Minimal automaton for multiplying and translating 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 a → ab, b → ba. The numbers in T are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set mT + 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 mT + r for any positive integers p and m and any remainder r ∈ {0, . . . , 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 mT + r.