Improving Formal Theorem Proving with LLMs: A Hierarchical Decomposition Approach Using Reinforcement Learning

Core Concepts

This paper proposes a novel reinforcement learning method, Proof Decomposer (ProD), which enhances the ability of Large Language Models (LLMs) to generate formal proofs by encouraging them to decompose complex theorems into simpler, provable lemmas in a hierarchical manner.

Abstract

Bibliographic Information:

Dong, K., Mahankali, A., & Ma, T. (2024). Formal Theorem Proving by Rewarding LLMs to Decompose Proofs Hierarchically. arXiv preprint arXiv:2411.01829.

Research Objective:

This research aims to improve the ability of LLMs to generate formal proofs in a more natural and challenging setup where directly relevant lemmas are not provided, requiring the model to exhibit stronger planning and decomposition capabilities.

Methodology:

The authors propose Proof Decomposer (ProD), an RL-based training algorithm that encourages LLMs to decompose theorems into lemmas, prove them individually, and then utilize these proven lemmas to prove the original theorem. The model is trained using a reward mechanism inspired by how mathematicians work, rewarding the model for proposing and proving correct and novel lemmas even if the original theorem remains unproven.

Key Findings:

• ProD-RL, the model trained with the proposed RL algorithm, outperforms the model trained with supervised fine-tuning (SFT) on both in-distribution (AFP test set) and out-of-distribution (AFP 2023 test set) datasets.
• ProD-RL demonstrates the ability to propose and prove novel lemmas not present in the training dataset, with 37.7% of the lemmas in the training replay buffer being newly proposed.
• The improvement of ProD-RL is primarily observed in proving theorems of low-to-medium difficulty, suggesting the need for further research to tackle more complex theorems.

Main Conclusions:

The study demonstrates the effectiveness of using RL with hierarchical lemma decomposition to enhance the formal theorem proving capabilities of LLMs. The proposed method encourages the model to learn a more natural and generalizable approach to theorem proving, moving beyond reliance on pre-existing lemmas.

Significance:

This research contributes to the field of automated theorem proving by presenting a novel approach that leverages the power of LLMs while addressing the limitations of previous methods. The ability to generate proofs in a more human-like manner holds significant potential for advancing the field.

Limitations and Future Research:

• The model's improvement over baseline methods is less significant when tested on datasets with a larger distribution shift from the training data, indicating a need for improved robustness.
• The proposed method primarily shows improvements in proving theorems of low-to-medium difficulty, suggesting the need for further research to scale the approach for more complex mathematical concepts.

Stats

37.7% of the lemmas proved during training are not in the dataset.
ProD-RL improves the pass rate from 40.8% to 45.5% on the AFP test set.
ProD-RL improves the pass rate from 36.5% to 39.5% on an out-of-distribution test set (AFP 2023).

Quotes

Key Insights Distilled From

Formal Theorem Proving by Rewarding LLMs to Decompose Proofs Hierarchically

by Kefan Dong, ... at arxiv.org 11-05-2024

https://arxiv.org/pdf/2411.01829.pdf

Deeper Inquiries

How can the proposed method be adapted to handle different formal proof systems beyond Isabelle, and what challenges might arise in such adaptations?

Adapting ProD to different formal proof systems like Lean, Coq, or HOL4 presents both opportunities and challenges.
Opportunities:

Generalization of hierarchical decomposition: The core principle of ProD, encouraging LLMs to decompose proofs hierarchically, is system-agnostic. This suggests that the approach could be beneficial across different proof systems.
Cross-system learning: Training on datasets from multiple proof systems could lead to models with a broader understanding of formal reasoning, potentially improving generalization and transfer learning.
Challenges:

Syntax and semantics: Each proof system has its own syntax and semantics. Adapting ProD would require tailoring the parsing of proof scripts, the design of the <invoke> mechanism, and the interaction with the specific proof verifier.
Proof style and granularity: Proof styles and the level of detail expected by different systems vary significantly. ProD's training would need to account for these differences to generate proofs acceptable to the target system.
Availability of tools and resources: The success of ProD relies on tools like Sledgehammer for premise selection and proof automation. The availability and effectiveness of analogous tools for other systems would be crucial.
Addressing the challenges:

System-specific fine-tuning: Fine-tuning the LLM on a corpus of proofs from the target system would be essential to adapt to its specificities.
Abstract syntax tree (AST) representation: Using an AST representation of proofs could provide a more system-agnostic way to handle syntax and semantics, facilitating adaptation.
Transfer learning: Leveraging pre-trained models like Llemma and fine-tuning them on the target system could reduce the amount of system-specific data required.

