Order Independence in Asynchronous Cellular Automata

TL;DR

This paper studies asynchronous cellular automata on circular graphs, identifying 104 out of 256 rules that behave consistently regardless of update order, thus revealing key properties of their dynamics.

Contribution

It classifies pi-independence for all Wolfram rules on circular graphs, extending previous classifications and providing new insights into their dynamics.

Findings

104 Wolfram rules are pi-independent for all n>3

Reproves and extends previous classification of symmetric rules

Provides insights into the dynamics of asynchronous cellular automata

Abstract

A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions F_Y, and a permutation pi of the vertex set (or more generally, a word w over the vertex set) that describes the order in which these local functions are to be applied. In this article we investigate the special case where Y is a circular graph with n vertices and all of the local functions are identical. The 256 possible local functions are known as Wolfram rules and the resulting sequential dynamical systems are called finite asynchronous elementary cellular automata, or ACAs, since they resemble classical elementary cellular automata, but with the important distinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said to be pi-independent if the set of periodic states does not depend on the choice of pi, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005 Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with this property. In addition to reproving and extending this earlier result, our proofs of pi-independence also provide significant insight into the dynamics of these systems.