Advertisement

Abstract

Parity games are 2-player games of perfect information and infinite duration that have important applications in automata theory and decision procedures (validity as well as model checking) for temporal logics. In this paper we investigate practical aspects of solving parity games. The main contribution is a suggestion on how to solve parity games efficiently in practice: we present a generic solver that intertwines optimisations with any of the existing parity game algorithms which is only called on parts of a game that cannot be solved faster by simpler methods. This approach is evaluated empirically on a series of benchmarking games from the aforementioned application domains, showing that using this approach vastly speeds up the solving process. As a side-effect we obtain the surprising observation that Zielonka’s recursive algorithm is the best parity game solver in practice.

Keywords

Model Check Recursive Algorithm Winning Strategy Strongly Connect Component Priority Propagation 
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.
    Antonik, A., Charlton, N., Huth, M.: Polynomial-time under-approximation of winning regions in parity games. Technical report, Dept. of Computer Science, Imperial College London (2006)Google Scholar
  2. 2.
    Arnold, A., Vincent, A., Walukiewicz, I.: Games for synthesis of controllers with partial observation. Theor. Comput. Sci. 303(1), 7–34 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Emerson, E.A., Jutla, C.S.: Tree automata, μ-calculus and determinacy. In: Proc. 32nd Symp. on Foundations of Computer Science, San Juan, Puerto Rico, pp. 368–377. IEEE, Los Alamitos (1991)Google Scholar
  4. 4.
    Jurdziński, M.: Deciding the winner in parity games is in UP ∩co-UP. Inf. Process. Lett. 68(3), 119–124 (1998)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. In: Proc. ACM-SIAM Symp. on Discrete Algorithms, SODA 2006, pp. 114–123. ACM/SIAM (2006) (to appear)Google Scholar
  7. 7.
    Kähler, D., Wilke, T.: Complementation, disambiguation, and determinization of Büchi automata unified. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 724–735. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Lange, M.: Solving parity games by a reduction to SAT. In: Majumdar, R., Jurdziński, M. (eds.) Proc. Int. Workshop on Games in Design and Verification, GDV 2005 (2005)Google Scholar
  9. 9.
    Piterman, N.: From nondeterministic Büchi and Streett automata to deterministic parity automata. In: Proc. 21st Symp. on Logic in Computer Science, LICS 2006, pp. 255–264. IEEE Computer Society, Los Alamitos (2006)Google Scholar
  10. 10.
    Safra, S.: On the complexity of ω-automata. In: Proc. 29th Symp. on Foundations of Computer Science, FOCS 1988, pp. 319–327. IEEE, Los Alamitos (1988)Google Scholar
  11. 11.
    Schewe, S.: Solving parity games in big steps. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 449–460. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Schewe, S.: An optimal strategy improvement algorithm for solving parity and payoff games. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 369–384. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Stevens, P., Stirling, C.: Practical model-checking using games. In: Steffen, B. (ed.) TACAS 1998. LNCS, vol. 1384, pp. 85–101. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  14. 14.
    Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)Google Scholar
  15. 15.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Computing 1, 146–160 (1972)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    van de Pol, J., Weber, M.: A multi-core solver for parity games. Electr. Notes Theor. Comput. Sci. 220(2), 19–34 (2008)CrossRefGoogle Scholar
  17. 17.
    Vöge, J., Jurdziński, 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)CrossRefGoogle Scholar
  18. 18.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. TCS 200(1–2), 135–183 (1998)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Oliver Friedmann
    • 1
  • Martin Lange
    • 1
  1. 1.Dept. of Computer ScienceUniversity of MunichGermany

Personalised recommendations