The impact of alphabet size on pattern complexity of maxmin-$$\omega$$ cellular automata

Abstract

AbstractWe present an analysis of an additive cellular automaton (CA) under asynchronous dynamics. The asynchronous scheme is maxmin-$$\omega$$ω, a deterministic system, introduced in our previous work with a binary alphabet. Extending this work, we study the impact of a larger alphabet, which also allows a meaningful inference of the behaviour of the resultant CA from the asymptotic behaviour of the maxmin-$$\omega$$ω update system. Far from being a straightforward positive correlation between complexity and alphabet size, we show that there is a region of $$\omega$$ω and alphabet size where complexity of CA is maximal. Thus, despite employing a fixed CA rule, the complexity of this CA can be controlled by $$\omega$$ω and alphabet size. The main message is that the effect of maxmin-$$\omega$$ω updating on the state of a network can be well understood, especially if the state alphabet is counter-intuitively large.