Complexity of Linear Subsequences of $k$-Automatic Sequences

TL;DR

This paper investigates the state complexity of automata recognizing relations and operations on $k$-automatic sequences, linking subword complexity with linear subsequences and addressing recent open questions.

Contribution

It introduces new bounds and relationships between subword and state complexity of $k$-automatic sequences, resolving a recent open problem by Zantema and Bosma.

Findings

Established a relationship between subword complexity and state complexity of linear subsequences.

Resolved a recent open question about linear subsequences in most-significant-digit-first format.

Analyzed the complexity of automata construction using B"uchi arithmetic.

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.