Non-Cartesian Guarded Recursion with Daggers

TL;DR

This paper extends guarded recursion to dagger categories to model reversible programming, providing a categorical framework for languages with higher-order features and symmetric pattern-matching.

Contribution

It introduces a novel categorical construction for guarded recursion within dagger categories, enabling formal semantics for reversible programming languages.

Findings

Constructed a suitable category for guarded recursion in dagger categories.

Analyzed the interaction between the construction and dagger rig structures.

Demonstrated the model's applicability to higher-order reversible programming languages.

Abstract

Guarded recursion is a framework allowing for a formalisation of streams in classical programming languages. The latter take their semantics in cartesian closed categories. However, some programming paradigms do not take their semantics in a cartesian setting; this is the case for concurrency, reversible and quantum programming for example. In this paper, we focus on reversible programming through its categorical model in dagger categories, which are categories that contain an involutive operator on morphisms. After presenting classical guarded recursion, we show how to introduce this framework into dagger categories with sufficient structure for data types, also called dagger rig categories. First, given an arbitrary category, we build another category shown to be suitable for guarded recursion in multiple ways, via enrichment and fixed point theorems. We then study the interaction between this construction and the structure of dagger rig categories, aiming for reversible programming. Finally, we show that our construction is suitable as a model of higher-order reversible programming languages, such as symmetric pattern-matching.