanathan, M.: A counterexample-guided abstraction-refinement framework for Markov decision processes. ACM TOCL 12(1), 1–45 (2010)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aljazzar, H., Leue, S.: Directed explicit state-space search in the generation of counterexamples for stochastic model checking. IEEE Trans. on Software Engineering 36(1), 37–60 (2010)CrossRefGoogle Scholar
  6. 6.
    Han, T., Katoen, J.-P., Damman, B.: Counterexample generation in probabilistic model checking. IEEE Trans. on Software Engineering 35(2), 241–257 (2009)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    Günther, M., Schuster, J., Siegle, M.: Symbolic calculation of k-shortest paths and related measures with the stochastic process algebra tool Caspa. In: Proc. of DYADEM-FTS, pp. 13–18. ACM Press (2010)Google Scholar
  10. 10.
    Jansen, N., Ábrahám, E., Katelaan, J., Wimmer, R., Katoen, J.-P., Becker, B.: Hierarchical Counterexamples for Discrete-Time Markov Chains. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 443–452. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  11. 11.
    Kattenbelt, M., Huth, M.: Verification and refutation of probabilistic specifications via games. In: Proc. of FSTTCS. LIPIcs, vol. 4, pp. 251–262. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)Google Scholar
  12. 12.
    Schmalz, M., Varacca, D., Völzer, H.: Counterexamples in Probabilistic LTL Model Checking for Markov Chains. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 587–602. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Fecher, H., Huth, M., Piterman, N., Wagner, D.: PCTL model checking of Markov chains: Truth and falsity as winning strategies in games. Performance Evaluation 67(9), 858–872 (2010)CrossRefGoogle Scholar
  14. 14.
    Wimmer, R., Becker, B., Jansen, N., Ábrahám, E., Katoen, J.-P.: Minimal critical subsystems as counterexamples for ω-regular DTMC properties. In: Brandt, J., Schneider, K. (eds.) Proc. of MBMV. Kovač-Verlag (2012)Google Scholar
  15. 15.
    de Moura, L.M., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Communication of the ACM 54(9), 69–77 (2011)CrossRefGoogle Scholar
  16. 16.
    Dutertre, B., de Moura, L.M.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley (1986)Google Scholar
  18. 18.
    Kwiatkowska, M.Z., 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
  19. 19.
    Reiter, M.K., Rubin, A.D.: Crowds: Anonymity for web transactions. ACM Trans. on Information and System Security 1(1), 66–92 (1998)CrossRefGoogle Scholar
  20. 20.
    Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. Information and Computation 88(1), 60–87 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    de Moura, L.M., 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ralf Wimmer
    • 1
  • Nils Jansen
    • 2
  • Erika Ábrahám
    • 2
  • Bernd Becker
    • 1
  • Joost-Pieter Katoen
    • 2
  1. 1.Albert-Ludwigs-UniversityFreiburgGermany
  2. 2.RWTH Aachen UniversityGermany

Personalised recommendations