A natural class of cellular automata containing fractional multiplication automata, Rule 30, and others

TL;DR

This paper introduces a new class of cellular automata called rapidly left expansive automata, encompassing fractional multiplication automata and Rule 30, with theoretical foundations based on aperiodicity and distribution modulo 1.

Contribution

The paper defines a broad class of cellular automata, generalizing previous specific automata, and provides theoretical insights into their properties using aperiodicity and distribution modulo 1.

Findings

Includes fractional multiplication automata and Rule 30 within the class

Establishes a connection between automata behavior and aperiodicity

Links automata properties to distribution modulo 1 theory

Abstract

We define the class of rapidly left expansive cellular automata, which contains fractional multiplication automata, Wolfram's Rule 30, and many others. The definition has been shaped by a proposition of Jen on aperiodicity of columns in space-time diagrams of certain cellular automata, which generalizes to this new class. We also present results that originate from the theory of distribution modulo 1.