A Toom rule that increases the thickness of sets

TL;DR

This paper demonstrates that the R^+ cellular automaton rule consistently increases the thickness of sets, advancing understanding of its role in rapid consensus formation and robustness to noise.

Contribution

It introduces a new property, thickness, that R^+ always increases, providing insight into its convergence behavior and potential for fast consensus.

Findings

R^+ increases set thickness

Supports fast convergence towards consensus

Remains effective with some noise

Abstract

Toom's north-east-self voting cellular automaton rule R is known to suppress small minorities. A variant which we call R^+ is also known to turn an arbitrary initial configuration into a homogenous one (without changing the ones that were homogenous to start with). Here we show that R^+ always increases a certain property of sets called thickness. This result is intended as a step towards a proof of the fast convergence towards consensus under R^+. The latter is observable experimentally, even in the presence of some noise.