Rigorous Function Calculi in Ariadne

2023-06-30

arXiv.org

AI Summary

This paper addresses the problem of rigorous computability for real-valued functions over Euclidean domains in calculus. Methodologically, it introduces a unified four-layer information model—exact, effective, verified, and approximate—grounded on polynomial primitives and integrating interval analysis, affine arithmetic, and symbolic-numeric hybrid computation. The framework is implemented in the open-source tool Ariadne, featuring a C++ core with Python bindings. Its key contribution is the first unified realization of both mathematical rigor and practical computability for function calculus within a single framework. Experimental results demonstrate end-to-end verified error bounds for solving nonlinear algebraic equations and initial-value problems of ordinary differential equations. Moreover, Ariadne successfully generates mathematically guaranteed state-evolution envelopes for multiple hybrid systems verification tasks, significantly enhancing the reliability of dynamic system analysis.

Abstract

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.

Problem

• Rigorous calculus for Euclidean functions
• Solution of algebraic and differential equations
• Analysis of hybrid systems

Innovation

• Rigorous calculus for Euclidean functions
• Polynomial function models for computations
• Solution of algebraic and differential equations