Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-19T15:04:06.807Z Has data issue: false hasContentIssue false
  • English
  • Français

A short proof of the strong normalization of classical natural deduction with disjunction

Published online by Cambridge University Press:  12 March 2014

René David  and
Karim Nour
René David
Affiliation:
Lama – Equipe De Logique, Université De Chambéry, 73376 Le Bourget Du Lac, France, E-mail: david@univ-savoie.fr
Karim Nour
Affiliation:
Lama – Equipe De Logique, Université De Chambéry, 73376 Le Bourget Du Lac, France, E-mail: nour@univ-savoie.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1277 - 1288
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Andou, Y., Church-Rosser property of a simple reduction for full first-order classical natural deduction. Annals of Pure and Applied Logic, vol. 119 (2003), pp. 225–237.Google Scholar
[2] Barbanera, F. and Berardi, S., A symmetric lambda calculus for “classical” program extraction, Proceedings of theoretical aspects of computer software, tacs '94 (Hagiya, M. and Mitchell, J.C., editors), Lecture Notes in Computer Science, vol. 789, Springer Verlag, 1994, pp. 495–515.Google Scholar
[3] Bittar, E. T., Strong Normalisation Proofs for Cut-Elimination in Gentzen's Sequent Calculi, Logic, algebra and computer science, vol. 46, Banach Center Publications, 1999, pp. 179–225.Google Scholar
[4] Constable, R. and Murthy, C., Finding computational content in classical proofs, Logical frame works (Huet, G. and Plotkin, G., editors), Cambridge University Press, 1991, pp. 341–362.CrossRefGoogle Scholar
[5] David, R., Normalization without reducibility, Annals of Pure and Applied Logic, vol. 107 (2001), pp. 121–130.Google Scholar
[6] David, R. and Guillaume, B., Strong normalization of the typed λws-calculus, Accepted in the CSL'03 colloquium.Google Scholar
[7] David, R. and Nour, K., A short proof of the strong normalization of the simply typed λμ-calculus, Schedae Informaticae, vol. 12 (2003), To appear.Google Scholar
[8] de Groote, P., On the strong normalization of natural deduction with permutation-conversions, In 10th International Conference on Typed Lambda Calculi and Applications, TLCA '95, Lecture Notes in Computer Science, vol. 902, Springer Verlag, 1995, pp. 201–205.Google Scholar
[9] de Groote, P., Strong normalization of classical natural deduction with disjunction, In 5th International Conference on Typed Lambda Calculi and Applications, TLCA '01, Lecture Notes in Computer Science, vol. 2044, Springer Verlag, 2001, pp. 182–196.Google Scholar
[10] Girard, J.-Y., A new constructive logic: classical logic, Mathematical Structures in Computer Science, vol. 1 (1991), pp. 255–296.Google Scholar
[11] Joachimski, F. and Matthes, R., Short proofs of normalization for the simply-typed λ-calculus, permutative conversions and Gödel's T, Archive for Mathematical Logic, vol. 42 (2003), pp. 59–87.Google Scholar
[12] Krivine, J.-L., Classical logic, storage operators and 2nd order lambda-calculus, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 53–78.Google Scholar
[13] Levy, J.J., Reductions correctes et optimales dans le lambda-calcul, Ph.D. thesis, Paris 7, 1978.Google Scholar
[14] Murthy, C.R., An evaluation semantics for classical proofs, Proceedings of the sixth annual IEEE symposium on logic in computer science, 1991, pp. 96–107.Google Scholar
[15] Murthy, C.R., A computational analysis of Girard's translation and LC, Proceedings of the seventh annual IEEE symposium on logic in computer science, 1992, pp. 90–101.Google Scholar
[16] Nour, K. and Saber, K., A semantical proof of the strong normalization theorem of full propositionnal classical natural deduction, Manuscript, 2003.Google Scholar
[17] Parigot, M., λμ-calculus: An algorithmic interpretation of classical natural deduction. Logic programming and automated reasoning (St. Petersburg, 1992), Lecture Notes in Computer Science, vol. 624, Springer Verlag, 1992, pp. 190–201.Google Scholar
[18] Parigot, M., Proofs of strong normalisation for second order classical natural deduction, this Journal, vol. 62 (1997), no. 4, pp. 1461–1479.Google Scholar
[19] Prawitz, D., Natural deduction. A proof-theoretical study, Almqvist & Wiksell, Stockholm, 1965.Google Scholar
[20] Prawitz, D., Ideas and results in proof-theory, Proceedings of the second Scandinavian logic symposium (Fenstad, J.E., editor), North-Holland, 1971, pp. 237–309.Google Scholar
[21] Pym, D. and Ritter, E., On the semantics of classical disjunction. Journal of Pure and Applied Algebra , to appear.Google Scholar
[22] Rehof, N. J. and Sorensen, M. H., The λΔ-calculus, Proceedings of the international symposium on theoretical aspects of computer software, TACS'94, Lecture Notes in Computer Science, vol. 789, Springer Verlag, 1994, pp. 516–542.Google Scholar
[23] Ritter, E., Pym, D., and Wallen, L., On the intuitionistic force of classical search. Theoretical Computer Science, vol. 232 (2000), pp. 299–333.Google Scholar
[24] Ritter, E., Proof-terms for classical and intuitionistic resolution, Journal of Logic and Computation, vol. 10 (2000), no. 2, pp. 173–207.CrossRefGoogle Scholar
[25] Seldin, J., On the proof theory of the intermediate logic MH, this Journal, vol. 51 (1986), no. 3, pp. 626–647.Google Scholar
[26] Stalmarck, G., Normalization theorems for full first order classical natural deduction, this Journal, vol. 56 (1991), no. 1, pp. 129–149.Google Scholar
[27] van Daalen, D., The language theory of automath, Ph.D. thesis, Eindhoven, 1977.Google Scholar