Interaction graphs: Graphings

In two previous papers [33], [37], we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard [16]. The strength of our approach lies in the fact that we interpret proofs by simpler structures - graphs - than Girard's constructions, while generalising the latter since they can be recovered as special cases of our setting. This third paper extends this approach by considering a generalisation of graphs named graphings, which is in some way a geometric realisation of a graph on a measured space. This very general framework leads to a number of new models of multiplicative-additive linear logic which generalise 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.