Verifying Numerical Methods with Isabelle/HOL

2025-11-25

AI Summary

Machine learning and cyber-physical systems increasingly demand numerically sound, formally verified computational methods. Method: This paper introduces the first end-to-end verifiable numerical methods framework built on Isabelle/HOL and ITrees. It enables users to declaratively specify numerical programs—including variants and invariants—using a high-level, user-friendly specification language; formal verification is automated via ITrees-based operational semantics and the HOL-Analysis library. Contribution/Results: We extend the formalization of Taylor’s theorem with higher-order derivatives and the Peano remainder form, strengthening the underlying mathematical foundation. Leveraging code generation, we fully verify the bisection and fixed-point iteration methods and produce certified, executable code. Our framework significantly improves both the efficiency and practical applicability of formal verification for numerical algorithms.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• 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

Modern machine learning pipelines are built on numerical algorithms. Reliable numerical methods are thus a prerequisite for trustworthy machine learning and cyber-physical systems. Therefore, we contribute a framework for verified numerical methods in Isabelle/HOL based on ITrees. Our user-friendly specification language enables the direct declaration of numerical programs that can be annotated with variants and invariants for reasoning about correctness specifications. The generated verification conditions can be discharged via automated proof methods and lemmas from the HOL-Analysis library. The ITrees foundation interacts with Isabelle's code generator to export source code. This provides an end-to-end path from formal specifications with machine-checked guarantees to executable sources. We illustrate the process of modelling numerical methods and demonstrate the effectiveness of the verification by focusing on two well-known methods, the bisection method and the fixed-point iteration method. We also contribute crucial extensions to the libraries of formalised mathematics required for this objective: higher-order derivatives and Taylor's theorem in Peano form. Finally, we qualitatively evaluate the use of the framework for verifying numerical methods.

Problem

• Verifying numerical algorithms for machine learning reliability
• Developing formal specification framework for numerical programs
• Providing machine-checked correctness guarantees for numerical methods

Innovation

• Framework for verified numerical methods using Isabelle/HOL
• User-friendly specification language with invariants annotation
• Automated proof methods with HOL-Analysis library integration