The Effect of Tossing Coins in Omega-Automata

  • Christel Baier
  • Nathalie Bertrand
  • Marcus Größer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5710)

Abstract

In this paper we provide a summary of the fundamental properties of probabilistic automata over infinite words. Such probabilistic automata are a variant of standard automata with Büchi or other ω-regular acceptance conditions, such as Rabin, Streett, parity or Müller, where the nondeterministic choices are resolved probabilistically. Acceptance of an infinite input word can be defined in different ways: by requiring that (i) almost all runs are accepting, or (ii) the probability for the accepting runs is positive, or (iii) the probability measure of the accepting runs is beyond a certain threshold. Surprisingly, even the qualitative criteria (i) and (ii) yield a different picture concerning expressiveness, efficiency, and decision problems compared to the nondeterministic case.

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References

  1. 1.
    Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: Proc. of the 39th Symposium on Foundations of Computer Science (FOCS 1998). IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  2. 2.
    Baier, C., Bertrand, N., Größer, M.: On decision problems for probabilistic büchi automata. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 287–301. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Grösser, M.: Recognizing ω-regular languages with probabilistic automata. In: Proc. of the 20th IEEE Symposium on Logic in Computer Science (LICS 2005), pp. 137–146. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  4. 4.
    Chadha, R., Sistla, A.P., Viswanathan, M.: On the expressiveness and complexity of randomization in finite state monitors. In: Proc. of the 23rd IEEE Symposium on Logic in Computer Science (LICS 2008), pp. 18–29. IEEE Computer Society Press, Los Alamitos (2008)CrossRefGoogle Scholar
  5. 5.
    Chadha, R., Sistla, A.P., Viswanathan, M.: Power of randomization in automata on infinite strings. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 229–243. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Chatterjee, K., Doyen, L., Henzinger, T.A., Raskin, J.-F.: Algorithms for ω-regular games with imperfect information. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 287–302. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Freivalds, R.: Probabilistic two-way machines. In: Gruska, J., Chytil, M.P. (eds.) MFCS 1981. LNCS, vol. 118, pp. 33–45. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  9. 9.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  10. 10.
    Größer, M.: Reduction Methods for Probabilistic Model Checking. PhD thesis, Technical University Dresden, Faculty for Computer Science (2008)Google Scholar
  11. 11.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proc. of the 38th Symposium on Foundations of Computer Science (FOCS 1997), pp. 66–75. IEEE Computer Society Press, Los Alamitos (1997)Google Scholar
  12. 12.
    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)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Paz, A.: Some aspects of probabilistic automata. Information and Control 9 (1966)Google Scholar
  14. 14.
    Paz, A.: Introduction to probabilistic automata. Academic Press Inc., London (1971)MATHGoogle Scholar
  15. 15.
    Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Safra, S.: On the complexity of ω-automata. In: Proc. of the 29th Symposium on Foundations of Computer Science (FOCS 1988), pp. 319–327. IEEE Computer Society Press, Los Alamitos (1988)Google Scholar
  17. 17.
    Safra, S., Vardi, M.Y.: On ω-automata and temporal logic. In: Proc. of the 21st ACM Symposium on Theory of Computing (STOC 1989), pp. 127–137. ACM, New York (1989)Google Scholar
  18. 18.
    Thomas, W.: Languages, automata, and logic. Handbook of formal languages 3, 389–455 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Proc. of the 1st IEEE Symposium on Logic in Computer Science (LICS 1986), pp. 332–345. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Christel Baier
    • 1
  • Nathalie Bertrand
    • 2
  • Marcus Größer
    • 1
  1. 1.Faculty of Computer ScienceTechnische Universität DresdenGermany
  2. 2.INRIA Rennes Bretagne AtlantiqueFrance

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