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Strong Normalisation of Cut-Elimination in Classical Logic

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Book cover Typed Lambda Calculi and Applications (TLCA 1999)

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Abstract

In this paper a strongly normalizing cut-elimination procedure is presented for classical logic. The procedure adapts the standard cut transformations, see for example [12]. In particular our cutelimination procedure requires no special annotations on formulae. We design a term calculus for a variant of Kleene’s sequent calculus G3 via the Curry-Howard correspondence and the cut-elimination steps are given as rewrite rules. In the strong normalization proof we adapt the symmetric reducibility candidates developed by Barbanera and Berardi.

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References

  1. F. Barbanera and S. Berardi. A Symmetric Lambda Calculus for “Classical” Program Extraction. In Proc. of Theoretical Aspects of Computer Software, volume 789 of LNCS, pages 495–515. Springer Verlag, 1994.

    Google Scholar 

  2. H. Barendregt. The Lambda Calculus-Its Syntax and Semantics, volume 103 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, 1981.

    Google Scholar 

  3. G. Bierman. Some Lectures on Proof Theory. Notes from course given at PUC-Rio, November 1997.

    Google Scholar 

  4. E. T. Bittar. Strong Normalisation Proofs for Cut Elimination in Gentzen’s Sequent Calculi. In Proc. of the Symposium: Logic, Algebra and Computer Science, Warsaw, December 1996.

    Google Scholar 

  5. E. A. Cichon, M. Rusinowitch, and S. Selhab. Cut Elimination and Rewriting: Termination Proofs. Theoretical Computer Science, 1996. (to appear).

    Google Scholar 

  6. V. Danos, J.-B. Joinet, and H. Schellinx. A New Deconstructive Logic: Linear Logic. Journal of Symbolic Logic, 62(3):755–807, Sept. 1997.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. G. Dragalin. Mathematical Intuitionism, volume 67. American Mathematical Society, Providence, Rhode Island, 1988.

    MATH  Google Scholar 

  8. R. Dyckhoff and L. Pinto. Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic. Studia Logica, 60:107–118, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Gallier. Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi. Theoretical Computer Science, 110(2):249–239, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Gentzen. Untersuchungen über das logische Schlieffen I and II. Mathematische Zeitschrift, 39:176–210, 405–431, 1935.

    Article  MathSciNet  Google Scholar 

  11. J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50:1–102, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.

    Google Scholar 

  13. E. H. Hauesler and L. C. Pereira. Gentzen’s Second Consistency Proof and Strong Cut-Elimination. Logique and Analyse, 154:95–111, 1996.

    MathSciNet  Google Scholar 

  14. H. Herbelin. A λ-Calculus Structure Isomorphic to Sequent Calculus Structure. In Proc. of the Conference on Computer Science Logic, volume 933 of LNCS, pages 67–75. Springer Verlag, 1995.

    Chapter  Google Scholar 

  15. J.-B. Joinet, H. Schellinx, and L. Tortora de Falco. SN and CR for Free-Style LK tq: Linear Decorations and Simulation of Normalisation. Technical Report, May 1998.

    Google Scholar 

  16. S. C. Kleene. Introduction to Metamathematics. North-Holland Publishing Company, Amsterdam, 1952.

    MATH  Google Scholar 

  17. M. Parigot. λμ-Calculus: An Algorithmic Interpretation of Classical Logic. In Proc. of the Int. Conference on Logic Programming and Automated Deduction, volume 624 of LNCS, pages 190–201. Springer Verlag, 1992.

    Google Scholar 

  18. A. S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1996.

    Google Scholar 

  19. C. Urban. First-Year Report: Computational Content of Classical Proofs. Technical Report, 1997.

    Google Scholar 

  20. J. Zucker. The Correspondance Between Cut-Elimination and Normalisation. Annals of Mathematical Logic, 7:1–112, 1974.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Cambridge Computer Laboratory, Cambridge, UK

    C. Urban & G. M. Bierman

Authors
  1. C. Urban
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  2. G. M. Bierman
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Editor information

Editors and Affiliations

  1. CNRS - UPR9016, Institut de Mathématiques de Luminy, 163 Avenue de Luminy, case 907, F-13288, Marseille cedex 9, France

    Jean-Yves Girard

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Urban, C., Bierman, G.M. (1999). Strong Normalisation of Cut-Elimination in Classical Logic. In: Girard, JY. (eds) Typed Lambda Calculi and Applications. TLCA 1999. Lecture Notes in Computer Science, vol 1581. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48959-2_26

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  • DOI: https://doi.org/10.1007/3-540-48959-2_26

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65763-7

  • Online ISBN: 978-3-540-48959-7

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