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arXiv · 2506.08233

Approximate Axiomatization for Differentially-Defined Functions

André Platzer·Long Qian
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Abstract

This article establishes a complete approximate axiomatization for the real-closed field $\mathbb{R}$ expanded with all differentially-defined functions, including special functions such as $\sin(x), \cos(x), e^x, \dots$. Every true sentence is provable up to some numerical approximation, and the truth of such approximations converge under mild conditions. Such an axiomatization is a fragment of the axiomatization for differential dynamic logic, and is therefore a finite extension of the axiomatization of real-closed fields. Furthermore, the numerical approximations approximate formulas containing special function symbols by $\text{FOL}_{\mathbb{R}}$ formulas, improving upon earlier decidability results only concerning closed sentences.

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André Platzer, Long Qian. 2025-06-09. Approximate Axiomatization for Differentially-Defined Functions. https://arxiv.org/abs/2506.08233

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