String Attractors of Simple-Parry Automatic Sequences Explored

Core Concepts

String attractors of prefixes of k-bonacci words related to simple Parry numbers are explored.

Abstract

The content delves into the concept of string attractors within combinatorics on words, focusing on specific families of sequences. It discusses the relationship between string attractors and numeration systems, highlighting the importance of fractional power prefixes in determining string attractors. The structure and properties of these sequences are analyzed, providing insights into their composition and relevance in mathematical contexts. Introduction: String attractors introduced by Kempa and Prezza in 2018 for data compression. Questions related to string attractors drawing attention from various scientific fields. Preliminaries: Definitions related to words, morphisms, and fixed points discussed. Various types of Lyndon words explored with their significance in combinatorics on words. Link with Numeration Systems: Dumont-Thomas numeration systems associated with prolongable morphisms explained. Greedy representations in numeration systems discussed along with their relevance. String Attractors of the Prefixes: Concept of string attractors defined within the context of finite words. Propositions and theorems presented regarding string attractors and their determinants. Fractional Power Prefixes and Anti-Lyndon Words: Analysis of fractional power prefixes and their relation to anti-Lyndon words provided. Data Extraction: No key metrics or figures were identified in the content provided.

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String attractors of some simple-Parry automatic sequences

by France Gheer... at arxiv.org 03-25-2024

https://arxiv.org/pdf/2302.13647.pdf

Deeper Inquiries

How do string attractors impact data compression algorithms?

String attractors play a significant role in data compression algorithms by providing a way to capture all distinct factors within a finite word. By identifying the positions in a word that enable the representation of all unique factors, string attractors help optimize the compression process. This concept has been utilized in text compression algorithms to improve efficiency and reduce redundancy in data storage.

What implications do Lyndon words have in combinatorics on words?

Lyndon words are primitive words that hold special significance in combinatorics on words. They are essential for understanding factorizations of free monoids and have applications in various areas such as pattern matching, coding theory, and algorithmic complexity. Lyndon words form complete factorizations of free monoids, making them valuable tools for analyzing word structures and properties.

How can the concept of greedy representations be applied beyond numeration systems?

The concept of greedy representations extends beyond numeration systems to other mathematical contexts where optimization plays a crucial role. In combinatorics on words, greedy representations can be used to efficiently encode information and analyze sequences or patterns. By selecting the largest possible term at each step of decomposition, greedy representations help simplify complex processes and enhance computational efficiency across different domains like cryptography, sequence analysis, and algorithm design.