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A Maximal Entropy Stochastic Process for a Timed Automaton,

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7966)

Abstract

Several ways of assigning probabilities to runs of timed automata (TA) have been proposed recently. When only the TA is given, a relevant question is to design a probability distribution which represents in the best possible way the runs of the TA. This question does not seem to have been studied yet. We give an answer to it using a maximal entropy approach. We introduce our variant of stochastic model, the stochastic process over runs which permits to simulate random runs of any given length with a linear number of atomic operations. We adapt the notion of Shannon (continuous) entropy to such processes. Our main contribution is an explicit formula defining a process Y * which maximizes the entropy. This formula is an adaptation of the so-called Shannon-Parry measure to the timed automata setting. The process Y * has the nice property to be ergodic. As a consequence it has the asymptotic equipartition property and thus the random sampling w.r.t. Y * is quasi uniform.

Keywords

  • Probability Density Function
  • Model Check
  • Maximal Entropy
  • Strong Connectivity
  • Time Automaton

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

An extended version of the present paper containing detailed proofs and examples is available on-line http://hal.archives-ouvertes.fr/hal-00808909 .

The support of Agence Nationale de la Recherche under the project EQINOCS (ANR-11-BS02-004) is gratefully acknowledged.

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References

  1. Algoet, P.H., Cover, T.M.: A sandwich proof of the Shannon-McMillan-Breiman theorem. The Annals of Probability 16(2), 899–909 (1988)

    MathSciNet  MATH  CrossRef  Google Scholar 

  2. Alur, R., Bernadsky, M.: Bounded model checking for GSMP models of stochastic real-time systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 19–33. Springer, Heidelberg (2006)

    CrossRef  Google Scholar 

  3. Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking for probabilistic real-time systems. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, Springer, Heidelberg (1991)

    Google Scholar 

  4. Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126, 183–235 (1994)

    MathSciNet  MATH  CrossRef  Google Scholar 

  5. Asarin, E., Basset, N., Béal, M.-P., Degorre, A., Perrin, D.: Toward a timed theory of channel coding. In: Jurdziński, M., Ničković, D. (eds.) FORMATS 2012. LNCS, vol. 7595, pp. 27–42. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  6. Asarin, E., Degorre, A.: Volume and entropy of regular timed languages: Analytic approach. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 13–27. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  7. Asarin, E., Degorre, A.: Volume and entropy of regular timed languages: Discretization approach. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 69–83. Springer, Heidelberg (2009)

    CrossRef  Google Scholar 

  8. Baier, C., Bertrand, N., Bouyer, P., Brihaye, T., Größer, M.: Probabilistic and topological semantics for timed automata. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 179–191. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  9. Basset, N., Asarin, E.: Thin and thick timed regular languages. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 113–128. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  10. Bernadsky, M., Alur, R.: Symbolic analysis for GSMP models with one stateful clock. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 90–103. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

  11. Bertrand, N., Bouyer, P., Brihaye, T., Markey, N.: Quantitative model-checking of one-clock timed automata under probabilistic semantics. In: QEST, pp. 55–64. IEEE Computer Society (2008)

    Google Scholar 

  12. Billingsley, P.: Probability and measure, vol. 939. Wiley (2012)

    Google Scholar 

  13. Bouyer, P., Brihaye, T., Jurdziński, M., Menet, Q.: Almost-sure model-checking of reactive timed automata. QEST 2012, 138–147 (2012)

    Google Scholar 

  14. Cover, T.M., Thomas, J.A.: Elements of information theory, 2nd edn. Wiley (2006)

    Google Scholar 

  15. David, A., Larsen, K.G., Legay, A., Mikučionis, M., Poulsen, D.B., van Vliet, J., Wang, Z.: Statistical model checking for networks of priced timed automata. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 80–96. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  16. Kempf, J.-F., Bozga, M., Maler, O.: As soon as probable: Optimal scheduling under stochastic uncertainty. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 385–400. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  17. Krasnosel’skij, M.A., Lifshits, E.A., Sobolev, A.V.: Positive Linear Systems: the Method of Positive Operators. Heldermann Verlag, Berlin (1989)

    MATH  Google Scholar 

  18. Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press (1995)

    Google Scholar 

  19. Lothaire, M.: Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications). Cambridge University Press, New York (2005)

    CrossRef  Google Scholar 

  20. Parry, W.: Intrinsic Markov chains. Transactions of the American Mathematical Society, 55–66 (1964)

    Google Scholar 

  21. Shannon, C.E.: A mathematical theory of communication. Bell Sys. Tech. J. 27, 379–423, 623–656 (1948)

    Google Scholar 

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Basset, N. (2013). A Maximal Entropy Stochastic Process for a Timed Automaton,. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7966. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39212-2_9

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  • DOI: https://doi.org/10.1007/978-3-642-39212-2_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39211-5

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