Advertisement

Small Progress Measures for Solving Parity Games

  • Marcin Jurdziński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1770)

Abstract

In this paper we develop a new algorithm for deciding the winner in parity games, and hence also for the modal μ-calculus model checking. The design and analysis of the algorithm is based on a notion of game progress measures: they are witnesses for winning strategies in parity games. We characterize game progress measures as pre-fixed points of certain monotone operators on a complete lattice. As a result we get the existence of the least game progress measures and a straightforward way to compute them. The worst-case running time of our algorithm matches the best worst-case running time bounds known so far for the problem, achieved by the algorithms due to Browne et al., and Seidl. Our algorithm has better space complexity: it works in small polynomial space; the other two algorithms have exponential worst-case space complexity.

Keywords

Model Check Monotone Operator Complete Lattice Winning Strategy Kripke Structure 
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]
    A. Browne, E. M. Clarke, S. Jha, D. E. Long, and W. Marrero. An improved algorithm for the evaluation of fixpoint expressions. Theoretical Computer Science, 178(1–2):237–255, May 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    E. A. Emerson and C. S. Jutla. Tree automata, mu-calculus and determinacy (Extended abstract). In Proceedings of 32nd Annual Symposium on Foundations of Computer Science, pages 368–377. IEEE Computer Society Press, 1991.Google Scholar
  3. [3]
    E. A. Emerson, C. S. Jutla, and A. P. Sistla. On model-checking for fragments of μ-calculus. In Costas Courcoubetis, editor, Computer Aided Verification, 5th International Conference, CAV’93, volume 697 of LNCS, pages 385–396, Elounda, Greece, June/July 1993. Springer-Verlag.Google Scholar
  4. [4]
    E. Allen Emerson and Charanjit S. Jutla. The complexity of tree automata and logics of programs. In Proceedings of 29th Annual Symposium on Foundations of Computer Science, pages 328–337, White Plains, New York, 24–26 October 1988. IEEE Computer Society Press.CrossRefGoogle Scholar
  5. [5]
    E. Allen Emerson and Chin-Laung Lei. Efficient model checking in fragments of the propositional mu-calculus (Extended abstract). In Proceedings, Symposium on Logic in Computer Science, pages 267–278, Cambridge, Massachusetts, 16–18 June 1986. IEEE.Google Scholar
  6. [6]
    Marcin Jurdziński. Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters, 68(3):119–124, November 1998.CrossRefMathSciNetGoogle Scholar
  7. [7]
    Nils Klarlund. Progress measures for complementation of ω-automata with applications to temporal logic. In 32nd Annual Symposium on Foundations of Computer Science, pages 358–367, San Juan, Puerto Rico, 1–4 October 1991. IEEE.Google Scholar
  8. [8]
    Nils Klarlund. Progress measures, immediate determinacy, and a subset construction for tree automata. Annals of Pure and Applied Logic, 69(2–3):243–268, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Nils Klarlund and Dexter Kozen. Rabin measures and their applications to fairness and automata theory. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 256–265, Amsterdam, The Netherlands, 15–18 July 1991. IEEE Computer Society Press.Google Scholar
  10. [10]
    Orna Kupferman and Moshe Y. Vardi. Weak alternating automata and tree automata emptiness. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, pages 224–233, Dallas, Texas, USA, 23–26 May 1998. ACM Press.Google Scholar
  11. [11]
    Xinxin Liu, C. R. Ramakrishnan, and Scott A. Smolka. Fully local and efficient evaluation of alternating fixed points. In Bernhard Steffen, editor, Tools and Algorithms for Construction and Analysis of Systems, 4th International Conference, TACAS’ 98, volume 1384 of LNCS, pages 5–19, Lisbon, Portugal, 28 March–4 April 1998. Springer.CrossRefGoogle Scholar
  12. [12]
    Robert McNaughton. Infinite games played on finite graphs. Annals of Pure and Applied Logic, 65(2):149–184, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. W. Mostowski. Games with forbidden positions. Technical Report 78, University of Gdańsk, 1991.Google Scholar
  14. [14]
    Amir Pnueli and Roni Rosner. On the synthesis of a reactive module. In Conference Record of the 16th Annual ACM Symposium on Principles of Programming Languages (POPL’ 89), pages 179–190, Austin, Texas, January 1989. ACM Press.Google Scholar
  15. [15]
    Helmut Seidl. Fast and simple nested fixpoints. Information Processing Letters, 59(6):303–308, September 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Colin Stirling. Local model checking games (Extended abstract). In Insup Lee and Scott A. Smolka, editors, CONCUR’95: Concurrency Theory, 6th International Conference, volume 962 of LNCS, pages 1–11, Philadelphia, Pennsylvania, 21–24 August 1995. Springer-Verlag.Google Scholar
  17. [17]
    Robert S. Streett and E. Allen Emerson. An automata theoretic decision procedure for the propositional mu-calculus. Information and Computation, 81(3):249–264, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Igor Walukiewicz. Pushdown processes: Games and model checking. In Thomas A. Henzinger and Rajeev Alur, editors, Computer Aided Verification, 8th International Conference, CAV’96, volume 1102 of LNCS, pages 62–74. Springer-Verlag, 1996. Full version available through http://zls.mimuw.edu.pl/~igw.Google Scholar
  19. [19]
    Wiesław Zielonka. Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science, 200:135–183, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Uri Zwick and Mike Paterson. The complexity of mean payoff games on graphs. Theoretical Computer Science, 158:343–359, 1996.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Marcin Jurdziński
    • 1
  1. 1.BRICS, Department of Computer ScienceUniversity of AarhusAarhus CDenmark

Personalised recommendations