Synchronous and asynchronous updating in cellular automata

We analyze the properties of a synchronous and of various asynchronous meth ods to iterate cellular automata. Asynchronous methods in which the time va riable is not explicitly defined, operate by specifying an updating order o f the cells. The statistical properties of this order have significant cons equences for the dynamics and the patterns generated by the cellular automa ta. Stronger correlations between consecutive steps in the updating order r esult in more, artificial structure in the patterns. Among these step-drive n methods, using random choice with replacement to pick the next cell for u pdating, yields results that are least influenced by the updating method. W e also analyse a time-driven method in which the state transitions of singl e cells are governed by a probability per unit time that determines an expo nential distribution of the waiting time until the next transition. The sta tistical properties of this method are completely independent of the size o f the grid. Consecutive updating steps therefore show no correlation at all . The stationary states of a cellular automaton do not depend on whether a synchronous or asynchronous updating method is used. Their basins of attrac tion might, however, be vastly different under synchronous and asynchronous iteration. Cyclic dynamics occur only with synchronous updating. (C) 1999 Elsevier Science Ireland Ltd. All rights reserved.