Büchi automata recognizing sets of reals definable in first-order logic with addition and order

This work considers weak deterministic B\"uchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time whether a set of reals recognized by a given minimal weak deterministic RNA is ${FO}[\mathbb R;+,<,1]$-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an existential (respectively, existential-universal) ${FO}[\mathbb R;+,<,1]$-formula which defines the set of reals recognized by the automaton. It is also shown that techniques given by Muchnik and by Honkala for automata over vector of natural numbers also works on vector of real numbers. It implies that some problems such as deciding whether a set of tuples of reals $R\subseteq\mathbb R^{d}$ is a subsemigroup of $(\mathbb R^{d},+)$ or is ${FO}[\mathbb R;+,<,1]$-definable is decidable.