State Complexity of the Multiples of the Thue-Morse Set
Abstract
The Thue-Morse set 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 abbabaabbaababba..., which is the fixed point starting with a of the word morphism sending a to ab and b to ba. The numbers in T are sometimes called the evil numbers. We obtain an exact formula for the state complexity (i.e. the number of states of its minimal automaton) of the multiplication by a constant of the Thue-Morse set with respect to any integer 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 mT for any positive integers m and p. The used 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 some multiple of the Thue-Morse set.
Keywords
Cite
@article{arxiv.1909.07676,
title = {State Complexity of the Multiples of the Thue-Morse Set},
author = {Émilie Charlier and Célia Cisternino and Adeline Massuir},
journal= {arXiv preprint arXiv:1909.07676},
year = {2019}
}
Comments
In Proceedings GandALF 2019, arXiv:1909.05979. Short paper. A full version of this paper appears as arXiv:1903.06114