用编译器当奖励信号，高效搜索数学定理证明路径
Reward-Oracle MCTS for Formal Theorem Proving: Sample-Efficient Search and the Need for Kernel-Level Proof Auditing
- 将编译器输出转为数值奖励，避免冗余错误信息干扰
- 在迷你数学竞赛题上达87.1%正确率，远超基线方法
- 发现模型依赖未完成代码却通过验证，需底层审计
大型语言模型在形式化定理证明中仍面临搜索空间过大、效率低的问题。现有树搜索方法要么直接将编译器错误信息注入生成上下文，增加上下文占用；要么使用非标准评估协议，难以与基准方法比较。本文提出三角色蒙特卡洛树搜索（MCTS）框架，将 Lean 4 编译器仅作为奖励代理，利用编译结果作为标量信号进行 UCB 引导的树更新，不将错误内容输入生成上下文。框架分解为生成器、分解器和评判器三个角色。在涵盖竞赛数学与物理的四个基准（MiniF2F、PutnamBench、LeanPhysBench、PhysLeandata）上，使用三种证明器模型在标准证明尝试预算（PAB@16 到 PAB@256）下评估。该方法在 MiniF2F 上以 Goedel-Prover-V2-8B 达到 87.1% 正确率，在 PutnamBench 以 PAB@32 解决 26/659 道题，优于基线采样方法的 18/659。通过逐公理层级的全面审计，发现 DeepSeek-Prover-V2-7B 在 PutnamBench 上生成的证明虽通过编译和 sorry-token 检查，却依赖 sorryAx。审计移除了在 PAB@32 与 PAB@128 下各 4 和 8 个此类证明，以及在 MCTS 中分别 11 和 19 个。这些结果并非由搜索过程导致，而是说明必须进行内核级审计以实现真正可信的编译验证。
原文摘要 · Abstract (English)
Formal theorem proving with large language models remains challenging due to the difficulty of navigating large proof search spaces efficiently. Existing tree search approaches either feed verbose compiler error messages directly into the generation context, increasing context usage during search, or employ non-standard evaluation protocols that prevent direct comparison with established baselines. We propose a three-role Monte Carlo Tree Search (MCTS) framework that treats the Lean 4 compiler purely as a reward oracle using compiler output as a scalar signal for UCB-guided tree updates without feeding error content into the generation context. Our framework decomposes proof search into three roles: a generator for proof attempts, a decomposer for subgoal decomposition, and a critic for subgoal quality evaluation. We evaluate across 4 benchmarks spanning competition mathematics and physics (MiniF2F, PutnamBench, LeanPhysBench, PhysLeandata) with three prover models at standard proof attempt budgets (PAB@16 to PAB@256). Our method achieves 87.1\% on MiniF2F with Goedel-Prover-V2-8B at PAB@256 and solves 26/659 PutnamBench problems at PAB@32 surpassing base sampling 18/659 at same proof attempt budget. Through an exhaustive axiom-level audit of every compiled proof, we further identify reward hacking in search-based theorem proving: DeepSeek-Prover-V2-7B produces proofs on PutnamBench that pass compilation and the standard sorry-token scan while depending on sorryAx. The audit removes 4 and 8 such proofs from whole-proof sampling at PAB@32 and PAB@128, and 11 and 19 from MCTS. We do not attribute these counts to the search procedure; we report them to establish that kernel-level auditing is necessary for compiler-verified evaluation.
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