On decidability of monadic logic of order over the naturals extended by monadic predicates

摘要

A fundamental result of Buechi 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, <, P_1,..., P_n). 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 second-order 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.