Algebraic Theory of Patterns as Generalized Symmetries

TL;DR

This paper develops an algebraic framework based on future equivalence to analyze and generalize symmetries and patterns in discrete systems and spacetime, extending traditional symmetry concepts.

Contribution

It introduces a novel algebraic theory of patterns using future equivalence, including new local representations for spacetime patterns, and clarifies challenges in generalizing symmetries.

Findings

Unique minimal semiautomaton from future equivalence in discrete systems

Generalization of translation symmetry to partial and hidden symmetries

New local representations for spacetime patterns that are faithful generators

Abstract

We generalize the exact predictive regularity of symmetry groups to give an algebraic theory of patterns, building from a core principle of future equivalence. For topological patterns in fully-discrete one-dimensional systems, future equivalence uniquely specifies a minimal semiautomaton. We demonstrate how the latter and its semigroup algebra generalizes translation symmetry to partial and hidden symmetries. This generalization is not as straightforward as previously considered. Here, though, we clarify the underlying challenges. A stochastic form of future equivalence, known as predictive equivalence, captures distinct statistical patterns supported on topological patterns. Finally, we show how local versions of future equivalence can be used to capture patterns in spacetime. As common when moving to higher dimensions, there is not a unique local approach, and we detail two local representations that capture different aspects of spacetime patterns. A previously-developed local spacetime variant of future equivalence captures patterns as generalized symmetries in higher dimensions, but we show this representation is not a faithful generator of its spacetime patterns. This motivates us to introduce a local representation that is a faithful generator, but we demonstrate that it no longer captures generalized spacetime symmetries. Taken altogether, building on future equivalence, the theory defines and quantifies patterns present in a wide range of classical field theories.