Symbolic Model Checking of Tense Logics on Rational Kripke Models

  • Wilmari Bekker
  • Valentin Goranko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5489)

Abstract

We introduce the class of rational Kripke models and study symbolic model checking of the basic tense logic Kt and some extensions of it on that class. Rational Kripke models are based on (generally infinite) rational graphs, with vertices labeled by the words in some regular language and transitions recognized by asynchronous two-head finite automata, also known as rational transducers. Every atomic proposition in a rational Kripke model is evaluated in a rational set of states. We show that every formula of Kt has an effectively computable rational extension in every rational Kripke model, and therefore local model checking and global model checking of Kt in rational Kripke models are decidable. These results are lifted to a number of extensions of Kt. We study and partly determine the complexity of the model checking procedures.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berstel, J.: Transductions and Context-Free Languages. Teubner Studienbücher Informatik. B.G. Teubner, Stuttgart (1979)Google Scholar
  2. 2.
    Biere, A., Cimatti, A., Clarke, E., Strichman, O., Zhu, Y.: Bounded model checking. Advances in Computers 58, 118–149 (2003)Google Scholar
  3. 3.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. CUP (2001)Google Scholar
  4. 4.
    Blackburn, P.: Representation, reasoning, and relational structures: a hybrid logic manifesto. Logic Journal of the IGPL 8(3), 339–365 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blumensath, A.: Automatic structures. Diploma thesis, RWTH Aachen (1999)Google Scholar
  6. 6.
    Blumensath, A., Grädel, E.: Automatic structures. In: Abadi, M. (ed.) Proc. LICS 2000, pp. 51–62 (2000)Google Scholar
  7. 7.
    Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 403–418. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)Google Scholar
  9. 9.
    Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, New York (1974)MATHGoogle Scholar
  10. 10.
    Elgot, C., Mezei, J.: On relations defined by finite automata. IBM J. of Research and Development 9, 47–68 (1965)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Esparza, J., Kučera, A., Schwoon, S.: Model-Checking LTL with Regular Valuations for Pushdown Systems. In: Kobayashi, N., Pierce, B.C. (eds.) TACS 2001. LNCS, vol. 2215, pp. 316–339. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Frougny, C., Sakarovitch, J.: Synchronized rational relations of finite and infinite words. Theor. Comput. Sci. 108(1), 45–82 (1993)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Johnson, J.H.: Rational equivalence relations. Theor. Comput. Sci. 47(3), 39–60 (1986)MATHCrossRefGoogle Scholar
  14. 14.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1994)Google Scholar
  15. 15.
    Kesten, Y., Maler, O., Marcus, M., Pnueli, A., Shahar, E.: Symbolic model checking with rich assertional languages. Theor. Comput. Sci. 256(1-2), 93–112 (2001)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kuper, G.M., Vardi, M.Y.: On the complexity of queries in the logical data model. In: Gyssens, M., Van Gucht, D., Paredaens, J. (eds.) ICDT 1988. LNCS, vol. 326, pp. 267–280. Springer, Heidelberg (1988)Google Scholar
  17. 17.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An automata-theoretic approach to branching-time model checking. Journal of the ACM 47(2), 312–360 (2000)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Martin, J.C.: Introduction to Languages and the Theory of Computation, 3rd edn., pp. 186–189. McGraw-Hill, Inc., New York (2002)Google Scholar
  19. 19.
    Meyer, A.R., Stockmeyer, L.J.: Word problems requiring exponential time: Preliminary report. In: Aho, A.V., et al. (eds.) Proc. STOC 1973, pp. 1–9 (1973)Google Scholar
  20. 20.
    Morvan, C.: On Rational Graphs. In: Tiuryn, J. (ed.) FOSSACS 2000. LNCS, vol. 1784, pp. 252–266. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  21. 21.
    Reisig, W.: Petri nets: An Introduction. Springer, New York (1985)MATHGoogle Scholar
  22. 22.
    Thomas, W.: Constructing infinite graphs with a decidable mso-theory. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003)Google Scholar
  23. 23.
    Vardi, M.: An automata-theoretic approach to linear temporal logic. In: Moller, F., Birtwistle, G. (eds.) Logics for Concurrency: Structure versus Automata (8th Banff Higher Order Workshop). LNCS, vol. 1043, pp. 238–266. Springer, Heidelberg (1996)Google Scholar
  24. 24.
    Walukiewicz, I.: Model checking CTL properties of pushdown systems. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 127–138. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Wilmari Bekker
    • 1
    • 2
  • Valentin Goranko
    • 2
  1. 1.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  2. 2.School of MathematicsUniversity of the WitwatersrandWitsSouth Africa

Personalised recommendations