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
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.
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.
G. Bierman. Some Lectures on Proof Theory. Notes from course given at PUC-Rio, November 1997.
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.
E. A. Cichon, M. Rusinowitch, and S. Selhab. Cut Elimination and Rewriting: Termination Proofs. Theoretical Computer Science, 1996. (to appear).
V. Danos, J.-B. Joinet, and H. Schellinx. A New Deconstructive Logic: Linear Logic. Journal of Symbolic Logic, 62(3):755–807, Sept. 1997.
A. G. Dragalin. Mathematical Intuitionism, volume 67. American Mathematical Society, Providence, Rhode Island, 1988.
R. Dyckhoff and L. Pinto. Cut-Elimination and a Permutation-Free Sequent Calculus for Intuitionistic Logic. Studia Logica, 60:107–118, 1998.
J. Gallier. Constructive Logics. Part I: A Tutorial on Proof Systems and Typed λ-Calculi. Theoretical Computer Science, 110(2):249–239, 1993.
G. Gentzen. Untersuchungen über das logische Schlieffen I and II. Mathematische Zeitschrift, 39:176–210, 405–431, 1935.
J.-Y. Girard. Linear Logic. Theoretical Computer Science, 50:1–102, 1987.
J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1989.
E. H. Hauesler and L. C. Pereira. Gentzen’s Second Consistency Proof and Strong Cut-Elimination. Logique and Analyse, 154:95–111, 1996.
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.
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.
S. C. Kleene. Introduction to Metamathematics. North-Holland Publishing Company, Amsterdam, 1952.
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.
A. S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1996.
C. Urban. First-Year Report: Computational Content of Classical Proofs. Technical Report, 1997.
J. Zucker. The Correspondance Between Cut-Elimination and Normalisation. Annals of Mathematical Logic, 7:1–112, 1974.
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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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