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arXiv · 2307.06851

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

Tomáš Gonda·Tobias Reinhart·Sebastian Stengele·Gemma De les Coves
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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.

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BibTeXRIS

Tomáš Gonda, Tobias Reinhart, Sebastian Stengele, Gemma De les Coves. 2023-06-30. A Framework for Universality in Physics, Computer Science, and Beyond. https://doi.org/10.46298/compositionality-6-3

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