Model completeness and relative decidability

We study the implications of model completeness a theory for effectiveness presentations models that theory. It is immediate computable $$\mathcal {A}$$ computably enumerable, complete theory, entire elementary diagram $$E(\mathcal {A})$$ must be decidable. prove indeed c.e. T if and only there uniform procedure succeeds in deciding from atomic $$\varDelta (\mathcal all countable T. Moreover, every presentation single isomorphism type has this property relative decidability, then with uniformly an expansion $$(\mathcal {A},\mathbf {a})$$ by finitely many new constants.