arXiv:2505.02735cs.AIcs.LG2025-05被引 43

构建首个大规模形式化数学推理基准,评估大模型在严谨证明中的真实能力。

FormalMATH: Benchmarking Formal Mathematical Reasoning of Large Language Models

  • 用人类协作的自动形式化流水线,高效生成5560个正式数学题
  • 最强模型仅16.46%正确率,且严重依赖代数、回避微积分
  • 发现自然语言提示反而干扰形式证明,揭示人类推理的噪声问题

形式化数学推理仍是人工智能的关键挑战,现有基准在规模与覆盖范围上存在局限。为此,我们提出 FormalMATH,一个包含5,560个经形式验证的数学问题的大规模 Lean4 基准,涵盖从高中奥数到本科级定理的多个领域(如代数、应用数学、微积分、数论和离散数学)。为降低手动形式化的低效性,我们引入一种新型人机协同自动形式化流水线,包括:(1) 专用大语言模型用于命题自动形式化,(2) 多模型语义验证,(3) 利用现成的基于 LLM 的证明器进行否定式反证过滤策略。该方法在保留72.09%原始陈述的前提下,显著降低了专家标注成本,同时保证与原自然语言问题的一致性。对前沿基于 LLM 的定理证明器的评估显示其存在显著局限:即使最强模型在实际采样预算下也仅达16.46%成功率,表现出明显的领域偏差(如在代数中表现优异,但在微积分中失败),并过度依赖简化自动化策略。值得注意的是,我们发现链式思考场景中,自然语言解题引导与证明成功率呈反向关系,表明人类编写的非形式化推理在形式化环境中反而引入噪声。我们认为 FormalMATH 为形式化数学推理提供了可靠基准。

原文摘要 · Abstract (English)

Formal mathematical reasoning remains a critical challenge for artificial intelligence, hindered by limitations of existing benchmarks in scope and scale. To address this, we present FormalMATH, a large-scale Lean4 benchmark comprising 5,560 formally verified problems spanning from high-school Olympiad challenges to undergraduate-level theorems across diverse domains (e.g., algebra, applied mathematics, calculus, number theory, and discrete mathematics). To mitigate the inefficiency of manual formalization, we introduce a novel human-in-the-loop autoformalization pipeline that integrates: (1) specialized large language models (LLMs) for statement autoformalization, (2) multi-LLM semantic verification, and (3) negation-based disproof filtering strategies using off-the-shelf LLM-based provers. This approach reduces expert annotation costs by retaining 72.09% of statements before manual verification while ensuring fidelity to the original natural-language problems. Our evaluation of state-of-the-art LLM-based theorem provers reveals significant limitations: even the strongest models achieve only 16.46% success rate under practical sampling budgets, exhibiting pronounced domain bias (e.g., excelling in algebra but failing in calculus) and over-reliance on simplified automation tactics. Notably, we identify a counterintuitive inverse relationship between natural-language solution guidance and proof success in chain-of-thought reasoning scenarios, suggesting that human-written informal reasoning introduces noise rather than clarity in the formal reasoning settings. We believe that FormalMATH provides a robust benchmark for benchmarking formal mathematical reasoning.

形式化推理大模型评测数学证明LLM

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