[Paper Review] A Computer Verified Theory of Compact Sets

Russell O’Connor|arXiv (Cornell University)|Jun 19, 2008

Computability, Logic, AI Algorithms6 references3 citations

TL;DR

This paper presents a computer-verified theory of compact sets in constructive mathematics by defining them as the completion of finite sets under the Hausdorff metric in complete metric spaces. The approach enables efficient, formally verified computation of uniformly continuous functions, resulting in provably correct plots through formal verification in a proof assistant.

ABSTRACT

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact sets as the completion of the space of finite sets under the Hausdorff metric. This definition allowed me to quickly develop a computer verified theory of compact sets. I applied this theory to compute provably correct plots of uniformly continuous functions.

Motivation & Objective

• To develop a formally verified theory of compact sets within constructive mathematics.
• To provide a computationally efficient representation of compact subsets in complete metric spaces, such as the plane.
• To enable the generation of provably correct visualizations of uniformly continuous functions using formal methods.
• To integrate compact set theory into a proof assistant framework for reliable computational mathematics.
• To establish a foundation for verified computer graphics and numerical computation in constructive analysis.

Proposed method

• Defining compact sets as the metric completion of the space of finite sets under the Hausdorff distance.
• Leveraging existing formalizations of complete metric spaces to build the theory of compact sets.
• Using a proof assistant to verify all properties and constructions of compact sets.
• Applying the theory to compute uniformly continuous functions with formal correctness guarantees.
• Representing compact sets via finite approximations that converge under the Hausdorff metric.
• Ensuring all operations on compact sets are computable and formally verified within the system.

Experimental results

Research questions

• RQ1 How can compact sets be formally defined in a constructive and computationally meaningful way?
• RQ2 Can the Hausdorff metric on finite sets be used to constructively define compact sets via completion?
• RQ3 What is the role of formal verification in ensuring correctness of plots of continuous functions?
• RQ4 How can compact set theory be integrated into a proof assistant for reliable computation?
• RQ5 What are the computational and logical properties of compact sets under this formalization?

Key findings

• The theory of compact sets is successfully formalized using the completion of finite sets under the Hausdorff metric.
• The formalization enables the computation of provably correct plots of uniformly continuous functions.
• All constructions and proofs are verified within a proof assistant, ensuring logical correctness.
• The approach provides an efficient and constructive representation of compact subsets in complete metric spaces.
• The method supports algorithmic manipulation and visualization of compact sets with formal guarantees.
• The framework demonstrates the feasibility of integrating verified analysis into computer graphics and numerical computation.