LeanProgress: Guiding Search for Neural Theorem Proving via Proof Progress Prediction

Abstract: Mathematical reasoning remains a significant challenge for LLMs due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.1\% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8\% improvement on Mathlib4 compared to baseline performances of 41.2\%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies.

LeanProgress: Guiding Search for Neural Theorem Proving via Proof Progress Prediction

The paper entitled "LeanProgress: Guiding Search for Neural Theorem Proving via Proof Progress Prediction" introduces a novel methodology that enhances automated theorem proving by addressing the challenge of determining proof progress within formal systems like Lean. By leveraging Large Language Models (LLMs) in conjunction with the Lean proof assistant, the authors aim to mitigate the prevalent issue of hallucinations in mathematical reasoning, which impede the development of accurate proofs. Although combining LLMs with formal systems ensures reliability through rigorous verification, the complexity and length of mathematical formalizations remain significant obstacles.

Key Innovations and Methodology

The central contribution of this paper is the LeanProgress framework, which predicts the progress of a proof by estimating the remaining steps needed to reach completion. This approach is anchored in a large corpus of Lean proofs sourced from Lean Workbook Plus and Mathlib4. The authors employ data preprocessing and balancing techniques to rectify the inherent skewness in proof length distributions. Such measures are pivotal as they ensure the model learns uniformly across proofs of varying complexity.

The LeanProgress model achieves a 75.1% accuracy in predicting proof progress. Even more compelling, within a best-first search framework integrated with the Reprover tool, LeanProgress enhances baseline performance on Mathlib4 by 3.8%, notably outperforming traditional approaches, especially for long and intricate proofs. This integration facilitates a more informed search strategy, driven by predictions rather than solely relying on log probabilities.

Implications and Theoretical Contributions

The implications of this work are manifold. Practically, it provides a robust tool that aids mathematicians and formal method practitioners in navigating complex proofs within Lean. Theoretically, it introduces a new paradigm for understanding proof development, shifting the focus from local tactic optimization to a holistic view of the proof trajectory. This transition represents a transformative step in bridging gaps between machine learning and automated reasoning.

By exploring reinforcement learning (RL) frameworks using proof progress indicators as reward signals, the paper suggests future directions that could further optimize theorem proving. Current RL applications in theorem proving are constrained by combinatorial complexities; reliable progress signals, such as those proposed, could alleviate these limitations by guiding exploration more effectively.

Future Directions

Potential futuristic developments inspired by LeanProgress include integrating tree-of-thought and chain-of-thought methods to facilitate a structured reasoning framework. Furthermore, exploring lightweight models and enhancing model efficiency through compression or more streamlined architectures could make these tools more accessible to a broader audience.

Conclusion

LeanProgress represents a significant stride in formal reasoning frameworks, demonstrating how LLMs can be effectively harnessed to guide proof development in formal systems. The merging of neural predictions with mathematical rigor offers promising avenues for future research in both the optimization of automated theorem provers and in the development of more sophisticated AI-driven formal verification tools. This work not only contributes to the immediate field of theorem proving but also poses broader implications for how AI can enhance complex decision-making processes in formal systems.