Automorphissm Classification of Cellular Automata

Abstract

A new classification of arbitrary cellular automata (CA for short) in Z d is studied considering the set (group) of all permutations of the neighborhood ν and state set Q. Two CA (Z d, Q, f A, ν A) and (Z d, Q, f B, ν B) are called automorphisc, if there is a pair of permutations (π, p) of ν and Q, respectively, such that (f B, ν B) = (p −1 f A πp, ν A π), where ν π denotes a permutation of ν and f π denotes a permutation of arguments of local function f corresponding to ν π. This automorphissm naturally induces a classification of CA, such that it generally preserves the global properties of CA up to permutation. As a typical example of the theory, the local functions of 256 ECA (1-dimensional 3-nearest neighbors 2-states CA) are classified into 46 classes. We also give a computer test of surjectivity, injecitivity and reversibility of the classes.