Discrete Models of Traffic Flow (3-3)

Cellular Automaton and traffic Flow 3

In the type of cellular automaton we have just treated, one-step evolution of a given cell is determined from three cells within it nearest neighbor including itself. Such type of automaton has the law of evolution specified by the table

Here, eight numbers b_{0, ... ,} b₇ take the value 0 or 1, and assigning all eight specifies a rule. Let us write the number generated by regarding b_i as eight digit binaries as B. Namely

B is an integer in the range between 0 and 255, and since each B represents different binary paterns of b_{0、... 、}b₇ , specifying a B uniquely determins a rule of two-states three-neighbor cellular automata. For example, the previous rule for the simple traffic flow model is given by

thus could be named as "rule 184".

Note also that eight digits B exaust all possible two-states three-neighbor cellular automata, whose number is 256.

All 256 cellular automata has different temporal evolution, but they are classified into four categories -- That is the finding of Wolfram, then a young particle physisist (who now is an IT company CEO).

According to him, the type I is the "death" which eventually die into all zero or all 1 states. The type II is the "temporal crystal" whose spatial pattern of states repeats in time. The type II is "chaos" in which seemingly random pattern is generated without repetition. Then come the final type, the type IV, which Wolfram could not classify into any of previous three. It is regarded as being on the border line between the type II and III, and the term "edge of chaos" was coined. He speculated this type IV being the prototype category of complex systems made of simple rule that include the life. This lead to the fin de siecle fashion of "the Science of Complexity".