Mapping the Cellular Automata Rule Spaces
Columbus State University
With the revitalization of the research on cellular automata in the 1980s by Stephen Wolfram, many branches of science have invested time and effort in trying to understand these simple yet powerful systems. Their versatility and simplicity are contrasted with the complexity that many of them exhibit. Stephen Wolfram classifed these automata by their behavior. Namely, class 1 for those that reach a fixed point, class 2 for those with periodic behavior, class 3 for the ones with random behavior, and class 4 for interesting automata that show complex behavior. Other researchers have proposed further decomposition of these classes, but these have remain almost unchanged since their inception. A given nontrivial rule space has rules that belong to one of these classes. An interesting question that comes to mind is: Is there a parameter or parameters in the rules that will provide a control to “smoothly” change from one class of behavior to another? I am in pursuit of such a parameter.
The presentation summarizes the results of research performed during the NKS Summer School 04. A map was created for the rule space k = 2, r = 1½ as shown in Figure 1. It also presents some more recent results based on experiments performed in the last few weeks. Using a different mapping, the contiguous areas of the same class are more evident and hence seems to cluster similar class behavior. Figure 2 shows a partial map of the elementary cellular automata rule space where you can see the regions of similar class behavior. Figure 3 shows a partial mapping of the rule space k = 2, r = 1½. This is work that started in the NKS 2003 and NKS 2004 conferences.