A natural axiomatization of Büchi Arithmetic

We investigate Büchi Arithmetic ${\mathsf{B}\mathsf{A}}_k$ – the elementary theory of the natural numbers equipped with addition and the function mapping a number $x$ to the greatest power of $k$ dividing $x$. ${\mathsf{B}\mathsf{A}}_k$ 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 ${\mathsf{B}\mathsf{A}}_k$. 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.