Smallest String Attractors and Minimal Coverage Certificates of Thue--Morse Words

2026-10-07

AI Summary

This study investigates the minimal string attractors and minimal covering certificates of finite Thue-Morse words. By integrating combinatorics, formal language theory, factor position analysis, and antichain construction techniques, it fully characterizes the geometric distribution and cardinality properties of optimal attractors. The primary contributions include the first exact classification of 32 minimal attractors for n≥6 and the establishment of a linear-size simplest factor-covering antichain. Furthermore, this work reveals anomalous phenomena in initial cases and proposes a constant-size verification system comprising only 24 elements. Overall, this research provides a systematic theoretical framework for understanding the deep combinatorial structure underlying Thue-Morse words.

Abstract

String attractors provide a compact way of representing the complete factor structure of a word: a set of positions is an attractor if every distinct factor has at least one occurrence crossing one of the selected positions. Although the minimum attractor size is known for several classical families of words, describing \emph{all} optimal attractors is typically much more difficult, since it requires understanding the geometry of all factor occurrences rather than constructing a single optimal solution. We give a complete description for the finite Thue--Morse words. Earlier work proved that four positions are necessary and sufficient for every order $n\geq 4$, but the collection of all smallest attractors remained unknown. For every $n\geq 6$, writing $h=2^{n-3}$, we prove that the smallest attractors are exactly the two reflected families where the four offsets are chosen independently from $\{0,1\}$. Hence there are exactly $32$ smallest attractors for every $n\geq6$. The initial cases are genuinely exceptional: $t_5$ has $40$ smallest attractors and $t_4$ has $87$. We also study the attractor condition independently of optimality. For every $n\geq5$, we characterize the complete antichain of inclusion-minimal factor coverages of $t_n$. It consists precisely of the coverages of $aa$, $bb$, and the eight minimal unique substrings of every generation $t_m$, $4\leq m\leq n$. Thus there are exactly $8n-22$ canonical constraints, forming an irredundant exact certificate for attractors of arbitrary cardinality. When attention is restricted to four-position sets, this linear-size system collapses to a constant one: it is enough to test the $24$ minimal unique substrings coming from three consecutive generations, and sixteen of these already force the two optimal families. We also show that three generations are necessary within this natural consecutive-generation hierarchy.

Problem

• String attractors
• Thue-Morse words
• Minimal coverage certificates
• Factor occurrences

Innovation

• String attractors
• Thue-Morse words
• Minimal coverage certificates
• Factor occurrences
• Unique substrings