Characteristic representation of elementary cellular automata

Abstract

We propose a characteristic representation of one-dimensional and two-state, three-neighbor cellular automaton rules, which can be interpreted as a numerical description of Wolfram's classification scheme. Each rule can be considered to behave, in many situations, like some other rules. The similarity of behavior is quantified. The quantification is done by creating a function designed to give an average weight of configurations after many time steps, and then comparing this function, in a precise way, to some other rules by the value of coefficients expanded by the rule functions. We call such an expansion form of the function characteristic representation. The simulation result of the representation shows that Class IV rules are distinguished from Class III rules by the similarity to Class II rules. Criticality of Class IV rules is displayed by functions linearly combining typical Class II and III rule functions.