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
• $rule: S\rightarrow$
• 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.