Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions

Hera Brown; Jakub Konieczny

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arXiv · 2508.19206

Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions

Hera Brown·Jakub Konieczny

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Abstract

We study the extension of Presburger arithmetic by the class of sub-polynomial Hardy field functions, and show the majority of these extensions to be undecidable. More precisely, we show that the theory $\mathrm{Th}(\mathbb{Z}; <, +, \lfloor f \rceil)$, where $f$ is a Hardy field function and $\lfloor \cdot \rceil$ the nearest integer operator, is undecidable when $f$ grows polynomially faster than $x$. Further, we show that when $f$ grows sub-linearly quickly, but still as fast as some polynomial, the theory $\mathrm{Th}(\mathbb{Z}; <, +, \lfloor f \rceil)$ is undecidable.

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Hera Brown, Jakub Konieczny. 2025-08-26. Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions. https://arxiv.org/abs/2508.19206

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