Orbits of one-dimensional cellular automata induced by symmetry transformations

Abstract

Using a group-theoretic approach, a method for determining the equivalence classes (also called orbits) of the set of rules of one-dimensional cellular automata induced by the symmetry operations of reflection and permutation and their product is presented. Orbits are classified by their isomorphism type. Results for the number of orbits and the number of orbits by type for state sets of size two and three are included.