Quantitative Equality in Substructural Logic via Lipschitz Doctrines

TL;DR

This paper introduces a categorical framework for interpreting equality as a distance in substructural logics, using Lipschitz doctrines to model resource-sensitive similarity measures.

Contribution

It develops a novel categorical semantics for quantitative equality in substructural logics, extending to richer systems like full Linear Logic with quantifiers.

Findings

Introduces Lipschitz doctrines for categorical semantics of distances

Provides a sound and complete deductive calculus for quantitative equality

Constructs examples based on metric spaces and quantitative realisability

Abstract

Substructural logics naturally support a quantitative interpretation of formulas, as they are seen as consumable resources. Distances are the quantitative counterpart of equivalence relations: they measure how much two objects are similar, rather than just saying whether they are equivalent or not. Hence, they provide the natural choice for modelling equality in a substructural setting. In this paper, we develop this idea, using the categorical language of Lawvere's doctrines. We work in a minimal fragment of Linear Logic enriched by graded modalities, which are needed to write a resource sensitive substitution rule for equality, enabling its quantitative interpretation as a distance. We introduce both a deductive calculus and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. The study of 2-categorical properties of Lipschitz doctrines provides us with a universal construction, which generates examples based for instance on metric spaces and quantitative realisability. Finally, we show how to smoothly extend our results to richer substructural logics, up to full Linear Logic with quantifiers.