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 exponential-free conclusions. 1

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 non-locality is the contraction rule. We show that the contraction rule can be restricted to the atomic one, provided we employ deep-inference, i.e., to allow rules to apply anywhere inside logical expressions. However, the use of deep-inference and the asymmetry of the logic give rise to the context-dependency 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 deep-inference. This system is shown to be sound and complete with respect to Gentzen's LJ and an equivalent notion of cut-elimination is proved.

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