Approximate Axiomatization for Differentially-Defined Functions

TL;DR

This paper develops a complete approximate axiomatization for real-closed fields extended with all differentially-defined functions, enabling provable numerical approximations of true sentences involving special functions.

Contribution

It introduces a finite extension of real-closed field axioms that can approximate truths involving differential functions, enhancing decidability and proof capabilities.

Findings

Every true sentence is provable up to a numerical approximation.

Approximations converge under mild conditions.

Improves decidability results for formulas with special functions.

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.