Interaction Graphs: Full Linear Logic

TL;DR

This paper extends interaction graphs to model full linear logic with second-order quantification by incorporating continuous sets of states, advancing the quantitative dynamic models of linear logic.

Contribution

It introduces a novel interpretation of interaction graphs using continuous state sets, enabling the modeling of full linear logic with second-order quantification.

Findings

Models full linear logic with second-order quantification.

Uses measure-theoretic finite sets of states for interpretation.

Extends previous discrete interaction graph frameworks.

Abstract

Interaction graphs were introduced as a general, uniform, construction of dynamic models of linear logic, encompassing all "Geometry of Interaction" (GoI) constructions introduced so far. This series of work was inspired from Girard's hyperfinite GoI, and develops a quantitative approach that should be understood as a dynamic version of weighted relational models. Until now, the interaction graphs framework has been shown to deal with exponentials for the constrained system ELL (Elementary Linear Logic) while keeping its quantitative aspect. Adapting older constructions by Girard, one can clearly define "full" exponentials, but at the cost of these quantitative features. We show here that allowing interpretations of proofs to use continuous (yet finite in a measure-theoretic sense) sets of states, as opposed to earlier Interaction Graphs constructions were these sets of states were discrete (and finite), provides a model for full linear logic with second order quantification.