TY - JOUR
T1 - On decidability of monadic logic of order over the naturals extended by monadic predicates
AU - Rabinovich, Alexander
PY - 2007/6
Y1 - 2007/6
N2 - A fundamental result of Büchi states that the set of monadic second-order formulas true in the structure (Nat, <) is decidable. A natural question is: what monadic predicates (sets) can be added to (Nat, <) while preserving decidability? Elgot and Rabin found many interesting predicates P for which the monadic theory of (Nat,<,P) is decidable. The Elgot and Rabin automata theoretical method has been generalized and sharpened over the years and their results were extended to a variety of unary predicates. We give a sufficient and necessary model-theoretical condition for the decidability of the monadic theory of (Nat, <, P1,..., Pn). We reformulate this condition in an algebraic framework and show that a sufficient condition proposed previously by O. Carton and W. Thomas is actually necessary. A crucial argument in the proof is that monadic secondorder logic has the selection and the uniformization properties over the extensions of (Nat, <) by monadic predicates. We provide a self-contained proof of this result.
AB - A fundamental result of Büchi states that the set of monadic second-order formulas true in the structure (Nat, <) is decidable. A natural question is: what monadic predicates (sets) can be added to (Nat, <) while preserving decidability? Elgot and Rabin found many interesting predicates P for which the monadic theory of (Nat,<,P) is decidable. The Elgot and Rabin automata theoretical method has been generalized and sharpened over the years and their results were extended to a variety of unary predicates. We give a sufficient and necessary model-theoretical condition for the decidability of the monadic theory of (Nat, <, P1,..., Pn). We reformulate this condition in an algebraic framework and show that a sufficient condition proposed previously by O. Carton and W. Thomas is actually necessary. A crucial argument in the proof is that monadic secondorder logic has the selection and the uniformization properties over the extensions of (Nat, <) by monadic predicates. We provide a self-contained proof of this result.
UR - http://www.scopus.com/inward/record.url?scp=84855204422&partnerID=8YFLogxK
U2 - 10.1016/j.ic.2006.12.004
DO - 10.1016/j.ic.2006.12.004
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AN - SCOPUS:84855204422
SN - 0890-5401
VL - 205
SP - 870
EP - 889
JO - Information and Computation
JF - Information and Computation
IS - 6
ER -