Skip to main content
Springer Nature Link
Log in

The Descriptive Complexity of Parity Games

  • Conference paper
Computer Science Logic (CSL 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5213))

Included in the following conference series:

Abstract

We study the logical definablity of the winning regions of parity games. For games with a bounded number of priorities, it is well-known that the winning regions are definable in the modal μ-calculus. Here we investigate the case of an unbounded number of priorities, both for finite game graphs and for arbitrary ones. In the general case, winning regions are definable in guarded second-order logic (GSO), but not in least-fixed point logic (LFP). On finite game graphs, winning regions are LFP-definable if, and only if, they are computable in polynomial time, and this result extends to any class of finite games that is closed under taking bisimulation quotients.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Abiteboul, S., Vardi, M.Y., Vianu, V.: Fixpoint logics, relational machines, and computational complexity. J. ACM 44(1), 30–46 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abiteboul, S., Vianu, V.: Computing with first-order logic. Journal of Computer and System Sciences 50(2), 309–335 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arnold, A.: The mu-calculus alternation-depth is strict on binary trees. RAIRO Informatique Théorique et Applications 33, 329–339 (1999)

    Article  MATH  Google Scholar 

  4. Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: Dag-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Berwanger, D., Grädel, E.: Fixed-point logics and solitaire games. Theory of Computing Systems 37, 675–694 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berwanger, D., Grädel, E.: Entanglement - A measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)

    Google Scholar 

  7. Bradfield, J.: The modal μ-calculus alternation hierarchy is strict. Theoretical Computer Science 195, 133–153 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Courcelle, B.: The monadic second-order logic of graphs XIV: Uniformly sparse graphs and edge set quantifications. Theoretical Computer Science 299, 1–36 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dawar, A., Gurevich, Y.: Fixed point logics. Bulletin of Symbolic Logic 8, 65–88 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)

    MATH  Google Scholar 

  11. Emerson, A., Jutla, C.: Tree automata, mu-calculus and determinacy. In: Proc. 32nd IEEE Symp. on Foundations of Computer Science, pp. 368–377 (1991)

    Google Scholar 

  12. Grädel, E., et al.: Finite Model Theory and Its Applications. Springer, Heidelberg (2007)

    Book  MATH  Google Scholar 

  13. Grädel, E., Hirsch, C., Otto, M.: Back and forth between guarded and modal logics. ACM Transactions on Computational Logic 3, 418–463 (2002)

    Article  MathSciNet  Google Scholar 

  14. Grädel, E., Walukiewicz, I.: Positional determinacy of games with infinitely many priorities. Logical Methods in Computer Science (2006)

    Google Scholar 

  15. Hunter, P.: Complexity and Infinite Games on Finite Graphs. PhD thesis, University of Cambridge (2007)

    Google Scholar 

  16. Jurdziński, M.: Deciding the winner in parity games is in UP ∩ Co-UP. Information Processing Letters 68, 119–124 (1998)

    Article  MathSciNet  Google Scholar 

  17. Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) STACS 2000. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  18. Jurdziński, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. In: Proceedings of ACM-SIAM Proceedings on Discrete Algorithms, SODA 2006, pp. 117–123 (2006)

    Google Scholar 

  19. Mostowski, A.: Games with forbidden positions. Technical Report Tech. Report 78, University of Gdansk (1991)

    Google Scholar 

  20. Obdrzálek, J.: Fast mu-calculus model checking when tree-width is bounded. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 80–92. Springer, Heidelberg (2003)

    Google Scholar 

  21. Obdrzálek, J.: DAG-width - connectivity measure for directed graphs. In: Proceedings of ACM-SIAM Proceedings on Discrete Algorithms, SODA 2006, pp. 814–821 (2006)

    Google Scholar 

  22. Obdrzálek, J.: Clique-width and parity games. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 54–68. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  23. Otto, M.: Bisimulation-invariant Ptime and higher-dimensional mu-calculus. Theoretical Computer Science 224, 237–265 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Stirling, C.: Bisimulation, model checking and other games. Notes for the Mathfit instructional meeting on games and computation. Edinburgh (1997)

    Google Scholar 

  25. Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theoretical Computer Science 200, 135–183 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Cambridge Computer Laboratory, Cambridge, CB3 0FD, UK

    Anuj Dawar

  2. Mathematische Grundlagen der Informatik, RWTH Aachen University, Germany

    Erich Grädel

Authors
  1. Anuj Dawar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erich Grädel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michael Kaminski Simone Martini

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dawar, A., Grädel, E. (2008). The Descriptive Complexity of Parity Games. In: Kaminski, M., Martini, S. (eds) Computer Science Logic. CSL 2008. Lecture Notes in Computer Science, vol 5213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87531-4_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-87531-4_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-87530-7

  • Online ISBN: 978-3-540-87531-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation