The syntactic side of autonomous categories enriched over generalised metric spaces

TL;DR

This paper develops a new V-equational deductive system for linear lambda calculus, establishing a categorical equivalence with autonomous categories enriched over generalized metric spaces, applicable to various computational models.

Contribution

It introduces a novel V-equational framework and proves its soundness, completeness, and categorical equivalence to enriched autonomous categories, extending to affine settings and diverse computational contexts.

Findings

V-equational deductive system for linear lambda calculus

Categorical equivalence with autonomous categories enriched over generalized metric spaces

Applicability to real-time, probabilistic, and quantum computing models

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 we go further than this, by showing that linear {\lambda}-theories based on this V-equational system form a category that is equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. We additionally show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.