THE CHURCH SYNTHESIS PROBLEM WITH PARAMETERS 1

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THE CHURCH SYNTHESIS PROBLEM WITH PARAMETERS 1

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Nov 14, 2007 - As implementations, Church considers functions from the set {0,1}Ï of Ï-strings over. {0,1} to ... In this paper we consider natural generalizations of the Church ...... mentioned in a later book by BÃ¼chi and Siefkes [BS73], which ...

Logical Methods in Computer Science Vol. 3 (4:9) 2007, pp. 1–24 www.lmcs-online.org

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THE CHURCH SYNTHESIS PROBLEM WITH PARAMETERS ALEXANDER RABINOVICH Sackler Faculty of Exact Sciences, Tel Aviv University, Israel 69978. e-mail address: [email protected] 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¨ uchi 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 hNat ,