A Small Sampling of Rules

We conclude our brief tour of one-dimensional CA by mentioning a few special properties of some selected one-dimensional rules. We will have occasion to refer back, more than once, to some of the behaviors discussed here, after the appropriate set of formal tools has been presented in later chapters. 

While none of the 256 possible radius r = 1 binary valued CA are believed to be capable of universal computation, the rule whose long time behavior has proven to be the most difficult to understand fully is rule R22  

The two-point spatial correlation function C2(r) (equation 3.26) does not decay exponentially with distance r, but appears instead to decay more slowly a.s exp(—Q- /r). 

The decay of the spatial block entropy, which gives the amount of information contained in a block of N contiguous site values ai.aN needed to predict the value (Jn+i is considerably slower than [block-length), and is therefore indicative of very long and complex correlations we will come back to this point later in chapter 4, following our discussion of dynamical system theory. 

29 Space-time pattern of one-dimensional elementary rule R22. 

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