Order Independence in Asynchronous Cellular Automata
Matthew Macauley, Jon McCammond and Henning S. Mortveit
A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertex-indexed list of local functions ¡Y, and a word w 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 w-independent if the set of periodic states does not depend on the choice of w, and our main result is that for all n>3 exactly 104 of the 256 Wolfram rules give rise to an w-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 w-independence also provide significant insight into the dynamics of these systems.