On the Morse-Hedlund complexity gap

TL;DR

This paper provides a concise, self-contained proof that the subword complexity of any language either remains bounded or grows at least linearly, extending Morse and Hedlund's 1938 result to a broader context.

Contribution

It offers a new, compact proof of Ehrenfeucht and Rozenberg's 1982 theorem on the complexity gap for languages, simplifying understanding of this fundamental property.

Findings

Subword complexity of languages is either bounded or linearly growing.

The paper presents a self-contained proof of the complexity gap theorem.

Extends Morse-Hedlund's result from words to general languages.

Abstract

In 1938, Morse and Hedlund proved that the subword complexity function of an infinite word is either bounded or at least linearly growing. In 1982, Ehrenfeucht and Rozenberg proved that this gap property holds for the subword complexity function of any language. The aim of the present paper is to present a self-contained, compact proof of Ehrenfeucht and Rozenberg's result.