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SN and CR for free-style LKtq: linear decorations and simulation of normalization

Published online by Cambridge University Press:  12 March 2014

Jean-Baptiste Joinet ,
Harold Schellinx  and
Lorenzo Tortora De Falco
Jean-Baptiste Joinet
Affiliation:
Équipe Preuves Programmes et Systémes, Université Denis Diderot, Case 7014, 2, Place Jussieu, 75251 Paris Cedex 05, France, E-mail: [email protected]
Harold Schellinx
Affiliation:
110, Avenue de Paris 94300, Vincennes, France, E-mail: [email protected]
Lorenzo Tortora De Falco
Affiliation:
Università Roma Tre, Via Ostiense, 234, 00146, Roma, Italy, E-mail: [email protected]
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Abstract

The present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.

The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.

We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.

We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e.. by only one specific combination, and structural rules.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 162 - 196
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[1]Danos, V., La logique Linéaire appliquée à l'étude de divers processus de normalisation (principalement du λ-calcul), Ph.D. thesis, Université Paris VII, 06 1990.Google Scholar
[2]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), Springer Verlag, 1993, Lecture Notes in Computer Science 713, Proceedings of the Third Kurt Gödel Colloquium, Brno, Czech Republic, 08 1993, pp. 159–171.CrossRefGoogle Scholar
[3]Danos, V., Joinet, J.-B., and Schellinx, H., LKT and LKQ: sequentcalculi 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, London Mathematical Society Lecture Note Series 222, Proceedings of the 1993 Workshop on Linear Logic, Cornell Univesity, Ithaca, pp. 211–224.CrossRefGoogle Scholar
[4]Danos, Vincent, Joinet, Jean-Baptiste, and Schellinx, Harold, A new deconstructive logic: linear logic, this Journal, vol. 62 (1997), no. 3, pp. 755–807.Google Scholar
[5]Fleury, A. and Retoré, C., The mix rule, Mathemathical Structures in Computer Science, vol. 2 (1994), pp. 273–285.CrossRefGoogle Scholar
[6]Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39, pp. 176–210, 405–431.CrossRefGoogle Scholar
[7]Girard, J.-Y., Linear logic. Theoretical Computer Science, vol. 50 (1987), pp. 1–102.CrossRefGoogle Scholar
[8]Girard, J.-Y., Quantifiers in linear logic II, Nuoviproblemi delta logica e della filosofia della scienza, volume II (Corsi, G. and Sambin, G., editors), CLUEB, Bologna(Italy), 1991, Proceedings of the conference with the same name, Viareggio, 8-13 gennaio 1990.Google Scholar
[9]Joinet, J.-B., Etude de la normalisation du calcul des séquents classique à travers la logique linéaire, Ph.D. thesis, Université Paris VII, January 1993.Google Scholar
[10]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., editor), Springer Verlag, 1996, Lecture Notes in Artificial Intelligence 1071, Proceedings of the 5th International Workshop Tableaux, Terrasini, Italy, 05 1996, pp. 226–243.CrossRefGoogle Scholar
[11]Kreisel, G., A survey of proof theory, 2, Proceedings of the second Scandinavian logic colloquium (Fenstad, J. E., editor). Studies in Logic and the Foundations of Mathematics, North-Holland, pp. 109–170.Google Scholar
[12]Régnier, L., λ-Calculet réseaux, Ph.D. thesis, Université Paris VII, 1992.Google Scholar
[13]Schellinx, H., The noble art of linear decorating, ILLC Dissertation Series, 1994-1, Institute for Language, Logic and Computation, University of Amsterdam, 1994.Google Scholar
[14]de Falco, L. Tortora, Additives of linear logic and normalization - part 2: the additive standardization theorem, Theoretical Computer Science, (2000), submitted. The extended abstract already appeared in vol 3 of Electronic Notes of Theoretical Computer Science, 1996.Google Scholar
[15]de Falco, L. Tortora, Réseaux, cohérence et expériences obsessionnelles, Ph.D. thesis, Université Paris VII, 01 2000.Google Scholar