A Proof Procedure for Hybrid Logic with Binders, Transitivity and Relation Hierarchies

  • Marta Cialdea Mayer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7898)

Abstract

A tableau calculus constituting a decision procedure for hybrid logic with the converse modalities, the global ones and a restricted use of the binder has been defined in a previous paper. This work shows how to extend such a calculus to multi-modal logic equipped with two features largely used in description logics, i.e. transitivity and relation inclusion assertions. An implementation of the proof procedure is also briefly presented, along with the results of some preliminary experiments.

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References

  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., Heguiabehere, J.: hGen: A random CNF formula generator for hybrid languages. In: Methods for Modalities 3 (M4M-3), Nancy, France (2003)Google Scholar
  3. 3.
    Areces, C., ten Cate, B.: Hybrid logics. In: Handbook of Modal Logics, pp. 821–868. Elsevier (2007)Google Scholar
  4. 4.
    Blackburn, P., Seligman, J.: Hybrid languages. Journal of Logic, Language and Information 4, 251–272 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bolander, T., Blackburn, P.: Terminating tableau calculi for hybrid logics extending K. Electronic Notes in Theoretical Computer Science 231, 21–39 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cerrito, S., Cialdea Mayer, M.: An efficient approach to nominal equalities in hybrid logic tableaux. Journal of Applied Non-classical Logics 20(1-2), 39–61 (2010)Google Scholar
  7. 7.
    Cerrito, S., Cialdea Mayer, M.: Nominal substitution at work with the global and converse modalities. In: Advances in Modal Logic, vol. 8, pp. 57–74. College Publications (2010)Google Scholar
  8. 8.
    Cerrito, S., Cialdea Mayer, M.: A tableaux based decision procedure for a broad class of hybrid formulae with binders. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS, vol. 6793, pp. 104–118. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Cerrito, S., Cialdea Mayer, M.: A tableau based decision procedure for a fragment of hybrid logic with binders. Journal of Automated Reasoning (2012) (published online, to appear on paper)Google Scholar
  10. 10.
    Cialdea Mayer, M.: Tableaux for multi-modal hybrid logic with binders, transitive relations and relation hierarchies. Technical Report RT-DIA-199-2012, Dipartimento di Informatica e Automazione, Università di Roma Tre (2012)Google Scholar
  11. 11.
    Cialdea Mayer, M., Cerrito, S.: Herod and Pilate: two tableau provers for basic hybrid logic. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 255–262. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Grädel, E.: On the restraining power of guards. Journal of Symbolic Logic 64, 1719–1742 (1998)CrossRefGoogle Scholar
  13. 13.
    Horrocks, I., Sattler, U.: A description logic with transitive and inverse roles and role hierarchies. Journal of Logic and Computation 9(3), 385–410 (1999)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Horrocks, I., Sattler, U.: A tableau decision procedure for \(\mathcal{SHOIQ}\). Journal of Automated Reasoning 39(3), 249–276 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kaminski, M., Schneider, S., Smolka, G.: Terminating tableaux for graded hybrid logic with global modalities and role hierarchies. Logical Methods in Computer Science 7(1) (2011)Google Scholar
  16. 16.
    Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. Journal of Logic, Language and Information 18(4), 437–464 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Mundhenk, M., Schneider, T.: Undecidability of multi-modal hybrid logics. Electronic Notes in Theoretical Computer Science 174(6), 29–43 (2007)CrossRefGoogle Scholar
  18. 18.
    Mundhenk, M., Schneider, T., Schwentick, T., Weber, V.: Complexity of hybrid logics over transitive frames. Journal of Applied Logic 8(4), 422–440 (2010)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Szwast, W., Tendera, L.: On the decision problem for the guarded fragment with transitivity. In: Proc. of the 16th Symposium on Logic in Computer Science (LICS), pp. 147–156 (2001)Google Scholar
  20. 20.
    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
  21. 21.
    Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 140–145. Springer, Heidelberg (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marta Cialdea Mayer
    • 1
  1. 1.Università di Roma TreItaly

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