On asynchronous cellular automata

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v(1),...,v(n)} and where each vertex v(i) has a state x(vi) in a finite field K, (b) a sequence of functions (F-vi,Y)(i), and (c) a word w = (w(1),...,w(k)), where each w(i) is a vertex in Y. The function F-vi,Y updates the state of vertex v(i) as a function of the state of v(i) and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [FY, w] Pi(k)(i=1) F-wi : K-n --> K-n. In the particular case of ACA, the graph is the circle graph on n vertices (Y Circ(n)), and all the maps F-vi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ(4). Finally, we analyze invertible ACA and offer an interpretation Of S-35 as the group of an SDS over the three-dimensional cube with local functions induced by nor(3) + nand(3).