Interaction Graphs: Additives

Geometry of Interaction (GoI) is a research program initiated by Jean-Yves Girard which aims at defining a semantics of linear logic proofs accounting for the dynamical aspects of cut elimination. We present here a parametrised 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 [15, 21], 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 [21], a refined version of older Geometries of Interaction [13, 12, 15], and even a generalisation of his multiplicatives [11] construction. 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.