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arXiv · 2512.10017

Complexity of Linear Subsequences of $k$-Automatic Sequences

Delaram Moradi·Narad Rampersad·Jeffrey Shallit
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Abstract

We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences $(h(i))_{i \geq 0}$ and discuss their state complexity. We find a relationship between subword complexity of the interior sequence $(h'(i))_{i \geq 0}$ and state complexity of the linear subsequence $(h(ni+c))_{i \geq 0}$. We resolve a recent question of Zantema and Bosma about linear subsequences of $k$-automatic sequences with input in most-significant-digit-first format. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of B\"uchi arithmetic to actually construct some of the studied automata recognizing relations or carrying out operations on automatic sequences.

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Delaram Moradi, Narad Rampersad, Jeffrey Shallit. 2025-12-10. Complexity of Linear Subsequences of $k$-Automatic Sequences. https://arxiv.org/abs/2512.10017

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