Approximate Axiomatization for Differentially-Defined Functions

2025-06-09

AI Summary

This paper addresses first-order formulas over the real closed field ℝ containing transcendental functions (e.g., sin, cos, eˣ) and introduces the first complete numerical approximation axiomatization for them. Methodologically, it integrates axioms of real closed fields, a fragment of differential dynamic logic, and an ε-approximation–based model-theoretic semantics, enabling approximate provability of non-closed formulas within first-order logic. The main contributions are threefold: (1) It breaks the traditional limitation that decidability results apply only to closed sentences; (2) It establishes that every true formula is ε-provable for any precision ε > 0, with its approximate truth value converging to the exact truth value as ε → 0; (3) It provides a theoretically sound foundation—balancing expressiveness and computability—for formal verification and automated reasoning of hybrid systems involving special functions.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Security and Privacy: Data transparency and provenance

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 $ ext{FOL}_{mathbb{R}}$ formulas, improving upon earlier decidability results only concerning closed sentences.

Problem

• Axiomatizing real-closed fields with differentially-defined functions
• Proving true sentences via numerical approximation convergence
• Approximating special function formulas using FOL_R formulas

Innovation

• Complete approximate axiomatization for real-closed fields
• Numerical approximations for special function symbols
• Finite extension of differential dynamic logic