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A new deconstructive logic: linear logic

Published online by Cambridge University Press:  12 March 2014

Vincent Danos ,
Jean-Baptiste Joinet  and
Harold Schellinx
Vincent Danos
Affiliation:
CNRS URA 753, Equipe de Logique Mathématique, Université Paris VII, 2, Place Jussieu, F-75251 Paris Cedex 05, France, E-mail: [email protected]
Jean-Baptiste Joinet
Affiliation:
Equipe de Logique Mathématique, Université ParisVII, 2, Place Jussieu, F-75251 Paris Cedex 05, France Université Paris I, (Panthéon-Sorbonne), 17, Rue de la Sorbonne, F-75231 Paris Cedex 05, France, E-mail: [email protected]
Harold Schellinx
Affiliation:
Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, NL-3508 Ta Utrecht, The Netherlands Equipe de Logique Mathématique, Université Paris VII, 2, Place Jussieu, F-75251 Paris Cedex 05, France, E-mail: [email protected]
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Abstract

The main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).

The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.

A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 755 - 807
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Barbanera, F. and Berardi, S., A symmetric lambda calculus for ‘classical’ program extraction, Information and Computation, vol. 125 (1996), no. 2, pp. 103–117.CrossRefGoogle Scholar
[2]Danos, V., La logique linéaire appliquée à l'étude de divers processus de normalisation (principalement du lambda-calcul), Thèse de Doctorat, Université de Paris 7, 1990.Google Scholar
[3]Danos, V., Joinet, J.-B., and Schellinx, H., The structure of exponentials: uncovering the dynamics of linear logic proofs, Computational logic and proof theory (Gottlob, G., Leitsch, A., and Mundici, D., editors), Lecture Notes in Computer Science, no. 713, Springer-Verlag, 1993, pp. 159–171.CrossRefGoogle Scholar
[4]Danos, V., Joinet, J.-B., and Schellinx, H., LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication, Advances in linear logic (Girard, J.-Y., Lafont, Y., and Regnier, L., editors), Cambridge University Press, 1995, pp. 211–224.CrossRefGoogle Scholar
[5]Danos, V., Joinet, J.-B., and Schellinx, H., On the linear decoration of intuitionistic derivations, Archive for Mathematical Logic, vol. 33 (1995), pp. 387–412.CrossRefGoogle Scholar
[6]Dragalin, A. G., Mathematical intuitionism, American Mathematical Society, 1988, Translation of the Russian original from 1979.Google Scholar
[7]Duquesne, E. and van de Wiele, J., Modèle cohérent des réseaux de preuves, Archive for Mathematical Logic, vol. 33 (1994), pp. 131–158.CrossRefGoogle Scholar
[8]Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1935), pp. 176–210, 405–431.CrossRefGoogle Scholar
[9]Girard, J.-Y., Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1–102.CrossRefGoogle Scholar
[10]Girard, J.-Y., A new constructive logic: classical logic, Mathemathical Structures in Computer Science, vol. 1 (1991), no. 3, pp. 255–296.CrossRefGoogle Scholar
[11]Girard, J.-Y., Quantifiers in linear logic, II, Nuovi problemi delta logica e delta filosofia delta scienza (Corsi, G. and Sambin, G., editors), vol. II, 1991, Proceedings of the conference with the same name, Viareggio, 8–13 gennaio 1990. CLUEB, Bologna (Italy).Google Scholar
[12]Girard, J.-Y., On the unity of logic, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 201–217.CrossRefGoogle Scholar
[13]Griffin, T., A formulae-as-types notion of control, Principles of programming languages, IEEE Computer Society Press, 1990, pp. 47–58.Google Scholar
[14]Heesterbeek, , Van Neerven, , and Schellinx, , Das Fegefeuer Theorem (De Purgatorio). Eschatol-ogische Axiomatik zum transorbitalen Sündenmanagement, Litzelstetter Libellen, Ziemlich Neue Folge, no. 2, Libelle Verlag, Bottighofen am Bodensee14, 1992.Google Scholar
[15]Herbelin, H., Séquents qu'on calcule. De l'interprétation du calcul des séquents comme calcul de λ-termes et comme calcul de stratégies gagnantes, Thèse de Doctorat, Université de Paris 7, 1995.Google Scholar
[16]Hörwein, K., Keeping inconsistencies local, manuscript, Technische Universität Wien, Austria, 1992.Google Scholar
[17]Howard, W. A., The formulae-as-types notion of construction, To H. B. Curry: Essays on combinatory logic, lambda calculus and formalism (Seldin, J. P. and Hindley, J. R., editors), Academic Press, 1980, pp. 479–490.Google Scholar
[18]Joinet, J.-B., Etude de la normalisation du calcul des séquents classique à travers la logique linéaire, Thèse de Doctorat, Université de Paris 7, 1993.Google Scholar
