The Monadic Theory of Toric Words

For which unary predicates $P_1, \ldots, P_m$ is the MSO theory of the structure $\langle \mathbb{N}; <, P_1, \ldots, P_m \rangle$ decidable? We survey the state of the art, leading us to investigate combinatorial properties of almost-periodic, morphic, and toric words. In doing so, we show that if each $P_i$ can be generated by a toric dynamical system of a certain kind, then the attendant MSO theory is decidable.