Axiomatizations of Presburger Arithmetic With Predicates For Powers

Abstract

We give a complete first-order axiomatization of the structure $(\mathbb{Z},+,({\ell }^{\mathbb{N}}{)}_{\ell \in L})$, where $L\subseteq {\mathbb{Z}}_{\ge 2}$ is a set of pairwise multiplicatively independent integers and ${\ell }^{\mathbb{N}}=\{{\ell }^n:n\in \mathbb{N}\}$. Using recent work of Karimov et al., we obtain that this axiomatization is computable for $|L|=2$, which proves that $(\mathbb{Z},+,k^{\mathbb{N}},{\ell }^{\mathbb{N}})$ is decidable for $k,\ell \in {\mathbb{Z}}_{\ge 2}$. Furthermore, we give an axiomatization of the universal theory of $(\mathbb{Z},+,<,({\ell }^{\mathbb{N}}{)}_{\ell \in L})$.