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This work considers weak deterministic B\"uchi automata reading encodings ofnon-negative reals in a fixed base. A Real Number Automaton is an automatonwhich recognizes all encoding of elements of a set of reals. It is explainedhow to decide in linear time whether a set of reals recognized by a givenminimal 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 bythe automaton. It is also shown that techniques given by Muchnik and by Honkalafor automata over vector of natural numbers also works on vector of realnumbers. It implies that some problems such as deciding whether a set of tuplesof reals $R\subseteq\mathbb R^{d}$ is a subsemigroup of $(\mathbb R^{d},+)$ oris ${FO}[\mathbb R;+,<,1]$-definable is decidable.

B\"uchi automata recognizing sets of reals definable in first-order logic with addition and order