arXiv cs.AI· Alexander Bastounis, Fabian Circelli, Anders C. Hansen·· 10 天前AI 评分62
论文：Lean 验证 AI 自动形式化无法保证自然语言证明正确
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs
AI 导读
arXiv 论文指出，AI 自动形式化把自然语言数学文本翻译成 Lean 后，即使形式化证明可被机械验证，也不能保证原始自然语言论证正确。作者证明，为做到语义忠实的翻译所需的歧义消解问题在可解性复杂度指数（SCI）层级中任意高（SCI = ∞），比包括停机问题（SCI = 1）在内的任何计算问题都更难。
正文 · 原文
Abstract:Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.
| Comments: | 25 pages, 4 Figures |
| Subjects: | Analysis of PDEs (math.AP); Artificial Intelligence (cs.AI); Logic (math.LO) |
| MSC classes: | 35Q30, 03Dxx (primary) and 68V20, 68Txx, 03B65 (secondary) |
| Cite as: | arXiv:2610.08144 [math.AP] |
| (or arXiv:2610.08144v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2610.08144 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Alexander Bastounis [view email]
[v1]
Tue, 6 Oct 2026 10:58:01 UTC (1,080 KB)
来源：arXiv cs.AI · arxiv.org