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What Is a Theory?

  • Gilles Dowek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2285)

Abstract

Deduction modulo is a way to express a theory using computation rules instead of axioms.We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones.We show that all theories in propositional calculus can be expressed in this framework and that cuts can always be eliminated with such theories.

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References

  1. 1.
    S.C. Bailin.A normalization theorem for set theory. The Journal of Symbolic Logic, 53, 3, 1988, pp.673–695.Google Scholar
  2. 2.
    M. Crabbé.Non-normalisation de la théorie de Zermelo. Manuscript, 1974.Google Scholar
  3. 3.
    M. Crabbé.Stratification and cut-elimination. The Journal of Symbolic Logic, 56, 1991, pp.213–226.Google Scholar
  4. 4.
    G. Do wek. Axioms vs. rewrite rules: from completeness to cut elimination. H. Kirchner and Ch. Ringeissen (Eds.), Frontiers of Combining Systems, Lecture Notes in Artificial Intelligence 1794, Springer-Verlag, 2000, pp.62–72.Google Scholar
  5. 5.
    G.Do wek.About folding-unfolding cuts and cuts modulo. Journal of Logic and Computation 11, 3, 2001, pp.419–429.Google Scholar
  6. 6.
    G.Do wek.Confluence as a cut elimination property. Workshop on Logic, Language, Information and Computation, 2001.Google Scholar
  7. 7.
    G.Do wek, Th. Hardin, and C. Kirchner.Theorem proving modulo. Journal of Automated Reasoning (to appear). Rapport de Recherche INRIA 3400, 1998.Google Scholar
  8. 8.
    G. Dowek and B. Werner.Proof normalization modulo. Types for proofs and programs, T. Altenkirch, W. Naraschewski, and B. Rues (Eds.), Lecture Notes in Computer Science 1657, Springer-Verlag, 1999, pp.62–77. Rapport de Recherche 3542, INRIA, 1998.CrossRefGoogle Scholar
  9. 9.
    J. Ekman. Normal proofs in set theory. Doctoral thesis, Chalmers University of Technology and University of Göteborg, 1994.Google Scholar
  10. 10.
    L. Hallnäs. On normalization of proofs in set theory. Doctoral thesis, University of Stockholm, 1983.Google Scholar
  11. 11.
    L. Hallnäs and P. Schroeder-Heister.A proof-theoretic approach to logic programming. I. Clauses as rules. Journal of Logic and Computation 1, 2, 1990, pp.261–283. II. Programs as definitions. Journal of Logic and Computation 1, 5, 1991, pp.635–660.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    D.E. Knuth and P.B. Bendix.Simple word problems in universal algebras. J. Leech (Ed.), Computational Problems in Abstract Algebra, Pergamon Press, 1970, pp.263–297.Google Scholar
  13. 13.
    R. McDowell and D. Miller.Cut-Elimination for a logic with definitions and induction. Theoretical Computer Science 232, 2000, pp.91–119.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Negri and J. Von Plato.Cut elimination in the presence of axioms. The Bulletin of Symbolic Logic, 4, 4, 1998, pp.418–435.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    D. Prawitz. Natural deduction, a proof-theoretical study. Almqvist & Wiksell, 1965.Google Scholar
  16. 16.
    P. Schroeder-Heister. Cut elimination in logics with definitional reflection. D. Pearce and H. Wansing (Eds.), Nonclassical Logics and Information Processing, Lecture Notes in Computer Science 619, Springer-Verlag, 1992, pp.146–171.Google Scholar
  17. 17.
    P. Schroeder-Heister.Rules of definitional reflection. Logic in Computer Science, 1993, pp.222–232.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Gilles Dowek
    • 1
  1. 1.INRIA-RocquencourtLe Chesnay CedexFrance

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