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Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

Henryk Fukś, Yucen Jin

TL;DR

This work addresses why local structure theory can yield exact or highly accurate predictions for certain CA by analyzing a four-input number-conserving CA with a blocking word, $100$, and Wolfram number $W(f)=56528$. The authors derive exact preimage structures and density polynomials, and show that the local structure approximation reproduces the steady-state probabilities of blocks of length up to 3, with a rigorous fixed-point and stability analysis. They formulate a finite-dimensional recurrence for key block densities that converges to the exact steady state, and provide numerical evidence corroborating the analytic results. The study highlights the combined role of number conservation and blocking words in suppressing long-range correlations, suggesting a path to identifying other solvable CA rules with similarly predictable behavior.

Abstract

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely-dimensional system. While it is well known that this approximation works surprisingly well for some cellular automata, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.

Approximating dynamics of a number-conserving cellular automaton by a finite-dimensional dynamical system

TL;DR

This work addresses why local structure theory can yield exact or highly accurate predictions for certain CA by analyzing a four-input number-conserving CA with a blocking word, , and Wolfram number . The authors derive exact preimage structures and density polynomials, and show that the local structure approximation reproduces the steady-state probabilities of blocks of length up to 3, with a rigorous fixed-point and stability analysis. They formulate a finite-dimensional recurrence for key block densities that converges to the exact steady state, and provide numerical evidence corroborating the analytic results. The study highlights the combined role of number conservation and blocking words in suppressing long-range correlations, suggesting a path to identifying other solvable CA rules with similarly predictable behavior.

Abstract

The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely-dimensional system. While it is well known that this approximation works surprisingly well for some cellular automata, it is still not clear why it is the case, and which CA rules have this property. In order to shed some light on this problem, we present an example of a four input CA for which probabilities of occurrence of short blocks of symbols can be computed exactly. This rule is number conserving and possesses a blocking word. Its local structure approximation correctly predicts steady-state probabilities of small length blocks, and we present a rigorous proof of this fact, without resorting to numerical simulations. We conjecture that the number-conserving property together with the existence of the blocking word are responsible for the observed perfect agreement between the finite-dimensional approximation and the actual infinite-dimensional dynamical system.
Paper Structure (6 sections, 5 theorems, 47 equations, 4 figures)

This paper contains 6 sections, 5 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Exact results: preimage sets
  4. Exact results: Block probabilities
  5. Local structure approximation
  6. Conclusions

Key Result

Proposition 1

$\mathbf{f}^{-n}(100)$ has the form and $\mathbf{f}^{-n}(00100)$ has the form where $*$ is an arbitrary element in $\{0,1\}$.

Figures (4)

  • Figure 1: Spatiotemporal pattern generated by rule 56528, using lattice of 100 sites with periodic boundaries. Black squares represent 1s and white squares represent 0s. Time (consecutive iterations) proceeds downwards.
  • Figure 2: Examples of preimages of 101. Top line in each diagram represents string of length $3\cdot 3+3=12$, followed by its three consecutive images under $\mathbf{f}$. Irrelevant symbols are represented by dots.
  • Figure 3: Comparison of the exact values of $P_{\infty}(00)$, $P_{\infty}(000)$ and $P_{\infty}(010)$ (denoted by "$+$") with values of $x_\infty$, $y_\infty$, $z_\infty$ obtained by iterating eqs. (\ref{['xyzeq']}) numerically (denoted by continuous line).
  • Figure 4: Consecutive images (dark red) of a neighbourhoood of the fixed point (black ellipse) under the map defined by eqs. (\ref{['xyzeq-reducd']}), for $p=0.3$. The initial point $(x_0, z_0)$ is shown as small black diamond. The fixed point $(x^\star, z^\star)$ is represented by small red circle, located at the intersection of two invariant manifolds, shown as dotted and dashed lines (blue and green).

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Proposition 4