Tractable Transformations from Modal Provability Logics into First-Order Logic

  • Stéephane Demri
  • Rajeev Goré
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)


We define a class of modal logics LF by uniformly extending a class of modal logics L. Each logic L is characterised by a class of first-order definable frames, but the corresponding logic LF is sometimes characterised by classes of modal frames that are not first-order definable. The class LF includes provability logics with deep arithmetical interpretations. Using Belnap’s proof-theoretical framework Display Logic we characterise the “pseudo-displayable” subclass of LF and show how to define polynomial-time transformations from each such LF into the corresponding L, and hence into first-order classical logic. Theorem provers for classical first-order logic can then be used to mechanise deduction in these “psuedo-displayable second order” modal logics.


Modal Logic Structural Connective Atomic Proposition Axiom Scheme Tractable Transformation 
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.


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  1. AEH90.
    Y. Auffray, P. Enjalbert, and J.-J. Herbrard. Strategies for modal resolution: results and problems. Journal of Automated Reasoning, 6:1–38, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Avr84.
    A. Avron. On modal systems having arithmetical interpretations. Journal of Symbolic Logic, 49(3):935–942, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Bel82.
    N. Belnap. Display logic. Journal of Philosophical Logic, 11:375–417, 1982.MathSciNetzbMATHGoogle Scholar
  4. BG86.
    M. Borga and P. Gentilini. On the proof theory of the modal logic Grz. Zeitschrift füur Mathematik Logik und Grundlagen der Mathematik, 32:145–148, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  5. BH94.
    Ph. Balbiani and A. Herzig. A translation from the modal logic of provability into K4. Journal of Applied Non-Classical Logics, 4:73–77, 1994.zbMATHMathSciNetGoogle Scholar
  6. Boo93.
    G. Boolos. The Logic of Provability. Cambridge University Press, 1993.Google Scholar
  7. DG99a.
    S. Demri and R. Goré. An O((n.log n)3)-time transformation from Grz into decidable fragments of classical first-order logic. In Selected papers of FTP’98, Vienna. LNAI, Springer-Verlag, 1999. to appear.Google Scholar
  8. DG99b.
    S. Demri and R. Goré. Theoremhood preserving maps as a characterisation of cut elimination for provability logics. Technical report, A.R.P., A.N.U., 1999. Forthcoming.Google Scholar
  9. dMP95.
    G. d’Agostino, A. Montanari, and A. Policriti. A set-theoretical translation method for polymodal logics. Journal of Automated Reasoning, 15:317–337, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Fit83.
    M. Fitting. Proof methods for modal and intuitionistic logics. D. Reidel Publishing Co., 1983.Google Scholar
  11. GHM98.
    H. Ganzinger, U. Hustadt, and R. Meyer, C. Schmidt. A resolution-based decision procedure for extensions of K4. In 2nd Workshop on Advances in Modal Logic (AiML’98), Uppsala, Sweden, October 1998. to appear.Google Scholar
  12. Gor99.
    R. Goré.Tableaux methods for modal and temporal logics. In M. d’Agostino, D. Gabbay, R. Hähnle, and J. Posegga, editors, Handbook of Tableaux Methods. Kluwer, Dordrecht, 1999. To appear.Google Scholar
  13. Her89.
    A. Herzig. Raisonnement automatique en logique modale et algorithmes d’unification. PhD thesis, Université P. Sabatier, Toulouse, 1989.Google Scholar
  14. HS97.
    U. Hustadt and R. Schmidt. On evaluating decision procedures for modal logic. In IJCAI-15, pages 202–207. Morgan Kaufmann, 1997.Google Scholar
  15. Kra96.
    M. Kracht. Power and weakness of the modal display calculus. In H. Wansing, editor, Proof theory of modal logic, pages 93–121. Kluwer Academic Publishers, 1996.Google Scholar
  16. Lei81.
    D. Leivant. On the proof theory of the modal logic for arithmetical provability. Journal of Symbolic Logic, 46(3):531–538, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Mas94.
    F. Massacci. Strongly analytic tableaux for normal modal logics. In A. Bundy, editor, CADE-12, pages 723–737. LNAI 814, July 1994.Google Scholar
  18. Min88.
    G. Mints. Gentzen-type and resolution rules part I: propositional logic. In P. Martin-Löf and G. Mints, editors, International Conference on Computer Logic, Tallinn, pages 198–231. LNCS 417, 1988.Google Scholar
  19. Mor76.
    Ch. Morgan. Methods for automated theorem proving in non classical logics. IEEE Transactions on Computers, 25(8):852–862, 1976.zbMATHCrossRefGoogle Scholar
  20. NS98.
    A. Nonnengart and A. Szalas. A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In E. Or lowska, editor, Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa, pages 89–108. Physica Verlag, 1998.Google Scholar
  21. Ohl88.
    H.J. Ohlbach. A resolution calculus for modal logics. In CADE-9, pages 500–516. LNCS 310, 1988.Google Scholar
  22. Ohl98.
    H.J. Ohlbach. Combining Hilbert style and semantic reasoning in a resolution framework. In C. Kirchner and H. Kirchner, editors, CADE-15, Lindau, Germany, pages 205–219. LNAI 1421, 1998.Google Scholar
  23. Pap94.
    Ch. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, 1994.Google Scholar
  24. Sah75.
    H. Sahlqvist. Completeness and correspondence in the first and second order semantics for modal logics. In S. Kanger, editor, 3rd Scandinavian Logic Symposium, pages 110–143. North Holland, 1975.Google Scholar
  25. Sch99.
    R. Schmidt. Decidability by resolution for propositional modal logics. Journal of Automated Reasoning, 1999. To appear.Google Scholar
  26. SV80.
    G. Sambin and S. Valentini. A modal sequent calculus for a fragment of arithmetic. Studia Logica, 39:245–256, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  27. SV82.
    G. Sambin and S. Valentini. The modal logic of provability. The sequential approach. Journal of Philosophical Logic, 11:311–342, 1982.CrossRefMathSciNetzbMATHGoogle Scholar
  28. Val83.
    S. Valentini. The modal logic of provability: cut-elimination. Journal of Philosophical Logic, 12:471–476, 1983.CrossRefMathSciNetzbMATHGoogle Scholar
  29. Wan94.
    H. Wansing. Sequent calculi for normal modal propositional logics. Journal of Logic and Computation, 4(2):125–142, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Wan98.
    H. Wansing. Displaying Modal Logic, volume 3 of Trends in Logic. Kluwer Academic Publishers, Dordrecht, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Stéephane Demri
    • 1
  • Rajeev Goré
    • 2
  1. 1.Laboratoire LEIBNIZ — C.N.R.S.GrenobleFrance
  2. 2.Automated Reasoning Project and Dept. of Computer ScienceAustralian National UniversityCanberraAustralia

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