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Nested Reachability Approximation for Discrete-Time Markov Chains with Univariate Parameters

  • Conference paper
Automated Technology for Verification and Analysis (ATVA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 8837))

Abstract

As models of real-world stochastic systems usually contain inaccurate information, probabilistic model checking for models with open or undetermined parameters has recently aroused research attention. In this paper, we study a kind of parametric variant of Discrete-time Markov Chains with uncertain transition probabilities, namely Parametric Markov Chains (PMCs), and probabilistic reachability properties with nested PCTL probabilistic operators. Such properties for a PMC with a univariate parameter define univariate real functions, called reachability functions, that map the parameter to reachability probabilities. An interesting application of these functions is sensitivity and robustness analysis of probabilistic model checking. However, a pitfall of computing the closed-form expression of a reachability function is the possible dynamism of its constraint set and target set. We pursue interval approximations for reachability functions with high accuracy. In particular, for reachability functions involving only single-nested probabilistic operators, we provide an efficient algorithm to compute their approximations. We demonstrate the applicability of our approach with a case study on a NAND multiplexing unit.

This work is partly supported by Grant R-252-000-458-133 from Singapore Ministry of Education Academic Research Fund.

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References

  1. Sen, K., Viswanathan, M., Agha, G.: Model-checking markov chains in the presence of uncertainties. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 394–410. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Chatterjee, K., Sen, K., Henzinger, T.A.: Model-checking ω-regular properties of interval markov chains. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 302–317. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  3. Puggelli, A., Li, W., Sangiovanni-Vincentelli, A.L., Seshia, S.A.: Polynomial-time verification of PCTL properties of mDPs with convex uncertainties. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 527–542. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  4. Benedikt, M., Lenhardt, R., Worrell, J.: LTL model checking of interval markov chains. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 32–46. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  5. Daws, C.: Symbolic and parametric model checking of discrete-time markov chains. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 280–294. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  6. Hahn, E., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. International Journal on Software Tools for Technology Transfer 13(1), 3–19 (2011)

    Article  Google Scholar 

  7. Lanotte, R., Maggiolo-Schettini, A., Troina, A.: Parametric probabilistic transition systems for system design and analysis. Formal Aspects of Computing 19(1), 93–109 (2007)

    Article  MATH  Google Scholar 

  8. Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6, 102–111 (1994)

    Article  Google Scholar 

  9. Chen, T., Han, T., Kwiatkowska, M.Z.: On the complexity of model checking interval-valued discrete time markov chains. Inf. Process. Lett. 113(7), 210–216 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Su, G., Rosenblum, D.S.: Asymptotic bounds for quantitative verification of perturbed probabilistic systems. In: Groves, L., Sun, J. (eds.) ICFEM 2013. LNCS, vol. 8144, pp. 297–312. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  11. Su, G., Rosenblum, D.S.: Perturbation analysis of stochastic systems with empirical distribution parameters. In: Proceeding of the 36th International Conference on Software Engineering, ICSE 2014 (2014)

    Google Scholar 

  12. Kwiatkowska, M., Norman, G., Parker, D.: The PRISM benchmark suite. In: Proc. 9th International Conference on Quantitative Evaluation of SysTems (QEST 2012), pp. 203–204. IEEE CS Press (2012)

    Google Scholar 

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

    Google Scholar 

  14. Chen, T., Feng, Y., Rosenblum, D.S., Su, G.: Perturbation analysis in verification of discrete-time markov chains. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 218–233. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  15. Norman, G., Parker, D., Kwiatkowska, M., Shukla, S.: Evaluating the reliability of NAND multiplexing with PRISM. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 24(10), 1629–1637 (2005)

    Article  Google Scholar 

  16. 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 

  17. MATLAB: version 8.0. The MathWorks Inc., Natick, Massachusetts (R2012b)

    Google Scholar 

  18. Hahn, E.M., Han, T., Zhang, L.: Synthesis for PCTL in parametric Markov decision processes. In: NASA Formal Methods, pp. 146–161 (2011)

    Google Scholar 

  19. Jansen, N., Corzilius, F., Volk, M., Wimmer, R., Ábrahám, E., Katoen, J.-P., Becker, B.: Accelerating parametric probabilistic verification. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 404–420. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  20. Filieri, A., Ghezzi, C., Tamburrelli, G.: Run-time efficient probabilistic model checking. In: Proceedings of the 33rd International Conference on Software Engineering, ICSE 2011, pp. 341–350. ACM, New York (2011)

    Google Scholar 

  21. Kozine, I., Utkin, L.V.: Interval-valued finite markov chains. Reliable Computing 8(2), 97–113 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: International conference on Logics in Computer Science (LICS), pp. 266–277 (1991)

    Google Scholar 

  23. Ghorbal, K., Duggirala, P.S., Kahlon, V., Ivančić, F., Gupta, A.: Efficient probabilistic model checking of systems with ranged probabilities. In: Finkel, A., Leroux, J., Potapov, I. (eds.) RP 2012. LNCS, vol. 7550, pp. 107–120. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  24. Schweitzer, P.J.: Perturbation theory and finite Markov chains. Journal of Applied Probability 5(2), 401–413 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cho, G.E., Meyer, C.D.: Comparison of perturbation bounds for the stationary distribution of a Markov chain. Linear Algebra Appl. 335, 137–150 (2000)

    Article  MathSciNet  Google Scholar 

  26. Solan, E., Vieille, N.: Perturbed Markov chains. J. Applied Prob. 40(1), 107–122 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heidergott, B.: Perturbation analysis of Markov chains. In: 9th International Workshop on Discrete Event Systems, WODES 2008, pp. 99–104 (2008)

    Google Scholar 

  28. Murdock, J.A.: Perturbation: Theory and Method. John Wiley & Sons, Inc. (1991)

    Google Scholar 

  29. Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 3rd edn. Addison-Wesley Longman Publishing Co., Inc, Boston (2006)

    Google Scholar 

  30. Verification, Model Checking, and Abstract Interpretation (2004)

    Google Scholar 

  31. Agrawal, M., Akshay, S., Genest, B., Thiagarajan, P.S.: Approximate verification of the symbolic dynamics of markov chains. In: International conference on Logics in Computer Science (LICS), pp. 55–64 (2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, School of Computing, National University of Singapore, Singapore

    Guoxin Su & David S. Rosenblum

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  1. Guoxin Su
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  2. David S. Rosenblum
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Editors and Affiliations

  1. NICTA, Kensington, NSW, Australia

    Franck Cassez

  2. Départament d’Informatique, Université Libre de Bruxelles (U.L.B.), Belgium

    Jean-François Raskin

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Su, G., Rosenblum, D.S. (2014). Nested Reachability Approximation for Discrete-Time Markov Chains with Univariate Parameters. In: Cassez, F., Raskin, JF. (eds) Automated Technology for Verification and Analysis. ATVA 2014. Lecture Notes in Computer Science, vol 8837. Springer, Cham. https://doi.org/10.1007/978-3-319-11936-6_26

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  • DOI: https://doi.org/10.1007/978-3-319-11936-6_26

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11935-9

  • Online ISBN: 978-3-319-11936-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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