Tableau Calculi for the Logics of Finite k-Ary Trees

  • Mauro Ferrari
  • Camillo Fiorentini
  • Guido Fiorino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2381)

Abstract

We present tableau calculi for the logics Dk (k≥2) semantically characterized by the classes of Kripke models built on finite k-ary trees. Our tableau calculi use the signs T and F, some tableau rules for Intuitionistic Logic and two rules formulated in a hypertableau fashion. We prove the Soundness and Completeness Theorems for our calculi. Finally, we use them to prove the main properties of the logics Dk, in particular their constructivity and their decidability.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Avellone, M. Ferrari, and P. Miglioli. Duplication-free tableau calculi and related cut-free sequent calculi for the interpolable propositional intermediate logics. Logic Journal of the IGPL, 7(4):447–480, 1999.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Avellone, P. Miglioli, U. Moscato, and M. Ornaghi. Generalized tableau systems for intermediate propositional logics. In D. Galmiche, editor, Proceedings of the 6th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods: Tableaux’ 97, volume 1227 of LNAI, pages 43–61. Springer-Verlag, 1997.Google Scholar
  3. 3.
    A. Avron. Hypersequents, logical consequence and intermediate logics for concurrency. Annals for Mathematics and Artificial Intelligence, 4:225–248, 1991.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. Baaz and C. G. Fermüller. Analytic calculi for projective logics. Lecture Notes in Computer Science, 1617:36–50, 1999.Google Scholar
  5. 5.
    A. Chagrov and M. Zakharyaschev. Modal Logic. Oxford University Press, 1997.Google Scholar
  6. 6.
    A. Ciabattoni and M. Ferrari. Hypertableau and path-hypertableau calculi for some families of intermediate logics. In R. Dyckhoff, editor, TABLEAUX 2000, Automated Reasoning with Analytic Tableaux and Related Methods, volume 1947 of LNAI, pages 160–174. Springer-Verlag, 2000.Google Scholar
  7. 7.
    A. Ciabattoni and M. Ferrari. Hypersequent calculi for some intermediate logics with bounded Kripke models. Journal of Logic and Computation, 11(2):283–294, 2001.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    R. Dyckhoff. A deterministic terminating sequent calculus for Gödel-Dummett logic. Logic Journal of the IGPL, 7(3):319–326, 1999.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Ferrari and P. Miglioli. Counting the maximal intermediate constructive logics. Journal of Symbolic Logic, 58(4):1365–1401, 1993.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    G. Fiorino. An O(n log n)-space decision procedure for the propositional Dummett Logic. Journal of Automated Reasoning, 27(3):297–311, 2001.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    G. Fiorino. Space-efficient Decision Procedures for Three Interpolable Propositional Intermediate Logics. Journal of Logic and Computation, To appear.Google Scholar
  12. 12.
    M.C. Fitting. Intuitionistic Logic, Model Theory and Forcing. North-Holland, 1969.Google Scholar
  13. 13.
    D.M. Gabbay and D.H.J. De Jongh. A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property. Journal of Symbolic Logic, 39:67–78, 1974.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    P. Hájek. Metamathematics of fuzzy logic. Kluwer, 1998.Google Scholar
  15. 15.
    D. Pearce. A new logical characterization of stable models and answer sets. In J. Dix, L.M. Pereira, and T. Przymusinski, editors, Non-Monotonic Extensions of Logic Programming, volume 1216 of LNAI, pages 57–70. Springer-Verlag, 1997.Google Scholar
  16. 16.
    D. Pearce. Stable inference as intuitionistic validity. Journal of Logic Programming, 38(1):79–91, 1999.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Mauro Ferrari
    • 1
  • Camillo Fiorentini
    • 1
  • Guido Fiorino
    • 2
  1. 1.Dipartimento di Scienze dell’InformazioneUniversitá degli Studi di MilanoMilanoItaly
  2. 2.CRIIUniversità dell’InsubriaVareseItaly

Personalised recommendations