Advertisement

Goal-Directed Calculi for Gödel-Dummett Logics

  • George Metcalfe
  • Nicola Olivetti
  • Dov Gabbay
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2803)

Abstract

In this work we present goal-directed calculi for the Gödel-Dummett logic LC and its finite-valued counterparts, LC n (n ≥ 2). We introduce a terminating hypersequent calculus for the implicational fragment of LC with local rules and a single identity axiom. We also give a labelled goal-directed calculus with invertible rules and show that it is co-NP. Finally we derive labelled goal-directed calculi for LC n .

Keywords

Gödel Logics Intermediate Logics Fuzzy Logics Hypersequents Goal-Directed Calculi 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aguzzoli, S., Gerla, B.: Finite-valued reductions of infinite-valued logics. Archive for Mathematical Logic 41(4), 361–399 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    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
  3. 3.
    Avron, A.: Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Artificial Intelligence 4(3–4), 225–248 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Avron, A., Konikowska, B.: Decomposition Proof Systems for Gödel-Dummett Logics. Studia Logica 69(2), 197–219 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baaz, M., Fermüller, C.G.: Analytic calculi for projective logics. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 36–50. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Baaz, M., Fermüller, C.G., Salzer, G.: Automated deduction for many-valued logics. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. II, ch. 20, pp. 1355–1402. Elsevier Science B.V., Amsterdam (2001)CrossRefGoogle Scholar
  7. 7.
    Dummett, M.: A propositional calculus with denumerable matrix. Journal of Symbolic Logic 24, 97–106 (1959)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dunn, J.M., Meyer, R.K.: Algebraic completeness results for Dummett’s LC and its extensions. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 17, 225–230 (1971)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dyckhoff, R.: A deterministic terminating sequent calculus for Gödel-Dummett logic. Logic Journal of the IGPL 7(3), 319–326 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fiorino, G.: An o(nlog n)-space decision procedure for the propositional Dummett logic. Journal of Automated Reasoning 27(3), 297–311 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gabbay, D.: Semantical Investigations in Heyting’s Intuitionistic Logic. Reidel, Dordrecht (1981)MATHGoogle Scholar
  12. 12.
    Gabbay, D., Olivetti, N.: Goal-directed Proof Theory. Kluwer Academic Publishers, Dordrecht (2000)MATHGoogle Scholar
  13. 13.
    Gabbay, D., Olivetti, N.: Goal oriented deductions. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosopophical Logic, 2nd edn., vol. 9, pp. 199–285. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  14. 14.
    Gödel, K.: Zum intuitionisticschen Aussagenkalkül. Anzeiger Akademie der Wissenschaften Wien, mathematisch-naturwiss. Klasse 32, 65–66 (1932)Google Scholar
  15. 15.
    Hähnle, R.: Automated Deduction in Multiple-Valued Logics. Oxford University Press, Oxford (1993)MATHGoogle Scholar
  16. 16.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht (1998)MATHGoogle Scholar
  17. 17.
    Larchey-Wendling, D.: Combining proof-search and counter-model construction for deciding gödel-dummett logic. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, p. 94. Springer, Heidelberg (2002)Google Scholar
  18. 18.
    Lifschitz, V., Pearce, D., Valverde, A.: Strongly equivalent logic programs. ACM Transactions on Computational Logic 2(4), 526–541 (2001)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Metcalfe, G., Olivetti, N., Gabbay, D.: Sequent and hypersequent calculi for abelian and Łukasiewicz logics (submitted)Google Scholar
  20. 20.
    Miller, D., Nadathur, G., Pfenning, F., Scedrov, A.: Uniform proofs as a foundation for logic programming. Annals of Pure and Applied Logic 51, 125–157 (1991)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Sonobe, O.: A Gentzen-type formulation of some intermediate propositional logics. Journal of Tsuda College 7, 7–14 (1975)MathSciNetGoogle Scholar
  22. 22.
    Visser, A.: On the completeness principle: a study of provability in heyting’s arithmetic. Annals of Mathematical Logic 22, 263–295 (1982)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • George Metcalfe
    • 1
  • Nicola Olivetti
    • 2
  • Dov Gabbay
    • 1
  1. 1.Department of Computer ScienceKing’s College LondonStrand, LondonUK
  2. 2.Department of Computer ScienceUniversity of TurinTurinItaly

Personalised recommendations