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Decidability of Sturmian Words in Presburger Arithmetic


Core Concepts
The author demonstrates the decidability of the first-order theory of Sturmian words over Presburger arithmetic, showcasing the uniform ω-automaticity of their expansions.
Abstract
The content explores the decidable nature of Sturmian words' first-order theory over Presburger arithmetic. It highlights the use of a general adder to recognize addition in Ostrowski numeration systems, leading to automatic reproval and new results about Sturmian words. The paper extends results on quadratic numbers to all Sturmian characteristic words, proving their decidability. It showcases how classical theorems about Sturmian words can now be automatically proven by theorem-provers in seconds. The study also reveals potential applications and outlines a more general theorem that encompasses previous results. Furthermore, it introduces #-binary coding and #-Ostrowski representations to encode continued fractions and real numbers uniquely. The content emphasizes the ω-regularity and bijectivity of these encodings, ensuring accurate representation and decoding capabilities for various mathematical structures. Overall, the article delves into the intricate world of Sturmian words, providing insights into their decidability and expanding on novel encoding techniques for precise numerical representations.
Stats
Let X ∈ N be written uniquely as X = ∑(n=0)^(∞) bn+1qn. Let α be irrational with a unique continued fraction expansion [a0; a1, ...]. The k-th difference βk is defined as βk = qkα - pk. For x ∈ Iα: x = ∑(k=0)^(∞) bk+1βk. For x ∈ Iα: x = [b1b2...bN+1]α. Let w ∈ R: w can be written uniquely as ∑(k=0)^(∞) bk+1βk. Let X ⊆ Rn be aligned: zero-closure(X) is ω-regular. Afin := {(v, w): v ∈ R, w ∈ Afin_v} is ω-regular. Av := {(v, w): v ∈ R, w ∈ Av} is ω-regular. Zv : Afin_v → N and Ov : Av → Iα(v) are bijective functions.
Quotes

Key Insights Distilled From

by Philipp Hier... at arxiv.org 03-06-2024

https://arxiv.org/pdf/2102.08207.pdf
Decidability for Sturmian words

Deeper Inquiries

How does the concept of #-binary coding enhance numerical representations beyond traditional methods

The concept of #-binary coding introduces a unique way to encode numerical representations by combining binary encoding with the symbol #. This method allows for a more structured and aligned representation of numbers, especially when dealing with irrational or quadratic numbers. By incorporating the # symbol into the binary encoding, it provides a clear delineation between different parts of the representation, making it easier to identify patterns and relationships within the encoded data. One key advantage of #-binary coding is its ability to represent continued fraction expansions in a concise and organized manner. The alignment constraints imposed by the presence of # symbols ensure that each part of the representation corresponds accurately to its position in the overall structure. This alignment facilitates comparisons between different representations and enables efficient processing and analysis of numerical data. Furthermore, #-binary coding enhances numerical representations beyond traditional methods by offering a standardized format that can be easily interpreted and manipulated algorithmically. The structured nature of this encoding scheme simplifies operations such as comparison, addition, and subtraction on encoded numbers, leading to more efficient computational processes in various mathematical contexts.

What implications do the findings on Sturmian word decidability have for computational complexity theory

The findings on Sturmian word decidability have significant implications for computational complexity theory, particularly in relation to automata theory and formal languages. Decidability results play a crucial role in understanding the limits of computation within specific mathematical structures like Sturmian words. In this context, demonstrating that certain classes of Sturmian words have decidable first-order logical theories opens up avenues for exploring their properties computationally. The ability to automatically decide combinatorial statements about all Sturmian words based on their characteristic features provides insights into their structural characteristics without requiring intricate manual proofs. From a computational complexity perspective, these results indicate that certain decision problems related to Sturmian words can be efficiently solved using automated theorem-proving techniques or algorithms based on ω-automatic structures. Understanding which properties are decidable within this framework helps delineate boundaries for tractable computations involving complex symbolic sequences like Sturmian words.

How can the study's encoding techniques be applied to other mathematical structures or problem domains

The study's innovative encoding techniques can be applied across various mathematical structures or problem domains where precise representation is essential for analysis or computation. Number Theory: The #-Ostrowski-representations introduced in the study could find applications in number theory research where accurate encodings are required for continued fractions or real numbers with specific properties. Data Compression: The concept of #-binary coding could be utilized in data compression algorithms where structured representations are needed for efficient storage and retrieval. Pattern Recognition: Applying similar alignment constraints from #-binary coding could enhance pattern recognition algorithms by providing clearer distinctions between different components within datasets. Automata Theory: Techniques used in representing Sturmian words through automata-inspired methods could inspire new approaches towards recognizing patterns or regularities within other types of symbolic sequences. By leveraging these novel encoding methodologies developed through studying Sturmian words' decidability, researchers can explore diverse areas where precise yet manageable numerical representations are vital for advancing computational analyses and problem-solving strategies across multiple disciplines."
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