Model Checking Probabilistic Timed Automata with One or Two Clocks

  • Marcin Jurdziński
  • François Laroussinie
  • Jeremy Sproston
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4424)


Probabilistic timed automata are an extension of timed automata with discrete probability distributions. We consider model-checking algorithms for the subclasses of probabilistic timed automata which have one or two clocks. Firstly, we show that Pctl probabilistic model-checking problems (such as determining whether a set of target states can be reached with probability at least 0.99 regardless of how nondeterminism is resolved) are PTIME-complete for one clock probabilistic timed automata, and are EXPTIME-complete for probabilistic timed automata with two clocks. Secondly, we show that the model-checking problem for the probabilistic timed temporal logic Ptctl is EXPTIME-complete for one clock probabilistic timed automata. However, the corresponding model-checking problem for the subclass of Ptctl which does not permit both (1) punctual timing bounds, which require the occurrence of an event at an exact time point, and (2) comparisons with probability bounds other than 0 or 1, is PTIME-complete.


  1. 1.
    Abdulla, P.A., et al.: Decidability and complexity results for timed automata via channel machines. In: Caires, L., et al. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1089–1101. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    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, pp. 115–136. Springer, Heidelberg (1991)Google Scholar
  3. 3.
    Alur, R., Courcoubetis, C., Dill, D.L.: Model-checking in dense real-time. Inf. and Comp. 104(1), 2–34 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Alur, R., Dill, D.L.: A theory of timed automata. Theo. Comp. Sci. 126(2), 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baier, C., et al.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. on Soft. Enginee. 29(6), 524–541 (2003)CrossRefGoogle Scholar
  6. 6.
    Baier, C., Kwiatkowska, M.: Model checking for a probabilistic branching time logic with fairness. Distributed Computing 11(3), 125–155 (1998)CrossRefGoogle Scholar
  7. 7.
    Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)Google Scholar
  8. 8.
    Clarke, E.M., Grumberg, O., Peled, D.: Model checking. MIT Press, Cambridge (1999)Google Scholar
  9. 9.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    de Alfaro, L.: Formal verification of probabilistic systems. PhD thesis, Stanford University, Department of Computer Science (1997)Google Scholar
  11. 11.
    de Alfaro, L.: Temporal logics for the specification of performance and reliability. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 165–176. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. 12.
    Hansson, H.A., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)zbMATHCrossRefGoogle Scholar
  13. 13.
    Jensen, H.E.: Model checking probabilistic real time systems. In: Proc. of the 7th Nordic Work. on Progr. Theory, pp. 247–261. Chalmers Institute of Technology (1996)Google Scholar
  14. 14.
    Kwiatkowska, M., et al.: Performance analysis of probabilistic timed automata using digital clocks. Formal Meth. in Syst. Design 29, 33–78 (2006)zbMATHCrossRefGoogle Scholar
  15. 15.
    Kwiatkowska, M., et al.: Automatic verification of real-time systems with discrete probability distributions. Theo. Comp. Sci. 286, 101–150 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Laroussinie, F., Markey, N., Oreiby, G.: Model checking timed ATL for durational concurrent game structures. In: Asarin, E., Bouyer, P. (eds.) FORMATS 2006. LNCS, vol. 4202, pp. 245–259. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Laroussinie, F., Markey, N., Schnoebelen, P.: Model checking timed automata with one or two clocks. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 387–401. Springer, Heidelberg (2004)Google Scholar
  18. 18.
    Laroussinie, F., Markey, N., Schnoebelen, P.: Efficient timed model checking for discrete-time systems. Theo. Comp. Sci. 353(1–3), 249–271 (2005)MathSciNetGoogle Scholar
  19. 19.
    Laroussinie, F., Sproston, J.: Model checking durational probabilistic systems. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 140–154. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Laroussinie, F., Sproston, J.: State explosion in almost-sure probabilistic reachability. To appear in IPL (2007)Google Scholar
  21. 21.
    Lasota, S., Walukiewicz, I.: Alternating timed automata. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 299–314. Springer, Heidelberg (2005)Google Scholar
  22. 22.
    Papadimitriou, C., Tsitsiklis, J.: The complexity of Markov decision processes. Mathematics of Operations Research 12(3), 441–450 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Stoelinga, M.: Alea Jacta est: Verification of probabilistic, real-time and parametric systems. PhD thesis, Institute for Computing and Information Sciences, University of Nijmegen (2002)Google Scholar
  24. 24.
    Tripakis, S., Yovine, S., Bouajjani, A.: Checking timed Büchi automata emptiness efficiently. Formal Meth. in Syst. Design 26(3), 267–292 (2005)zbMATHCrossRefGoogle Scholar
  25. 25.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. of the 16th An. Symp. on Foundations of Computer Science (FOCS’85), pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Marcin Jurdziński
    • 1
  • François Laroussinie
    • 2
  • Jeremy Sproston
    • 3
  1. 1.Department of Computer Science, University of Warwick, Coventry CV4 7ALUK
  2. 2.Lab. Spécification & Verification, ENS Cachan – CNRS UMR 8643France
  3. 3.Dipartimento di Informatica, Università di Torino, 10149 TorinoItaly

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