A Study of Continuous Vector Representationsfor Theorem Proving

TL;DR

This paper introduces a reversible vector encoding for mathematical formulas that preserves logical structure and properties, enabling improved machine learning applications in theorem proving.

Contribution

It develops a novel encoding method with decoders that preserve structural and logical properties of formulas, including complex operations like unifiability.

Findings

Encoding preserves tree structure and symbols

Effective for premise selection in theorem proving

Applicable to complex logical operations

Abstract

Applying machine learning to mathematical terms and formulas requires a suitable representation of formulas that is adequate for AI methods. In this paper, we develop an encoding that allows for logical properties to be preserved and is additionally reversible. This means that the tree shape of a formula including all symbols can be reconstructed from the dense vector representation. We do that by training two decoders: one that extracts the top symbol of the tree and one that extracts embedding vectors of subtrees. The syntactic and semantic logical properties that we aim to reserve include both structural formula properties, applicability of natural deduction steps, and even more complex operations like unifiability. We propose datasets that can be used to train these syntactic and semantic properties. We evaluate the viability of the developed encoding across the proposed datasets as well as for the practical theorem proving problem of premise selection in the Mizar corpus.