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Solving the initial value problem for cellular automata by pattern decomposition

Henryk Fukś

TL;DR

The paper addresses solving the initial-value problem for cellular automata by a pattern-decomposition approach, focusing on a nontrivial perturbation of Rule 140, Rule 156. It develops a framework where an unperturbed rule with a known IVP solution and a mild perturbation are combined to yield an explicit formula for the nth iterate, $[F^n_{156}(x)]_i=[F^n_{140}(x)]_i+D+B_1+B_2+S_1+S_2$, and proves correctness via a sequence of lemmas and ab initio verification. This explicit solution enables a complete probabilistic analysis: the density of ones after $n$ steps for random initial configurations is given by a closed form that exhibits $(-1)^n$ oscillations due to blinkers, along with exact short-block probabilities. The work demonstrates the method's potential generality for almost equicontinuous CA and discusses how the choice of perturbation influences tractability, illustrating both the promise and the limitations of pattern-decomposition strategies in CA IVP problems.

Abstract

For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after $n$ iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.

Solving the initial value problem for cellular automata by pattern decomposition

TL;DR

The paper addresses solving the initial-value problem for cellular automata by a pattern-decomposition approach, focusing on a nontrivial perturbation of Rule 140, Rule 156. It develops a framework where an unperturbed rule with a known IVP solution and a mild perturbation are combined to yield an explicit formula for the nth iterate, , and proves correctness via a sequence of lemmas and ab initio verification. This explicit solution enables a complete probabilistic analysis: the density of ones after steps for random initial configurations is given by a closed form that exhibits oscillations due to blinkers, along with exact short-block probabilities. The work demonstrates the method's potential generality for almost equicontinuous CA and discusses how the choice of perturbation influences tractability, illustrating both the promise and the limitations of pattern-decomposition strategies in CA IVP problems.

Abstract

For many cellular automata, it is possible to express the state of a given cell after iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.
Paper Structure (7 sections, 5 theorems, 104 equations, 6 figures)

This paper contains 7 sections, 5 theorems, 104 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Rule 140
  4. Rule 156
  5. Proof of correctness of the solution formula
  6. Probabilistic solution
  7. Conclusions

Key Result

theorem 1

For elementary cellular automaton rule 140, for any $x\in \{0,1\}^{\mathbb{Z}}$ and $n>1$, the state of the $j$-th cell after $n$ iterations of the rule starting from $x$ is given by

Figures (6)

  • Figure 1: Spatiotemporal patterns of rule 140 (left) and rule 156 (right).
  • Figure 2: Spatiotemporal pattern of rule 156 (top) decomposed into six elements. Initial configuration is shown in black color in the decomposed elements.
  • Figure 3: Three initial configurations producing blinkers of even type.
  • Figure 4: Graph of the amplitude of density's oscillations as a function of the initial density $p$.
  • Figure 5: Spatiotemporal patterns of rule 156 generated starting from random initial configurations with three different values of the initial density $p$. Sites in state 1 are black in the initial string, red in odd blinkers and green in even blinkers. Other sites in state 1 are blue. Lattice of 120 sites with periodic boundaries.
  • ...and 1 more figures

Theorems & Definitions (8)

  • theorem 1
  • theorem 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof