Advertisement

Strategy Improvement for Stochastic Rabin and Streett Games

  • Krishnendu Chatterjee
  • Thomas A. Henzinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4137)

Abstract

A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as Rabin or Streett objectives. These games are NP-complete and coNP-complete, respectively. The value of the game for a player at a state s given an objective Φ is the maximal probability with which the player can guarantee the satisfaction of Φ from s. We present a strategy-improvement algorithm to compute values in stochastic Rabin games, where an improvement step involves solving Markov decision processes (MDPs) and nonstochastic Rabin games. The algorithm also computes values for stochastic Streett games but does not directly yield an optimal strategy for Streett objectives. We then show how to obtain an optimal strategy for Streett objectives by solving certain nonstochastic Streett games.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bjorklund, H., Sandberg, S., Vorobyov, S.: A discrete subexponential algorithms for parity games. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 663–674. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: Trading memory for randomness. In: QEST, pp. 206–217. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  3. 3.
    Chatterjee, K., de Alfaro, L., Henzinger, T.A.: The Complexity of Stochastic Rabin and Streett Games. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 878–890. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Chatterjee, K., Henzinger, T.A.: Strategy Improvement and Randomized Subexponential Algorithms for Stochastic Parity Games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 512–523. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Chatterjee, K., Jurdziński, M., Henzinger, T.A.: Simple Stochastic Parity Games. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 100–113. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Condon, A.: The complexity of stochastic games. Information and Computation 96, 203–224 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Condon, A.: On algorithms for simple stochastic games. In: Advances in Computational Complexity Theory. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 13, pp. 51–73. AMS (1993)Google Scholar
  8. 8.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University (1997)Google Scholar
  9. 9.
    de Alfaro, L., Henzinger, T.A.: Concurrent ω-regular games. In: LICS, pp. 141–154. IEEE Computer Society, Los Alamitos (2000)Google Scholar
  10. 10.
    Emerson, E.A., Jutla, C.: The complexity of tree automata and logics of programs. In: FOCS, pp. 328–337. IEEE Computer Society, Los Alamitos (1988)Google Scholar
  11. 11.
    Hoffman, A., Karp, R.: On nonterminating stochastic games. Management Science 12, 359–370 (1966)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Kupferman, O., Vardi, M.Y.: Weak alternating automata and tree-automata emptiness. In: STOC, pp. 224–233. ACM Press, New York (1998)Google Scholar
  13. 13.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer, Heidelberg (1992)Google Scholar
  14. 14.
    Piterman, N., Pnueli, A.: Faster solutions of Rabin and Streett games. In: LICS, IEEE Computer Society, Los Alamitos (to appear, 2006)Google Scholar
  15. 15.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: POPL, pp. 179–190. ACM, New York (1989)Google Scholar
  16. 16.
    Ramadge, P.J., Wonham, W.M.: Supervisory control of a class of discrete-event processes. SIAM J. Control and Optimization 25, 206–230 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Thomas, W.: Languages, automata, and logic. In: Beyond Words. Handbook of Formal Languages, vol. 3, pp. 389–455. Springer, Heidelberg (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Thomas A. Henzinger
    • 1
    • 2
  1. 1.UC BerkeleyUSA
  2. 2.EPFLSwitzerland

Personalised recommendations