Model checking of probabilistic and nondeterministic systems

  • Andrea Bianco
  • Luca de Alfaro
Temporal Logies and Verification Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1026)


The temporal logics pCTL and pCTL* have been proposed as tools for the formal specification and verification of probabilistic systems: as they can express quantitative bounds on the probability of system evolutions, they can be used to specify system properties such as reliability and performance. In this paper, we present model-checking algorithms for extensions of pCTL and pCTL* to systems in which the probabilistic behavior coexists with nondeterminism, and show that these algorithms have polynomial-time complexity in the size of the system. This provides a practical tool for reasoning on the reliability and performance of parallel systems.


Model Check Temporal Logic Temporal Formula Nondeterministic Choice Finite Markov Chain 
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  1. 1.
    R. Alur, C. Courcoubetis, and D. Dill. Verifying automata specifications of probabilistic real-time systems. In Real Time: Theory in Practice, Lecture Notes in Computer Science 600, pages 28–44. Springer-Verlag, 1992.Google Scholar
  2. 2.
    A. Aziz, V. Singhal, F. Balarin, R.K. Brayton, and A.L. Sangiovanni-Vincentelli. It usually works: The temporal logic of stochastic systems. In Computer Aided Verification, 7th International Workshop, volume 939 of Lect. Notes in Comp. Sci. Springer-Verlag, 1995.Google Scholar
  3. 3.
    E. Chang, Z. Manna, and A. Pnueli. The safety-progress classification. In Logic, Algebra, and Computation, NATO ASI Series, Subseries F: Computer and System Sciences. Springer-Verlag, Berlin, 1992.Google Scholar
  4. 4.
    E.M. Clarke, E.A. Emerson, and A.P. Sistla. Automatic verification of finite state concurrent systems using temporal logic. In Proc. 10th ACM Symp. Princ. of Prog. Lang., 1983.Google Scholar
  5. 5.
    C. Courcoubetis and M. Yannakakis. Verifying temporal properties of finite-state probabilistic programs. In Proc. 29th IEEE Symp. Found. of Comp. Sci., 1988.Google Scholar
  6. 6.
    E.A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume E, chapter 16, pages 995–1072. Elsevier Science Publishers (North-Holland), Amsterdam, 1990.Google Scholar
  7. 7.
    E.A. Emerson and C.L. Lei. Modalities for model checking: Branching time strikes back. In Proc. 12th ACM Symp. Princ. of Prog. Lang., pages 84–96, 1985.Google Scholar
  8. 8.
    E.A. Emerson and A.P. Sistla. Deciding branching time logic. In Proc. 16th ACM Symp. Theory of Comp., pages 14–24, 1984.Google Scholar
  9. 9.
    H. Hansson. Time and Probabilities in Formal Design of Distributed Systems. Real-Time Safety Critical Systems. Elsevier, 1994.Google Scholar
  10. 10.
    H. Hansson and B. Jonsson. A framework for reasoning about time and reliability. In Proc. of Real Time Systems Symposium, pages 102–111. IEEE, 1989.Google Scholar
  11. 11.
    H. Hansson and B. Jonsson. A logic for reasoning about time and probability. Formal Aspects of Computing, 6(5):512–535, 1994.Google Scholar
  12. 12.
    S. Hart and M. Sharir. Probabilistic temporal logic for finite and bounded models. In Proc. 16th ACM Symp. Theory of Comp., pages 1–13, 1984.Google Scholar
  13. 13.
    J.G. Kemeny, J.L. Snell, and A.W. Knapp. Denumerable Markov Chains. D. Van Nostrand Company, 1966.Google Scholar
  14. 14.
    D. Lehman and S. Shelah. Reasoning with time and chance. Information and Control, 53(3):165–198, 1982.Google Scholar
  15. 15.
    O. Lichtenstein, A. Pnueli, and L. Zuck. The glory of the past. In Proc. Conf. Logics of Programs, volume 193 of Lect. Notes in Comp. Sci., pages 196–218. Springer-Verlag, 1985.Google Scholar
  16. 16.
    O. Maler and A. Pnueli. Tight bounds on the complexity of cascaded decomposition of automata. In Proc. 31th IEEE Symp. Found. of Comp. Sci., pages 672–682, 1990.Google Scholar
  17. 17.
    Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer-Verlag, New York, 1991.Google Scholar
  18. 18.
    A. Pnueli. On the extremely fair treatment of probabilistic algorithms. In Proc. 15th ACM Symp. Theory of Comp., pages 278–290, 1983.Google Scholar
  19. 19.
    A. Pnueli and L. Zuck. Probabilistic verification by tableaux. In Proc. First IEEE Symp. Logic in Comp. Sci., pages 322–331, 1986.Google Scholar
  20. 20.
    A. Pnueli and L.D. Zuck. Probabilistic verification. Information and Computation, 103:1–29, 1993.Google Scholar
  21. 21.
    A. Schrijver. Theory of Linear and Integer Programming. J. Wiley & Sons, 1987.Google Scholar
  22. 22.
    M.Y. Vardi. Automatic verification of probabilistic concurrent finite-state systems. In Proc. 26th IEEE Symp. Found. of Comp. Sci., pages 327–338, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Andrea Bianco
    • 1
  • Luca de Alfaro
    • 2
  1. 1.Politecnico di TorinoItaly
  2. 2.Stanford UniversityUSA

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