Interaction Graphs: Additives

TL;DR

This paper introduces a novel parametrized Geometry of Interaction model for MALL that uses finite graphs, solves previous issues with additive connectives, and relates to Girard's constructions through a key geometric property.

Contribution

It presents the first finite, denotational semantics for MALL with additives using a parametrized graph-based approach and introduces the trefoil property as a unifying geometric principle.

Findings

Finite graph-based semantics for MALL with additives.

Introduction of the trefoil property as a core geometric principle.

Connection of the model to Girard's GoI constructions.

Abstract

Geometry of Interaction (GoI) is a kind of semantics of linear logic proofs that aims at accounting for the dynamical aspects of cut-elimination. We present here a parametrized construction of a Geometry of Interaction for Multiplicative Additive Linear Logic (MALL) in which proofs are represented by families of directed weighted graphs. Contrarily to former constructions dealing with additive connectives, we are able to solve the known issue of obtaining a denotational semantics for MALL by introducing a notion of observational equivalence. Moreover, our setting has the advantage of being the first construction dealing with additives where proofs of MALL are interpreted by finite objects. The fact that we obtain a denotational model of MALL relies on a single geometric property, which we call the trefoil property, from which we obtain, for each value of the parameter, adjunctions. We then proceed to show how this setting is related to Girard's various constructions: particular choices of the parameter respectively give a combinatorial version of his latest GoI, a refined version of older Geometries of Interaction based on nilpotency. This shows the importance of the trefoil property underlying our constructions since all known GoI construction to this day rely on particular cases of it.