State Complexity of the Multiples of the Thue-Morse Set

Émilie Charlier
Célia Cisternino
Adeline Massuir

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.

In Jérôme Leroux and Jean-Francois Raskin: Proceedings Tenth International Symposium on Games, Automata, Logics, and Formal Verification (GandALF 2019), Bordeaux, France, 2-3rd September 2019, Electronic Proceedings in Theoretical Computer Science 305, pp. 34–49.
Short paper
Published: 18th September 2019.

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