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Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions

Authors Hera Brown , Jakub Konieczny

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LIPIcs.STACS.2026.21.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
Keywords
  • Arithmetic theories
  • Hardy fields
  • Undecidability

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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 Th(ℤ; < , +, ⌊f⌉), where f is a Hardy field function and ⌊⋅⌉ 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 Th(ℤ; < , +, ⌊f⌉) is undecidable.

Cite As Get BibTex

Hera Brown and Jakub Konieczny. Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 21:1-21:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.21

Author Details

Hera Brown
  • Department of Computer Science, University of Oxford, UK
Jakub Konieczny
  • Department of Mathematics, Kyiv School of Economics, Ukraine
  • Department of Computer Science, University of Oxford, UK

Funding

  • Konieczny, Jakub: supported by UKRI Fellowship EP/X033813/1.

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