On Decision Problems for Probabilistic Büchi Automata

  • Christel Baier
  • Nathalie Bertrand
  • Marcus Größer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4962)

Abstract

Probabilistic Büchi automata (PBA) are finite-state acceptors for infinite words where all choices are resolved by fixed distributions and where the accepted language is defined by the requirement that the measure of the accepting runs is positive. The main contribution of this paper is a complementation operator for PBA and a discussion on several algorithmic problems for PBA. All interesting problems, such as checking emptiness or equivalence for PBA or checking whether a finite transition system satisfies a PBA-specification, turn out to be undecidable. An important consequence of these results are several undecidability results for stochastic games with incomplete information, modelled by partially-observable Markov decision processes and ω-regular winning objectives. Furthermore, we discuss an alternative semantics for PBA where it is required that almost all runs for an accepted word are accepting, which turns out to be less powerful, but has a decidable emptiness problem.

References

  1. [BG05]
    Baier, C., Größer, M.: Recognizing ω-regular languages with probabilistic automata. In: Proc. 20th IEEE Symp. on Logic in Computer Science (LICS 2005), pp. 137–146. IEEE Computer Society Press, Los Alamitos (2005)CrossRefGoogle Scholar
  2. [BRV04]
    Bustan, D., Rubin, S., Vardi, M.: Verifying ω-regular properties of Markov chains. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 189–201. Springer, Heidelberg (2004)Google Scholar
  3. [Cas98]
    Cassandra, A.R.: A survey of POMD applications. Presented at the AAAI Fall Symposium (1998), http://pomdp.org/pomdp/papers/applications.pdf
  4. [CY95]
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)MATHCrossRefMathSciNetGoogle Scholar
  5. [dA99]
    de Alfaro, L.: The verification of probabilistic systems under memoryless partial-information policies is hard. In: Proc. Workshop on Probabilistic Methods in Verification (ProbMiV 1999), Birmingham University, Research Report CSR-99-9, pp. 19–32 (1999)Google Scholar
  6. [GD07]
    Giro, S., D’Argenio, P.R.: Quantitative model checking revisited: neither decidable nor approximable. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 179–194. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. [GTW02]
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  8. [HSP83]
    Hart, S., Sharir, M., Pnueli, A.: Termination of probabilistic concurrent programs. ACM Transactions on Programming Languages and Systems 5(3), 356–380 (1983)MATHCrossRefGoogle Scholar
  9. [KSK66]
    Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov chains. D. Van Nostrand Co (1966)Google Scholar
  10. [Kul95]
    Kulkarni, V.G.: Modeling and Analysis of Stochastic Systems. Chapman & Hall, Boca Raton (1995)MATHGoogle Scholar
  11. [MHC03]
    Madani, O., Hanks, S., Condon, A.: On the undecidability of probabilistic planning and related stochastic optimization problems. Artificial Intelligence 147(1–2), 5–34 (2003)MATHMathSciNetGoogle Scholar
  12. [Paz71]
    Paz, A.: Introduction to probabilistic automata. Academic Press Inc., London (1971)MATHGoogle Scholar
  13. [PP04]
    Perrin, D., Pin, J.-É.: Infinite Words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  14. [Put94]
    Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley and Sons, Chichester (1994)MATHGoogle Scholar
  15. [Rab63]
    Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)CrossRefGoogle Scholar
  16. [Rei84]
    Reif, J.H.: The complexity of two-player games of incomplete information. Journal of Computer System Sciences 29(2), 274–301 (1984)MATHCrossRefMathSciNetGoogle Scholar
  17. [Saf88]
    Safra, S.: On the complexity of omega-automata. In: Proc. 29th Symposium on Foundations of Computer Science (FOCS 1988), pp. 319–327. IEEE Computer Society Press, Los Alamitos (1988)CrossRefGoogle Scholar
  18. [Ste94]
    Stewart, W.J.: Introduction to the numerical solution of Markov Chains. Princeton University Press, Princeton (1994)MATHGoogle Scholar
  19. [Tho90]
    Thomas, W.: Automata on infinite objects. In: Handbook of Theoretical Computer Science, vol. B, ch. 4, pp. 133–191. Elsevier, Amsterdam (1990)Google Scholar
  20. [Var85]
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state programs. In: Proc. 26th Symposium on Foundations of Computer Science (FOCS 1985), pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christel Baier
    • 1
  • Nathalie Bertrand
    • 1
    • 2
  • Marcus Größer
    • 1
  1. 1.Technische Universität DresdenGermany
  2. 2.IRISA/INRIA RennesFrance

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