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March 2002 SN and {CR} for free-style {${\bf LK}\sp {tq}$}: linear decorations and simulation of normalization
Jean-Baptiste Joinet, Harold Schellinx, Lorenzo Tortora de Falco
J. Symbolic Logic 67(1): 162-196 (March 2002). DOI: 10.2178/jsl/1190150036

Abstract

The present report is a, somewhat lengthy, addendum to "A new deconstructive logic: linear logic," 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 interpre- tation 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.

Citation

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Jean-Baptiste Joinet. Harold Schellinx. Lorenzo Tortora de Falco. "SN and {CR} for free-style {${\bf LK}\sp {tq}$}: linear decorations and simulation of normalization." J. Symbolic Logic 67 (1) 162 - 196, March 2002. https://doi.org/10.2178/jsl/1190150036

Information

Published: March 2002
First available in Project Euclid: 18 September 2007

zbMATH: 1015.03053
MathSciNet: MR1889543
Digital Object Identifier: 10.2178/jsl/1190150036

Subjects:
Primary: 03F05
Secondary: 03B10, 03B35, 03F07, 03F52

Rights: Copyright © 2002 Association for Symbolic Logic

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Vol.67 • No. 1 • March 2002
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