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An Axiomatic Approach to Structural Rules for Locative Linear Logic

Jean-Marc Andreoli
This paper proposes a generic, axiomatic framework to express and study structural rules in resource conscious logics derived from Linear Logic. The proposed axioms aim at capturing minimal concepts, operations and relations in order to build an inference system which extends that of linear logic by the introduction of structure and structural rules, but still preserves in a very natural way the essential properties of any logical inference system: Cut elimination and Focussing. We consider here finite but unbounded structures, built over finite subsets of a potentially infinite set of abstract "places" (eg. the natural numbers). The set of places is "isotropic", in that no single place has a distinguished role in the structures, and each structure can be translated into an isomorphic structure on any subset of same cardinality as its support set. The essential role of these places in the definition of Logic has been shown in "Locus Solum" [6], and leads to a locative reading of the traditional logical concepts (formulas, sequents, proofs), which is a adopted here. All the logical connectives (multiplicatives, additives, exponentials) are expressed here in a locative manner.
To appear in Linear Logic in Computer Science