Alchemy: Amplifying Theorem-Proving Capability through Symbolic Mutation

TL;DR

This paper introduces Alchemy, a framework that synthetically enlarges formal theorem datasets via symbolic mutation, significantly boosting neural theorem proving performance on benchmarks.

Contribution

Alchemy is a novel data synthesis method that increases theorem corpus size by an order of magnitude, enhancing neural theorem prover training and performance.

Findings

Theorem corpus increased from 110k to 6M using Alchemy.

Achieved 4.70% performance improvement on Leandojo benchmark.

Gained 2.47% performance boost on miniF2F out-of-distribution benchmark.

Abstract

Formal proofs are challenging to write even for experienced experts. Recent progress in Neural Theorem Proving (NTP) shows promise in expediting this process. However, the formal corpora available on the Internet are limited compared to the general text, posing a significant data scarcity challenge for NTP. To address this issue, this work proposes Alchemy, a general framework for data synthesis that constructs formal theorems through symbolic mutation. Specifically, for each candidate theorem in Mathlib, we identify all invocable theorems that can be used to rewrite or apply to it. Subsequently, we mutate the candidate theorem by replacing the corresponding term in the statement with its equivalent form or antecedent. As a result, our method increases the number of theorems in Mathlib by an order of magnitude, from 110k to 6M. Furthermore, we perform continual pretraining and supervised finetuning on this augmented corpus for large language models. Experimental results demonstrate the effectiveness of our approach, achieving a 4.70% absolute performance improvement on Leandojo benchmark. Additionally, our approach achieves a 2.47% absolute performance gain on the out-of-distribution miniF2F benchmark based on the synthetic data.To provide further insights, we conduct a comprehensive analysis of synthetic data composition and the training paradigm, offering valuable guidance for developing a strong theorem prover.