Rich Sequences and Decidability of Arithmetic Theories

Abstract: We develop a new framework for proving the undecidability of first-order theories of structures of the form $\langle \mathbb{N}; +, P \rangle$, $\langle \mathbb{N}; <, f \rangle$, and $\langle \mathbb{N}; +, f\rangle$, where $P \subseteq \mathbb{N}$ and $f \colon \mathbb{N} \to \mathbb{N}$. It is based on the recent proof of Hieronymi and Schulz that the first-order theory of $$\langle \mathbb{N}; +, {2ⁿ \colon n \in \mathbb{N}}, {3ⁿ \colon n \in \mathbb{N}}\rangle$$ is undecidable, and capable of transforming various randomness results about integer sequences into undecidability proofs. We apply our method to a large class of integer linear recurrence sequences, as well as various special functions, in particular showing that the first-order theories of $$\langle \mathbb{N}; +, {u_n \colon n \in \mathbb{N}} \cap \mathbb{N}\rangle$$, $\langle\mathbb{N}; <, n \mapsto \max{0,u_n}\rangle$, and $\langle \mathbb{N}; <, φ\rangle$ are undecidable, where $(u_n)_{n\in\mathbb{N}}$ is any integer LRS with exactly two non-repeated dominant roots satisfying a non-degeneracy assumption, and $φ$ is Euler's totient function.