Bounded-Variable Fragments of Hybrid Logics

  • Thomas Schwentick
  • Volker Weber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


Hybrid logics extend modal logics by first-order concepts, in particular they allow a limited use of variables. Unfortunately, in general, satisfiability for hybrid formulas is undecidable and model checking is PSPACE-hard. It is shown here that on the linear frame (ω, < ), the restriction to one name, although expressively complete, has EXPSPACE-complete satisfiability and polynomial time model-checking.

For the upper bound, a result of independent interest is found: Non-emptiness for alternating two-way Büchi automata with one pebble is EXPSPACE-complete.


Model Check Temporal Logic Kripke Structure Hybrid Logic Model Check Problem 
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.


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  1. 1.
    Areces, C., Blackburn, P., Marx, M.: A road-map on complexity for hybrid logics. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 307–321. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Areces, C., Blackburn, P., Marx, M.: The computational complexity of hybrid temporal logics. Logic Journal of the IGPL 8(5), 653–679 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Areces, C., Blackburn, P., Marx, M.: Hybrid logics: Characterization, interpolation and complexity. Journal of Symbolic Logic 66(3), 977–1010 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy. In: Proc. of 32nd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 368–377. IEEE Computer Society Press, Los Alamitos (1991)CrossRefGoogle Scholar
  5. 5.
    Franceschet, M., de Rijke, M.: Model checking hybrid logics (with an application to semistructured data). Journal of Applied Logic (2005)Google Scholar
  6. 6.
    Franceschet, M., de Rijke, M., Schlingloff, B.-H.: Hybrid logics on linear structures: Expressivity and complexity. In: 10th TIME / 4th ICTL, pp. 192–202. IEEE Computer Society Press, Los Alamitos (2003)Google Scholar
  7. 7.
    Gabbay, D.M.: The declarative past and imperative future: Executable temporal logic for interactive systems. In: Banieqbal, B., Pnueli, A., Barringer, H. (eds.) Temporal Logic in Specification. LNCS, vol. 398, pp. 409–448. Springer, Heidelberg (1989)Google Scholar
  8. 8.
    Gabbay, D.M., et al.: On the temporal basis of fairness. In: Conference Record of the 7th Annual ACM Symposium on Principles of Programming Languages (POPL’80), pp. 163–173. ACM Press, New York (1980)Google Scholar
  9. 9.
    Globerman, N., Harel, D.: Complexity results for two-way and multi-pebble automata and their logics. Theoretical Computer Science 169(2), 161–184 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Goralčik, P., Goralčiková, A., Koubek, V.: Alternation with a pebble. Information Processing Letters 38(1), 7–13 (1991)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Jiang, T., Ravikumar, B.: A note on the space complexity of some desicion problems for finite automata. Information Processing Letters 40, 25–31 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kamp, H.: Tense logic and the Theory of Linear Order. PhD thesis, University of California Los Angeles (1968)Google Scholar
  13. 13.
    Kupferman, O., Piterman, N., Vardi, M.Y.: Extended temporal logic revisited. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 519–535. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Mostowski, A.W.: Games with forbidden positions. Technical report, Uniwersytet Gdanski, Instytut Matematyki (1991)Google Scholar
  15. 15.
    Mundhenk, M., et al.: Complexity of hybrid logics over transitive frames. In: Proc. of M4M-4. Informatik-Berichte, vol. 194, pp. 62–78. Humboldt-Universität, Berlin (2005)Google Scholar
  16. 16.
    Sattler, U., Vardi, M.Y.: The hybrid μ-calculus. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 76–91. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Stockmeyer, L.J.: The complexity of decision problems in automata theory and logic. PhD thesis, MIT (1974)Google Scholar
  18. 18.
    ten Cate, B., Franceschet, M.: On the complexity of hybrid logics with binders. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 339–354. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    van Emde Boas, P.: The convenience of tilings. In: Complexity, Logic, and Recursion Theory. Lecture Notes in Pure and Applied Mathematics, vol. 187, pp. 331–363. Marcel Dekker, New York (1997)Google Scholar
  20. 20.
    Vardi, M.Y.: Nontraditional applications of automata theory. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 575–597. Springer, Heidelberg (1994)Google Scholar
  21. 21.
    Vardi, M.Y.: An automata-theoretic approach to linear temporal logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency. LNCS, vol. 1043, pp. 238–266. Springer, Heidelberg (1996)Google Scholar
  22. 22.
    Vardi, M.Y.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  23. 23.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wolper, P., Vardi, M.Y., Sistla, A.P.: Reasoning about infinite computation paths (extended abstract). In: 24th Annual Symposium on Foundations of Computer Science (FOCS), pp. 185–194. IEEE Computer Society Press, Los Alamitos (1983)Google Scholar
  25. 25.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata and infinite trees. Theoretical Computer Science 200, 135–183 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas Schwentick
    • 1
  • Volker Weber
    • 1
  1. 1.Fachbereich Informatik, Universität DortmundGermany

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