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Abstract

In this paper it is shown that deciding the winner of a parity game is in LogCFL, if the underlying graph has bounded tree-width, and in LogDCFL, if the tree-width is at most 2. It is also proven that parity games of bounded clique-width can be solved in LogCFL via a log-space reduction to the bounded tree-width case, assuming that a k-expression for the parity game is part of the input.

Keywords

Outgoing Edge Tree Decomposition Parse Tree Boundary Vertex Tree Game 
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.
    Stirling, C.: Local model checking games. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 1–11. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  2. 2.
    Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters 68(3), 119–124 (1998)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Obdržálek, 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)CrossRefGoogle Scholar
  4. 4.
    Obdržálek, J.: Clique-width and parity games. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 54–68. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Fearnley, J., Schewe, S.: Time and parallelizability results for parity games with bounded treewidth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 189–200. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Sudborough, I.H.: On the tape complexity of deterministic context-free languages. Journal of the Association for Computing Machinery 25(3), 405–414 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Göller, S., Lohrey, M.: Fixpoint logics over hierarchical structures. Theory Comput. Syst. 48(1), 93–131 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Vollmer, H.: Introduction to Circuit Complexity. Texts in Theoretical Computer Science. Springer (1999)Google Scholar
  9. 9.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy (extended abstract). In: 32nd Annual Symposium on Foundations of Computer Science, pp. 368–377. IEEE Computer Society (1991)Google Scholar
  10. 10.
    Walukiewicz, I.: Monadic second order logic on tree-like structures. In: Puech, C., Reischuk, R. (eds.) STACS 1996. LNCS, vol. 1046, pp. 401–413. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  11. 11.
    Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle. In: 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 143–152 (2010)Google Scholar
  12. 12.
    Friedmann, O., Lange, M.: Solving parity games in practice. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 182–196. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Courcelle, B.: Graphs as relational structures: An algebraic and logical approach. In: Ehrig, H., Kreowski, H.-J., Rozenberg, G. (eds.) Graph Grammars 1990. LNCS, vol. 532, pp. 238–252. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Downey, M.R.R.G., Fellows: Parameterized Complexity. Monographs in Computer Science. Springer (1999)Google Scholar
  17. 17.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Applied Mathematics 101(1-3), 77–114 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width is NP-complete. SIAM J. Discrete Math. 23(2), 909–939 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Lohrey, M.: On the parallel complexity of tree automata. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 201–215. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games: A Guide to Current Research. LNCS, vol. 2500. Springer, Heidelberg (2002)Google Scholar
  21. 21.
    Seese, D.: Tree-partite graphs and the complexity of algorithms. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 412–421. Springer, Heidelberg (1985)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.University of SiegenSiegenGermany

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