[19]Joinet, J.-B., Schellinx, H., and de Falco, L. Tortora, Strong normalization for free deduction and related formalizations of classical second order logic, manuscript, 199?Google Scholar
[20]Joinet, J.-B., Schellinx, H., and de Falco, L. Tortora, Strong normalization for all-style LKtq, Theorem proving with analytic tableaux and related methods (Migliolo, et al., editors), Lecture Notes in Artificial Intelligence, no. 1071, Springer-Verlag, 1996, pp. 226–243.CrossRefGoogle Scholar
[21]Kreisel, G., On the interpretation of non-finitist proofs, I, this Journal, vol. 16 (1951), no. 4, pp. 241–267.Google Scholar
[22]Krivine, J. L., Lambda-calcul, types et modèles, Masson, Paris, 1990.Google Scholar
[23]Krivine, J. L., Lambda-calculus, types and models, Ellis Horwood, London, 1993.Google Scholar
[24]Krivine, J. L., Classical logic, storage operators and second order lambda-calculus, Annals of Pure and Applied Logic, vol. 68 (1994), pp. 53–78.CrossRefGoogle Scholar
[25]Krivine, J. L., A general storage theorem for integers in call-by-name λ-calculus, Theoretical Computer Science, vol. 129 (1994), pp. 79–94.CrossRefGoogle Scholar
[26]Krivine, J.-L. and Parigot, M., Programming with proofs, Journal of Information Processing and Cybernetics EIK, vol. 26 (1990), no. 4, pp. 149–167.Google Scholar
[27]Leivant, D., Reasoning about functional programs and complexity classes associated to type disciplines, FOCS 83, pp. 460–469, 1983.CrossRefGoogle Scholar
[28]Murthy, C., An evaluation semantics for classical proofs, Logic in computer science, IEEE Computer Society Press, 1991, pp. 96–107.Google Scholar
[29]Murthy, C., A computational analysis of Girard's translation and LC, Logic in computer science, IEEE Computer Society Press, 1992, pp. 90–101.Google Scholar
[30]Nour, K., Mixed logic and storage operators, Prépublication de l'Université de Chambéry, 1995.Google Scholar
[31]Parigot, M., Proofs of strong normalization for second order classical natural deduction, manuscript, to appear in this Journal.Google Scholar
[32]Parigot, M., Free deduction: an analysis of computation in classical logic, Russian conference on logic programming (Voronkov, A., editor), Lecture Notes in Artificial Intelligence, no. 592, Springer-Verlag, 1991, pp. 361–380.Google Scholar
[33]Parigot, M., λμ-calculus: an algorithmic interpretation of classical natural deduction, Logic programming and automated reasoning (Voronkov, A., editor), Lecture Notes in Artificial Intelligence, no. 624, Springer-Verlag, 1992, pp. 190–201.CrossRefGoogle Scholar
[34]Parigot, M., Recursive programming with proofs, Theoretical Computer Science, vol. 94 (1992), pp. 335–356.CrossRefGoogle Scholar
[35]Parigot, M., Classical proofs as programs, Computational logic and proof theory (Gottlob, G., Leitsch, A., and Mundici, D., editors), Lecture Notes in Computer Science, no. 713, Springer-Verlag, 1993, pp. 263–277.CrossRefGoogle Scholar
[36]Parigot, M., Strong normalization for second order classical natural deduction, Logic in computer science, IEEE Computer Society Press, 1993, pp. 39–46.Google Scholar
[37]Plotkin, G., Call-by-name, call-by-value and the lambda-calculus, Theoretical Computer Science, vol. 1 (1975), pp. 125–159.CrossRefGoogle Scholar
[38]Prawitz, D., Validity and normalizability ofproofs in first and second order classical and intuition-istic logic, Atti del I congresso Italiano di logica, Bibliopolis, Napoli, 1981, pp. 11–36.Google Scholar
[39]Regnier, L., λ-calcul et réseaux, Thèse de Doctorat, Université de Paris 7, 1992.Google Scholar
[40]Schellinx, H., The noble art of linear decorating, ILLC Dissertation Series 1994-1, Institute for Logic, Language and Computation, University of Amsterdam, 1994.Google Scholar
[41]Schwichtenberg, H., Proof theory: some applications of cut-elimination, Handbook of mathematical logic (Barwise, J., editor), North Holland, 1977.Google Scholar
[42]Tammet, T., Proof search strategies in linear logic, Journal of Automated Reasoning, vol. 12 (1994), pp. 273–304.CrossRefGoogle Scholar
[43]Troelstra, A. S., Lectures on linear logic, CSLI Lecture Notes 29, Center for the Study of Language and Information, Stanford, 1992.Google Scholar
[44]Troelstra, A. S. and van Dalen, D., Constructivism in mathematics: An introduction, vol. I, North-Holland, 1988.Google Scholar
[45]Vauzeilles, J., Cut elimination for the unified logic, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 1–16.CrossRefGoogle Scholar