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.

翻译：我们证明了，给定一个自动序列 $\bf s$ 和一个常数 $c$，判定 $\bf s$ 的所有前缀是否都存在一个大小 $\leq c$ 的字符串吸引子是可判定的。利用基于此结果的判定过程，我们证明了长度 $\geq 2$ 的倍周期序列的所有前缀都存在一个大小为 $2$ 的字符串吸引子。我们还证明了其他序列（包括 Thue-Morse 序列和 Tribonacci 序列）的类似结果。我们还为不同类型序列的字符串吸引子大小提供了通用的上界和下界。例如，如果 $\bf s$ 具有有限出现常数，则 ${\bf s}[0..n-1]$ 存在一个大小为 $O(\log n)$ 的字符串吸引子。如果进一步 $\bf s$ 是线性递归的，则 ${\bf s}[0..n-1]$ 存在一个大小为 $O(1)$ 的字符串吸引子。对于自动序列，${\bf s}[0..n-1]$ 的最小字符串吸引子的大小要么是 $\Theta(1)$，要么是 $\Theta(\log n)$，并且哪种情况发生是可判定的。最后，我们以关于贪婪字符串吸引子的一些评述作为结束。