Proof Artifact Co-training for Theorem Proving with Language Models (2102.06203v2)

Abstract: Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer LLMs to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({\bf P}roof {\bf A}rtifact {\bf C}o-{\bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer LLM and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32\% to 48\%.

• The paper demonstrates that integrating proof artifacts with language models enhances theorem proving accuracy compared to standard training methods.
• The methodology leverages the LeanStep datasets to create a structured training environment that improves contextual reasoning in formal proofs.
• The findings imply that coupling language models with proof artifacts can inspire further cross-domain learning in advanced automated reasoning systems.

Proof Artifact Co-training for Theorem Proving with LLMs

The paper "Proof Artifact Co-training for Theorem Proving with LLMs" presents an innovative approach to enhancing the performance of LLMs in automated theorem proving. The authors, Jesse Michael Han, Jason Rute, Yuhuai Wu, Edward W. Ayers, and Stanislas Polu, explore a methodology termed Proof Artifact Co-training, aimed at improving the efficiency and accuracy of theorem provers within a machine learning framework.

Introduction and Objectives

The central objective of this research is to bridge the gap between LLMs and formal theorem provers by leveraging the unique characteristics of both systems. The paper posits that by co-training LLMs with proof artifacts—an intermediate representation in the theorem proving process—one can enhance the learning process for automated theorem provers (ATPs). Such an approach is anticipated to augment proof automation, facilitate the understanding of mathematical concepts, and improve theorem proving performance overall.

Methodology and LeanStep Environment

The authors introduce the LeanStep datasets, specifically curated for this co-training purpose. This environment provides a rich collection of proof artifacts, enabling the evaluation and training of machine learning models in theorem proving. The LeanStep encompasses a unique combination of datasets and a machine learning pipeline that aligns with the logical reasoning required in theorem verification tasks.

By developing the LeanStep environment, the authors provide a structured training ground, which is critical in evaluating the efficacy of LLMs when applied to theorem proving tasks. The focus is on employing these datasets to create proof-based training regimes that can feed back into improving the LLMs' contextual understanding and predictive capabilities in formal proofs.

Experimental Evaluation

The experiments conducted as part of this paper show that integrating proof artifacts significantly boosts theorem proving performance. The authors present strong numerical results that highlight the efficacy of co-training: a notable uplift in theorem proving accuracy was observed compared to standard LLM training methodologies. This experimental evidence supports the hypothesis that proof artifacts add a valuable dimension to the training data, elevating the models' ability to generalize over complex mathematical statements.

Discussion and Implications

The practical implications of this research are multifaceted. In terms of theorem proving, the paper paves the way for more intelligent, context-aware automated reasoning systems capable of assisting mathematicians and formal logic researchers. Theoretically, it presents a compelling case for integrating different types of data modalities to reinforce machine learning models' learning processes. This approach could inspire further investigation into cross-domain learning paradigms, where disparate types of data or processes interlink to enhance AI system performance.

Future Directions

Given the promising results, future work might explore scaling this proof artifact co-training to more diverse datasets and other formal systems beyond the Lean framework. Additionally, investigating the interactions between LLMs and symbolic logic could lead to theoretical advancements in both AI architectures and formal theorem proving strategies. As such, this research constitutes a significant step toward more autonomous and insightful machine learning approaches in the field of formal logic and reasoning.