TLDR: PROOF COMPASS is a new AI method that combines large language models (LLMs) with specialized theorem provers to solve complex math problems more efficiently. It guides provers with natural language strategies and helps them break down problems by analyzing failed attempts to find useful intermediate steps. This approach achieves better accuracy with significantly fewer computational resources, making advanced theorem proving more accessible.
Formal mathematical reasoning, the process of translating mathematical proofs into machine-verifiable forms, is a critical step towards building more reliable mathematical knowledge. Historically, this process has been incredibly labor-intensive, demanding deep expertise in both mathematics and specialized formal verification tools like Isabelle, HOL Light, and Lean.
Recent advancements in Large Language Models (LLMs) have shown great promise in transforming this landscape. However, existing approaches often face limitations: large general-purpose models struggle with the precise syntax required by proof assistants, while smaller, specialized models, though good at generating syntactically correct proofs, are limited in their broader mathematical reasoning abilities. Training large, specialized models also demands significant computational resources, creating a high barrier to entry for many researchers.
Addressing these challenges, a new methodology called PROOF COMPASS has been introduced. This innovative hybrid approach aims to achieve remarkable computational efficiency and improved accuracy by strategically guiding existing specialized prover methods with a Large Language Model, all without requiring any additional model training. This makes cutting-edge research in automated theorem proving more accessible.
PROOF COMPASS works by leveraging the strengths of both general-purpose LLMs and specialized provers. Specifically, it directs the DeepSeek-Prover-v1.5-RL (DSP-v1.5) model, enhancing its theorem-proving capabilities in Lean 4 through two main mechanisms.
How PROOF COMPASS Works
First, the LLM utilizes its superior natural language understanding and mathematical reasoning to produce a detailed natural language proof. This proof is then summarized and provided to DSP-v1.5, guiding it during the formal proof generation process. This “informal proof guidance” helps the specialized prover by giving it a high-level strategy to follow.
Second, inspired by methods that use intermediate goals or lemmas to structure proof searches, PROOF COMPASS employs the LLM to analyze the specialized model’s failed attempts. From these failures, the LLM extracts relevant and correct intermediate steps, known as lemmas, formatted as “have” statements in Lean. This “lemma-based guidance” effectively breaks down the original complex problem into smaller, more manageable sub-problems for the specialized prover, allowing it to tackle problems that were previously intractable on its own.
The design of PROOF COMPASS is highly modular. The guiding LLM can be easily swapped out, allowing the system to benefit from future model improvements. Furthermore, its core principles are adaptable for use with other specialized provers, making it a versatile framework.
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Impressive Results and Efficiency
The effectiveness of PROOF COMPASS was evaluated on the miniF2F benchmark, a collection of formal mathematics problems. The results are compelling: PROOF COMPASS achieved a Pass@128 success rate of 55.3%, which not only outperforms the standalone DSP-v1.5’s Pass@128 rate of 51.6% but also matches the performance DSP-v1.5 achieves with a significantly larger budget of 3200 attempts (54.9%). This demonstrates a remarkable 25-fold reduction in computational attempts while maintaining or improving accuracy.
Even with a highly constrained budget of just 32 attempts, PROOF COMPASS achieved a success rate of 52.5%, surpassing the standalone prover’s performance at 128 attempts (51.6%). This highlights the significant resource efficiency gained through this synergistic approach.
An ablation study further confirmed the individual contributions of both the informal proof guidance and the lemma-based guidance. Both components were found to independently improve performance, and their combination yielded the best overall results, underscoring their complementary nature.
In conclusion, PROOF COMPASS represents a significant step forward in automated theorem proving. By intelligently combining the high-level reasoning capabilities of general-purpose LLMs with the precision of specialized provers, it offers a more resource-efficient and accessible path towards developing more capable theorem-proving systems. For more details, you can read the full research paper available at arXiv.
