Church Synthesis Problem with Parameters

The following problem is known as the Church Synthesis problem: Input: an ${\mathit{MLO}}$ formula ψ(X,Y). Task: Check whether there is an operator Y=F(X) such that$$Nat \models \forall X \psi(X,F(X))$$ and if so, construct this operator. Buchi and Landweber proved that the Church synthesis problem is decidable; moreover, they proved that if there is an operator F which satisfies ([1]), then ([1]) can be satisfied by the operator defined by a finite state automaton. We investigate a parameterized version of the Church synthesis problem. In this version ψ might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of $\langle{\mathit{Nat},<,P}\rangle$ is decidable. We also show that the Buchi-Landweber theorem can be extended only to ultimately periodic parameters.