Solving Counter Parity Games

  • Dietmar Berwanger
  • Łukasz Kaiser
  • Simon Leßenich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


We study a class of parity games equipped with counters that evolve according to arbitrary non-negative affine functions. These games capture several cost models for dynamic systems from the literature. We present an elementary algorithm for computing the exact value of a counter parity game, which both generalizes previous results and improves their complexity. To this end, we introduce a class of ω-regular games with imperfect information and imperfect recall, solve them using automata-based techniques, and prove a correspondence between finite-memory strategies in such games and strategies in counter parity games.


Model Check Online Algorithm Regular Language Imperfect Information Winning Strategy 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aminof, B., Kupferman, O., Lampert, R.: Formal Analysis of Online Algorithms. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 213–227. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Berwanger, D., Kaiser, Ł., Leßenich, S.: Imperfect recall and counter games. Research Report LSV-11-20, LSV, ENS Cachan, France (2011),
  3. 3.
    Brázdil, T., Forejt, V., Krcál, J., Kretínský, J., Kucera, A.: Continuous-time stochastic games with time-bounded reachability. In: Proc. of FSTTCS 2009, pp. 61–72 (2009)Google Scholar
  4. 4.
    Chatterjee, K., Doyen, L., Henzinger, T.A., Raskin, J.-F.: Generalized mean-payoff and energy games. In: Proc. of FSTTCS 2010, pp. 505–516 (2010)Google Scholar
  5. 5.
    Chatterjee, K., Henzinger, T.A., Horn, F.: The Complexity of Request-Response Games. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 227–237. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  6. 6.
    de Alfaro, L., Faella, M., Henzinger, T.A., Majumdar, R., Stoelinga, M.: Model checking discounted temporal properties. Theoretical Computer Science 345(1), 139–170 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fischer, D., Grädel, E., Kaiser, Ł.: Model checking games for the quantitative μ-calculus. Theory Comput. Syst. 47(3), 696–719 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fischer, D., Kaiser, Ł.: Model Checking the Quantitative μ-Calculus on Linear Hybrid Systems. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 404–415. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Horn, F., Thomas, W., Wallmeier, N.: Optimal Strategy Synthesis in Request-Response Games. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 361–373. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Kaiser, Ł., Leßenich, S.: Counting μ-calculus on structured transition systems. In: Proc. of CSL 2012. LIPIcs, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany (to appear, 2012)Google Scholar
  11. 11.
    Kupferman, O., Piterman, N., Vardi, M.Y.: From liveness to promptness. Formal Methods in System Design 34(2), 83–103 (2009)zbMATHCrossRefGoogle Scholar
  12. 12.
    Thomas, W.: Infinite Games and Verification (Extended Abstract of a Tutorial). In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 58–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Wan, H.: Upper bounds for Ramsey numbers R(3, 3,…, 3) and Schur numbers. Journal of Graph Theory 26(3), 119–122 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Zimmermann, M.: Optimal bounds in parametric LTL games. In: Proc. of GandALF 2011, pp. 146–161 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dietmar Berwanger
    • 1
  • Łukasz Kaiser
    • 2
  • Simon Leßenich
    • 1
    • 3
  1. 1.LSVCNRS & ENS CachanFrance
  2. 2.LIAFACNRS & Université Paris Diderot – Paris 7France
  3. 3.Mathematische Grundlagen der InformatikRWTH AachenGermany

Personalised recommendations