Finite horizon analysis of Markov Chains with the Murϕ verifier

  • Giuseppe Della PennaEmail author
  • Benedetto Intrigila
  • Igor Melatti
  • Enrico Tronci
  • Marisa Venturini Zilli
Special section on Recent Advances in Hardware Verification


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 ProbabilisticMurϕ). 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).


Automatic verification Model checking Markov chains Probabilistic model checking Probabilistic verification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bahar, R.I., Frohm, E.A., Gaona, C.M., Hachtel, G.D., Macii, E., Pardo, A., Somenzi, F.: Algebraic decision diagrams and their applications. In: ICCAD ’93: Proceedings of the 1993 IEEE/ACM International Conference on Computer-Aided Design, pp. 188–191. IEEE Computer Society Press, Los Alamitos, CA, USA (1993)Google Scholar
  2. 2.
    Baier, C., Clarke, E.M., Hartonas-Garmhausen, V., Kwiatkowska, M., Ryan, M.: Symbolic model checking for probabilistic processes. In: Degano, P., Gorrieri, P., Marchetti-Spaccamela, A. (eds.) Automata, Languages and Programming, 24th International Colloquium, ICALP’97, Bologna, Italy, Proceedings, vol. 1256 of Lecture Notes in Computer Science, pp. 430–440. Springer, Berlin (1997)Google Scholar
  3. 3.
    Behrends, E.: Introduction to Markov Chains, Vieweg. Germany (2000)zbMATHGoogle Scholar
  4. 4.
    Bianco, de Alfaro: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) Foundations of Software Technology and Theoretical Computer Science, 15th Conference, Bangalore, India, Proceedings, vol. 1026 of Lecture Notes in Computer Science, pp. 499–513. Springer, Berlin (1995)Google Scholar
  5. 5.
    Bobbio, A., Ciancamerla, E., Franceschinis, G., Gaeta, R., Minichino, M., Portinale, L.: Methods of increasing modelling power for safety analysis, applied to a turbine digital control system. In: Anderson, S., Bologna, S., Felici, M. (eds.) Computer Safety, Reliability and Security 21st International Conference, SAFECOMP 2002, Catania, Italy, Proceedings, vol. 2434 of Lecture Notes in Computer Science, pp. 212–223. Springer, Berlin (2002)Google Scholar
  6. 6.
    Bobbio, A., Ciancamerla, E., Gribaudo, M., Horvath, A., Minichino, M., Tronci, E.: Model Checking based on fluid petri nets for the temperature control system of the icaro co-generative Planti. In: Anderson, S., Bologna, S., Felici, M. (eds.) Computer Safety, Reliability and Security, 21st International Conference, SAFECOMP 2002, Catania, Italy, Proceedings, vol. 2434 of Lecture Notes in Computer Science, pp. 273–283. Springer, Berlin (2002)Google Scholar
  7. 7.
    Bobbio, A., Bologna, S., Minichino, M., Ciancamerla, E., Incalcaterra, P., Kropp, C., Tronci, E.: Advanced techniques for safety analysis applied to the gas turbine control system of Icaro co generative plant. In: Proceedings of X Convegno TESEC, Genova, Italy (2001)Google Scholar
  8. 8.
    Bryant, R.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. C-35 (8), 677–691 (1986)Google Scholar
  9. 9.
    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)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Clarke, E.M., McMillan, K.L., Zhao, X., Fujita, M., Yang, J.: 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
  11. 11.
    Courcoubetis, C., Yannakakis, M.: Verifying temporal properties of finite-state probabilistic programs. In: Proceedings of the IEEE Conference on Decision and Control, pp. 338–345. IEEE Press, Piscataway, NJ (1988)Google Scholar
  12. 12.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM. 42(4), 857–907 (1995)CrossRefMathSciNetGoogle Scholar
  13. 13.
    CUDD Web Page: (2004)
  14. 14.
    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
  15. 15.
    Della Penna, G., Intrigila, B., Melatti, I., Minichino, M., Ciancamerla, E., Parisse, A., Tronci, E., Venturini Zilli, M.: Automatic verification of a turbogas control system with the murφ verifier. In: Maler, O., Pnueli, A. (eds.) Hybrid Systems: Computation and Control, 6th International Workshop, HSCC 2003 Prague, Czech Republic, Proceedings, vol. 2623 of Lecture Notes in Computer Science, pp. 141–155. Springer, Berlin (2003)Google Scholar
