Axiomatizations of Presburger Arithmetic With Predicates For Powers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914343945240576 |
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| author | Hieronymi, Philipp Reitmeir, Michael Wang, Xiaoduo |
| author_facet | Hieronymi, Philipp Reitmeir, Michael Wang, Xiaoduo |
| contents | 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 $(\mathbb{Z},+,<,(\ell^{\mathbb{N}})_{\ell\in L})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19602 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Axiomatizations of Presburger Arithmetic With Predicates For Powers Hieronymi, Philipp Reitmeir, Michael Wang, Xiaoduo Logic 03B25, 03C10 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 $(\mathbb{Z},+,<,(\ell^{\mathbb{N}})_{\ell\in L})$. |
| title | Axiomatizations of Presburger Arithmetic With Predicates For Powers |
| topic | Logic 03B25, 03C10 |
| url | https://arxiv.org/abs/2602.19602 |