Abstract
Computing the reachability probability in infinite state probabilistic models has been the topic of numerous works. Here we introduce a new property called divergence that when satisfied allows to compute reachability probabilities up to an arbitrary precision. One of the main interest of divergence is that our algorithm does not require the reachability problem to be decidable. Then we study the decidability of divergence for probabilistic versions of pushdown automata and Petri nets where the weights associated with transitions may also depend on the current state. This should be contrasted with most of the existing works that assume weights independent of the state. Such an extended framework is motivated by the modeling of real case studies. Moreover, we exhibit some divergent subclasses of channel systems and pushdown automata, particularly suited for specifying open distributed systems and networks prone to performance collapsing in order to compute the probabilities related to service requirements.
This work has been supported by ANR project BRAVAS (ANR-17-CE40-0028) with Alain Finkel and ANR project MAVeriQ (ANR-20-CE25-0012) with Serge Haddad.
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Notes
- 1.
Surprisingly, in 1972, to the best of our knowledge, Santos gave the first definition of probabilistic pushdown automata [18] that did not open up a new field of research at the time.
References
Abdulla, P.A., Bertrand, N., Rabinovich, A.M., Schnoebelen, P.: Verification of probabilistic systems with faulty communication. Inf. Comput. 202(2), 141–165 (2005)
Abdulla, P.A., Henda, N.B., Mayr, R.: Decisive Markov chains. Log. Methods Comput. Sci. 3(4) (2007)
Baier, C., Katoen, J.: Principles of Model Checking. MIT Press, Heidelberg (2008)
Brázdil, T., Chatterjee, K., Kucera, A., Novotný, P., Velan, D., Zuleger, F.: Efficient algorithms for asymptotic bounds on termination time in VASS. In: Dawar, A., Grädel, E. (eds.) Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, pp. 185–194. ACM (2018)
Brázdil, T., Esparza, J., Kiefer, S., Kucera, A.: Analyzing probabilistic pushdown automata. Formal Methods Syst. Des. 43(2), 124–163 (2013)
Brázdil, T., Esparza, J., Kucera, A.: Analysis and prediction of the long-run behavior of probabilistic sequential programs with recursion (extended abstract). In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 521–530. IEEE Computer Society (2005)
Brázdil, T., Kiefer, S., Kučera, A.: Efficient analysis of probabilistic programs with an unbounded counter. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 208–224. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_18
Brázdil, T., Kiefer, S., Kucera, A.: Efficient analysis of probabilistic programs with an unbounded counter. J. ACM 61(6), 41:1–41:35 (2014)
Brázdil, T., Kiefer, S., Kucera, A., Novotný, P., Katoen, J.: Zero-reachability in probabilistic multi-counter automata. In: Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS 2014, pp. 22:1–22:10. ACM (2014)
Esparza, J., Kucera, A., Mayr, R.: Model checking probabilistic pushdown automata. In: Proceedings of 19th IEEE Symposium on Logic in Computer Science (LICS), pp. 12–21. IEEE Computer Society (2004)
Esparza, J., Kucera, A., Mayr, R.: Quantitative analysis of probabilistic pushdown automata: expectations and variances. In: Proceedings of the 20th IEEE Symposium on Logic in Computer Science (LICS), pp. 117–126. IEEE Computer Society (2005)
Esparza, J., Kucera, A., Mayr, R.: Model checking probabilistic pushdown automata. Logical Methods Comput. Sci. 2(1) (2006)
Finkel, A., Haddad, S., Ye, L.: About decisiveness of dynamic probabilistic models. CoRR abs/2305.19564 (2023). https://doi.org/10.48550/arXiv.2305.19564, to appear in CONCUR’23
Finkel, A., Haddad, S., Ye, L.: Introducing divergence for infinite probabilistic models. CoRR abs/2308.08842 (2023). https://doi.org/10.48550/arXiv.2308.08842
Iyer, P., Narasimha, M.: Probabilistic lossy channel systems. In: Bidoit, M., Dauchet, M. (eds.) CAAP 1997. LNCS, vol. 1214, pp. 667–681. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0030633
Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains, 2nd edn. Springer, Heidelberg (1976). https://doi.org/10.1007/978-1-4684-9455-6
Kucera, A., Esparza, J., Mayr, R.: Model checking probabilistic pushdown automata. Log. Methods Comput. Sci. 2(1) (2006). https://doi.org/10.2168/LMCS-2(1:2)2006
Santos, E.S.: Probabilistic grammars and automata. Inf. Control. 21(1), 27–47 (1972). https://doi.org/10.1016/S0019-9958(72)90026-5
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: Proceedings of the 26th Annual Symposium on Foundations of Computer Science, pp. 327–338. IEEE Computer Society (1985)
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Finkel, A., Haddad, S., Ye, L. (2023). Introducing Divergence for Infinite Probabilistic Models. In: Bournez, O., Formenti, E., Potapov, I. (eds) Reachability Problems. RP 2023. Lecture Notes in Computer Science, vol 14235. Springer, Cham. https://doi.org/10.1007/978-3-031-45286-4_10
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