Decidability of Extensions of Presburger Arithmetic by Hardy Field Functions

2025-08-26

AI Summary

This paper investigates the decidability of extensions of Presburger arithmetic (Th(ℤ; <, +)) by a Hardy field function f, i.e., Th(ℤ; <, +, ⌊f⌉), focusing on subpolynomial-growth f. Using methods from model theory and computability theory—leveraging the nearest-integer operator ⌊·⌉ and the ordered additive structure—we systematically characterize how the growth rate of f precisely affects decidability. Our main contribution is the first exact undecidability threshold: the expanded theory is undecidable whenever f grows strictly faster than every polynomial, or when it lies strictly between some polynomial and a linear function—i.e., satisfies x^ε ≲ f(x) ≲ x^{1−δ} for some ε, δ > 0. This reveals the critical mechanism underlying the collapse of decidability within the subpolynomial regime and establishes a key logical stratification criterion for expansions of arithmetic.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 $mathrm{Th}(mathbb{Z}; <, +, lfloor f ceil)$, where $f$ is a Hardy field function and $lfloor cdot ceil$ 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 $mathrm{Th}(mathbb{Z}; <, +, lfloor f ceil)$ is undecidable.

Problem

• Studying decidability of Presburger arithmetic with Hardy field functions
• Showing undecidability for functions growing polynomially faster than x
• Proving undecidability for sub-linear functions with polynomial growth

Innovation

• Extending Presburger arithmetic with Hardy functions
• Using nearest integer operator for function discretization
• Analyzing decidability for sub-polynomial growth rates