Using Large Language Models as Assistants for Theorem Proving in Lean

Core Concepts

Large language models can effectively assist humans in proving theorems in the Lean proof assistant by suggesting tactics, searching for complete proofs, and selecting relevant premises.

Abstract

The paper introduces Lean Copilot, a framework for running large language model (LLM) inference natively in the Lean proof assistant. Using Lean Copilot, the authors build three proof automation tools:

suggest_tactics: This tool uses an LLM to generate tactic suggestions for the current proof goal. It checks the generated tactics to see if they lead to errors, successfully finish the proof, or change the goal without errors. The valid suggestions are displayed in the Lean IDE.

search_proof: This tool combines the LLM-generated tactics from suggest_tactics with Lean's existing rule-based proof search tool, aesop. It allows the proof search to adaptively select tactics based on the current goal, rather than using a fixed rule set.

select_premises: This tool uses an LLM-based retrieval system to select the most relevant premises (lemmas or facts) for the current proof goal. It annotates the selected premises with their type information and definitions to help users decide which ones to use.

The authors evaluate these tools on a set of theorems from the "Mathematics in Lean" book. The results show that the LLM-based search_proof tool can prove more theorems autonomously and requires fewer manually-entered tactics compared to the rule-based aesop tool and the single-step suggest_tactics tool.
The paper also introduces two general-purpose interfaces in Lean Copilot for text-to-text generation and text-to-vector encoding, which allow users to bring their own LLMs and use them natively in Lean. This enables further development of LLM-based proof automation tools and facilitates the integration of LLMs into the Lean ecosystem.

Stats

The paper reports the following key statistics:

On average, the search_proof tool requires 1.02 manually-entered tactics, compared to 3.62 for aesop and 2.72 for suggest_tactics.
The search_proof tool can prove 64% of the theorems autonomously, significantly higher than aesop (12%) and suggest_tactics (34%).
On average, search_proof can automate 81.2% of the proof steps, compared to 48.6% for suggest_tactics and 35.2% for aesop.

Quotes

"Lean Copilot provides a general framework for running LLM inference in Lean through FFI, either natively via CTranslate2 or on a server."
"Search_proof augments aesop's rule set with goal-dependent tactics generated by suggest_tactics in each step. It allows the rule set to be customized for every goal, which makes aesop substantially more flexible."
"Lean Copilot provides a general interface between LLMs and Lean, allowing users to bring their own models and run either using local text-to-text generators and text-to-vector encoders, or on a server process."

Key Insights Distilled From

Towards Large Language Models as Copilots for Theorem Proving in Lean

by Peiyang Song... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12534.pdf

Deeper Inquiries

How can the LLM-based proof automation tools be further improved to handle more complex or domain-specific theorems

To improve LLM-based proof automation tools for handling more complex or domain-specific theorems, several strategies can be implemented:

Fine-tuning on Domain-Specific Data: Training the LLMs on a more extensive and diverse dataset specific to the domain of interest can enhance their understanding of complex theorems and specialized terminology.

Ensemble Models: Combining multiple LLMs with different architectures or training methodologies can provide a more comprehensive coverage of theorem proving strategies, improving the overall performance on complex theorems.

Interactive Learning: Implementing a feedback loop where the LLMs interact with human experts to learn from their reasoning processes can help the models adapt to more intricate theorem proving scenarios.

Customized Rule Sets: Allowing users to customize the rule sets used by the LLMs during proof search can enable tailored strategies for handling domain-specific theorems effectively.

Incorporating Mathematical Structures: Enhancing the LLMs' understanding of mathematical structures and relationships can aid in tackling complex theorems that require advanced mathematical reasoning.

What are the potential limitations or failure modes of using LLMs for theorem proving, and how can they be addressed

While LLMs offer significant potential for theorem proving, there are some limitations and failure modes that need to be addressed:

Limited Generalization: LLMs may struggle with novel or unseen theorems that deviate significantly from the training data. Continual training on diverse datasets and fine-tuning on specific domains can mitigate this limitation.

Lack of Explainability: LLMs often lack transparency in their decision-making process, making it challenging for users to understand why certain proofs are generated. Developing methods for explaining the reasoning behind LLM-generated proofs can enhance trust and usability.

Bias and Error Propagation: Biases present in the training data can lead to biased or incorrect proofs generated by LLMs. Regular bias audits, diverse dataset curation, and bias mitigation techniques are essential to address this issue.

Resource Intensive: LLMs can be computationally expensive, especially for large-scale theorem proving tasks. Optimizing model architectures, leveraging efficient inference techniques, and utilizing hardware acceleration can help mitigate resource constraints.

How can the integration of LLMs and proof assistants like Lean be leveraged to enhance the overall mathematical reasoning capabilities of AI systems

The integration of LLMs and proof assistants like Lean can significantly enhance the mathematical reasoning capabilities of AI systems in the following ways:

Automated Theorem Proving: LLMs can automate routine proof steps, suggest tactics, and assist in proof search, reducing the cognitive load on mathematicians and accelerating the theorem proving process.

Enhanced Data-Driven Insights: By leveraging the vast amount of formalized mathematical knowledge in proof assistants, LLMs can learn from structured data, improving their understanding of mathematical concepts and reasoning strategies.

Human-AI Collaboration: The collaboration between mathematicians and LLMs can lead to synergistic problem-solving approaches, where human intuition and creativity are combined with the computational power and efficiency of AI systems.

Continuous Learning: LLMs integrated into proof assistants can continuously learn from new proofs, feedback from users, and evolving mathematical theories, enhancing their adaptability and performance in theorem proving tasks.