On monadic theories of monadic predicates

Abstract

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 of the natural numbers and the monadic theory of the binary tree, i.e., of the two-successor structure . We consider expansions of these structures by a monadic predicate P. It is known that the monadic theory of is decidable iff the weak monadic theory is, and that for recursive P this theory is in, i.e. of low degree in the arithmetical hierarchy. We show that there are structures for which the first result fails, and that there is a recursive P such that the monadic theory of is -hard. © 2010 Springer-Verlag Berlin Heidelberg.