On the Decidability of Monadic Theories of Arithmetic Predicates
Valérie Berthé, Toghrul Karimov, Joris Nieuwveld, Joël Ouaknine, Mihir Vahanwala, James Worrell
TL;DR
This work studies the decidability of monadic second-order theories for expansions ⟨N;<,P1,...,Pd⟩ with arithmetic predicates drawn from linear recurrences and exponential sequences. It develops a unified automata-dynamics framework that connects MSO decidability to automata on characteristic words, reductions to order words, and torus-based dynamical systems, augmented by number-theoretic tools such as Baker’s theorem and Schanuel’s conjecture. The authors prove unconditional and conditional decidability results for several predicate families (e.g., 2^N with Fib; 2^N with 3^N and 6^N; 4^N with N^2) and relate more complex expansions to base-b representations of algebraic numbers, yielding equivalences with base expansions of numbers like sqrt(2)-1. They introduce notions of effectively procyclic and effectively sparse predicates, and show how transductions and cutting-sequence theory enable reductions from characteristic words to order words, enabling broad decidability results under plausible conjectures and conditions.
Abstract
We investigate the decidability of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N};<,P_1, \ldots,P_d \rangle$, for various unary predicates $P_1,\ldots,P_d \subseteq \mathbb{N}$. We focus in particular on 'arithmetic' predicates arising in the study of linear recurrence sequences, such as fixed-base powers $k^{\mathbf{N}} = \{k^n : n \in \mathbb{N}\}$, $k$-th powers $\mathbf{N}^k = \{n^k : n \in \mathbb{N}\}$, and the set of terms of the Fibonacci sequence $\mathsf{Fib} = \{0,1,2,3,5,8,13,\ldots\}$ (and similarly for other linear recurrence sequences having a single, non-repeated, dominant characteristic root). We obtain several new unconditional and conditional decidability results, a select sample of which are the following: $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathsf{Fib} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 6^{\mathbf{N}} \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, 3^{\mathbf{N}}, 5^{\mathbf{N}} \rangle$ is decidable assuming Schanuel's conjecture; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 4^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is decidable; $\bullet$ The MSO theory of $\langle \mathbb{N};<, 2^{\mathbf{N}}, \mathbf{N}^2 \rangle$ is Turing-equivalent to the MSO theory of $\langle \mathbb{N};<,S \rangle$, where $S$ is the predicate corresponding to the binary expansion of $\sqrt{2}$. (As the binary expansion of $\sqrt{2}$ is widely believed to be normal, the corresponding MSO theory is in turn expected to be decidable.) These results are obtained by exploiting and combining techniques from dynamical systems, number theory, and automata theory.