Orbits of One-Dimensional Cellular Automata Induced by Symmetry Transformations

TL;DR

This paper introduces a group-theoretic method to classify one-dimensional cellular automata rules into equivalence classes based on symmetry transformations, providing counts of these classes for small state sets.

Contribution

It presents a novel algebraic approach to determine automata rule equivalence classes induced by symmetry operations, including reflection and permutation.

Findings

Number of orbits for binary and ternary state sets calculated

Classification of orbits by isomorphism type provided

Method facilitates understanding of automata rule symmetries

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.