LTL-Model-Checking via Model Composition

  • Ingo Felscher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7550)

Abstract

We develop a composition technique for linear time logic (LTL) over ordered disjoint sums of words. This technique allows to reduce the validity of an LTL formula in the sum to LTL formulas over the components. It is known that for first order logic (FO) and even its three variable fragment FO3 the number of formulas generated for the components is at least non-elementary in the size of the input formula. We show that for LTL – expressively equivalent to FO logic – over an ordered disjoint sum of words the number of formulas for the components can be kept at an at most exponential growth in the size of the input formula.

Keywords

model-checking linear time logic model composition 

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References

  1. 1.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press (2008)Google Scholar
  2. 2.
    Chang, C.C., Keisler, H.J.: Model Theory. North Holland, Amsterdam (1990)MATHGoogle Scholar
  3. 3.
    Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Model Theory Makes Formulas Large. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 913–924. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Ehrenfeucht: An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae 49, 129–141 (1961)Google Scholar
  5. 5.
    Feferman, S., Vaught, R.: The first-order properties of products of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)MathSciNetMATHGoogle Scholar
  6. 6.
    Felscher, I.: The Compositional Method and Regular Reachability. Electronic Notes in Theoretical Computer Science 223, 103–117 (2008)CrossRefGoogle Scholar
  7. 7.
    Felscher, I., Thomas, W.: Compositionality and Reachability with Conditions on Path Lengths. Int. Journal of Foundations of Computer Science 20(5), 851–868 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fraïssé: Sur quelques classifications des systèmes de relations. Publications Scientifiques de l’université d’Alger 1(A), 35–182 (1954)Google Scholar
  9. 9.
    Göller, S., Jung, J.C., Lohrey, M.: The complexity of decomposing modal and first-order theories. In: Twenty-Seventh Annual ACM/IEEE Symposium on Logic In Computer Science (LICS 2012), Dubrovnik, Croatia, June 25-28 (2012)Google Scholar
  10. 10.
    Hodges, W.: Model Theory, Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)Google Scholar
  11. 11.
    Makowsky, J.A.: Algorithmic Uses of the Feferman-Vaught Theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Mostowski, A.: On Direct Products of Theories. The Journal of Symbolic Logic 17(1), 1–31 (1952)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Rabinovich, A.: On Compositionality and its Limitations. ACM Transactions on Computational Logic 8(1) (January 2007)Google Scholar
  14. 14.
    Thomas, W.: Ehrenfeucht Games, the Composition Method, and the Monadic Theory of Ordinal Words. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 118–143. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Wöhrle, S., Thomas, W.: Model Checking Synchronized Products of Infinite Transition Systems. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 2–11. IEEE Computer Society, Washington, DC (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ingo Felscher
    • 1
  1. 1.RWTH Aachen UniversityAachenGermany

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