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Limit dynamics of elementary cellular automaton 18

Hervé Sabrié, Ilkka Törmä

TL;DR

This work analyzes the long-term behavior of elementary CA rule $18$ by examining three limit notions: the limit set $8Omega}(f_{18})$, the generic limit set $ ilde{8omega}(f_{18})$, and the $8Omega_}(f_{18})$ (the $$-limit set). It shows that defect dynamics are governed by kinks, and provides a complete characterization of configurations with up to two kinks in the generic limit set, proving that $ ilde{8omega}(f_{18})$ contains all kinkless words, all one-kink words, and a precisely described family of two-kink words $P$. The paper then proves that the three limit sets are distinct by exhibiting words present in $8Omega}(f_{18})$ but not in $ ilde{8omega}(f_{18})$ or any $8Omega_}(f_{18})$, thereby highlighting nontrivial separations between limit notions. Overall, it offers a strategy for using the generic limit set to address conjectures about particle densities in CA and clarifies the intricate defect dynamics that govern rule $18$.

Abstract

We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and $μ$-limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.

Limit dynamics of elementary cellular automaton 18

TL;DR

This work analyzes the long-term behavior of elementary CA rule by examining three limit notions: the limit set , the generic limit set , and the (the -limit set). It shows that defect dynamics are governed by kinks, and provides a complete characterization of configurations with up to two kinks in the generic limit set, proving that contains all kinkless words, all one-kink words, and a precisely described family of two-kink words . The paper then proves that the three limit sets are distinct by exhibiting words present in but not in or any , thereby highlighting nontrivial separations between limit notions. Overall, it offers a strategy for using the generic limit set to address conjectures about particle densities in CA and clarifies the intricate defect dynamics that govern rule .

Abstract

We study the the asymptotic dynamics of elementary cellular automaton 18 through its limit set, generic limit set and -limit set. The dynamics of rule 18 are characterized by persistent local patterns known as kinks. We characterize the configurations of the generic limit set containing at most two kinks. As a corollary, we show that the three limit sets of rule 18 are distinct.
Paper Structure (25 sections, 20 theorems, 5 equations, 11 figures, 1 table)

This paper contains 25 sections, 20 theorems, 5 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Limit sets and variants
  4. Defect dynamics of rule 18
  5. The generic limit set of rule 18
  6. Kinkless words
  7. Words with one kink
  8. Words with two kinks
  9. Case 0: $|w| \leq 7$.
  10. Case 1: $w = 11 0^{2k} 1$ for some $k \in \mathbb{N}$.
  11. Case 2: $w = 1101 0^{2k} 1$ for some $k \in \mathbb{N}$.
  12. Case 3: $w = 11 (01)^k 0^{2\ell} 1$ for some $k, \ell \geq 1$.
  13. Case 4: $\ell \geq 2$.
  14. Case 5: $\ell \leq 1$ and $k \geq 1$.
  15. Case 6: $\ell = 1$ and $k = 0$.
  16. ...and 10 more sections

Key Result

Lemma 2

Let $f : A^\mathbb{Z} \to A^\mathbb{Z}$ be a cellular automaton and $w \in A^*$. Then $w$ occurs in the generic limit set $\Tilde{\omega}(f)$ if and only if there exists a "seed" word $s \in A^*$ and a position $i \in \mathbb{Z}$ such that for all $u, v \in A^*$, there are infinitely many $n \in \ma

Figures (11)

  • Figure 1: An example run of rule 18.
  • Figure 2: The word $u_1$ above contains an even number of kinks, hence $f(u_1)$ is not a kink. The word $u_2$ below contains an odd number of kinks, hence $f(u_2)$ is a kink.
  • Figure 3: Two kinks collide and annihilate each other, but it is possible to avoid collision by making the leftmost kink longer.
  • Figure 4: We can see here that $w = 0010101100101$ is stable, $f(w)$ and $f^2(w)$ are stable and $f^3(w) = 1001011$. We can deduce that $\Sigma(1001011) = f^3(\Sigma(w)) \subseteq f^3(\Sigma(1001011))$.
  • Figure 5: We can see here that $w = 1010010110100$ is stable, $f(w)$ and $f^2(w)$ are stable and $f^3(w) = 1101001$. We can deduce that $\Sigma(1101001) = f^3(\Sigma(w)) \subseteq f^3(\Sigma(1001011))$.
  • ...and 6 more figures

Theorems & Definitions (39)

  • Definition 1: MIL85KM00
  • Lemma 2: TOR20
  • Lemma 3: Proposition 6.2 of DJEGUI19
  • Definition 4
  • Proposition 5: JEN90
  • Definition 6
  • Definition 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 29 more