Dynamics Groups of Asynchronous Cellular Automata

TL;DR

This paper classifies the periodic states and describes the dynamics groups of pi-independent asynchronous cellular automata, revealing their structure as quotients of Coxeter groups and connecting dynamical systems with algebraic combinatorics.

Contribution

It provides a classification of periodic states and characterizes the dynamics groups of pi-independent asynchronous cellular automata, linking them to Coxeter groups.

Findings

104 of 256 cellular automaton rules are pi-independent

Dynamics groups are quotients of Coxeter groups

Classification of periodic states of these systems

Abstract

We say that a finite asynchronous cellular automaton (or more generally, any sequential dynamical system) is pi-independent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible 256 cellular automaton rules are pi-independent. In this article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and algebraic combinatorics in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.