Joris Nieuwveld

Valérie Berthé, Toghrul Karimov, Joris Nieuwveld et al.

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.

Piotr Bacik, Joris Nieuwveld, Joël Ouaknine et al.

We consider expansions of Presburger arithmetic with families of monadic polynomial predicates. (Examples of such predicates are the set of perfect squares, or the set of integers of the form $2n^3-5n+3$, etc.) Although the full attendant first-order theories are well known to be undecidable, very little is known when one restricts the number of variables. In the case of single-variable theories, we obtain positive results for the following two families of predicates: (i) for perfect fixed powers, decidability ofthe corresponding theory follows from the solvability of hyperellipticDiophantine equations; and (ii) for polynomials of degree at most three, we establish decidability by relying on the low genus of the resulting algebraic curves. Finally, we discuss limitations and hardness results (via encodings of longstanding open Diophantine problems) as soon as any of the above restrictions are lifted.