Automaticity of spacetime diagrams generated by cellular automata on commutative monoids

TL;DR

This paper extends the understanding of spacetime diagrams of cellular automata, showing that fractal-like patterns and regularities occur under more general algebraic structures and initial conditions than previously known.

Contribution

It demonstrates that the regularity of spacetime diagrams persists when the algebraic structure is a commutative monoid and initial configurations are k-automatic.

Findings

Spacetime diagrams of cellular automata can exhibit fractal structures.

Regularity properties hold under relaxed algebraic conditions.

k-automatic initial configurations lead to predictable patterns.

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.