Autoformalisation cannot guarantee correct natural language mathematical proofs
Navier-Stokes lost in translation: Why Lean verification of AI autoformalisation does not guarantee correct natural language proofs
Artificial Intelligence
Summary
Mathematics written in everyday language can be hard to translate into computer-checked formal code without losing or changing meaning. The authors show that this translation problem is extremely difficult—even harder than famous computer problems like the Halting problem. They give examples where AI wrongly translated math proofs, including a famous fluid dynamics problem, making the computer-checked proofs unreliable as evidence of the original argument's correctness.
What this means in practice
- •For software verification engineers: Recognize that fully reliable automatic translation of complex natural language proofs into formal code is not currently achievable, guiding careful human oversight in formal verification workflows.
- •For mathematical ai developers: Develop AI tools for autoformalisation with improved handling of semantic ambiguities, informed by the theoretical limits highlighted by the authors.
A theory result. No direct application yet.
Authors
Alexander Bastounis, Fabian Circelli, Anders C. Hansen
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.