🤖 AI Summary
This paper investigates the decidability of the monadic second-order (MSO) theory of the natural number structure ⟨ℕ; <, P₁, …, Pₘ⟩, where each unary predicate Pᵢ is generated by a toric dynamical system. The study establishes, for the first time, a rigorous connection between such toric-generation and MSO decidability. Employing an integrated methodology drawing from model theory (MSO logic), combinatorics on words, dynamical systems, and algebraic ergodic theory, the authors prove that if each Pᵢ arises from a toric system satisfying a specific uniform distribution condition, then the associated MSO theory is decidable. This result substantially extends the known classes of decidable structures—from almost periodic and morphic sequences—to a broader family of aperiodic, higher-dimensional dynamically generated sequences. It provides novel sufficient conditions and a unifying theoretical framework for the logical decidability of infinite discrete structures.
📝 Abstract
For which unary predicates $P_1, ldots, P_m$ is the MSO theory of the structure $langle mathbb{N};<, P_1, ldots, P_m
angle$ 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.
