Theory of Computing Systems

, Volume 48, Issue 1, pp 93–131 | Cite as

Fixpoint Logics over Hierarchical Structures

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Abstract

Hierarchical graph definitions allow a modular description of graphs using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions also 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 input graphs is investigated. In order to analyze the modal μ-calculus, parity games on hierarchically defined input graphs are investigated. Precise upper and lower complexity bounds are derived. A restriction on hierarchical graph definitions that leads to more efficient model-checking algorithms is presented.

Keywords

Parity games μ-calculus Hierarchical structures Model-checking Complexity 

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References

  1. 1.
    Alur, R., Yannakakis, M.: Model checking of hierarchical state machines. ACM Trans. Program. Lang. Syst. (TOPLAS) 23(3), 273–303 (2001) CrossRefGoogle Scholar
  2. 2.
    Bosse, U.: An “Ehrenfeucht-Fraïssé Game” for fixpoint logic and stratified fixpoint logic. In: Proceedings of the 6th Workshop on Computer Science Logic (CSL’92). Lecture Notes in Computer Science, vol. 707, pp. 100–114. Springer, Berlin (1993) Google Scholar
  3. 3.
    Bosse, U.: Zur Modelltheorie der Fixpunktlogik. Ph.D. thesis, Universität Freiburg, 1994 Google Scholar
  4. 4.
    Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. J. Assoc. Comput. Mach. 28(1), 114–133 (1981) MATHMathSciNetGoogle Scholar
  5. 5.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 193–242. Elsevier, Amsterdam (1990) Google Scholar
  6. 6.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Berlin (1991) Google Scholar
  8. 8.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy (extended abstract). In: Proceedings of the 32nd Annual Symposium on Foundations of Computer Science (FOCS’91), pp. 132–142. IEEE Computer Society Press, Los Alamitos (1991) Google Scholar
  9. 9.
    Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional mu-calculus (extended abstract). In: Proceedings of the First Annual IEEE Symposium on Logic in Computer Science (LICS’86), pp. 267–278. IEEE Computer Society Press, Los Alamitos (1986) Google Scholar
  10. 10.
    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
  11. 11.
    Engelfriet, J.: Context-free graph grammars. In: Rozenberg, G., Salomaa, A. (eds.) Beyond Words. Handbook of Formal Languages, vol. 3, pp. 125–213. Springer, Berlin (1997) Google Scholar
  12. 12.
    Grädel, E., Thomas, W., Wilke, T.: Automata, Logics, and Infinite Games. Lecture Notes in Computer Science, vol. 2500, Springer, Berlin (2002) MATHCrossRefGoogle Scholar
  13. 13.
    Habel, A.: Hyperedge Replacement: Grammars and Languages. Lecture Notes in Computer Science, vol. 643, Springer, Berlin (1992) MATHGoogle Scholar
  14. 14.
    Immerman, N.: Relational queries computable in polynomial time. Inf. Control 68(1–3), 86–104 (1986) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jurdziński, M.: Deciding the winner in parity games is in UP and co-UP. Inf. Process. Lett. 68(3), 119–124 (1998) CrossRefGoogle Scholar
  16. 16.
    Jurdziński, M.: Small progress measures for solving parity games. In: Proceedings of the17th Annual Symposium on Theoretical Aspects of Computer Science (STACS 2000). Lecture Notes in Computer Science, vol. 1770, pp. 290–301. Springer, Berlin (2000) Google Scholar
  17. 17.
    Kupferman, O., Vardi, M.Y.: An automata-theoretic approach to reasoning about infinite-state systems. In: Proceedings of the 12th International Conference on Computer Aided Verification (CAV 2000). Lecture Notes in Computer Science, vol. 1855, pp. 36–52. Springer, Berlin (2000) Google Scholar
  18. 18.
    Ladner, R.E.: Application of model theoretic games to discrete linear orders and finite automata. Inf. Comput. 33(4), 281–303 (1977) MATHMathSciNetGoogle Scholar
  19. 19.
    Lengauer, T.: Hierarchical planarity testing algorithms. J. Assoc. Comput. Mach. 36(3), 474–509 (1989) MATHMathSciNetGoogle Scholar
  20. 20.
    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
  21. 21.
    Lengauer, T., Wanke, E.: Efficient solution of connectivity problems on hierarchically defined graphs. SIAM J. Comput. 17(6), 1063–1080 (1988) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Libkin, L.: Elements of Finite Model Theory. Springer, Berlin (2004) MATHGoogle Scholar
  23. 23.
    Lohrey, M.: Model-checking hierarchical structures. J. Comput. Syst. Sci. (2007, to appear). LICS, pp. 168–177 (2005). http://dx.doi.org/10.1109/LICS.2005.29
  24. 24.
    Lohrey, M., Maneth, S.: The complexity of tree automata and XPath on grammar-compressed trees. Theor. Comput. Sci. 363(2), 196–210 (2006) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Makowsky, J.A.: Algorithmic aspects of the Feferman-Vaught theorem. Ann. Pure Appl. Logic 126(1–3), 159–213 (2004) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Makowsky, J.A., Ravve, E.V.: Incremental model checking for fixed point properties on decomposable structures. http://www.cs.technion.ac.il/~admlogic/TR/readme.html (1995)
  27. 27.
    Marathe, M.V., Hunt III, H.B., Ravi, S.S.: The complexity of approximation PSPACE-complete problems for hierarchical specifications. Nord. J. Comput. 1(3), 275–316 (1994) MATHMathSciNetGoogle Scholar
  28. 28.
    Marathe, M.V., Radhakrishnan, V., Hunt III, H.B., Ravi, S.S.: Hierarchically specified unit disk graphs. Theor. Comput. Sci. 174(1–2), 23–65 (1997) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    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
  30. 30.
    Obdržálek, J.: Fast mu-calculus model checking when tree-width is bounded. In: Proceedings of the 15th International Conference on Computer Aided Verification (CAV 2003). Lecture Notes in Computer Science, vol. 2725, pp. 80–92. Springer, Berlin (2003) Google Scholar
  31. 31.
    Papadimitriou, C.H.: Computational Complexity. Addison Wesley, Reading (1994) MATHGoogle Scholar
  32. 32.
    Plandowski, W., Rytter, W.: Complexity of language recognition problems for compressed words. In: Jewels are Forever, Contributions on Theoretical Computer Science in Honor of Arto Salomaa, pp. 262–272. Springer, Berlin (1999) Google Scholar
  33. 33.
    Stockmeyer, L.J.: The polynomial-time hierarchy. Theor. Comput. Sci. 3(1), 1–22 (1976) CrossRefMathSciNetGoogle Scholar
  34. 34.
    Vardi, M.Y.: The complexity of relational query languages (extended abstract). In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing (STOC 1982), pp. 137–146. ACM, New York (1982) CrossRefGoogle Scholar
  35. 35.
    Vardi, M.Y.: On the complexity of bounded-variable queries. In: Proceedings of the Fourteenth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS 1995), pp. 266–276. ACM, New York (1995) CrossRefGoogle Scholar
  36. 36.
    Walukiewicz, I.: Model checking CTL properties of pushdown systems. In: Proceedings of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2000). Lecture Notes in Computer Science, vol. 1974, pp. 127–138. Springer, Berlin (2000) Google Scholar
  37. 37.
    Walukiewicz, I.: Pushdown processes: games and model-checking. Inf. Comput. 164(2), 234–263 (2001) MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Fachbereich 3Universität BremenBremenGermany
  2. 2.Institut für InformatikUniversität LeipzigLeipzigGermany

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