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On the Convergence of Elementary Cellular Automata under Sequential Update Modes

Isabel Donoso-Leiva, Eric Goles, Martín Ríos-Wilson, Sylvain Sené

TL;DR

This paper performs a theoretical analysis of the sequential convergence of elementary cellular automata that have at least one fixed point, and classify these rules according to whether all initial configurations converge under all, some, one or none sequential update modes.

Abstract

In this paper, we perform a theoretical analysis of the sequential convergence of elementary cellular automata that have at least one fixed point. Our aim is to establish which elementary rules always reach fixed points under sequential update modes, regardless of the initial configuration. In this context, we classify these rules according to whether all initial configurations converge under all, some, one or none sequential update modes, depending on if they have fixed points under synchronous (or parallel) update modes.

On the Convergence of Elementary Cellular Automata under Sequential Update Modes

TL;DR

This paper performs a theoretical analysis of the sequential convergence of elementary cellular automata that have at least one fixed point, and classify these rules according to whether all initial configurations converge under all, some, one or none sequential update modes.

Abstract

In this paper, we perform a theoretical analysis of the sequential convergence of elementary cellular automata that have at least one fixed point. Our aim is to establish which elementary rules always reach fixed points under sequential update modes, regardless of the initial configuration. In this context, we classify these rules according to whether all initial configurations converge under all, some, one or none sequential update modes, depending on if they have fixed points under synchronous (or parallel) update modes.

Paper Structure

This paper contains 17 sections, 25 theorems, 14 equations, 3 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Structure of the Paper
  4. Preliminaries
  5. Elementary Cellular Automata and Sequential Update Modes
  6. Dynamical Systems
  7. Results
  8. Rules for which Theorem \ref{['thm:DAS']} holds.
  9. Rules that admit a Covering of Sequential Update Modes
  10. Rules with no Covering
  11. Notable Rules Outside the Theorem
  12. Restriction to Rings of Even Size
  13. Restriction to Rings of Size Multiple of $3$
  14. Without Covering of Sequential Update Modes
  15. Rule $37$
  16. ...and 2 more sections

Key Result

Theorem 1

Let $f$ be an ECA. Then there exists (at least) one asynchronous update mode $\mu$ for $f$ such that $f_{\mu}$ converges to fixed points if any of the following condition is verified:

Figures (3)

  • Figure 1: Space-time diagrams (time going downward) of configuration $0100$ following Rule $45$ under: (a) the update mode $\mu_{1} = (0,1,2,3)$, (b) the update mode $\mu_{2} = (2,0,3,1)$, (c) the update mode $\mu_{3} = (3,2,1,0)$. Configurations colored in gray (resp. black) represent the ones obtained at substeps (resp. steps).
  • Figure 2: Space-time diagrams (time going downward) of configuration $10010000$ following Rule $28$ (a) and Rule $29$ (b) under the update mode $\mu = (7,6,5,4,3,2,1,0)$. We are only showing the resulting configurations after full steps have been completed. In this example, $01$ is a wall for both Rule $28$ and Rule $29$. The example starts with two isles of $0$s separated by one $1$. A new isle is created at time step $2$ for case (a) and at time step $1$ for case (b).
  • Figure 3: Space-time diagrams (time going downward) of configuration $01010011$, in (a) and (b), and $11011100$, in (c) and (d), following Rule $138$ under: (a) and (c) the update mode $\mu_{1} = (9,8,7,6,5,4,3,2,1,0)$, (b) the update mode $\mu_{2} = (1,3,9,8,0,2,4,5,6,7)$, (d) the update mode $\mu_{3} = (5,4,3,1,0,2,6,7,8,9)$. Configurations colored in gray (resp. black) represent the ones obtained at substeps (resp. steps). Update mode $\mu_1$ is the one defined in Theorem \ref{['thm:universal']}, while update modes $\mu_2$ and $\mu_3$ are obtained following the instructions given by the proof of Theorem \ref{['thm:DAS']} in Sethi2016.

Theorems & Definitions (58)

  • Definition 1: Sequential update modes
  • Definition 2: Wall and isle
  • Definition 3: Universal update mode
  • Definition 4: Covering
  • Definition 5: $X$-Universal update mode
  • Definition 6: $X$-Covering
  • Theorem 1: das2013Sethi2016
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 48 more