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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: joinet@logique.jussieu.fr
Harold Schellinx
Affiliation:
110, Avenue de Paris 94300, Vincennes, France, E-mail: hars@logique.jussieu.fr
Lorenzo Tortora De Falco
Affiliation:
Università Roma Tre, Via Ostiense, 234, 00146, Roma, Italy, E-mail: tortora@logique.jussieu.fr
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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

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