TY - JOUR

T1 - The church synthesis problem with parameters

AU - Rabinovich, Alexander

N1 - Publisher Copyright:
© A. Rabinovich.

PY - 2007/11/14

Y1 - 2007/11/14

N2 - 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.

AB - 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.

KW - Decidability

KW - Monadic logic

KW - Synthesis problem

UR - http://www.scopus.com/inward/record.url?scp=70350362155&partnerID=8YFLogxK

U2 - 10.2168/LMCS-3(4:9)2007

DO - 10.2168/LMCS-3(4:9)2007

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AN - SCOPUS:70350362155

SN - 1860-5974

VL - 3

JO - Logical Methods in Computer Science

JF - Logical Methods in Computer Science

IS - 4

M1 - 9

ER -