Order Independence in Asynchronous Cellular Automata

A sequential dynamical system, or SDS, consists of an undirected graph Y , a vertexindexed list of local functions FY , and a word ω over the vertex set, containing each vertex at least once, 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 ω-independent if the set of periodic states does not depend on the choice of ω, and our main result is that for all n > 3 exactly 104 of the 256 Wolfram rules give rise to an ω-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 ω-independence also provide significant insight into the dynamics of these systems. Our main result, as recorded in Theorem 2.2, is a complete classification of the Wolfram rules that for all n > 3 lead to an ω-independent finite asynchronous elementary cellular automaton, or ACA. The structure of the article is relatively straightforward. The first two sections briefly describe how an ACA can be viewed as either a special type of sequential dynamical system or as a modified version of a classical elementary cellular automaton. These two sections also contain the background definitions and notations needed to carefully state our main result. Next, we introduce several new notations for Wolfram rules in order to make certain patterns easier to discern, and we significantly reduce the number of cases we need to consider by invoking the notion of dynamical equivalence. Sections 5 and 6 contain the heart of the proof. The former covers four large classes of rules whose members are ω-independent for similar reasons, and the latter finishes off three pairs of unusual cases that exhibit more intricate behavior requiring more delicate proofs. The final section contains remarks about directions for future research. 1. SEQUENTIAL DYNAMICAL SYSTEMS Cellular automata, or CAs, are discrete dynamical systems that have been thoroughly studied by both professional and amateur mathematicians. They are defined over regular grids of cells Date: July 16, 2007. 2000 Mathematics Subject Classification. 37B99,68Q80.