Interaction graphs: Graphings

Abstract

In two previous papers [Sei12a, Sei14b], we exposed a combinatorial approach to the program of Geometry of Interaction, a program initiated by Jean-Yves Girard [Gir89b]. 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 measure space. This very general framework leads to a number of new models of multiplicativeadditive 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.

Interaction Graphs

“…The following results were shown in our previous paper [Sei14c]. They ensure that the given definition of truth is coherent.…”

“…This generalization, or more precisely a fragment of it, already appeared in the author's PhD thesis[Sei12b].2 For technical reasons, we in fact consider monoids of Borel-preserving non-singular maps[Sei14c].…”

Perspectives on Interrogative Models of Inquiry

“…Interaction Graphs (Seiller 2014). From the notions of termination and execution, one derives a notion of orthogonality over the set of untyped paraproofs, i.e.…”

Coherent Interaction Graphs

“…One aspect which has received less attention is the construction of models of linear logic where morphisms are generalized programs on which a GoI-style execution procedure can be defined, and serves to compose morphisms. Girard's successive papers on GoI all investigate such models, using operator algebras as the spaces of generalized programs; recently, Seiller has simplified Girard's models using graphings (a measure-theoretic extension of graphs) instead of operators [24]. Note that discrete graphs suffice to obtain a model of linear logic without exponentials or quantifiers [21,23].The aforementioned models all start from an untyped universe of programs, and interpret formulae as specifications for the behavior of programs: they can be seen as realizability models with operators/graphs as realizers.…”