Self-stabilization of Circular Arrays of Automata

TL;DR

This paper strengthens previous results by explicitly proving that a small minority of automata in a finite circular array of simple two-state cellular automata will eventually vanish, demonstrating self-stabilization.

Contribution

It explicitly proves that a small minority of states in finite circular arrays of automata will disappear, extending prior infinite array results.

Findings

Small minority states vanish in finite circular arrays

Self-stabilization occurs in finite automata arrays

Strengthens previous infinite array results

Abstract

[Gacs, Kurdiumov, Levin, 78] proposed simple one-dimensional cellular automata with 2 states. In an infinite array they are self-stabilizing: if all but a finite minority of automata are in the same state, the minority states disappear. Implicit in the paper was a stronger result that a sufficiently small minority of states vanish even in a finite circular array. The following note makes this strengthening explicit.