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  • We show that it is decidable whether or not a relation on the reals definable in the structure [Formula: see text] can be defined in the structure [Formula: see text]. This result is achieved by obtaining a topological characterization of [Formula: see text]-definable relations in the family of [Formula: see text]-definable relations and then by following Muchnik’s approach of showing that this characterization can be expressed in the logic of [Formula: see text].
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