Interaction Graphs: Graphings

TL;DR

This paper introduces graphings, a generalized graph-based framework for the Geometry of Interaction, enabling new models of linear logic that overcome previous limitations, including handling second-order quantification.

Contribution

It extends previous graph-based approaches to Geometry of Interaction by introducing graphings, broadening the scope of models and applications in linear logic.

Findings

Developed a generalization of graphs called graphings.

Created new models of linear logic that include second-order quantification.

Demonstrated that these models generalize Girard's models and overcome their limitations.

Abstract

In two previous papers, we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard. The strength of our approach lies in the fact that we interpret proofs by simpler structures - graphs - than Girard's constructions, while generalizing the latter since they can be recovered as special cases of our setting. This third paper extends this approach by considering a generalization of graphs named graphings, which is in some way a geometric realization of a graph. This very general framework leads to a number of new models of multiplicative-additive linear logic which generalize Girard's geometry of interaction models and opens several new lines of research. As an example, we exhibit a family of such models which account for second-order quantification without suffering the same limitations as Girard's models.