Approximate Axiomatization for Differentially-Defined 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.