Order Independence in Asynchronous Cellular Automata
Abstract
A sequential dynamical system, or SDS, consists of an undirected graph Y, a vertexindexed 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 piindependent 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 piindependent 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 piindependence also provide significant insight into the dynamics of these systems.
 Publication:

arXiv eprints
 Pub Date:
 July 2007
 DOI:
 10.48550/arXiv.0707.2360
 arXiv:
 arXiv:0707.2360
 Bibcode:
 2007arXiv0707.2360M
 Keywords:

 Mathematics  Dynamical Systems;
 37B99;
 68Q80
 EPrint:
 18 pages. New version distinguishes between functions that are piindependent but not windependent