Quiescent String Dominance Parameter F and Classification of One-Dimensional Cellular Automata

Abstract

The mechanism which discriminates the pattern classes at the same $λ$, is found. It is closely related to the structure of the rule table and expressed by the numbers of the rules which break the strings of the quiescent states. It is shown that for the N-neighbor and K-state cellular automata, the class I, class II, class III and class IV patterns coexist at least in the range, $\frac{1}{K} \le λ\le 1-\frac{1}{K} $. The mechanism is studied quantitatively by introducing a new parameter $F$, which we call quiescent string dominance parameter. It is taken to be orthogonal to $λ$. Using the parameter F and $λ$, the rule tables of one dimensional 5-neighbor and 4-state cellular automata are classified. The distribution of the four pattern classes in ($λ$,F) plane shows that the rule tables of class III pattern class are distributed in larger $F$ region, while those of class II and class I pattern classes are found in the smaller $F$ region and the class IV behaviors are observed in the overlap region between them. These distributions are almost independent of $λ$ at least in the range $0.25 \leq λ\leq 0.75$, namely the overlapping region in $F$, where the class III and class II patterns coexist, has quite gentle $λ$ dependence in this $λ$ region. Therefore the relation between the pattern classes and the $λ$ parameter is not observed. PACS: 89.75.-k Complex Systems