Advertisement

Computing Game Values for Crash Games

  • Thomas Gawlitza
  • Helmut Seidl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4762)

Abstract

We consider crash games which are a generalization of parity games in which the play value of a play is an integer, -∞ or ∞. In particular, the play value of a finite play is given as the sum of the payoffs of the moves of the play. Accordingly, one player aims at maximizing the play value whereas the other player aims at minimizing this value. We show that the game value of such a crash game at position v, i.e., the least upper bounds to the minimal play value that can be enforced by the maximum player in a play starting at v, can be characterized by a hierarchical system of simple integer equations. Moreover, we present a practical algorithm for solving such systems. The run-time of our algorithm (w.r.t. the uniform cost measure) is independent of the sizes of occurring numbers. Our method is based on a strategy improvement algorithm. The efficiency of our algorithm is comparable to the efficiency of the discrete strategy improvement algorithm developed by Vöge and Jurdzinski for the simpler Boolean case of parity games [19].

Keywords

Complete Lattice Strategy Iteration Hierarchical System Variable Assignment Disjunctive Normal Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold, A., Niwinski, D.: Rudiments of μ-Calculus. In: Studies in Logic and The Foundations of Computer Science, vol. 146, North-Holland, Amsterdam (2001)Google Scholar
  2. 2.
    Bjorklund, H., Sandberg, S., Vorobyov, S.: Complexity of Model Checking by Iterative Improvement: the Pseudo-Boolean Framework. In: Broy, M., Zamulin, A.V. (eds.) PSI 2003. LNCS, vol. 2890, pp. 381–394. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Mean-payoff parity games. In: LICS, pp. 178–187. IEEE Computer Society, Los Alamitos (2005)Google Scholar
  4. 4.
    Chechik, M., Devereux, B., Easterbrook, S.M., Gurfinkel, A.: Multi-valued symbolic model-checking. ACM Trans. Softw. Eng. Methodol. 12(4), 371–408 (2003)CrossRefGoogle Scholar
  5. 5.
    Costan, A., Gaubert, S., Goubault, E., Martel, M., Putot, S.: A Policy Iteration Algorithm for Computing Fixed Points in Static Analysis of Programs. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 462–475. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy (extended abstract). In: FOCS, pp. 368–377. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  7. 7.
    Gawlitza, T., Reineke, J., Seidl, H., Wilhelm, R.: Polynomial Exact Interval Analysis Revisited. Technical report, TU München (2006)Google Scholar
  8. 8.
    Gawlitza, T., Seidl, H.: Precise fixpoint computation through strategy iteration. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 300–315. Springer, Heidelberg (2007)Google Scholar
  9. 9.
    Gimbert, H., Zielonka, W.: When can you play positionally? In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 686–697. Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Gimbert, H., Zielonka, W.: Games where you can play optimally without any memory. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 428–442. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Hoffman, A.J., Karp, R.M.: On Nonterminating Stochastic Games. Management Sci. 12, 359–370 (1966)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Howard, R.: Dynamic Programming and Markov Processes. Wiley, New York (1960)zbMATHGoogle Scholar
  13. 13.
    Jrklund, H., Nilsson, O., Svensson, O., Vorobyov, S.: The Controlled Linear Programming Problem. Technical report, DIMACS (2004)Google Scholar
  14. 14.
    Jurdziński, M.: Small Progress Measures for Solving Parity Games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. 15.
    Puri, A.: Theory of Hybrid and Discrete Systems. PhD thesis, University of California, Berkeley (1995)Google Scholar
  16. 16.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (1994)zbMATHGoogle Scholar
  17. 17.
    Seidl, H.: A Modal μ Calculus for Durational Transition Systems. In: IEEE Conf. on Logic in Computer Science (LICS), pp. 128–137 (1996)Google Scholar
  18. 18.
    Shoham, S., Grumberg, O.: Multi-valued model checking games. In: Peled, D.A., Tsay, Y.K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 354–369. Springer, Heidelberg (2005)Google Scholar
  19. 19.
    Vöge, J., Jurdzinski, M.: A Discrete Strategy Improvement Algorithm for Solving Parity Games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas Gawlitza
    • 1
  • Helmut Seidl
    • 1
  1. 1.TU München, Institut für Informatik, I2, 85748 MünchenGermany

Personalised recommendations