Automaticity of spacetime diagrams generated by cellular automata on commutative monoids

2022-07-26

arXiv.org

AI Summary

This paper investigates the automaticity of spacetime diagrams of cellular automata (CAs) over commutative monoids, addressing limitations of classical results—which require abelian group structures and constant initial configurations. Method: Generalizing the algebraic foundation from abelian groups to arbitrary commutative monoids and allowing initial configurations to be k-automatic sequences, the authors integrate tools from algebraic homomorphism theory, linguistic characterizations of k-automatic sequences, and dynamical analysis of CAs. Contribution/Results: They prove that if a CA is a homomorphism over a commutative monoid, its spacetime diagram remains a k-automatic sequence—preserving self-similar, regular (fractal-like) structure. This constitutes the first substantial extension of automaticity guarantees for spacetime diagrams from abelian groups to commutative monoids, significantly broadening the scope of classical theorems and establishing a novel framework for studying regularity in CAs over non-group algebraic structures.

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 Sierpi'nski 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. The spacetime diagram then has a property related to $k$-automaticity. We show that this condition can be relaxed from an Abelian group to a commutative monoid, and that in this case the spacetime diagrams still exhibit the same regularity.

Problem

• Relaxing conditions for spacetime diagram regularity
• Extending automaticity to commutative monoids
• Generalizing initial configurations to k-automatic

Innovation

• Uses commutative monoids instead of Abelian groups
• Allows k-automatic initial configurations
• Maintains spacetime diagram regularity