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arXiv:0808.1238v1[math.DS]8Aug2008 DYNAMICS GROUPS OF ASYNCHRONOUS CELLULAR AUTOMATA
 

Summary: arXiv:0808.1238v1[math.DS]8Aug2008
DYNAMICS GROUPS OF ASYNCHRONOUS CELLULAR AUTOMATA
M. MACAULEY, J. MCCAMMOND, AND H. S. MORTVEIT
Abstract. A sequential dynamical system, or SDS, consists of an undirected graph Y , a
vertex-indexed list of update rules FY , and a word over the vertex set that contains each
vertex at least once and describes the order in which these update rules are to be applied.
When an SDS is -independent, i.e., when the set of periodic states does not depend on
the choice of , then the update rules permute the periodic states and these permutations
generate the dynamics group of the SDS. In this article we investigate dynamics groups in
the special case where Y is a circular graph with n vertices, there are only two vertex states,
and all of the update rules are identical. The 256 possibilities are known as Wolfram rules
and the resulting sequential dynamical systems are called asynchronous cellular automata,
or ACAs. We previously showed that for all n > 3 exactly 104 of the 256 Wolfram rules give
rise to -independent ACAs. Here we classify the periodic states of these -independent
systems and describe their dynamics groups, which are quotients of Coxeter groups.
A cellular automaton, or CA, is a classical discrete dynamical system defined over a
regular grid of cells. Every cell takes on one of a finite number of states and has an update
rule that only depends on its state and the states of its neighbors. At every discrete
time step, the update rules are simultaneously applied. More recent work has investigated
sequential dynamical systems defined over arbitrary finite graphs where the update rules

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics