Backtracking Games and Inflationary Fixed Points

  • Anuj Dawar
  • Erich Grädel
  • Stephan Kreutzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)


We define a new class of games, called backtracking games. Backtracking games are essentially parity games with an additional rule allowing players, under certain conditions, to return to an earlier position in the play and revise a choice.

This new feature makes backtracking games more powerful than parity games. As a consequence, winning strategies become more complex objects and computationally harder. The corresponding increase in expressiveness allows us to use backtracking games as model checking games for inflationary fixed-point logics such as IFP or MIC. We identify a natural subclass of backtracking games, the simple games, and show that these are the “right” model checking games for IFP by a) giving a translation of formulae φ and structures \(\cal A\) into simple games such that \(\cal A \models \phi\) if, and only if, Player 0 wins the corresponding game and b) showing that the winner of simple backtracking games can again be defined in IFP.


Model Check Simple Game Winning Strategy Point Logic Game Graph 
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  1. 1.
    Dawar, A., Grädel, E., Kreutzer, S.: Inflationary fixed points in modal logic. ACM Transactions on Computational Logic, TOCL (2003) (accepted for publication)Google Scholar
  2. 2.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman and company, New York (1979) ISBN 0-7167-1044-7zbMATHGoogle Scholar
  3. 3.
    Grädel, E.: Finite model theory and descriptive complexity. In: Finite Model Theory and Its Applications, Springer, Heidelberg (2003) (to appear), See Google Scholar
  4. 4.
    Grädel, E., Kreutzer, S.: Will deflation lead to depletion? On non-monotone fixed-point inductions. In: IEEE Symp. of Logic in Computer Science, LICS (2003)Google Scholar
  5. 5.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  6. 6.
    Kreutzer, S.: Expressive equivalence of least and inflationary fixed-point logic. In: 17th Symp. on Logic in Computer Science (LICS), pp. 403 – 413 (2002)Google Scholar
  7. 7.
    Martin, D.: Borel determinacy. Annals of Mathematics 102, 336–371 (1975)CrossRefGoogle Scholar
  8. 8.
    Stirling, C.: Bisimulation, model checking and other games. Notes for the Mathfit instructional meeting on games and computation, Edinburgh (1997)Google Scholar
  9. 9.
    Walukiewicz, I.: Monadic second order logic on tree-like structures. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 401–414. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Anuj Dawar
    • 1
  • Erich Grädel
    • 2
  • Stephan Kreutzer
    • 3
  1. 1.University of Cambridge Computer LaboratoryCambridgeUK
  2. 2.Mathematische Grundlagen der InformatikAachen-University 
  3. 3.Logik in der InformatikHumboldt-UniversityBerlin

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