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Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30

E. Chan-López, A. Martín-Ruiz

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 $\mathcal{F}_2$, 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 $ε(m)=|S_m^{(30)}|-|S_m^{(22)}|$ 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.

Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30

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 over , the simplest rule combining full symmetry with genuine nonlinearity. Three closed-form results are established: a formula for the support-set cardinality, ; a two-step recursive construction of the support sets; and the continuous limit as a parabolic reaction--diffusion equation, . Rule~22 is then used as a symmetric reference for Rule~30. The symmetry-breaking deviation is empirically consistent with a power-law scaling of the form (), quantifying the cumulative effect of replacing the symmetric cubic with the asymmetric quadratic . A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile.

Paper Structure

This paper contains 12 sections, 5 theorems, 16 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Algebraic Normal Form and Symmetry
  3. Support Sets and the Cardinality Theorem
  4. Recursive Structure of the Support Sets
  5. Generating Polynomials
  6. The Continuous Limit
  7. Symmetry-Breaking Deviation
  8. A Mechanism for Apparent Randomness
  9. Left-Permutive Decomposition and the XOR Lemma
  10. Asymmetric Sensitivity Profile
  11. Connection to the Transport Term
  12. Conclusions and Open Problems

Key Result

Proposition 1

Rule 22 has full $S_3$ symmetry and is simultaneously left-permutive, right-permutive, and center-permutive. Rule 30, with ANF $g_{30}=a\oplus b\oplus c\oplus bc$, is left-permutive but lacks spatial symmetry. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Spatio-temporal evolution from a single seed for 64 generations. Rule 22 (left) produces a modified Sierpiński triangle with clusters of three consecutive active cells. Rule 150 (center) produces the classical Sierpiński triangle. Rule 30 (right) produces the well-known irregular pattern. The bilateral symmetry of Rules 22 and 150 contrasts with Rule 30's asymmetry.
  • Figure 2: Support-set cardinality $|S_m|$ on a logarithmic scale for $m=1,\ldots,64$. Red bars: odd $m$ (factor 3 from the least significant bit). Blue bars: even $m$ (factor 2 only). The pattern resets at each power of 2, where $|S_m|=1$.
  • Figure 3: Left: support cardinalities $|S_m^{(22)}|$ (closed form) and $|S_m^{(30)}|$ (empirical) overlaid. Right: symmetry-breaking deviation $\epsilon(m)$ on log--log axes with a fitted power law $\epsilon\sim m^{1.11}$ (log–log regression).
  • Figure 4: (a) Sensitivity profile of $\eta_{20}(0)$ for Rule 30 from random initial conditions. Blue bars (left, $j<0$): flat at $\approx 0.5$. Red bars (right, $j>0$): decaying. (b) Growth of total left and right sensitivity over time.
  • Figure 5: Statistical properties of Rule 30's center column from a single seed ($N=4096$). (a) Normalized block entropy $H_n/n$ remains near the maximal value 1 for small $n$. (b) Block complexity ratio $p(n)/2^n$ shows full complexity for $n\leq 6$.

Theorems & Definitions (7)

  • Definition 1
  • Proposition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • Theorem 3