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

TL;DR

This paper presents a novel approach to analyzing formal mathematical proofs by representing them as grammar-compressed proof terms, combining human and automated structuring methods, with practical implementation and experimental validation.

Contribution

It introduces a formal framework linking proof structuring and grammar-based compression, utilizing Metamath proofs as terms, and provides a practical toolkit for proof analysis.

Findings

Proofs can be effectively represented as grammar-compressed terms.

The approach reveals structural similarities between human and automated proofs.

Experimental results demonstrate the method's efficiency and potential for proof analysis.

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 on 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.