 
Summary: ORDER INDEPENDENCE IN ASYNCHRONOUS CELLULAR AUTOMATA
M. MACAULEY J. MCCAMMOND H.S. MORTVEIT
ABSTRACT. A sequential dynamical system, or SDS, consists of an undirected graph Y , a vertex
indexed list of local functions FY , and a word 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 re
sulting sequential dynamical systems are called finite asynchronous elementary cellular automata,
or ACAs, since they resemble classical elementary cellular automata, but with the important dis
tinction that the vertex functions are applied sequentially rather than in parallel. An ACA is said
to be independent if the set of periodic states does not depend on the choice of , and our main
result is that for all n > 3 exactly 104 of the 256 Wolfram rules give rise to an 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 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 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
