Confliction-Like Dynamics of Rule 20 ECA of Wolfram Class II

Abstract

In this paper, we examine rule 20 elementary cellular automaton of Wolfram class II and show that the rule has 2-step 2-right and 2-step 2-left shift dynamical subsystems, and combining these two subsystems moving in the opposite direction, we have a confliction-like dynamical sub-system which cannot be imagined from that the rule belongs to Wolfram class II. Furthermore we show $g^{2}_{20}=\max\{f_{20RE},f_{20LE}\}$ suggesting that the rule 20 is composed of two simpler cellular automata f _20RE and f _20LE corresponding to 2-step 2-right and 2-step 2-left shift dynamical subsystems, respectively. We also mention that there exist several rules of class II showing confliction-like dynamics similar to rule 20 but all of them have not yet been fully examined. Rule 14, one of them, especially shows an interesting movement called reversing and right shift.

From these observations we know that even a single cellular automaton has not necessarily only one dynamical property but also includes several entirely different subsystems and each of them emerges depending on patterns of initial configurations. And furthermore we may say that there exist some basic cellular automata and methods, perhaps Boolean functions, to combine them to generate other cellular automata.