A Functorial Perspective on (Multi)computational Irreducibility

Abstract

This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding category of (1-dimensional) cobordisms. We proceed to demonstrate that, by equipping both categories with a symmetric monoidal structure and considering the case of higher-dimensional cobordism categories, we obtain a natural extension of this formalism that serves also to encompass non-deterministic or ``multiway'' computations, in which one quantifies not only the irreducibility in the behavior of a single (deterministic) computation path, but in the branching and merging behavior of an entire ``multiway system'' of such paths too. We finally outline how, in the most general case, the resulting symmetric monoidal functor may be considered to be adjoint to the functor characterizing the Atiyah-Segal axiomatization of a functorial quantum field theory. Thus, we conclude by arguing that the irreducibility of (multi)computations may be thought of as being dual to the locality of time evolution in functorial approaches to quantum mechanics and quantum field theory. In the process, we propose an extension of the methods of standard (monoidal) category theory, in which morphisms are effectively equipped with intrinsic computational complexity data, together with an algebra for how those complexities compose (both in sequence and in parallel, subject to the monoidal structure).