Could the inclusion of a mechanism for evaluating the potential usefulness of a proposed lemma before attempting its proof lead to further performance improvements and address the limitations observed with complex theorems?

Yes, incorporating a mechanism to evaluate the potential usefulness of a proposed lemma before devoting resources to its proof could significantly enhance ProD's performance, particularly for complex theorems.
Benefits:

Reduced wasted effort: Currently, ProD attempts to prove all proposed lemmas, even those that might be irrelevant or even incorrect. Evaluating lemma usefulness beforehand would allow the model to focus on more promising candidates.
Improved exploration: By prioritizing potentially useful lemmas, the model could explore the proof search space more efficiently, potentially leading to solutions for more complex theorems.
Better lemma quality: A usefulness evaluation mechanism could guide the model towards proposing lemmas that are more likely to be relevant and contribute to the overall proof.
Implementation:

Learned lemma scoring function: Train a separate model to predict the likelihood of a proposed lemma being useful, given the theorem statement, context, and potentially even a partial proof.
Integration with RL: Incorporate the lemma score into the reward function, encouraging the model to propose and prove lemmas deemed useful by the scoring function.
Iterative refinement: The lemma scoring function could be iteratively refined using feedback from the proof verifier, improving its accuracy over time.
Addressing limitations with complex theorems:

Hierarchical decomposition: By focusing on proving useful lemmas, the model could build up a hierarchy of proven statements that contribute to the final proof, making complex theorems more manageable.
Targeted exploration: Evaluating lemma usefulness would guide the exploration towards more promising areas of the proof search space, increasing the chances of finding solutions for challenging theorems.

What are the implications of training LLMs on formal theorem proving for other domains that require complex reasoning and problem-solving, such as program synthesis or scientific discovery?

Training LLMs on formal theorem proving has significant implications for domains requiring complex reasoning and problem-solving, such as program synthesis and scientific discovery.
Transferable Skills:

Logical reasoning and deduction: Formal theorem proving hones an LLM's ability to reason logically, chain inferences, and construct sound arguments – skills crucial for program synthesis, where code must adhere to logical constraints, and scientific discovery, where hypotheses are evaluated based on evidence.
Hierarchical problem decomposition: ProD's approach of decomposing complex proofs into smaller, manageable lemmas could translate to breaking down complex problems in other domains, making them more tractable.
Symbolic manipulation and abstraction: Formal systems often involve manipulating abstract symbols and applying rules systematically. This ability could be valuable for program synthesis, where code involves symbolic representations, and scientific discovery, where models and theories are often expressed symbolically.
Potential Applications:

Program synthesis: LLMs trained on theorem proving could generate code that is not only syntactically correct but also logically sound and verifiable against given specifications.
Scientific discovery: These models could assist in formulating hypotheses, designing experiments, and evaluating evidence to support or refute scientific claims.
Automated reasoning systems: LLMs could enhance existing automated reasoning systems by providing more human-like intuition and guidance in navigating complex search spaces.
Challenges and Future Directions:

Domain adaptation: While the core reasoning skills are transferable, adapting LLMs to specific domains like program synthesis or scientific discovery would require specialized training data and fine-tuning.
Explainability and trust: For these applications, it's crucial that the LLM's reasoning process is transparent and understandable to humans, requiring further research on explainable AI.
Integration with domain knowledge: Effectively leveraging LLMs in these domains would necessitate integrating them with existing domain knowledge bases and tools.