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Degrees of Lookahead in Regular Infinite Games

  • Michael Holtmann
  • Łukasz Kaiser
  • Wolfgang Thomas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6014)

Abstract

We study variants of regular infinite games where the strict alternation of moves between the two players is subject to modifications. The second player may postpone a move for a finite number of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This captures situations in distributed systems, e.g. when buffers are present in communication or when signal transmission between components is deferred. We distinguish strategies with different degrees of lookahead, among them being the continuous and the bounded lookahead strategies. In the first case the lookahead is of finite possibly unbounded size, whereas in the second case it is of bounded size. We show that for regular infinite games the solvability by continuous strategies is decidable, and that a continuous strategy can always be reduced to one of bounded lookahead. Moreover, this lookahead is at most doubly exponential in the size of the parity automaton recognizing the winning condition. We also show that the result fails for non-regular games where the winning condition is given by a context-free ω-language.

Keywords

Continuous Operator Winning Strategy Delay Operator Constant Delay Game Graph 
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.

References

  1. 1.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  2. 2.
    Büchi, J.R., Landweber, L.H.: Solving sequential conditions by finite-state strategies. Transactions of the AMS 138, 295–311 (1969)CrossRefGoogle Scholar
  3. 3.
    Walukiewicz, I.: Pushdown processes: Games and model checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 62–74. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Cachat, T.: Higher order pushdown automata, the caucal hierarchy of graphs and parity games. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 556–569. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Bouquet, A.J., Serre, O., Walukiewicz, I.: Pushdown games with unboundedness and regular conditions. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 88–99. Springer, Heidelberg (2003)Google Scholar
  6. 6.
    Moschovakis, Y.N.: Descriptive Set Theory. Studies in Logic and the Foundations of Mathematics, vol. 100. North-Holland Publishing Company, Amsterdam (1980)zbMATHGoogle Scholar
  7. 7.
    Trakhtenbrot, B.A., Barzdin, Y.M.: Finite Automata, Behavior and Synthesis. North Holland, Amsterdam (1973)zbMATHGoogle Scholar
  8. 8.
    Thomas, W., Lescow, H.: Logical specifications of infinite computations. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1993. LNCS, vol. 803, pp. 583–621. Springer, Heidelberg (1994)Google Scholar
  9. 9.
    Hosch, F.A., Landweber, L.H.: Finite delay solutions for sequential conditions. In: Nivat, M. (ed.) Automata, Languages and Programming, Paris, France, pp. 45–60. North-Holland, Amsterdam (1972)Google Scholar
  10. 10.
    Even, S., Meyer, A.: Sequential boolean equations. IEEE Transactions on Computers C-18, 230–240 (1969)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Perrin, D., Pin, J.: Semigroups and automata on infinite words. In: Fountain, J. (ed.) NATO Advanced Study Institute Semigroups, Formal Language and Groups, pp. 49–72. Kluwer Academic Publishers, Dordrecht (1995)Google Scholar
  12. 12.
    Pin, J.: Finite semigroups and recognizable languages: An introduction (1995)Google Scholar
  13. 13.
    Cohen, R.S., Gold, A.Y.: Omega-computations on deterministic pushdown machines. Journal of Computer and System Sciences 16, 275–300 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Finkel, O.: Topological properties of omega context-free languages. Theoretical Computer Science 262, 669–697 (2001)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michael Holtmann
    • 1
  • Łukasz Kaiser
    • 2
  • Wolfgang Thomas
    • 1
  1. 1.RWTH Aachen, Lehrstuhl für Informatik 7Aachen
  2. 2.RWTH Aachen, Mathematische Grundlagen der InformatikAachen

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