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Proofs of strong normalisation for second order classical natural deduction

Published online by Cambridge University Press:  12 March 2014

Michel Parigot
Michel Parigot*
Affiliation:
Equipe de Logique—CNRS UA 753, 45-55 5Ème Étage, Université Paris7, 2 Place Jussieu, 75251 Paris Cedex 05, France E-mail: parigot@logique.jussieu.fr
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Abstract

We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1461 - 1479
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Barendregt, H., The lambda-calculus, North-Holland, 1981.Google Scholar
[2]Berardi, S. and Barbanera, F., A symmetric lambda-calculus for “classical” program extraction, draft, 1994.CrossRefGoogle Scholar
[3]Coquand, T., A semantics of evidence for classical arithmetic, this Journal, vol. 60 (1995), pp. 325–337.Google Scholar
[4]Danos, V., Une application de la logique linéaire à l'étude des processus de normalisation, Ph.D. thesis, Université Paris 7, 1990.Google Scholar
[5]Danos, V., Joinet, J. B., and Schellinx, H., A new deconstructive logic: linear logic, to appear in this Journal.Google Scholar
[6]DeGroote, P., A CPS-translation of the λü-calculus, Proceedings of CAAP '94, Lecture Notes in Computer Science, no. 787, Springer-Verlag, 1994, pp. 85–99.Google Scholar
[7]Fortune, S., Leivant, D., and O'Donnell, M., The expressiveness of simple and second order type structures, Journal of the Association for Computing Machinery, vol. 30 (1983), pp. 151–185.CrossRefGoogle Scholar
[8]Gallier, J. H., On Girard's “candidats de réductibilités”, Logic and computer science (Odifreddi, P., editor), Academic Press, 1990, pp. 123–203.Google Scholar
[9]Girard, J. Y., Interprétation fonctionnelle et élimination des coupures de l'arithmétique d'ordre supérieure, Ph.D. thesis, Université Paris 7, 1972.Google Scholar
[10]Girard, J. Y., Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1–102.CrossRefGoogle Scholar
[11]Girard, J. Y., A new constructive logic: classical logic, Mathematical Structures in Computer Science, vol. 1 (1991), pp. 255–296.CrossRefGoogle Scholar
[12]Girard, J. Y., Lafont, Y., and Taylor, P., Proofs and types, Cambridge University Press, 1989.Google Scholar
[13]Griffin, T., A formulae-as-types notion of control, Proceedings of POPL, 1990, pp. 47–58.Google Scholar
[14]Krivine, J. L., Lambda-calcul, types et modèles, Masson, 1990.Google Scholar
[15]Murthy, C., Extracting constructive content from classical proofs, Ph.D. thesis, Cornell, 1990.Google Scholar
[16]Parigot, M., Internal labellings in lambda-calculus, Proceedings of MFCS (Bystrica, Banská, editor), Lecture Notes in Computer Science, no. 452, 1990, pp. 439–445.Google Scholar
[17]Parigot, M., Free deduction: an analysis of “computations” in classical logic, Proceedings of the Russian conference on logic programming, Lecture Notes in Computer Science, no. 592, Springer-Verlag, 1991, pp. 361–380.Google Scholar
[18]Parigot, M., λü-calculus: an algorithmic interpretation of classical natural deduction, Proceedings of the international conference on logic programming and automated reasoning, Lecture Notes in Computer Science, no. 624, Springer-Verlag, 1992, pp. 190–201.CrossRefGoogle Scholar
[19]Plotkin, G., Call-by-name, call-by-value and the λ-calculus, Theoretical Computer Science, vol. 1 (1975), pp. 125–159.CrossRefGoogle Scholar
[20]Rezus, A., Beyond BHK, draft, 1993.Google Scholar