Formal Language Constrained Reachability and Model Checking Propositional Dynamic Logics

  • Roland Axelsson
  • Martin Lange
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6945)

Abstract

We show interreducibility under (Turing) reductions of low polynomial degree between three families of problems parametrised by classes of formal languages: the problem of reachability in a directed graph constrained by a formal language, the problem of deciding whether or not the intersection of a language of some class with a regular language is empty, and the model checking problem for Propositional Dynamic Logic over some class of formal languages. This allows several decidability and complexity results to be transferred, mainly from the area of formal languages to the areas of modal logics and formal language constrained reachability.

Keywords

Model Check Formal Language Description Logic Regular Language Kripke Structure 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aho, A.V.: Indexed grammars - an extension of context-free grammars. J. ACM 15(4), 647–671 (1968)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. 36th Ann. ACM Symp. on Theory of Computing (STOC 2004), pp. 202–211. ACM Press, New York (2004)Google Scholar
  3. 3.
    Baader, F., Lutz, C., Turhan, A.-Y.: Small is again beautiful in description logics. KI – Künstliche Intelligenz (2010) (to appear)Google Scholar
  4. 4.
    Bar-Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Zeitschrift für Phonologie, Sprachwissenschaft und Kommunikationsforschung 14, 113–124 (1961)MathSciNetMATHGoogle Scholar
  5. 5.
    Barrett, C., Jacob, R., Marathe, M.: Formal-language-constrained path problems. SIAM Journal on Computing 30(3), 809–837 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.: Reasoning about Knowledge. MIT Press, Cambridge (1995)MATHGoogle Scholar
  7. 7.
    Fähndrich, M., Rehof, J.: Type-based flow analysis and context-free language reachability. Mathematical Structures in Computer Science 18(5), 823–894 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18, 194–211 (1979)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gazdar, G.: Applicability of indexed grammars to natural languages. In: Reyle, U., Rohrer, C. (eds.) Natural Language Parsing and Linguistic Theories, pp. 69–94. Reidel, Dordrecht (1988)CrossRefGoogle Scholar
  10. 10.
    De Giacomo, G., Lenzerini, M.: Boosting the correspondence between description logics and propositional dynamic logics. In: Proc. of the 12th National Conference on Artificial Intelligence (AAAI 1994), pp. 205–212. AAAI-Press/The MIT-Press (1994)Google Scholar
  11. 11.
    Harel, D., Kaminsky, M.: Strengthened results on nonregular PDL. Technical Report MCS99-13, Weizmann Institute of Science, Faculty of Mathematics and Computer Science (1999)Google Scholar
  12. 12.
    Harel, D., Pnueli, A., Stavi, J.: Propositional dynamic logic of nonregular programs. Journal of Computer and System Sciences 26(2), 222–243 (1983)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Harel, D., Raz, D.: Deciding properties of nonregular programs. SIAM J. Comput. 22(4), 857–874 (1993)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Harel, D., Singerman, E.: More on nonregular PDL: Finite models and Fibonacci-like programs. Information and Computation 128(2), 109–118 (1996)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hunt, H.B.: On the time and tape complexity of languages I. In: ACM (ed.) Conf. Rec. of 5th Annual ACM Symp. on Theory of Computing (STOC 1973), pp. 10–19. ACM Press, New York (1973)Google Scholar
  16. 16.
    Landweber, P.S.: Three theorems on phrase structure grammars of type 1. Inform. and Control 6, 131–136 (1963)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lange, M.: Alternating context-free languages and linear time μ-calculus with sequential composition. In: Proc. 9th Workshop on Expressiveness in Concurrency (EXPRESS 2002). ENTCS, vol. 68.2, pp. 71–87. Elsevier, Amsterdam (2002)Google Scholar
  18. 18.
    Lange, M.: Model checking propositional dynamic logic with all extras. Journal of Applied Logic 4(1), 39–49 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Löding, C., Lutz, C., Serre, O.: Propositional dynamic logic with recursive programs. J. Log. Algebr. Program 73(1-2), 51–69 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 422–435. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  21. 21.
    Moriya, E.: A grammatical characterization of alternating pushdown automata. TCS 67(1), 75–85 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nitta, N., Seki, H., Takata, Y.: Security verification of programs with stack inspection. In: SACMAT, pp. 31–40 (2001)Google Scholar
  23. 23.
    Okhotin, A.: Conjunctive grammars. Journal of Automata, Languages and Combinatorics 6(4), 519–535 (2001)MathSciNetMATHGoogle Scholar
  24. 24.
    Okhotin, A.: Boolean grammars. Information and Computation 194(1), 19–48 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Prendinger, H., Schurz, G.: Reasoning about action and change. A dynamic logic approach. Journal of Logic, Language and Information 5(2), 209–245 (1996)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Reps, T.: Shape analysis as a generalized path problem. In: Proc. ACM SIGPLAN Symp. on Partial Evaluation and Semantics-Based Program Manipulation, pp. 1–11 (1995)Google Scholar
  27. 27.
    Reps, T.W.: Program analysis via graph reachability. Information & Software Technology 40(11-12), 701–726 (1998)CrossRefGoogle Scholar
  28. 28.
    Tanaka, S., Kasai, T.: The emptiness problem for indexed language is exponential-time complete. Systems and Computers in Japan 17(9), 29–37 (2007)MathSciNetCrossRefGoogle Scholar
  29. 29.
    La Torre, S., Madhusudan, P., Parlato, G.: A robust class of context-sensitive languages. In: Proc. 22nd Conf. on Logic in Computer Science (LICS 2007), pp. 161–170. IEEE, Los Alamitos (2007)Google Scholar
  30. 30.
    Vijay-Shanker, K., Weir, D.J.: The equivalence of four extensions of context-free grammars. Mathematical Systems Theory 27, 27–511 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Roland Axelsson
    • 1
  • Martin Lange
    • 2
  1. 1.Dept. of Computer ScienceUniversity of MunichGermany
  2. 2.School of Electr. Eng. and Computer ScienceUniversity of KasselGermany

Personalised recommendations