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