Finite Horizon Analysis of Markov Chains with the Murϕ Verifier

  • Giuseppe Della Penna
  • Benedetto Intrigila
  • Igor Melatti
  • Enrico Tronci
  • Marisa Venturini Zilli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2860)


In this paper we present an explicit disk based verification algorithm for Probabilistic Systems defining discrete time/finite state Markov Chains. Given a Markov Chain and an integer k (horizon), our algorithm checks whether the probability of reaching an error state in at most k steps is below a given threshold.

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

We present 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 handle systems that are out of reach for PRISM, namely those involving arithmetic operations on the state variables (e.g. hybrid systems).


  1. 1.
    Baier, C., Clarke, E.M., Hartonas-Garmhausen, V., Kwiatkowska, M., Ryan, M.: Symbolic model checking for probabilistic processes. Automata, Languages and Programming, 430–440 (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) (August 1986)Google 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. Information and Computation, 98 (1992)Google Scholar
  6. 6.
    Clarke, E.M., McMillan, K.L., Zhao, X., Fujita, M., Yang, J.: Spectral transforms for large boolean functions with applications to technology mapping. In: Proc. 30th ACM/IEEE Design Automation Conference, pp. 54–60 (1993)Google Scholar
  7. 7.
  8. 8.
    Courcoubetis, C., Yannakakis, M.: Verifying temporal properties of finite-state probabilistic programs. In: Proc. of FOCS 1988, pp. 338–345. IEEE CS Press, Los Alamitos (1988)Google Scholar
  9. 9.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42(4), 857–907 (1995)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    de Alfaro, L.: Formal verification of performance and reliability of real-time systems. Technical report, Stanford University (1996)Google Scholar
  11. 11.
    Dill, D.L., Drexler, A.J., Hu, A.J., Yang, C.H.: Protocol verification as a hardware design aid. In: IEEE International Conference on Computer Design: VLSI in Computers and Processors, pp. 522–525 (1992)Google Scholar
  12. 12.
    Hansson, H.: Time and Probability in Formal Design of Distributed Systems. Elsevier, Amsterdam (1994)Google Scholar
  13. 13.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and probability. Formal Aspects of Computing 6, 512–535 (1994)MATHCrossRefGoogle Scholar
  14. 14.
    Hart, S., Sharir, M.: Probabilistic temporal logic for finite and bounded models. In: Proc. of 16th ACM Symposium on Theory of Computing, pp. 1–13. ACM, New York (1984)Google Scholar
  15. 15.
    Holzmann, G.J.: Design and Validation of Computer Protocols. Prentice Hall, New Jersey (1991)Google Scholar
  16. 16.
    Holzmann, G.J.: The spin model checker. IEEE Trans. on Software Engineering 23(5), 279–295 (1997)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Kemper, P. (ed.) Proc. Tools Session of Aachen 2001 International Multiconference on Measurement, Modelling and Evaluation of Computer-Communication Systems, September 2001, pp. 7–12 (2001); Available as Technical Report 760/2001, University of DortmundGoogle Scholar
  18. 18.
    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, p. 52. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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
  21. 21.
  22. 22.
    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 murphi verifier. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 141–155. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Pnueli, A., Zuck, L.: Verification of multiprocess probabilistic protocols. Distributed Computing 1(1), 53–72 (1986)MATHCrossRefGoogle Scholar
  24. 24.
    Pnueli, A., Zuck, L.: Probabilistic verification. Information and Computation 103, 1–29 (1993)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
  26. 26.
    Lynchand, N., Saias, I., Segala, R.: Proving time bounds for randomized distributed algorithms. In: Proc. 13th ACM Symposium on Principles of Distributed Computing, pp. 314–323 (1994)Google Scholar
  27. 27.
    Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 381–496. Springer, Heidelberg (1994)Google Scholar
  28. 28.
  29. 29.
    Stern, U., Dill, D.L.: Improved probabilistic verification by hash compaction. In: Camurati, P.E., Eveking, H. (eds.) CHARME 1995. LNCS, vol. 987, pp. 206–224. Springer, Heidelberg (1995)Google Scholar
  30. 30.
    Stern, U., Dill, D.L.: A new scheme for memory-efficient probabilistic verification. In: IFIP TC6/WG6.1 Joint International Conference on: Formal Description Techniques for Distributed Systems and Communication Protocols, and Protocol Specification, Testing, and Verification (1996)Google Scholar
  31. 31.
    Tronci, E., Della Penna, G., Intrigila, B., Venturini Zilli, M.: Exploiting transition locality in automatic verification. In: Margaria, T., Melham, T.F. (eds.) CHARME 2001. LNCS, vol. 2144, p. 259. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  32. 32.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. of FOCS 1985, pp. 327–338. IEEE CS Press, Los Alamitos (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

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 Informatica Università di Roma “La Sapienza”RomaItaly

Personalised recommendations