On Monadic Theories of Monadic Predicates

Pioneers of logic, among them J.R. Büchi, M.O. Rabin, S. Shelah, and Y. Gurevich, have shown that monadic second-order logic offers a rich landscape of interesting decidable theories. Prominent examples are the monadic theory of the successor structure S1 = (N,+1) of the natural numbers and the monadic theory of the binary tree, i.e., of the two-successor structure S2 = ({0, 1} ∗, ·0, ·1). We consider expansions of these structures by a monadic predicate P . It is known that the monadic theory of (S1, P ) is decidable iff the weak monadic theory is, and that for recursive P this theory is in ∆03, i.e. of low degree in the arithmetical hierarchy. We show that there are structures (S2, P ) for which the first result fails, and that there is a recursive P such that the monadic theory of (S2, P ) is Π 1 1 -hard.