Orbits of One-Dimensional Cellular Automata Induced by Symmetry Transformations

TL;DR

This work develops a group-theoretic framework to classify one-dimensional cellular automata rules under symmetry transformations—state permutation and reflection—by analyzing the action of the group $S_kR$ on the local rule space $L(k,n)$. Using Burnside’s lemma and a partition by orbit type, the authors derive explicit formulas for the number of orbits and their distribution by type for $k=2$ and $k=3$ across arbitrary neighbourhood sizes, detailing the degrees of orbits and offering concrete invariant-rule constructions. They validate the theory with a brute-force Python implementation, demonstrating alignment between analytic counts and exhaustive enumeration for small parameter regimes. The approach substantially reduces the effective CA rule space through symmetry, providing a rigorous foundation for studying CA dynamics, isomorphism types, and computational properties in discrete systems.

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.

Figures (2)

• Figure 1: The lattice of $S_2R$.
• Figure 2: The lattice of $S_3R$.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Proposition 3
• proof
• Proposition 4: Orbit-Stabilizer Theorem
• proof
• Proposition 5: Burnside's Lemma
• proof
• Lemma 6
• proof