1 research outputs found
Minimal automaton for multiplying and translating the Thue-Morse set
The Thue-Morse set is the set of those non-negative integers
whose binary expansions have an even number of . The name of this set comes
from the fact that its characteristic sequence is given by the famous
Thue-Morse word , which is the fixed point
starting with of the word morphism .
The numbers in are commonly called the {\em evil numbers}. We
obtain an exact formula for the state complexity of the set
(i.e.\ the number of states of its minimal automaton) with respect to any base
which is a power of . Our proof is constructive and we are able to
explicitly provide the minimal automaton of the language of all
-expansions of the set of integers for any positive
integers and and any remainder . The proposed
method is general for any -recognizable set of integers. As an application,
we obtain a decision procedure running in quadratic time for the problem of
deciding whether a given -recognizable set is equal to a set of the form
.Comment: arXiv admin note: substantial text overlap with arXiv:1909.07676,
arXiv:1903.0611