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 -