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 \({\cal S}_1 = (\mathbb{N}, +1)\) of the natural numbers and the monadic theory of the binary tree, i.e., of the two-successor structure \({\cal S}_2 = (\{0,1\}^*, \cdot 0, \cdot 1)\). We consider expansions of these structures by a monadic predicate P. It is known that the monadic theory of \(({\cal S}_1, P)\) is decidable iff the weak monadic theory is, and that for recursive P this theory is in \(\Delta^0_3\), i.e. of low degree in the arithmetical hierarchy. We show that there are structures \(({\cal S}_2, P)\) for which the first result fails, and that there is a recursive P such that the monadic theory of \(({\cal S}_2, P)\) is \(\Pi^1_1\)-hard.