On Reachability Games of Ordinal Length

  • Julien Cristau
  • Florian Horn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4910)


Games are a classical model in the synthesis of controllers in the open setting. In particular, games of infinite length can represent systems which are not expected to reach a correct state, but rather to handle a continuous stream of events. Yet, even longer sequences of events have to be considered when infinite sequences of events can occur in finite time — Zeno behaviours.

In this paper, we extend two-player games to this setting by considering plays of ordinal length. Our two main results are determinacy of reachability games of length less than ω ω on finite arenas, and the PSPACE-completeness of deciding the winner in such a game.


Memory State Winning Strategy Boolean Formula Limit Transition Tree Automaton 
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. [AD94]
    Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126(2), 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AM99]
    Asarin, E., Maler, O.: As soon as possible: Time optimal control for timed automata. In: Vaandrager, F.W., van Schuppen, J.H. (eds.) HSCC 1999. LNCS, vol. 1569, pp. 19–30. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. [BC01]
    Bruyère, V., Carton, O.: Automata on linear orderings. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 236–247. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. [BÉ02]
    Bloom, S.L., Ésik, Z.: Some remarks on regular words. Technical Report RS-02-39, 27 (September 2002)Google Scholar
  5. [Cac06]
    Cachat, T.: Controller synthesis and ordinal automata. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 215–228. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. [CdAH05]
    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)Google Scholar
  7. [Cho78]
    Choueka, Y.: Finite automata, definable sets, and regular expressions over ω n-tapes. Journal of Computer and System Sciences 17(1), 81–97 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [CJH03]
    Chatterjee, K., Jurdzinski, 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)Google Scholar
  9. [dAFH+03]
    de Alfaro, L., Faella, M., Henzinger, T.A., Majumdar, R., Stoelinga, M.: The element of surprise in timed games. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, Springer, Heidelberg (2003)Google Scholar
  10. [DN05]
    Demri, S., Nowak, D.: Reasoning about transfinite sequences. In: Peled, D.A., Tsay, Y.-K. (eds.) ATVA 2005. LNCS, vol. 3707, pp. 248–262. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. [EJ91]
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy. In: Proceedings of FOCS 1991, pp. 368–377. IEEE, Los Alamitos (1991)Google Scholar
  12. [EL85]
    Emerson, E.A., Lei, C.-L.: Modalities for model checking: Branching time strikes back. In: Proceedings of POPL 1985, pp. 84–96 (1985)Google Scholar
  13. [GTW02]
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  14. [HD05]
    Hunter, P., Dawar, A.: Complexity bounds for regular games. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 495–506. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. [JT07]
    Jurdziński, M., Trivedi, A.: Reachability-time games on timed automata. In: ICALP 2007. LNCS, vol. 4596, pp. 838–849. Springer, Heidelberg (2007)Google Scholar
  16. [Mar75]
    Martin, D.A.: Borel determinacy. Annals of Mathematics 102, 363–371 (1975)CrossRefMathSciNetGoogle Scholar
  17. [Tho95]
    Thomas, W.: On the synthesis of strategies in infinite games. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 1–13. Springer, Heidelberg (1995)Google Scholar
  18. [Zie98]
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theorical Computer Science 200(1-2), 135–183 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Julien Cristau
    • 1
  • Florian Horn
    • 1
    • 2
  1. 1.LIAFAUniversité Paris 7Paris 5France
  2. 2.Lehrstuhl für Informatik VII, RWTHAachenGermany

Personalised recommendations