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String attractors of some simple-Parry automatic sequences

France Gheeraert, Giuseppe Romana, Manon Stipulanti

TL;DR

This paper focuses on string attractors of prefixes of particular automatic infinite words related to simple-Parry numbers related to simple-Parry numbers, and describes string attractors of optimal size for a subfamily of these words.

Abstract

Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we obtain string attractors of prefixes of particular infinite words generalizing k-bonacci words (including the famous Fibonacci word) and related to simple Parry numbers. In fact, our description involves the numeration systems classically derived from the considered morphisms. This extends our previous work published in the international conference WORDS 2023.

String attractors of some simple-Parry automatic sequences

TL;DR

This paper focuses on string attractors of prefixes of particular automatic infinite words related to simple-Parry numbers related to simple-Parry numbers, and describes string attractors of optimal size for a subfamily of these words.

Abstract

Firstly studied by Kempa and Prezza in 2018 as the cement of text compression algorithms, string attractors have become a compelling object of theoretical research within the community of combinatorics on words. In this context, they have been studied for several families of finite and infinite words. In this paper, we obtain string attractors of prefixes of particular infinite words generalizing k-bonacci words (including the famous Fibonacci word) and related to simple Parry numbers. In fact, our description involves the numeration systems classically derived from the considered morphisms. This extends our previous work published in the international conference WORDS 2023.
Paper Structure (8 sections, 24 theorems, 38 equations, 2 figures, 3 tables)

This paper contains 8 sections, 24 theorems, 38 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Words
  4. Morphisms and fixed points of interest
  5. Link with numeration systems
  6. String attractors of the prefixes
  7. Fractional power prefixes and anti-Lyndon words
  8. Optimality of the string attractors

Key Result

Theorem 1

For every non-empty word $w\in A^*$, there exists a unique factorization $(\ell_1,\ldots, \ell_n)$ of $w$ into Lyndon words over $A$ such that $\ell_1 \geq\ell_2 \geq\cdots \geq\ell_n$.

Figures (2)

  • Figure 1: The automaton $\mathcal{A}_{\mu_c}$ for $c = 102$.
  • Figure 2: Representation of the proof of the second claim of Theorem \ref{['T:sa of prefixes']}. As we warned the reader before, elements in a string attractor are indexed starting at $1$ (in red), while indices of letters in $\mathbf{u}$ start at $0$.

Theorems & Definitions (57)

  • Theorem 1: Chen-Fox-Lyndon CFL58
  • Definition 1
  • Example 2
  • Definition 2
  • Example 3
  • Proposition 4
  • Remark 5
  • Proposition 6: Dumont-Thomas Dumont-Thomas-1989
  • Example 7
  • Remark 8
  • ...and 47 more