# Circularly squarefree words and unbordered conjugates: a new approach

## TL;DR

This paper introduces a new approach using automatic sequences, logic, and decision procedures to analyze circularly squarefree words and unbordered conjugates, providing simpler proofs and new classifications.

## Contribution

It offers a novel method for studying unbordered conjugates, completes the classification of binary words with maximum unbordered conjugates, and determines their expected number in random words.

## Key findings

- Complete classification of binary words with maximum unbordered conjugates
- Existence of words with any number of unbordered conjugates up to maximum
- Calculated the expected number of unbordered conjugates in random words

## Abstract

Using a new approach based on automatic sequences, logic, and a decision procedure, we reprove some old theorems about circularly squarefree words and unbordered conjugates in a new and simpler way. Furthermore, we prove three new results about unbordered conjugates: we complete the classification, due to Harju and Nowotka, of binary words with the maximum number of unbordered conjugates; we prove that for every possible number, up to the maximum, there exists a word having that number of unbordered conjugates, and finally, we determine the expected number of unbordered conjugates in a random word.