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Energy Parity Games

  • Krishnendu Chatterjee
  • Laurent Doyen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6199)

Abstract

Energy parity games are infinite two-player turn-based games played on weighted graphs. The objective of the game combines a (qualitative) parity condition with the (quantitative) requirement that the sum of the weights (i.e., the level of energy in the game) must remain positive. Beside their own interest in the design and synthesis of resource-constrained omega-regular specifications, energy parity games provide one of the simplest model of games with combined qualitative and quantitative objective. Our main results are as follows: (a) exponential memory is sufficient and may be necessary for winning strategies in energy parity games; (b) the problem of deciding the winner in energy parity games can be solved in NP ∩ coNP; and (c) we give an algorithm to solve energy parity by reduction to energy games. We also show that the problem of deciding the winner in energy parity games is polynomially equivalent to the problem of deciding the winner in mean-payoff parity games, which can thus be solved in NP ∩ coNP. As a consequence we also obtain a conceptually simple algorithm to solve mean-payoff parity games.

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References

  1. 1.
    Abadi, M., Lamport, L., Wolper, P.: Realizable and unrealizable specifications of reactive systems. In: Ronchi Della Rocca, S., Ausiello, G., Dezani-Ciancaglini, M. (eds.) ICALP 1989. LNCS, vol. 372, pp. 1–17. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  2. 2.
    Björklund, H., Sandberg, S., Vorobyov, S.G.: Memoryless determinacy of parity and mean payoff games: a simple proof. Theor. Comput. Sci. 310(1-3), 365–378 (2004)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bloem, R., Chatterjee, K., Henzinger, T.A., Jobstmann, B.: Better quality in synthesis through quantitative objectives. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 140–156. Springer, Heidelberg (2009)Google Scholar
  4. 4.
    Bouyer, P., Fahrenberg, U., Larsen, K.G., Markey, N., Srba, J.: Infinite runs in weighted timed automata with energy constraints. In: Cassez, F., Jard, C. (eds.) FORMATS 2008. LNCS, vol. 5215, pp. 33–47. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Büchi, J.R., Landweber, L.H.: Solving sequential conditions by finite-state strategies. SIAM Journal on Control and Optimization 25(1), 206–230 (1987)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chakrabarti, A., de Alfaro, L., Henzinger, T.A., Stoelinga, M.: Resource interfaces. In: Alur, R., Lee, I. (eds.) EMSOFT 2003. LNCS, vol. 2855, pp. 117–133. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Chaloupka, J., Brim, L.: Faster algorithm for mean-payoff games. In: Proc. of MEMICS, pp. 45–53. Nov. Press (2009)Google Scholar
  8. 8.
    Chatterjee, K., Henzinger, T.A., Jurdzinski, M.: Mean-payoff parity games. In: Proc. of LICS, pp. 178–187. IEEE Computer Society, Los Alamitos (2005)Google Scholar
  9. 9.
    Condon, A.: The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Doyen, L., Gentilini, R., Raskin, J.-F.: Faster pseudopolynomial algorithms for mean-payoff games. Technical Report 2009.120, Université Libre de Bruxelles (ULB), Bruxelles, Belgium (2009)Google Scholar
  11. 11.
    Emerson, E.A., Jutla, C.: Tree automata, mu-calculus and determinacy. In: Proc. of FOCS, pp. 368–377. IEEE, Los Alamitos (1991)Google Scholar
  12. 12.
    Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model-checking for fragments of μ-calculus. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 385–396. Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  14. 14.
    Gurevich, Y., Harrington, L.: Trees, automata, and games. In: Proc. of STOC, pp. 60–65. ACM Press, New York (1982)Google Scholar
  15. 15.
    McNaughton, R.: Infinite games played on finite graphs. Annals of Pure and Applied Logic 65(2), 149–184 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Pnueli, A., Rosner, R.: On the synthesis of a reactive module. In: Proc. of POPL, pp. 179–190 (1989)Google Scholar
  17. 17.
    Ramadge, P.J., Wonham, W.M.: Supervisory control of a class of discrete event processes. SIAM Journal on Control and Optimization 25(1), 206–230 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shapley, L.S.: Stochastic games. Proc. of the National Acadamy of Science USA 39, 1095–1100 (1953)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages, Beyond Words, ch.7, vol. 3, pp. 389–455. Springer, Heidelberg (1997)Google Scholar
  20. 20.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200, 135–183 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theor. Comput. Sci. 158(1&2), 343–359 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Krishnendu Chatterjee
    • 1
  • Laurent Doyen
    • 2
  1. 1.IST Austria, Institute of Science and TechnologyAustria
  2. 2.LSV, ENS Cachan & CNRSFrance

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