Abstract
Abstraction is a key technique to combat the state space explosion problem in model checking probabilistic systems. In this paper we present new ways to abstract Discrete Time Markov Chains (DTMCs), Markov Decision Processes (MDPs), and Continuous Time Markov Chains (CTMCs). The main advantage of our abstractions is that they result in abstract models that are purely probabilistic, which maybe more amenable to automatic analysis than models with both nondeterministic and probabilistic steps that typically arise from previously known abstraction techniques. A key technical tool, developed in this paper, is the construction of least upper bounds for any collection of probability measures. This upper bound construction may be of independent interest that could be useful in the abstract interpretation and static analysis of probabilistic programs.
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References
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Model checking continuous-time Markov chains. ACM TOCL 1, 162–170 (2000)
Baier, C., Haverkrot, B., Hermanns, H., Katoen, J.-P.: Efficient computation of time-bounded reachability probabilities in uniform continuous-time Markov decision processes. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 61–76. Springer, Heidelberg (2004)
Baier, C., Hermanns, H., Katoen, J.-P., Wolf, V.: Comparative branching-time semantics for Markov chains. Inf. and Comp. 200, 149–214 (2005)
Chadha, R., Viswanathan, M., Viswanathan, R.: Least upper bounds for probability measures and their applications to abstractions. Technical Report UIUCDCS-R-2008-2973, UIUC (2008)
D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)
D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and refinement strategies for probabilistic analysis. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002. LNCS, vol. 2399, pp. 57–76. Springer, Heidelberg (2002)
Desharnais, J.: Labelled Markov Processes. PhD thesis, McGill University (1999)
Fecher, H., Leucker, M., Wolf, V.: Don’t know in probabilistic systems. In: Proc. of SPIN, pp. 71–88 (2006)
Huth, M.: An abstraction framework for mixed non-deterministic and probabilistic systems. In: Validation of Stochastic Systems: A Guide to Current Research, pp. 419–444 (2004)
Huth, M.: On finite-state approximants for probabilistic computation tree logic. TCS 346, 113–134 (2005)
Jones, C.: Probabilistic Non-determinism. PhD thesis, University of Edinburgh (1990)
Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: Proc. of LICS, pp. 266–277 (1991)
Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for continuous-time Markov chains. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 311–324. Springer, Heidelberg (2007)
Kwiatkowska, M., Norman, G., Parker, D.: Game-based abstraction for Markov decision processes. In: Proc. of QEST, pp. 157–166 (2006)
McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Springer, Heidelberg (2004)
Monniaux, D.: Abstract interpretation of programs as Markov decision processes. Science of Computer Programming 58, 179–205 (2005)
Norman, G.: Analyzing randomized distributed algorithms. In: Validation of Stochastic Systems: A Guide to Current Research, pp. 384–418 (2004)
Rutten, J.M., Kwiatkowska, M., Norman, G., Parker, D.: Mathematical Techniques for Analyzing Concurrent and Probabilistic Systems. AMS (2004)
Saheb-Djahromi, N.: Probabilistic LCF. In: Winkowski, J. (ed.) MFCS 1978. LNCS, vol. 64, pp. 442–451. Springer, Heidelberg (1978)
Segala, R.: Probability and nondeterminism in operational models of concurrency. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 64–78. Springer, Heidelberg (2006)
Sen, K., Viswanathan, M., Agha, G.: Model checking Markov chains in the presence of uncertainties. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 394–410. Springer, Heidelberg (2006)
Wachter, B., Zhang, L., Hermanns, H.: Probabilistic model checking modulo theories. In: Proc. of QEST (2007)
Younes, H., Kwiatkowska, M., Norman, G., Parker, D.: Numerical vs. statistical probabilistic model checking: An empirical study. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 46–60. Springer, Heidelberg (2004)
Younes, H., Simmons, R.G.: Probabilistic verification of discrete event systems using acceptance sampling. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 223–235. Springer, Heidelberg (2002)
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Chadha, R., Viswanathan, M., Viswanathan, R. (2008). Least Upper Bounds for Probability Measures and Their Applications to Abstractions. In: van Breugel, F., Chechik, M. (eds) CONCUR 2008 - Concurrency Theory. CONCUR 2008. Lecture Notes in Computer Science, vol 5201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85361-9_23
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DOI: https://doi.org/10.1007/978-3-540-85361-9_23
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