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Geometry of Interaction III: Accommodating the Additives
 In: Advances in Linear Logic, LNS 222,CUP, 329–389
, 1995
"... The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution f ..."
Abstract

Cited by 29 (5 self)
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The paper expounds geometry of interaction, for the first time in the full case, i.e. for all connectives of linear logic, including additives and constants. The interpretation is done within a C ∗algebra which is induced by the rule of resolution of logic programming, and therefore the execution formula can be presented as a simple logic programming loop. Part of the data is public (shared channels) but part of it can be viewed as private dialect (defined up to isomorphism) that cannot be shared during interaction, thus illustrating the theme of communication without understanding. One can prove a nilpotency (i.e. termination) theorem for this semantics, and also its soundness w.r.t. a slight modification of familiar sequent calculus in the case of exponentialfree conclusions. 1
A Local System for Intuitionistic Logic: Preliminary Results
, 2005
"... This paper presents a system for intuitionistic logic in which all the rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of nonlocality is the contraction rule. We show th ..."
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Cited by 2 (0 self)
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This paper presents a system for intuitionistic logic in which all the rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. The main source of nonlocality is the contraction rule. We show that the contraction rule can be restricted to the atomic one, provided we employ deepinference, i.e., to allow rules to apply anywhere inside logical expressions. However, the use of deepinference and the asymmetry of the logic give rise to the contextdependency of the rules. We further show that this context dependency can be removed by introducing polarities into logical expressions. We present the system in the calculus of structures, a proof theoretic formalism which supports deepinference. This system is shown to be sound and complete with respect to Gentzen's LJ and an equivalent notion of cutelimination is proved.