Abstract
We give a discretization technique that allows one to check reachability properties in a family of continuous-state processes. We consider a sub-family of labelled Markov processes (LMP), whose transitions can be defined by uniform distributions, and simple reachability formulas.
The key of the discretization is the use of the mean-value theorem to construct, for a family of LMPs and reachability properties, a (finite) Markov decision process (MDP) equivalent to the initial (potentially infinite) LMP with respect to the formula. On the MDP obtained, we can apply known algorithms and tools for probabilistic systems with finite or countable state space. The MDP is constructed in such a way that the LMP satisfies the reachability property if and only if the MDP also satisfies it. Theoretically, our approach gives a precise final result. In practice, this is not the case, of course, but we bound the error on the formula with respect to the errors that can be introduced in the computation of the MDP. We also establish a bisimulation relation between the latter and the theoretical MDP.
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Notes
- 1.
See [4] for a comparison between hybrid systems and LMPs on \(\mathbb R\) with a finite list of transitions (similar but a little more general than LMP\(_U\)’s).
- 2.
The supremum is taken for every f over the interval of \(\mathcal {S}\) on which it is defined.
References
Abate, A., Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic model checking of labelled Markov processes via finite approximate bisimulations. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden. LNCS, vol. 8464, pp. 40–58. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06880-0_2
de Alfaro, L.: Formal verification of probabilistic systems. Ph.D. thesis, Stanford University. Technical Report STAN-CS-TR-98-1601 (1997)
Amin, S., Abate, A., Prandini, M., Lygeros, J., Sastry, S.: Reachability analysis for controlled discrete time stochastic hybrid systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 49–63. Springer, Heidelberg (2006). https://doi.org/10.1007/11730637_7
Assouramou, J., Desharnais, J.: Continuous time and/or continuous distributions. In: Aldini, A., Bernardo, M., Bononi, L., Cortellessa, V. (eds.) EPEW 2010. LNCS, vol. 6342, pp. 99–114. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15784-4_7
Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Software Eng. (2003)
Baier, C., Katoen, J.-P.: Principles of Model Checking (Representation and Mind Series). The MIT Press, Cambridge (2008)
Blute, R., Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. In: Proceedings of the Twelfth IEEE Symposium on Logic in Computer Science (LICS), Warsaw, Poland, Test-of-time award in 2017 (1997)
Comenetz, M.: Calculus: The Elements, 1st edn. World Scientific, Singapore (2002)
Danos, V., Desharnais, J., Panangaden, P.: Labelled Markov processes: stronger and faster approximations. Electron. Notes Theor. Comput. Sci. 87, 157–203 (2004)
Desharnais, J., Laviolette, F., Tracol, M.: Approximate analysis of probabilistic processes: logic, simulation and games. In: Fifth International Conference on the Quantitative Evaluaiton of Systems, QEST: 14–17 September 2008, Saint-Malo, France (2008)
Desharnais, J., Panangaden, P., Jagadeesan, R., Gupta, V.: Approximating labeled Markov processes. In: Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science, LICS 2000, p. 95 (2000)
Desharnais, J., Abbas, E., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labeled Markov processes. Inf. Comput. 184(1), 160–200 (2003)
Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects Comput. 6(5), 512–535 (1994)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_47
Kwiatkowska, M., Norman, G., Sproston, J.: Symbolic computation of maximal probabilistic reachability. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 169–183. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44685-0_12
LSFM: CISMO. http://www.ift.ulaval.ca/~jodesharnais/cismo/. Accessed 14 Aug 2018
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite state programs. In: Proceedings of the 26th Annual Symposium on Foundations of Computer Science, SFCS 1985, pp. 327–338. IEEE Computer Society, Washington, DC (1985). https://doi.org/10.1109/SFCS.1985.12
Acknowledgements
The authors thank the reviewers for their helpful comments. This research has been supported by NSERC grants RGPIN-239294 and RGPIN-262067.
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Kouko, G., Desharnais, J., Laviolette, F. (2019). Finite Approximation of LMPs for Exact Verification of Reachability Properties. In: Parker, D., Wolf, V. (eds) Quantitative Evaluation of Systems. QEST 2019. Lecture Notes in Computer Science(), vol 11785. Springer, Cham. https://doi.org/10.1007/978-3-030-30281-8_5
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