Coxeter Groups and Asynchronous Cellular Automata

TL;DR

This paper explores the connection between Coxeter groups and asynchronous cellular automata, revealing how group-theoretic structures inform the understanding of automaton dynamics and introducing new analytical tools.

Contribution

It introduces the state automaton of an ACA and links it to the root automaton of Coxeter groups, providing novel insights into their interplay and open problems.

Findings

The root automaton of a Coxeter group is part of the ACA's state automaton.

Group-theoretic methods can inform ACA dynamics analysis.

New approaches to open problems in ACA and Coxeter groups.

Abstract

The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.