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arXiv · 2605.28408

A natural axiomatization of B\"uchi Arithmetic

Konstantin Kovalyov
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Abstract

We investigate B\"uchi 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 B\"uchi-Bruy\`ere Theorem on the equivalence of definability in B\"uchi Arithmetic and recognizability by finite automata.

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Konstantin Kovalyov. 2026-05-27. A natural axiomatization of B\"uchi Arithmetic. https://arxiv.org/abs/2605.28408

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