Hacker News Reader

Well that happened already without AI to Mochizuki with his proposed solution to the abc conjecture.

So an LLM (or more realistically, a huge swarm of agents) should check his work, find a mistake or gap, or, if there is none, provide a formal verification.

I know it adds no value, but cannot resist telling I have thought that exact same analogy.

I would say this is why formal proofs (and things like the Lean 4 libs) are so important, so that you can deconstruct the tower provably back into pieces you can understand. It shouldn't be possible to construct a formal proof you cannot destructure like this.

As a (crude) analogy, it's a bit like how you can prove the healthiness of a git tree because it's a graph of content hashes and the tree graph pointers are part of the hash. Imagine this but with a tree of knowledge.

> Not necessarily applied to OpenAI's solution to Navier-Stokes, but what happens if and when an AI genuinely appears to solve an extremely difficult problem but humans cannot independently verify the solution because understanding the proof/argument requires intelligence the verifiers biologically don't have or the resources to afford to use automated tools?

That's what Lean is for. The OpenAI LLM agents first provided a proof in natural language. Since it may be hard for mathematicians to understand and check this proof, the agents then produced a formalization in Lean. Lean is an automated proof checker. It checks whether a formal proof is correct without the need for humans to understand the proof itself.

The only way the Lean proof could still be wrong is if the conjecture was formalized wrong via misleading definitions (if it doesn't say what it seems to say) or if there is some bug in Lean itself.

Sadly, as ideal as this seems, Lean has a history of kernel bugs that allow one to prove False.

It's unlikely to be the case here as instead of hillclimbing a Lean proof for validity it appears the proof was first constructed in English before being translated to Lean, which intuitively (hopefully) reduces the chance it exploits a bug.

Surely some understanding of the lean proof is required, to make sure it proves what it claims to prove. Otherwise, what happens if the LLM includes an underhanded addition to the lean code which leads it to output a false positive?

> Surely some understanding of the lean proof is required, to make sure it proves what it claims to prove.

Yes:

> The only way the Lean proof could still be wrong is if the conjecture was formalized wrong via misleading definitions (if it doesn't say what it seems to say)

However, it is much easier to manually check whether the statement of the conjecture was formalized correctly than to manually check the whole proof.

People also need to be cautious with potential adversarial proofs. Like don't decide to give money on a sure-bet thing, just because they have a Lean proof. Not saying that these AI labs would do this.

a^n ＋ b^n ＝ c＾ｎ

...(there are two different "n"s in the above https://unicodeplus.com/U+FF4E . In addition, the plus sign is: https://unicodeplus.com/U+FF0B . I tried to use another "n" as well: https://unicodeplus.com/U+1D5C7, but looks like HN strips it out, even though it looks identical to the ASCII "n" in the default font on my browser.)

> What happens if there's a long-play social engineering attack (like the attempted XZ takeover) of something upstream of a core tool (or its dependencies) for formal verification and we have no trusted computing base?

I don't really think the current LLMs have enough context window to plan and execute something like XZ takeover without a human carefully guiding it.

But if they do, formal verification is the least thing we need to worry about. Formally verifying pure math problems will generate negative financial value once A and O get IPOed. Plus Lean is a quite small project (thus the name 'lean'). It has virtually no dependency besides a C compiler.

Nothing happens I guess. If the AI can't communicate its work or apply it to anything, it's useless and funding for those experiments will quickly dry up.

P=/=NP