  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.
