Abstract
This paper presents a symbolic model checking algorithm for continuous-time Markov chains for an extension of the continuous stochastic logic CSL of Aziz et al [1]. The considered logic contains a time-bounded until-operator and a novel operator to express steadystate probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady state-operator) and a Volterra integral equation system for timebounded until. We propose a symbolic approximate method for solving the integrals using MTDDs (multi-terminal decision diagrams), a generalisation of MTBDDs. These new structures are suitable for numerical integration using quadrature formulas based on equally-spaced abscissas, like trapezoidal, Simpson and Romberg integration schemes.
The first and second author are sponsored by the DAAD-Project AZ 313-ARC-XII- 98/38 on stochastic modelling and verification.
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References
A. Aziz, K. Sanwal, V. Singhal and R. Brayton. Verifying continuous time Markov chains. In CAV, LNCS 1102, pp. 269–276, 1996.
A. Aziz, V. Singhal, F. Balarin, R. Brayton and A. Sangiovanni-Vincentelli. It usually works: the temporal logic of stochastic systems. In CAV, LNCS 939, pp. 155–165, 1995.
I. Bahar, E. Frohm, C. Gaona, G. Hachtel, E. Macii, A. Padro and F. Somenzi. Algebraic decision diagrams and their applications. Formal Methods in System Design, 10(2/3):171–206, 1997.
C. Baier. On algorithmic verification methods for probabilistic systems. Habilitation thesis (submitted), Univ. Mannheim, 1998.
C. Baier, E. Clarke, V. Hartonas-Garmhausen, M. Kwiatkowska, and M. Ryan. Symbolic model checking for probabilistic processes. In ICALP, LNCS 1256, pp. 430–440, 1997.
C. Baier and M. Kwiatkowska. Model checking for a probabilistic branching-time logic with fairness. Distr. Comp., 11(3): 125–155, 1998.
D. Beauquier and A. Slissenko. Polytime model checking for timed probabilistic computation tree logic. Acta Inf., 35: 645–664, 1998.
A. Bianco and L. de Alfaro. Model checking of probabilistic and nondeterministic systems. In FSTTCS, LNCS 1026, pp. 499–513, 1995.
R. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Trans. on Comp., C-35(8): 677–691, 1986.
W. Chan, R. Anderson, P. Beame, S. Burns, F. Modugno, D. Notkin and J. D. Reese. Model checking large software specifications. IEEE Trans. on Softw. Eng., 24(7): 498–519, 1998.
I. Christoff and L. Christoff. Reasoning about safety and liveness properties for probabilistic systems. In FSTTCS, LNCS 652, pp 342–355, 1992.
E. Clarke, M. Fujita, P. McGeer, J. Yang and X. Zhao. Multi-terminal binary decision diagrams: an efficient data structure for matrix representation. In Proc. IEEE Int. Workshop on Logic Synthesis, pp. 1–15, 1993.
E. Clarke, O. Grumberg and D. Long. Verification tools for finite-state concurrent programs. In A Decade of Concurrency, LNCS 803, pp. 124–175, 1993.
C. Courcoubetis and M. Yannakakis. Verifying temporal properties of finite-state probabilistic programs. In FOCS, pp. 338–345, 1988.
C. Courcoubetis and M. Yannakakis. The complexity of probabilistic verification. J. ACM, 42(4): 857–907, 1995.
L. de Alfaro. How to specify and verify the long-run average behavior of probabilistic systems. In LICS, 1998.
L. de Alfaro. Stochastic transition systems. In CONCUR, LNCS 1466, pp. 423–438, 1998.
G. Hachtel, E. Macii, A. Padro and F. Somenzi. Markovian analysis of large finitestate machines. IEEE Trans. on Comp. Aided Design of Integr. Circ. and Sys., 15(12): 1479–1493, 1996.
H. Hansson and B. Jonsson. A logic for reasoning about time and probability. Form. Asp. of Comp., 6: 512–535, 1994.
V. Hartonas-Garmhausen, S. Campos and E. M. Clarke. ProbVerus: probabilistic symbolic model checking. In Formal Methods for Real-Time and Probabilistic Systems (ARTS), LNCS 1601, pp. 96–110, 1999.
H. Hermanns. Interactive Markov Chains. Ph.D thesis, U. Erlangen-Nürnberg, 1998.
H. Hermanns and J.-P. Katoen. Automated compositional Markov chain generation for a plain-old telephone system. Sci. of Comp. Programming, 1999.
A. Pnueli, L. Zuck. Probabilistic verification. Inf. and Comp., 103,: 1–29, 1993.
W. Press, B. Flannery, S. Teukolsky and W. Vetterling. Numerical Recipes in C: The Art of Scientific Computing. Cambridge Univ. Press, 1989.
W. Stewart. Introduction to the Numerical Solution of Markov Chains. Princeton Univ. Press, 1994.
K. S. Trivedi, J. K. Muppala, S. P. Woolet, and B. R. Haverkort. Composite performance and dependability analysis. Performance Evaluation, 14: 197–215, 1992.
M. Y. Vardi. Automatic verification of probabilistic concurrent finite state programs. In FOCS, pp 327–338, 1985.
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Baier, C., Katoen, JP., Hermanns, H. (1999). Approximative Symbolic Model Checking of Continuous-Time Markov Chains. In: Baeten, J.C.M., Mauw, S. (eds) CONCUR’99 Concurrency Theory. CONCUR 1999. Lecture Notes in Computer Science, vol 1664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48320-9_12
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DOI: https://doi.org/10.1007/3-540-48320-9_12
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