On the Expressiveness of B\"uchi Arithmetic

TL;DR

This paper investigates the expressive power of B"uchi arithmetic, demonstrating that the existential fragment is less expressive than the full logic but still captures all polynomial growth regular languages.

Contribution

It establishes the expressive limits of the existential fragment and shows its completeness at the -level, expanding understanding of Bbchi arithmetic.

Findings

Existential fragment is less expressive than full Bbchi arithmetic.

The -fragment of Bbchi arithmetic is expressively complete.

Regular languages with polynomial growth are definable in the existential fragment.

Abstract

We show that the existential fragment of B\"uchi arithmetic is strictly less expressive than full B\"uchi arithmetic of any base, and moreover establish that its $\Sigma_2$-fragment is already expressively complete. Furthermore, we show that regular languages of polynomial growth are definable in the existential fragment of B\"uchi arithmetic.