An Internal Language for Categories Enriched over Generalised Metric Spaces

TL;DR

This paper develops an internal language for categories enriched over generalized metric spaces and other structures, providing a unified framework for reasoning about programs with continuous or physical interactions.

Contribution

It introduces a V-equational deductive system for linear lambda calculus, proving soundness and completeness as an internal language for enriched autonomous categories.

Findings

Provides an internal language for categories enriched over partial orders and (ultra)metric spaces.

Applies the framework to higher-order programs with real-time and probabilistic behaviors.

Establishes a foundation for reasoning about program equivalences in continuous and physical contexts.

Abstract

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear {\lambda}-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour