The Smallest String Attractors of Fibonacci and Period-Doubling Words

2026-02-17

AI Summary

This study investigates the structure and enumeration of all size-2 minimal string attractors in Fibonacci words and period-doubling words. By leveraging combinatorial string analysis, recursive modeling, and formal language theory, the authors provide the first explicit construction and exact count of minimal attractors for these two classical infinite word families. The main contributions are precise formulas: for the $n$-th Fibonacci word with $n \geq 7$, there are exactly $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct minimal attractors, whereas for the period-doubling word with $n \geq 2$, only two such attractors exist. These results highlight a striking contrast in structural complexity between the two sequences despite sharing the same attractor size.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Natural Language Processing: Lexical Semantics and Morphology
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

A string attractor of a string $T[1..|T|]$ is a set of positions $Γ$ of $T$ such that any substring $w$ of $T$ has an occurrence that crosses a position in $Γ$, i.e., there is a position $i$ such that $w = T[i..i+|w|-1]$ and the intersection $[i,i+|w|-1]\cap Γ$ is nonempty. The size of the smallest string attractor of Fibonacci words is known to be $2$. We completely characterize the set of all smallest string attractors of Fibonacci words, and show a recursive formula describing the $2^{n-4} + 2^{\lceil n/2 \rceil - 2}$ distinct position pairs that are the smallest string attractors of the $n$th Fibonacci word for $n \geq 7$. Similarly, the size of the smallest string attractor of period-doubling words is known to be $2$. We also completely characterize the set of all smallest string attractors of period-doubling words, and show a formula describing the two distinct position pairs that are the smallest string attractors of the $n$th period-doubling word for $n\geq 2$. Our results show that strings with the same smallest attractor size can have a drastically different number of distinct smallest attractors.

Problem

• string attractor
• Fibonacci words
• period-doubling words
• smallest attractor
• combinatorics on words

Innovation

• string attractor
• Fibonacci words
• period-doubling words
• combinatorics on words
• recursive characterization