State Complexity of the Multiples of the Thue-Morse Set

TL;DR

This paper derives an exact formula for the state complexity of multiplying the Thue-Morse set by a constant in base 2^p, providing explicit automata constructions and a decision procedure for recognizing multiples.

Contribution

It introduces a constructive method to compute minimal automata for multiples of the Thue-Morse set in powers of two, extending to any b-recognizable set.

Findings

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

Explicit minimal automaton construction for 2^p-expansions of mT

Quadratic time decision procedure for recognizing multiples of Thue-Morse

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.