A Compositional Account of Generalized Reversible Computing

TL;DR

This paper introduces a compositional framework for generalized reversible computing using category theory, providing formal tools to analyze physical and computational transformations and their entropy-related properties.

Contribution

It develops a novel categorical framework for reversible computing, integrating resource theories and copy-discard structures to analyze physical and computational processes.

Findings

Established conditions for determinism and partial invertibility.

Proved the fundamental theorem of generalized reversible computing.

Defined new entropy measures for these processes.

Abstract

We develop a compositional framework for generalized reversible computing using copy-discard categories and resource theories. We introduce partitioned matrices between partitioned sets as subdistribution matrices which preserve the equivalence relation of its domain. We model computational and physical transformations as subdistribution matrices over the category of sets and partitioned matrices on partitioned sets, respectively. We show that the interactions between the physical and computational transformations are governed by an aggregation functor whose functoriality and monoidality we deduce from general principles of the formal theory of monads. We study the associated copy-discard structures, in particular, general conditions for determinism and partial invertibility. We then define several notions of entropies that we use to state and prove the fundamental theorem of generalized reversible computing.