On Expansions of Monadic Second-Order Logic with Dynamical Predicates

Abstract

Expansions of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N} ; < \rangle$ have been a fertile and active area of research ever since the publication of the seminal papers of B\"uchi and Elgot & Rabin on the subject in the 1960s. In the present paper, we establish decidability of the MSO theory of $\langle \mathbb{N} ; <,P \rangle$, where $P$ ranges over a large class of unary ''dynamical'' predicates, i.e., sets of non-negative values assumed by certain integer linear recurrence sequences. One of our key technical tools is the novel concept of (effective) prodisjunctivity, which we expect may also find independent applications further afield.