Studies Of Sub-classes In The Cellular Automata For The Wolfram's Class Ⅳ

In this paper, we obtain the collision results of sub-classes of the Wolfram class IV and discuss the evolution stability of the Wolfram's classes and sub-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 denote the state of the cell at the time of, the mathematic expression of evolution rule is .So we can give the expression of the total- rule which is .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; (3) chaotic state; (4) complex state. The Wolfram class IV discussed in this paper is complex state. Through calculating one-dimensional Wolfram's total- rule with k=2, r=2, we get four patterns of the above four sub-classes. We only study the total-rule R=20 which belongs to fourth class. Through observing its patterns, we easily obtain six sub-classes of it and extract its gene-pieces and existing circumstances. In order to prove the exist of these sub-classes, we successfully reproduce them according to their gene-pieces and existing circumstances. We realize the collisions between any two sub-classes which we have already gotten, by means of constructing special initial condition. At the same time, we find the evolution expressions of these sub-classes.Third, we propose a conception of sub-classes and evolution stability for the Wolfram's Classes: At the certain moment in the evolution process, the CA may do the choosing, varying and crossing operation. This may change some values of the Cellular Automata, then the Cellular Automata will continue to evolve with the same evolution rule as before. If after the time of , the Cellular Automata's class or sub-class does not change, we call this class or sub-class evolution stability. Applying three operations of the above conception to the calculation, we can get the evolution results of the total- rule R=20. Through analyzing the patterns of evolution, we draw a conclusion: The Wolfram's class I, II, and III and IV are stable, but the sub-classes of The Wolfram's class IV are unstable.