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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.28408 |
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| _version_ | 1866911724649578496 |
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| author | Kovalyov, Konstantin |
| author_facet | Kovalyov, Konstantin |
| contents | We investigate Büchi Arithmetic $\mathsf{BA}_k$ -- the elementary theory of the natural numbers equipped with addition and the function mapping a number $x$ to the greatest power of $k$ dividing $x$. $\mathsf{BA}_k$ is known to be decidable and to enjoy a few important properties, in particular, a first-order structure is automatic iff it is interpretable in $\mathsf{BA}_k$. We propose a natural axiomatization of this theory based on a comprehension schema restricted to bounded formulas, interpreting natural numbers as finite (multi)sets of powers of $k$ via their base-$k$ expansions. The completeness proof for this axiomatization proceeds through a formalization of the Büchi-Bruyère Theorem on the equivalence of definability in Büchi Arithmetic and recognizability by finite automata. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28408 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A natural axiomatization of Büchi Arithmetic Kovalyov, Konstantin Logic We investigate Büchi Arithmetic $\mathsf{BA}_k$ -- the elementary theory of the natural numbers equipped with addition and the function mapping a number $x$ to the greatest power of $k$ dividing $x$. $\mathsf{BA}_k$ is known to be decidable and to enjoy a few important properties, in particular, a first-order structure is automatic iff it is interpretable in $\mathsf{BA}_k$. We propose a natural axiomatization of this theory based on a comprehension schema restricted to bounded formulas, interpreting natural numbers as finite (multi)sets of powers of $k$ via their base-$k$ expansions. The completeness proof for this axiomatization proceeds through a formalization of the Büchi-Bruyère Theorem on the equivalence of definability in Büchi Arithmetic and recognizability by finite automata. |
| title | A natural axiomatization of Büchi Arithmetic |
| topic | Logic |
| url | https://arxiv.org/abs/2605.28408 |