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.

翻译：对于哪些一元谓词 $P_1, \ldots, P_m$，结构 $\langle \mathbb{N}; <, P_1, \ldots, P_m \rangle$ 的 MSO（单子二阶逻辑）理论是可判定的？我们综述了当前研究进展，由此引致对几乎周期词、形态词和完全词组合性质的探究。在此过程中，我们证明：若每个 $P_i$ 可由特定类型的完全动力系统生成，则相应的 MSO 理论是可判定的。