A New Method Of Classifying The Elementary Cellular Automata

In this paper, we obtain a new method of classifying the elementary cellular automata in characteristic length and the relationship to the Wolfram's four classes. First, we simply review the history of the Cellular Automata. During Von Neumann invented the cellular automata initially to the Artificial Life is raising, there have been some theoretical studies and application in different fields. Second, we propose the mathematic definition of the Cellular Automata in detail. When Φ_j ( r,t+1) denote the state of the cell at the time of t +1, the mathematic expression of evolution rule is Φ_j ( r ,t+1)=Rj[Φ(r,t),Φ(r+δ₁ ,t),Φ(r+δ₂,t),...,Φ(r+δ_k,t)]. So we can give the expression of the total-rule which is Φ_j ( r,t+1)=R_j[Φ(r,t)+Φ(r+δ₁ ,t)+Φ(r+δ+₂,t)+...+Φ(r+δ_k,t)]. We introduce two kinds of neighbor relation used in the Cellular Automata: one is Von Neumann neighbor; another is Moore neighbor. By calculating most Cellular Automata rules, Wolfram found their long-term behaviors can be divided into four classes: (1) homogeneous state; (2) periodic structures;...