String Attractors for Automatic Sequences

TL;DR

This paper investigates the decidability and bounds of string attractors in automatic sequences, providing algorithms to determine attractor sizes and analyzing specific sequences like Thue-Morse and period-doubling.

Contribution

It introduces a decision procedure for string attractor sizes in automatic sequences and establishes bounds for various classes of sequences, including linearly recurrent and finite appearance constant sequences.

Findings

Decidability of whether all prefixes have small string attractors.

For certain sequences, all prefixes of length ≥ 2 have attractors of size 2.

Bounds on attractor sizes are established for different sequence classes.

Abstract

We show that it is decidable, given an automatic sequence $\bf s$ and a constant $c$, whether all prefixes of $\bf s$ have a string attractor of size $\leq c$. Using a decision procedure based on this result, we show that all prefixes of the period-doubling sequence of length $\geq 2$ have a string attractor of size $2$. We also prove analogous results for other sequences, including the Thue-Morse sequence and the Tribonacci sequence. We also provide general upper and lower bounds on string attractor size for different kinds of sequences. For example, if $\bf s$ has a finite appearance constant, then there is a string attractor for ${\bf s}[0..n-1]$ of size $O(\log n)$. If further $\bf s$ is linearly recurrent, then there is a string attractor for ${\bf s}[0..n-1]$ of size $O(1)$. For automatic sequences, the size of the smallest string attractor for ${\bf s}[0..n-1]$ is either $\Theta(1)$ or $\Theta(\log n)$, and it is decidable which case occurs. Finally, we close with some remarks about greedy string attractors.