Fife's Theorem Revisited

TL;DR

This paper presents a simpler proof of Fife's theorem on infinite binary overlap-free words and characterizes which of these words are 2-automatic, enhancing understanding of their structure and automaton representations.

Contribution

It provides a significantly simpler proof of Fife's theorem and characterizes 2-automatic overlap-free words.

Findings

Simplified proof of Fife's theorem

Complete characterization of 2-automatic overlap-free words

Enhanced understanding of automaton representations

Abstract

We give another proof of a theorem of Fife - understood broadly as providing a finite automaton that gives a complete description of all infinite binary overlap-free words. Our proof is significantly simpler than those in the literature. As an application we give a complete characterization of the overlap-free words that are 2-automatic.