Abstract

Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems. In this paper, the model-checking problem for the modal μ-calculus and (monadic) least fixpoint logic on hierarchically defined graphs is investigated. In order to analyze the modal μ-calculus, parity games on hierarchically defined graphs are studied.

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References

  1. 1.
    Alur, R., Yannakakis, M.: Model checking of hierarchical state machines. ACM Trans. Program. Lang. Syst. 23(3), 273–303 (2001)CrossRefGoogle Scholar
  2. 2.
    Dziembowski, S.: Bounded-variable fixpoint queries are PSPACE-complete. In: van Dalen, D., Bezem, M. (eds.) CSL 1996. LNCS, vol. 1258, pp. 89–105. Springer, Heidelberg (1997)Google Scholar
  3. 3.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy (extended abstract). In: Proc. FOCS 1991, pp. 132–142. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  4. 4.
    Emerson, E.A., Jutla, C.S., Sistla, A.P.: On model checking for the μ-calculus and its fragments. Theor. Comput. Sci. 258(1-2), 491–522 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional mu-calculus (extended abstract). In: Proc. LICS 1986, pp. 267–278. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  6. 6.
    Engelfriet, J.: Context-free graph grammars. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Beyond Words, vol. 3, pp. 125–213. Springer, Heidelberg (1997)Google Scholar
  7. 7.
    Göller, S., Lohrey, M.: Fixpoint logics on hierarchical structures. Tech. Rep. 2005/3, University of Stuttgart, Germany (2005), ftp.informatik.uni-stuttgart.de/pub/library/ncstrl.ustuttgart fi/TR-2005-04/Google Scholar
  8. 8.
    Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHCrossRefGoogle Scholar
  9. 9.
    Immerman, N.: Relational queries computable in polynomial time. Inf. Control 68(1-3), 86–104 (1986)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jurdziński, M.: Deciding the winner in parity games is in UP and co-UP. Inf. Process. Lett. 68(3), 119–124 (1998)CrossRefGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Lengauer, T.: Hierarchical planarity testing algorithms. J. Assoc. Comput. Mach. 36(3), 474–509 (1989)MATHMathSciNetGoogle Scholar
  13. 13.
    Lengauer, T., Wagner, K.W.: The correlation between the complexities of the nonhierarchical and hierarchical versions of graph problems. J. Comput. Syst. Sci. 44, 63–93 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lengauer, T., Wanke, E.: Efficient solution of connectivity problems on hierarchically defined graphs. SIAM J. Comput. 17(6), 1063–1080 (1988)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)MATHGoogle Scholar
  16. 16.
    Lohrey, M.: Model-checking hierarchical graphs. Tech. Rep. 2005/1, University of Stuttgart, Germany (2005), ftp.informatik.uni-stuttgart.de/pub/library/ncstrl.ustuttgart fi/TR-2005-1/Google Scholar
  17. 17.
    Lohrey, M.: Model-checking hierarchical structures. In: Proc. LICS 2005, pp. 168–177. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  18. 18.
    Marathe, M.V., Hunt III, H.B., Stearns, R.E., Radhakrishnan, V.: Approximation algorithms for PSPACE-hard hierarchically and periodically specified problems. SIAM J. Comput. 27(5), 1237–1261 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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
  20. 20.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  21. 21.
    Vardi, M.Y.: The complexity of relational query languages (extended abstract). In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. 137–146, vol. 888, pp. 137–146. Springer, Heidelberg (1995)Google Scholar
  22. 22.
    Vardi, M.Y.: On the complexity of bounded-variable queries. In: Proc. PODS 1995, pp. 266–276. ACM Press, New York (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Stefan Göller
    • 1
  • Markus Lohrey
    • 1
  1. 1.FMIUniversität StuttgartGermany

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