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 $(\mathbb{Z},+,<,(\ell^{\mathbb{N}})_{\ell\in L})$.

Figures (1)

• Figure 1: Illustration of the last part of the proof of Lemma \ref{['thm:algorithm']}

Theorems & Definitions (68)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 3.1: Mann property
• Lemma 3.2
• proof
• Definition 3.3: Mann axioms
• Definition 3.4: Karimov
• Lemma 3.5: Karimov