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Symbolic Counterexample Generation for Discrete-Time Markov Chains

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Formal Aspects of Component Software (FACS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 7684))

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Abstract

In this paper we investigate the generation of counterexamples for discrete-time Markov chains (DTMCs) and PCTL properties. Whereas most available methods use explicit representations for at least some intermediate results, our aim is to develop fully symbolic algorithms. As in most related work, our counterexample computations are based on path search. We first adapt bounded model checking as a path search algorithm and extend it with a novel SAT-solving heuristics to prefer paths with higher probabilities. As a second approach, we use symbolic graph algorithms to find counterexamples. Experiments show that our approaches, in contrast to other existing techniques, are applicable to very large systems with millions of states.

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References

  1. 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)

    Chapter  Google Scholar 

  2. Chadha, R., Viswanathan, M.: A counterexample-guided abstraction-refinement framework for Markov decision processes. ACM TOCL 12(1), 1–45 (2010)

    Article  MathSciNet  Google Scholar 

  3. Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  4. Baier, C., Katoen, J.P.: Principles of Model Checking. The MIT Press (2008)

    Google Scholar 

  5. 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)

    Chapter  Google Scholar 

  6. 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)

    Chapter  Google Scholar 

  7. 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 2010, pp. 13–18. ACM Press (2010)

    Google Scholar 

  8. Han, T., Katoen, J.P., Damman, B.: Counterexample generation in probabilistic model checking. IEEE Trans. on Software Engineering 35(2), 241–257 (2009)

    Article  Google Scholar 

  9. Kattenbelt, M., Huth, M.: Verification and refutation of probabilistic specifications via games. In: Proc. of FSTTCS 2009. LIPIcs, vol. 4, pp. 251–262. Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

    Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. 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)

    Chapter  Google Scholar 

  14. Wimmer, R., Jansen, N., Ábrahám, E., Becker, B., Katoen, J.-P.: Minimal Critical Subsystems for Discrete-Time Markov Models. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 299–314. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  15. Bryant, R.E.: Graph-based algorithms for boolean function manipulation. IEEE Trans. Computers 35(8), 677–691 (1986)

    Article  MATH  Google Scholar 

  16. Fujita, M., McGeer, P.C., Yang, J.C.Y.: Multi-terminal binary decision diagrams: An efficient data structure for matrix representation. Formal Methods in System Design 10(2/3), 149–169 (1997)

    Article  Google Scholar 

  17. Baier, C., Clarke, E.M., Hartonas-Garmhausen, V., Kwiatkowska, M.Z., Ryan, M.: Symbolic Model Checking for Probabilistic Processes. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 430–440. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  18. Parker, D.: Implementation of Symbolic Model Checking for Probabilistic Systems. PhD thesis, University of Birmingham (2002)

    Google Scholar 

  19. Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)

    Article  MATH  Google Scholar 

  20. Katoen, J.P., Zapreev, I.S., Hahn, E.M., Hermanns, H., Jansen, D.N.: The ins and outs of the probabilistic model checker MRMC. Performance Evaluation 68(2), 90–104 (2011)

    Article  Google Scholar 

  21. 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)

    Chapter  Google Scholar 

  22. Kwiatkowska, M., Norman, G., Parker, D.: The PRISM benchmark suite. In: Proc. of QEST. IEEE CS (September 2012)

    Google Scholar 

  23. Tseitin, G.S.: On the complexity of derivations in the propositional calculus. Studies in Mathematics and Mathematical Logic (Part II), 115–125 (1968)

    Google Scholar 

  24. Schmid, W.: Berechnung kürzester Wege in Straßennetzen mit Wegeverboten. PhD thesis, Universität Stuttgart, Fakultät für Bauingenieur- und Vermessungswesen (2000)

    Google Scholar 

  25. Somenzi, F.: Cudd: Cu decision diagram package release 2.4.1 (2005)

    Google Scholar 

  26. Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  27. Norman, G., Shmatikov, V.: Analysis of probabilistic contract signing. Journal of Computer Security 14(6), 561–589 (2006)

    Google Scholar 

  28. Reiter, M.K., Rubin, A.D.: Crowds: Anonymity for web transactions. ACM Trans. on Information and System Security 1(1), 66–92 (1998)

    Article  Google Scholar 

  29. Jansen, N., Ábrahám, E., Volk, M., Wimmer, R., Katoen, J.-P., Becker, B.: The COMICS Tool – Computing Minimal Counterexamples for DTMCs. In: Chakraborty, S., Mukund, M. (eds.) ATVA 2012. LNCS, vol. 7561, pp. 349–353. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. RWTH Aachen University, Germany

    Nils Jansen, Erika Ábrahám, Barna Zajzon & Joost-Pieter Katoen

  2. Albert-Ludwigs-University Freiburg, Germany

    Ralf Wimmer & Bernd Becker

  3. University of the Federal Armed Forces Munich, Germany

    Johann Schuster

Authors
  1. Nils Jansen
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  2. Erika Ábrahám
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  3. Barna Zajzon
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  4. Ralf Wimmer
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  5. Johann Schuster
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  6. Joost-Pieter Katoen
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  7. Bernd Becker
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Editor information

Editors and Affiliations

  1. NASA Ames Research Center, Mail Stop 269-2, 94035, Moffett Field, CA, USA

    Corina S. Păsăreanu

  2. INRIA Grenoble - Rhône-Alpes/CONVECS, 655, avenue de l’Europe, 38330, Montbonnot Saint-Martin, France

    Gwen Salaün

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Jansen, N. et al. (2013). Symbolic Counterexample Generation for Discrete-Time Markov Chains. In: Păsăreanu, C.S., Salaün, G. (eds) Formal Aspects of Component Software. FACS 2012. Lecture Notes in Computer Science, vol 7684. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35861-6_9

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  • DOI: https://doi.org/10.1007/978-3-642-35861-6_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35860-9

  • Online ISBN: 978-3-642-35861-6

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