The church synthesis problem with parameters

For a two-variable formula ψ(X, Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of an operator Y = F(X) such that ψ(X, F(X)) is universally valid over Nat. Büchi and Landweber proved that the Church synthesis problem is decidable; moreover, they showed that if there is an operator F that solves the Church Synthesis Problem, then it can also be solved by an operator defined by a finite state automaton or equivalently by an MLO formula. 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 (Formula Found) is decidable. We prove that the Büchi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the Büchi-Landweber theorem holds for the parameterized version of the Church synthesis problem.