The Church Synthesis Problem with Parameters

TL;DR

This paper explores a parameterized version of the Church Synthesis Problem within Monadic Logic of Order, establishing conditions for its computability and extending classical theorems to parameterized contexts.

Contribution

It introduces a parameterized framework for the Church Synthesis Problem and characterizes its computability based on the decidability of the monadic theory of the structure.

Findings

Church synthesis problem with parameters is computable iff the monadic theory is decidable.

Extension of the B"uchi-Landweber theorem to ultimately periodic parameters.

MLO-definability remains valid in the parameterized setting.

Abstract

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. B\"{u}chi 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 <Nat,<,P> is decidable. We prove that the B\"{u}chi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the B\"{u}chi-Landweber theorem holds for the parameterized version of the Church synthesis problem.