Shortest Repetition-Free Words Accepted by Automata

TL;DR

This paper investigates the minimal length of repetition-free words accepted by finite automata, providing bounds that are currently far apart, highlighting gaps in understanding of automata accepting such words.

Contribution

It offers new upper and lower bounds on the shortest repetition-free words accepted by automata, a problem with limited prior precise bounds.

Findings

Provided upper bounds on shortest accepted repetition-free words.

Established lower bounds demonstrating the minimal length can be large.

Highlighted the significant gap between known bounds.

Abstract

We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.