String Attractors for Automatic Sequences

12/12/2020 ∙ by Luke Schaeffer, et al. ∙ 0

We show that it is decidable, given an automatic sequence s and a constant c, whether all prefixes of s have a string attractor of size ≤ c. Using a decision procedure based on this result, we show that all prefixes of the period-doubling sequence of length ≥ 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 s has a finite appearance constant, then there is a string attractor for s[0..n-1] of size O(log n). If further s is linearly recurrent, then there is a string attractor for s[0..n-1] of size O(1). For automatic sequences, the size of the smallest string attractor for s[0..n-1] is either Θ(1) or Θ(log n), and it is decidable which case occurs. Finally, we close with some remarks about greedy string attractors.

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