Hypertableau and Path-Hypertableau Calculi for some Families of Intermediate Logics

  • Agata Ciabattoni
  • Mauro Ferrari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1847)


In this paper we investigate the tableau systems corresponding to hypersequent calculi. We call these systems hypertableau calculi. We define hypertableau calculi for some propositional intermediate logics. We then introduce path-hypertableau calculi which are simply defined by imposing additional structure on hypertableaux. Using path-hypertableaux we define analytic calculi for the intermediate logics Bdk, with k ≥1, which are semantically characterized by Kripke models of depth ≤k. These calculi are obtained by adding one more structural rule to the path-hypertableau calculus for Intuitionistic Logic.


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  1. 1.
    Avellone, A., Ferrari, M., Miglioli, P.: 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.
    Avron, A.: A constructive analysis of RM. J. Symbolic Logic 52, 939–951 (1987)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4, 225–248 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Avron, A.: The method of hypersequents in the proof theory of propositional nonclassical logics. In: Logic: from Foundations to Applications, European Logic Colloquium, pp. 1–32. Oxford University Press, Oxford (1996)Google Scholar
  5. 5.
    Baumgartner, P., Furbach, U., Niemelae, I.: Hyper Tableaux. In: Orłowska, E., Alferes, J.J., Moniz Pereira, L. (eds.) JELIA 1996. LNCS (LNAI), vol. 1126, pp. 1–17. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Baaz, M., Fermüller, C.G.: Analytic calculi for projective logics. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 36–51. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Chagrov, A., Zakharyaschev, M.: Modal Logic. Oxford University Press, Oxford (1997)MATHGoogle Scholar
  8. 8.
    Ciabattoni, A., Ferrari, M.: Hypersequent calculi for some intermediate logics with bounded Kripke models. Journal of Logic and Computation (to appear)Google Scholar
  9. 9.
    Ciabattoni, A., Gabbay, D.M., Olivetti, N.: Cut-free proof systems for logics of weak excluded middle. Soft Computing 2(4), 147–156 (1998)Google Scholar
  10. 10.
    Dyckhoff, R.: A deterministic terminating sequent calculus for Gödel-Dummett logic. Logic J. of the IGPL 7, 319–326 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fitting, M.C.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)MATHGoogle Scholar
  12. 12.
    Gabbay, D.M.: Semantical Investigations in Heyting’s Intuitionistic Logic. Reidel, Dordrecht (1983)Google Scholar
  13. 13.
    Gabbay, D.M., Olivetti, N.: Goal-Directed Proof Theory. Kluwer, Dordrecht (to appear)Google Scholar
  14. 14.
    Hähnle, R.: Automated Deduction in Multiple-valued Logics. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  15. 15.
    Maksimova, L.: Craig’s theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras. Algebra and Logic 16, 427–455 (1977)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Miglioli, P., Moscato, U., Ornaghi, M.: Avoiding duplications in tableau systems for intuitionistic logic and Kuroda logic. Logic J. of the IGPL 5(1), 145–167 (1997)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pottinger, G.: Uniform, cut-free formulation of T,S4 and S5 (abstract). J. Symbolic Logic 48, 900 (1983)Google Scholar
  18. 18.
    Jankov, V.: The calculus of the weak ”law of excluded middle”. Mathematics of the USSR 8, 648–658 (1968)Google Scholar
  19. 19.
    Rousseau, G.: Sequent in many valued logic, I and II. Fund. Math. 60 and 67, 23–33 (1967), 125–131 (1970)Google Scholar
  20. 20.
    Smullyan, R.M.: First-Order Logic. Springer, Berlin (1968)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Agata Ciabattoni
    • 1
  • Mauro Ferrari
    • 1
  1. 1.Dipartimento di Scienze dell’InformazioneUniversità degli Studi di MilanoMilanoItaly

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