  16. 16.
    Della Penna, G., Intrigila, B., Melatti, I., Tronci, E., Venturini Zilli, M.: Finite horizon analysis of markov chains with the Murφ verifier. In: Geist, D., Tronci, E. (eds.) Correct Hardware Design and Verification Methods, 12th IFIP WG 10.5 Advanced Research Working Conference, CHARME 2003, L’Aquila, Italy, Proceedings, vol. 2860 of Lecture Notes in Computer Science, pp. 394–409. Springer (2003)Google Scholar
  17. 17.
    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 and Processors, pp. 522–525. IEEE Computer Society, Washington, DC (1992)Google Scholar
  18. 18.
    ENEA: Proprietary ICARO Documentation (2001)Google Scholar
  19. 19.
    Hansson, H.: Time and Probability in Formal Design of Distributed Systems. Elsevier, Amsterdam (1994)Google Scholar
  20. 20.
    Hansson, H., Jonsson, B.: A logic for reasoning about time and probability. Formal Aspects Comput 6(5), 512–535 (1994)CrossRefGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Holzmann,G.J.: Design and Validation of Computer Protocols. Prentice-Hall, Upper Saddle River, NJ (1991)Google Scholar
  23. 23.
    Holzmann, G.J.: The spin model checker. IEEE Trans. Software Eng. 23(5), 279–295, (1997)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J.T., Harder, U. (eds.) Computer Performance Evaluation, Modelling Techniques and Tools 12th International Conference, TOOLS 2002, London, UK, Proceedings, vol. 2324 of Lecture Notes in Computer Science, pp. 200–204. Springer, Berlin (2002)Google Scholar
  25. 25.
    Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. In: Katoen, J.-P., Stevens, P. (eds.) Tools and Algorithms for the Construction and Analysis of Systems, 8th International Conference, TACAS 2002, Held as Part of the Joint European Conference on Theory and Practice of Software, ETAPS 2002, Grenoble, France, April 8–12, 2002, Proceedings, vol. 2280 of Lecture Notes in Computer Science, pp. 52–66. Springer, Berlin (2002)Google Scholar
  26. 26.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Lehmann, D., Rabin, M.: On the advantages of free choice: A symmetric fully distributed solution to the dining philosophers problem (extended abstract). In: Proceedings of 8th Symposium on Principles of Programming Languages, pp. 133–138 (1981)Google Scholar
  28. 28.
    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)Google Scholar
  29. 29.
  30. 30.
    Pnueli, A., Zuck, L.: Verification of multiprocess probabilistic protocols. Distrib. Comput. 1(1), 53–72 (1986)CrossRefGoogle Scholar
  31. 31.
    Pnueli, A., Zuck, L.D.: Probabilistic verification. Inf. Comput. 103(1), 1–29 (1993)CrossRefMathSciNetGoogle Scholar
  32. 32.
  33. 33.
    Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR ’94, Concurrency Theory, 5th International Conference, Uppsala, Sweden, Proceedings, vol. 836 of Lecture Notes in Computer Science, pp. 481–496. Springer, Berlin (1994)Google Scholar
  34. 34.
    SPIN Web Page: (2004)
  35. 35.
    Tronci, E., Della Penna, G., Intrigila, B., Venturini Zilli, M.: Exploiting transition locality in automatic verification. In: Margaria, T., Melham, T.F. (eds.) Correct Hardware Design and Verification Methods, 11th IFIP WG 10.5 Advanced Research Working Conference, CHARME 2001, Livingston, Scotland, UK, Proceedings, vol. 2144 of Lecture Notes in Computer Science, pp. 259–274. Springer, Berlin (2001)Google Scholar
  36. 36.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: 26th Annual Symposium on Foundations of Computer Science, pp. 327–338, IEEE CS Press, Portland, OR (1985)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Giuseppe Della Penna
    • 1
    Email author
  • Benedetto Intrigila
    • 3
  • Igor Melatti
    • 1
  • Enrico Tronci
    • 2
  • Marisa Venturini Zilli
    • 2
  1. 1.Dipartimento di InformaticaUniversità di L’AquilaCoppitoItaly
  2. 2.Dipartimento di InformaticaUniversità di Roma “La Sapienza”RomeItaly
  3. 3.Dipartimento di Matematica Pura ed ApplicataUniversità di Roma “Tor Vergata”RomeItaly

Personalised recommendations