Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30

TL;DR

This paper develops an algebraic framework for elementary cellular automata, focusing on Rule 22's symmetry and nonlinearity, and uses it to analyze Rule 30's complexity and randomness.

Contribution

It introduces a novel algebraic approach centered on symmetry to compare cellular automata, especially highlighting Rule 22 as a reference for understanding Rule 30.

Findings

Closed-form support-set cardinality formulas for Rule 22

Recursive construction of support sets

Rule 30's deviation from Rule 22 follows a power-law scaling

Abstract

A comparative algebraic framework for elementary cellular automata is developed, centered on the role of spatial symmetry. The primary object of study is Rule~22, the elementary cellular automaton with algebraic normal form $g(a,b,c)=a\oplus b\oplus c\oplus abc$ over $\mathbb{F}_2$, the simplest rule combining full $S_3$ symmetry with genuine nonlinearity. Three closed-form results are established: a formula for the support-set cardinality, $|S_m|=2^{\mathrm{popcount}(\lfloor m/2 \rfloor)}\cdot 3^{m\bmod 2}$; a two-step recursive construction of the support sets; and the continuous limit as a parabolic reaction--diffusion equation, $\partial_m u=u_{xx}+2u+u^3$. Rule~22 is then used as a symmetric reference for Rule~30. The symmetry-breaking deviation $\epsilon(m)=|S_m^{(30)}|-|S_m^{(22)}|$ is empirically consistent with a power-law scaling of the form $m^b$ ($b\approx 1.11$), quantifying the cumulative effect of replacing the symmetric cubic $abc$ with the asymmetric quadratic $bc$. A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile.