Minimal automaton for multiplying and translating the Thue-Morse set

TL;DR

This paper derives an explicit minimal automaton for the set of integers obtained by multiplying and translating the Thue-Morse set, providing a formula for its state complexity in powers of two and a decision procedure for recognizing such sets.

Contribution

It presents a constructive method to explicitly compute the minimal automaton for sets of the form mT+r in base powers of two, extending to general b-recognizable sets.

Findings

Exact formula for state complexity of mT+r in base 2^p

Explicit minimal automaton construction for these sets

Quadratic time decision procedure for recognizing such sets

Abstract

The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word ${\tt abbabaabbaababba\cdots}$, which is the fixed point starting with ${\tt a}$ of the word morphism ${\tt a\mapsto ab,b\mapsto ba}$. The numbers in $\mathcal{T}$ are commonly called the {\em evil numbers}. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e.\ the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m-1\}$. The proposed method is general for any $b$-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given $2^p$-recognizable set is equal to a set of the form $m\mathcal{T}+r$.