Causality in Linear Logic: full completeness and injectivity (unit-free multiplicative-additive fragment)

Abstract

Commuting conversions of Linear Logic induce a notion of dependencybetween rules inside a proof derivation: a rule depends on a previous rule whenthey cannot be permuted using the conversions. We propose a new interpretation ofproofs of Linear Logic ascausal invariantswhich capturesexactlythis dependency.We represent causal invariants using game semantics based on general eventstructures, carving out, inside the model of [6], a submodel of causal invariants.This submodel supports an interpretation of unit-free Multiplicative AdditiveLinear Logic with MIX (MALL−) which is (1)fully complete: every element ofthe model is the denotation of a proof and (2)injective: equality in the modelcharacterises exactly commuting conversions of MALL−. This improves over thestandard fully complete game semantics model of MALL−