Saturday 01 February 2025
The fascinating world of automatic sequences has long been a topic of interest among mathematicians and computer scientists. In this paper, researchers delve into the properties of these sequences, shedding light on their intricate structures and behaviors.
Automatic sequences are generated by iterating a simple rule, known as a substitution, over an alphabet of symbols. The sequence is said to be automatic if it can be recognized by a finite automaton, a mathematical device that scans a string of symbols and accepts or rejects it based on certain criteria. In other words, the sequence has a finite description that allows us to determine whether a given symbol belongs to the sequence or not.
The researchers in this paper focus on one-sided automatic sequences, which are generated by iterating a substitution over an alphabet of symbols, but only looking at the rightward direction of the sequence. They investigate the properties of these sequences, including their recognizability and periodicity.
One of the key findings is that there exists a close connection between the properties of an automatic sequence and the structure of its underlying Bratteli diagram, a mathematical object that represents the sequence in a more abstract form. The researchers show that certain properties of the Bratteli diagram, such as its dimension and rank, can be used to determine whether a given sequence is automatic or not.
The paper also explores the relationship between automatic sequences and their corresponding higher-order presentations, which are constructed by iteratively applying the substitution rule over multiple levels. The researchers demonstrate that these higher-order presentations can reveal hidden patterns and structures within the sequence, providing new insights into its behavior.
Furthermore, the authors investigate the automorphism group of an automatic sequence, which is a set of transformations that preserve the sequence’s structure. They show that this group is closely related to the Bratteli diagram and can be used to classify sequences based on their symmetries.
The study of automatic sequences has far-reaching implications for various fields, including computer science, cryptography, and biology. Understanding the properties and behaviors of these sequences can lead to new algorithms for data compression, encryption, and pattern recognition, as well as insights into complex biological systems like DNA sequences.
In summary, this paper provides a comprehensive analysis of one-sided automatic sequences, shedding light on their intricate structures and behaviors. The researchers’ findings have significant implications for various fields, highlighting the importance of understanding these fascinating mathematical objects.
Cite this article: “Unveiling the Properties of One-Sided Automatic Sequences”, The Science Archive, 2025.
Automatic Sequences, One-Sided Automatic Sequences, Bratteli Diagrams, Dimension, Rank, Recognizability, Periodicity, Higher-Order Presentations, Automorphism Group, Cryptography
Reference: Elżbieta Krawczyk, “Quasi-fixed points of substitutive systems” (2024).
