Quantitative Equality in Substructural Logic via Lipschitz Doctrines

2021-10-11

AI Summary

Existing notions of equivalence in substructural logics fail to adequately capture object similarity in resource-sensitive settings. Method: We introduce a metric-space-based quantitative equality semantics framework, centered on a novel Lipschitz doctrine categorical model that integrates Lawvere’s theory of metric spaces, fragments of linear logic, graded modal logic, and 2-categorical semantics. This yields resource-sensitive quantitative substitution principles and a general construction for metric spaces and quantitative realizers. Contribution/Results: We establish the first sound and complete axiomatic system for quantitative equality. Moreover, we extend this framework—previously limited to propositional fragments—to full first-order linear logic with quantifiers, thereby overcoming the inherent limitations of classical truth-functional equivalence in resource-aware reasoning. This advancement enables fine-grained, metrically grounded comparisons of computational objects while preserving resource constraints.

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.

Problem

• Substructural Logic
• Equivalence Handling
• Metric Space Instantiation

Innovation

• Lawvere doctrine
• Lipschitz doctrine
• resource-sensitive substitution