An Explicit Framework for Interaction Nets

TL;DR

This paper introduces an explicit, permutation-based framework for interaction nets, leveraging Geometry of Interaction concepts to clarify their structure and reduction, and proves strong confluence within this new formalism.

Contribution

It presents a novel permutation-based formalism for interaction nets, connecting them to proof-net quotients and providing a clearer, more explicit framework.

Findings

Proves strong confluence of reduction in the new framework

Shows interaction nets as quotients of generalized proof-nets

Provides an explicit permutation-based formalism

Abstract

Interaction nets are a graphical formalism inspired by Linear Logic proof-nets often used for studying higher order rewriting e.g. \Beta-reduction. Traditional presentations of interaction nets are based on graph theory and rely on elementary properties of graph theory. We give here a more explicit presentation based on notions borrowed from Girard's Geometry of Interaction: interaction nets are presented as partial permutations and a composition of nets, the gluing, is derived from the execution formula. We then define contexts and reduction as the context closure of rules. We prove strong confluence of the reduction within our framework and show how interaction nets can be viewed as the quotient of some generalized proof-nets.