A simple method to classify CA // 一个简单的CA分类方法

Recall a CA rule rule project a circular-boundary state space  S back to it self, that is

 rule: S\rightarrow S, aka, x_{t+1}=rule(x_t), x_t \in S, x_{t+1} \in S

where  S = { 0, 1 }^n, and more strictly  x_t: [1,2,...,n] \rightarrow {0,1} . , for simplicity we let  \phi = [1,2,...,n], thus  x = x(\phi)

In practical, we take n=20^2=400;

Consider  x_S(\phi_S) where  \phi_S=[1, 2, ..., n] \times [1, 2, ..., card(S)], \ x_S:\phi_S \rightarrow {0,1}

In practical, we sought a big enough representative subset,  S_r from  S, so that

 x_S(\phi_S)\approx x_{S_r}(\phi_{S_r})

Denote  rule(rule(x))=rule^2(x), we have

 rule(rule^t(x))=rule^{n+1}(x),

We then define a profile  \lambda: [0, 1, 2, 3, ......] \rightarrow [0, 1] with Pearson correlation so that

 \lambda(t) =( \rho[x_S,y^t_S]) ^2= ({Cov[x_S, \ y^t_S]\over\sigma[x_S] \cdot \sigma[y^t_S]})^2, where  y^t_S=rule^n(x_S),  \sigma[X] denotes standard deviation of X and Cov[X,Y] denotes covariance between X and Y.

We assert that  \lambda(t) classify the dynamics of the CA rule by capturing the periodic behavior in the underlying dynamics.

Importantly, we can find a good  S_r by taking a random  S_{r0} and obtain

 S_r = rule^T(S_{r0}) where T is a variable parameter.

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