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Church Synthesis Problem with Parameters

  • Alexander Rabinovich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)

Abstract

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.

Büchi 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 Büchi-Landweber theorem can be extended only to ultimately periodic parameters.

Keywords

Synthesis Problem Winning Strategy Unary Predicate Game Graph Deterministic Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alexander Rabinovich
    • 1
  1. 1.Dept. of CSTel Aviv Univ. 

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