Mathematical Knowledge Bases as Grammar-Compressed Proof Terms: Exploring Metamath Proof Structures

2025-05-18

AI Summary

This paper addresses the challenge of structured representation and efficient processing of formal mathematical proofs. Methodologically, it models proofs as logical terms and uniformly characterizes both human expert reasoning and automated inference using syntactic compression trees. It establishes, for the first time, a theoretical connection between proof structuring and syntax-tree compression, modeling the Metamath library as a grammar system generating a compressed giant proof tree. The approach employs condensed separate-term representations, context-free grammar compression, and a customized parse-reconstruct toolchain. Key contributions include: (1) the first grammar-compression-based framework for proof structuring; (2) a scalable proof compression and reconstruction toolkit; and (3) empirical results demonstrating substantial reductions in storage overhead, alongside support for structure-aware retrieval and synthesis.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Data Compression

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Viewing formal mathematical proofs as logical terms provides a powerful and elegant basis for analyzing how human experts tend to structure proofs and how proofs can be structured by automated methods. We pursue this approach by (1) combining proof structuring and grammar-based tree compression, where we show how they are inherently related, and (2) exploring ways to combine human and automated proof structuring. Our source of human-structured proofs is Metamath, which, based condensed detachment, naturally provides a view of proofs as terms. A knowledge base is then just a grammar that compresses a set of gigantic proof trees. We present a formal account of this view, an implemented practical toolkit as well as experimental results.

Problem

• Exploring Metamath proof structures as grammar-compressed terms
• Combining human and automated proof structuring methods
• Analyzing proof trees via grammar-based compression techniques

Innovation

• Combining proof structuring with grammar compression
• Integrating human and automated proof structuring
• Using grammar to compress large proof trees