A Functorial Perspective on (Multi)computational Irreducibility (2301.04690v1)

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 entiremultiway 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).

• The paper introduces a category-theoretic formalization of computational irreducibility by mapping data structures and elementary computations to cobordism categories.
• It extends this framework to multicomputational systems by employing a symmetric monoidal structure to capture branching and merging of computation paths.
• The work reveals a duality between computational complexity and quantum locality, suggesting new directions in theoretical computer science and quantum mechanics.

A Functorial Perspective on (Multi)computational Irreducibility

Overview

The paper by Jonathan Gorard proposes a novel formalization of computational irreducibility using category theory, particularly focusing on functorial correspondences between configurations of computational processes and cobordism categories. The core idea is to define computational irreducibility through exact functorial mappings from a category of data structures and elementary computations (e.g., Turing machine configurations) to a category of 1-dimensional cobordisms. This perspective naturally extends to higher-dimensional cobordism categories to account for non-deterministic or multiway computations — essentially branching and merging of computational paths. Computational irreducibility, as framed in this paper, is then seen as dual to the locality of time evolution in functorial approaches to quantum mechanics and quantum field theory.

Computational Irreducibility and Functoriality

The author begins by revisiting the concept of computational irreducibility, where the progression of a computational process cannot be predicted or shortcut beyond what is possible by directly simulating each step of the computation itself. In formal category-theoretic terms, data structures become objects and elementary computations become morphisms of a category. Computational irreducibility in this framework corresponds to the situation where these morphisms, decorated with computational complexity data, compose additively. In essence, a computational process is irreducible if simulating a sequence of processes concatenates their complexities linearly; this behavior directly links irreducibility to the functorial property of categorical maps.

Multicomputational Irreducibility

The paper extends the notion to multicomputational systems — models where multiple, branching computation paths are considered simultaneously. By equipping the category of computations with a symmetric monoidal structure (a tensor product), the concepts of branching and merging paths can be captured. Thus, multicomputational irreducibility becomes a measure of the additivity of time complexities under parallel compositions (tensor product composition). A symmetric monoidal functor characterizing this irreducibility maps the category of multicomputations to higher-dimensional cobordism categories, using constructs like branchial graphs to represent these composite structures.

Relating to Quantum Mechanics

Gorard highlights a significant connection between this framework and categorical quantum mechanics and quantum field theories. Just as computational irreducibility correlates with functorial mappings in computation, the locality of time evolution in quantum mechanics can be described as a functor from a category of 1-dimensional cobordisms to the category of vector spaces. By viewing this functoriality within adjunction theories, the paper reveals a duality between computational complexity and quantum locality, suggesting a broader theoretical landscape in which these concepts coexist.

Implications and Future Directions

The theoretical implications point towards exploring computational complexity within more complex algebraic structures than traditional complexity classes. The method proposes examining 'wilder’ or less structured classes, considering not just traditional computational advantages or limitations but integrating non-trivial equivalence and evolution complexities of computations in new ways. It also raises the prospect of characterizing 'irreversibility' and 'causal complexity' in computations, leveraging higher-dimensional algebraic structures and causal categories.

Moreover, extending this work to handle causality or glocal multiway systems — where computational tokens are managed at intermediate stages — promises intriguing avenues for exploring the intrinsic complexities of distributed computations. These directions suggest a convergence of theoretical computer science, physical theories of computation, and quantum mechanics in a unified conceptual and mathematical framework, offering new perspectives and tools for future investigations.