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Automaticity of spacetime diagrams generated by cellular automata on commutative monoids

Vincent Nesme

Abstract

It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is endowed with the structure of an Abelian group, provided the cellular automaton is a morphism with respect to this structure and the initial configuration has finite support. The spacetime diagram then has a property related to k-automaticity. We show that these conditions can be relaxed: the Abelian group can be a commutative monoid, the initial configuration can be k-automatic, and the spacetime diagrams still exhibit the same regularity.

Automaticity of spacetime diagrams generated by cellular automata on commutative monoids

Abstract

It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is endowed with the structure of an Abelian group, provided the cellular automaton is a morphism with respect to this structure and the initial configuration has finite support. The spacetime diagram then has a property related to k-automaticity. We show that these conditions can be relaxed: the Abelian group can be a commutative monoid, the initial configuration can be k-automatic, and the spacetime diagrams still exhibit the same regularity.
Paper Structure (13 sections, 16 theorems, 54 equations, 12 figures)

This paper contains 13 sections, 16 theorems, 54 equations, 12 figures.

Table of Contents

  1. Groups
  2. Aperiodic Monoids
  3. First Example
  4. Second Example
  5. General Aperiodic Case
  6. Free Commutative $(\mathfrak{i},\mathfrak{p})$-Monoids
  7. Third Example
  8. Monogenic Monoids
  9. Free Commutative $(\mathfrak{i},\mathfrak{p})$-Monoids
  10. General Case
  11. Automatic initial configuration
  12. Fourth Example
  13. General Case

Key Result

Theorem 1

If $(\Sigma,\cdot)$ is an abelian $p$-group, then the double sequence generated by a cellular automaton that is also an endomorphism of $\Sigma^\mathbb{Z}$, starting on a finite configuration, is $p$-automatic.

Figures (12)

  • Figure 1: Ten iterations of $F$ on the initial configuration $\bar{2}$. The top cell has coordinates $(0,0)$; time flows downwards. The neutral element 0 is not depicted.
  • Figure 2: Finite automaton describing the paths of computation.
  • Figure 3: For $0\leq i\leq j$, the cell $(i,j)$ contains $x$ iff $(i,j)\in\varphi(\mathcal{L}_x)$. The top cell has coordinates $(0,0)$.
  • Figure 4: Ten iterations of $G$ on the initial configuration $\bar{2}$. The top cell has coordinates $(0,0)$; time flows downwards. The neutral element $0$ is not depicted.
  • Figure 5: Finite automaton describing pairs of paths
  • ...and 7 more figures

Theorems & Definitions (27)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Definition 2
  • Proposition 2
  • proof
  • Theorem 2
  • Theorem 3
  • Proposition 3
  • proof
  • ...and 17 more