Probabilistic Timed Automata with Clock-Dependent Probabilities

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10506)

Abstract

Probabilistic timed automata are classical timed automata extended with discrete probability distributions over edges. We introduce clock-dependent probabilistic timed automata, a variant of probabilistic timed automata in which transition probabilities can depend linearly on clock values. Clock-dependent probabilistic timed automata allow the modelling of a continuous relationship between time passage and the likelihood of system events. We show that the problem of deciding whether the maximum probability of reaching a certain location is above a threshold is undecidable for clock-dependent probabilistic timed automata. On the other hand, we show that the maximum and minimum probability of reaching a certain location in clock-dependent probabilistic timed automata can be approximated using a region-graph-based approach.

This is a preview of subscription content, log in to check access

Notes

Acknowledgments

The inspiration for cdPTA arose from a discussion with Patricia Bouyer on the corner-point abstraction. Thanks also to Holger Hermanns, who expressed interest in a cdPTA-like formalism in a talk at Dagstuhl Seminar 14441.

References

  1. 1.
    Abate, A., Katoen, J., Lygeros, J., Prandini, M.: Approximate model checking of stochastic hybrid systems. Eur. J. Control 16(6), 624–641 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Akshay, S., Bouyer, P., Krishna, S.N., Manasa, L., Trivedi, A.: Stochastic timed games revisited. In: Proceedings of the 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). LIPIcs, vol. 58, pp. 8:1–8:14. Leibniz-Zentrum für Informatik (2016)Google Scholar
  3. 3.
    Akshay, S., Bouyer, P., Krishna, S.N., Manasa, L., Trivedi, A.: Stochastic timed games revisited. CoRR, abs/1607.05671 (2016)Google Scholar
  4. 4.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoret. Comput. Sci. 126(2), 183–235 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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). doi: 10.1007/978-3-642-24310-3_9 CrossRefGoogle Scholar
  6. 6.
    Bian, G., Abate, A.: On the relationship between bisimulation and trace equivalence in an approximate probabilistic context. In: Esparza, J., Murawski, A.S. (eds.) FoSSaCS 2017. LNCS, vol. 10203, pp. 321–337. Springer, Heidelberg (2017). doi: 10.1007/978-3-662-54458-7_19 CrossRefGoogle Scholar
  7. 7.
    Bouyer, P.: On the optimal reachability problem in weighted timed automata and games. In: Proceedings of the 7th Workshop on Non-Classical Models of Automata and Applications (NCMA 2015). books@ocg.at, vol. 318, pp. 11–36. Austrian Computer Society (2015)Google Scholar
  8. 8.
    Bouyer, P., Brinksma, E., Larsen, K.G.: Optimal infinite scheduling for multi-priced timed automata. Formal Methods Syst. Des. 32(1), 2–23 (2008)CrossRefMATHGoogle Scholar
  9. 9.
    D’Innocenzo, A., Abate, A., Katoen, J.: Robust PCTL model checking. In: Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2012), pp. 275–286. ACM (2012)Google Scholar
  10. 10.
    Gregersen, H., Jensen, H.E.: Formal design of reliable real time systems. Master’s thesis, Department of Mathematics and Computer Science, Aalborg University (1995)Google Scholar
  11. 11.
    Hahn, E.M.: Model checking stochastic hybrid systems. Ph.D. thesis, Universität des Saarlandes (2013)Google Scholar
  12. 12.
    Jurdziński, M., Laroussinie, F., Sproston, J.: Model checking probabilistic timed automata with one or two clocks. Log. Methods Comput. Sci. 4(3), 1–28 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    Kattenbelt, M., Kwiatkowska, M., Norman, G., Parker, D.: A game-based abstraction-refinement framework for Markov decision processes. Formal Methods Syst. Des. 36(3), 246–280 (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov Chains. Graduate Texts in Mathematics, 2nd edn. Springer, New York (1976)CrossRefMATHGoogle Scholar
  15. 15.
    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). doi: 10.1007/978-3-642-22110-1_47 CrossRefGoogle Scholar
  16. 16.
    Kwiatkowska, M., Norman, G., Parker, D., Sproston, J.: Performance analysis of probabilistic timed automata using digital clocks. Formal Methods Syst. Des. 29, 33–78 (2006)CrossRefMATHGoogle Scholar
  17. 17.
    Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theoret. Comput. Sci. 286, 101–150 (2002)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Minsky, M.: Computation: Finite and Infinite Machines. Prentice Hall International, Upper Saddle River (1967)MATHGoogle Scholar
  19. 19.
    Norman, G., Parker, D., Sproston, J.: Model checking for probabilistic timed automata. Formal Methods Syst. Des. 43(2), 164–190 (2013)CrossRefMATHGoogle Scholar
  20. 20.
    Puterman, M.L.: Markov Decision Processes. Wiley, Hoboken (1994)CrossRefMATHGoogle Scholar
  21. 21.
    Segala, R.: Modeling and verification of randomized distributed real-time systems. Ph.D. thesis, Massachusetts Institute of Technology (1995)Google Scholar
  22. 22.
    Sproston, J.: Probabilistic timed automata with clock-dependent probabilities. CoRR (2017)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversity of TurinTurinItaly

Personalised recommendations