String Attractors and Infinite Words

TL;DR

This paper explores the properties of string attractors in infinite words, examining their relationship with other complexity measures and introducing new attractor-based metrics for better classification of infinite sequences.

Contribution

It establishes connections between string attractor profiles and classical combinatorial measures, and introduces novel attractor-based complexity metrics for infinite words.

Findings

String attractor profiles relate to factor complexity and recurrence.

New attractor-based measures offer finer classification of infinite words.

The study enhances understanding of repetitiveness in infinite sequences.

Abstract

The notion of string attractor has been introduced in [Kempa and Prezza, 2018] in the context of Data Compression and it represents a set of positions of a finite word in which all of its factors can be "attracted". The smallest size $\gamma^*$ of a string attractor for a finite word is a lower bound for several repetitiveness measures associated with the most common compression schemes, including BWT-based and LZ-based compressors. The combinatorial properties of the measure $\gamma^*$ have been studied in [Mantaci et al., 2021]. Very recently, a complexity measure, called string attractor profile function, has been introduced for infinite words, by evaluating $\gamma^*$ on each prefix. Such a measure has been studied for automatic sequences and linearly recurrent infinite words [Schaeffer and Shallit, 2021]. In this paper, we study the relationship between such a complexity measure and other well-known combinatorial notions related to repetitiveness in the context of infinite words, such as the factor complexity and the recurrence. Furthermore, we introduce new string attractor-based complexity measures, in which the structure and the distribution of positions in a string attractor of the prefixes of infinite words are considered. We show that such measures provide a finer classification of some infinite families of words.