A Framework for Universality in Physics, Computer Science, and Beyond

TL;DR

This paper develops a categorical framework to unify the concept of universality across different domains like physics and computer science, revealing fundamental properties and limitations of universal systems.

Contribution

It introduces a general categorical framework for universality, connecting various notions and establishing conditions and limitations for universal models.

Findings

Universal spin models cannot be finite.

Universal Turing machines are non-trivial.

Universality and negation lead to unreachability.

Abstract

Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as instances universal Turing machines, universal spin models, NP completeness, top of a preorder, denseness of a subset, and more. By identifying necessary conditions for universality, we show that universal spin models cannot be finite. We also characterize when universality can be distinguished from a trivial one and use it to show that universal Turing machines are non-trivial in this sense. Our framework allows not only to compare universalities within each instance, but also instances themselves. We leverage a Fixed Point Theorem inspired by a result of Lawvere to establish that universality and negation give rise to unreachability (such as uncomputability). As such, this work sets the basis for a unified approach to universality and invites the study of further examples within the framework.