Bounded Probabilistic Model Checking with the Murφ Verifier

  • Giuseppe Della Penna
  • Benedetto Intrigila
  • Igor Melatti
  • Enrico Tronci
  • Marisa Venturini Zilli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3312)


In this paper we present an explicit verification algorithm for Probabilistic Systems defining discrete time/finite state Markov Chains. We restrict ourselves to verification of Bounded PCTL formulas(BPCTL), that is, PCTL formulas in which all Until operators arebounded, possibly with different bounds. This means that we consider only paths (system runs) of bounded length. Given a Markov Chain \({\cal M}\) and a BPCTL formula Φ, our algorithm checks if Φ is satisfied in \({\cal M}\). This allows to verify important properties, such as reliability in Discrete Time Hybrid Systems.

We present an implementation of our algorithm within a suitable extension of the Murφ verifier. We call FHP-Murφ (Finite Horizon Probabilistic Murφ) such extension of the Murφ verifier.

We give experimental results comparing FHP-Murφ with (a finite horizon subset of) PRISM, a state-of-the-art symbolic model checker for Markov Chains. Our experimental results show that FHP-Murφ can effectively handle verification of BPCTL formulas for systems that are out of reach for PRISM, namely those involving arithmetic operations on the state variables (e.g. hybrid systems).


Markov Chain Model Check Recursive Call Atomic Proposition Finite Horizon 
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.
    Baier, C., Clarke, E.M., Hartonas-Garmhausen, V., Kwiatkowska, M., 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)Google Scholar
  2. 2.
    Behrends, E.: Introduction to Markov Chains. Vieweg (2000)Google Scholar
  3. 3.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Bryant, R.: Graph-based algorithms for boolean function manipulation. IEEE Trans. on Computers C-35(8), 677–691 (1986)zbMATHCrossRefGoogle Scholar
  5. 5.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: 1020 states and beyond. Inf. Comput. 98(2), 142–170 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
  7. 7.
    Clarke, E.M., McMillan, K.L.: Spectral transforms for large boolean functions with applications to technology mapping. In: Proceedings of the 30th international on Design automation conference, pp. 54–60. ACM Press, New York (1993)Google Scholar
  8. 8.
    Courcoubetis, C., Yannakakis, M.: Verifying temporal properties of finite-state probabilistic programs. In: Proceedings of the IEEE Conference on Decision and Control, Piscataway, NJ, pp. 338–345. IEEE Press, Los Alamitos (1988)Google Scholar
  9. 9.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
  11. 11.
    de Alfaro, L.: Formal verification of performance and reliability of real-time systems. Technical Report STAN-CS-TR-96-1571, Stanford University (1996)Google Scholar
  12. 12.
    Dill, D.L., Drexler, A.J., Hu, A.J., Yang, C.H.: Protocol verification as a hardware design aid. In: Proceedings of the 1991 IEEE International Conference on Computer Design on VLSI in Computer & Processors, pp. 522–525. IEEE Computer Society, Los Alamitos (1992)CrossRefGoogle Scholar
  13. 13.
  14. 14.
    Jonsson, B., Hansson, H.: A logic for reasoning about time and probability. Formal Aspects of Computing 6(5), 512–535 (1994)zbMATHCrossRefGoogle Scholar
  15. 15.
    Hansson, H.: Time and Probability in Formal Design of Distributed Systems. Elsevier, Amsterdam (1994)Google Scholar
  16. 16.
    Hart, S., Sharir, M.: Probabilistic temporal logics for finite and bounded models. In: Proceedings of the sixteenth annual ACM symposium on Theory of computing, pp. 1–13. ACM Press, New York (1984)CrossRefGoogle Scholar
  17. 17.
    Hermanns, H., Katoen, J.-P., Meyer-Kayser, J., Siegle, M.: A tool for modelchecking markov chains. Software Tools for Technology Transfer 4(2), 153–172 (2003)CrossRefGoogle Scholar
  18. 18.
    Holzmann, G.J.: Design and Validation of Computer Protocols. Prentice Hall, New Jersey (1991)Google Scholar
  19. 19.
    Holzmann, G.J.: The spin model checker. IEEE Trans. on Software Engineering 23(5), 279–295 (1997)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, pp. 200–204. Springer, Heidelberg (2002)Google Scholar
  21. 21.
    Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 52–66. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lehmann, D., Rabin, M.: On the advantages of free choice: A symmetric fully distributed solution to the dining philosophers problem (extended abstract). In: Proc. 8th Symposium on Principles of Programming Languages, pp. 133–138 (1981)Google Scholar
  24. 24.
    Lynch, N., Saias, I., Segala, R.: Proving time bounds for randomized distributed algorithms. In: Proceedings of the thirteenth annual ACM symposium on Principles of distributed computing, pp. 314–323. ACM Press, New York (1994)CrossRefGoogle Scholar
  25. 25.
  26. 26.
    Della Penna, G., Intrigila, B., Melatti, I., Minichino, M., Ciancamerla, E., Parisse, A., Tronci, E., Zilli, M.V.: Automatic verification of a turbogas control system with the murϕ verifier. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 141–155. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  27. 27.
    Della Penna, G., Intrigila, B., Melatti, I., Tronci, E., Zilli, M.V.: G. Della Penna, B. Intrigila, I. Melatti, E. Tronci, and M. V. Zilli. Finite horizon analysis of markov chains with the murϕ verifier. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 394–409. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  28. 28.
    Della Penna, G., Intrigila, B., Melatti, I., Tronci, E., Zilli, M.V.: Finite horizon analysis of stochastic systems with the murϕ verifier. In: Blundo, C., Laneve, C. (eds.) ICTCS 2003. LNCS, vol. 2841, pp. 58–71. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  29. 29.
    Pnueli, A., Zuck, L.: Verification of multiprocess probabilistic protocols. Distrib. Comput. 1(1), 53–72 (1986)zbMATHCrossRefGoogle Scholar
  30. 30.
  31. 31.
    Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  32. 32.
    Spin web page:
  33. 33.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, pp. 327–338. IEEE CS Press, Los Alamitos (1985)CrossRefGoogle Scholar
  34. 34.
    Clarke, E.M., Hartonas-Garmhausen, V., Aguiar Campos, S.V.: Probverus: Probabilistic symbolic model checking. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, pp. 96–110. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Giuseppe Della Penna
    • 1
  • Benedetto Intrigila
    • 1
  • Igor Melatti
    • 1
  • Enrico Tronci
    • 2
  • Marisa Venturini Zilli
    • 2
  1. 1.Dip. di InformaticaUniversità di L’Aquila, CoppitoL’AquilaItaly
  2. 2.Dip. di InformaticaUniversità di Roma “La Sapienza”RomaItaly

Personalised recommendations