> The practical consequence: checking with an independent kernel still works, since it required two distinct bugs in two implementations, but users who rely on it need current versions of both.
Things like this aren't too surprising, given that even much simpler type checkers like Rust's have soundness issues occasionally. I think it's very important to view verified results not as an absolute and unbreakable guarantee, just an extraordinarily strong one where (1) the surface area for soundness issues has been painstakingly minimized and (2) any realized soundness issues are taken very seriously and fixed in short order.
Has there ever been a bug that allowed to prove a previously unproven statement, without allowing the user to prove "false" by exploiting the bug directly?
If every bug-exploiting proof would make it easy to prove false, putting a bounty on proving false could increase trust in the validity of verified but obscure Lean proofs.
I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system like Metamath?
That's with the original C verifier only. The actual database is cross-checked by 6 independent implementations. This is the whole point of Metamath: its kernel is so tiny that you can implement a verifier in a weekend.
Metamath isn't a silver bullet in the design space of formal proof tools, but I personally think it just about nails the metatheory we want. Maybe some explicit facility around definitions would be desireable.
A counterexample of the form "X cycles to X after N steps" is easy to check. A counterexample of the form "starting with X we keep going up forever" is hard to check in finite time.
This was never about the Collatz conjecture itself. If I understand the original discussion correctly (as of a few days ago, not sure if new stuff has come to light), everybody agreed that that framing was just a flashy gimmick. And some Lean maintainers were unhappy about it, since this framing just added noise to the reproducer; they would have preferred a simple proof of False. Nobody ever thought that the disproof might be real.
Trust in formal reasoning is necessarily always conditional. You have to start somewhere. The good thing about verified formal proofs is that the only way they can be in error is if the verifier is faulty. This drastically limits the possible reasons for error.
(In practice, there’s also the possible error that the proved formal statement means something different than what you thought it meant.)
"A soundness bug in the Lean kernel (#14576) was reported and fixed during the week of July 27. [...] On July 25, Ramana Kumar published a repository containing a sorry-free "disproof" of the Collatz conjecture, produced with AI assistance. It is not a valid proof because it exploits a bug in the kernel's handling of nested inductive types. On July 28, Kiran Gopinathan reduced it to a small proof of False and opened issue #14576."
The relevant kernel code isn't written by Claude though. According to git blame, 95% of the code in inductive.cpp is 5+ years old, with only ~3 hunks (totaling less than 30 lines) coming within the last 12 months including this fix. The Lean kernel in general does not seem to change very much, e.g. the last two years are very sparse in terms of activity with relatively contained changes when I examine the history (especially so when contrasted with the pace of the rest of the project).
Beyond that it looks like a pretty simple oversight. Coq and Isabelle have also had 'prove False' bugs, it isn't the end of the world. Stuff like this happens.
Yeah people don't seem to get that the whole point of having a tightly checked kernel is so you don't have to care so much about the rest of it. Tactic heavy proofs have been "slop" long before LLMs got involved, and they lean heavily on the kernel rejecting nonsense.
What is the error rate of human programmers? Anyone who tells you that either human or A.I. code is magically exempt from issues is selling you something.
Mmm there are humans who think they make very few mistakes. Ones who actually make few mistakes, not sure about that one. Could be a mistake that they catch themselves very quickly, but I just don’t think humans are very good at generating 100% reliable output on the first try anywhere close to most of the time.
> The practical consequence: checking with an independent kernel still works, since it required two distinct bugs in two implementations, but users who rely on it need current versions of both.
Things like this aren't too surprising, given that even much simpler type checkers like Rust's have soundness issues occasionally. I think it's very important to view verified results not as an absolute and unbreakable guarantee, just an extraordinarily strong one where (1) the surface area for soundness issues has been painstakingly minimized and (2) any realized soundness issues are taken very seriously and fixed in short order.
The implementation of the kernel is relatively trivial, it's an intentional design choice.
As claude would say, "relatively" is carrying a lot of weight in that frase. Definitely load-bearing — would not pass a quick smoke test.
Feels appropriate for bugs in a formal proof system:
> Beware of bugs in the above code; I have only proved it correct, not tried it.
-Knuth, 1977
Has there ever been a bug that allowed to prove a previously unproven statement, without allowing the user to prove "false" by exploiting the bug directly?
If every bug-exploiting proof would make it easy to prove false, putting a bounty on proving false could increase trust in the validity of verified but obscure Lean proofs.
Reminds me of this: https://mathoverflow.net/questions/513742/are-we-stuck-with-...
I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system like Metamath?
Took me about 90 seconds to find a Metamath implementation bug that apparently allowed proving something that shouldn't be provable: https://github.com/metamath/metamath-exe/issues/184
That's with the original C verifier only. The actual database is cross-checked by 6 independent implementations. This is the whole point of Metamath: its kernel is so tiny that you can implement a verifier in a weekend.
Metamath isn't a silver bullet in the design space of formal proof tools, but I personally think it just about nails the metatheory we want. Maybe some explicit facility around definitions would be desireable.
And everyone who's been into this stuff for a while has had their prediction come through.
If AI is water, Lean is the pipe and collatz is a clog on one end, then surely we'll find the cracks.
Isn’t a disproof of the Collatz conjecture easy to check as it should just be a counterexample? Or is the proof not constructive?
A counterexample of the form "X cycles to X after N steps" is easy to check. A counterexample of the form "starting with X we keep going up forever" is hard to check in finite time.
And still that requires X to not be particularly large. It could conceivably be in ballpark of BB(40).
This feels like saying there are probably hundreds of stars in the universe
You, sir, have a very different definition of "not particularly large" than I do!
To be fair, BB(40) is smaller than almost all positive integers.
It’s practically zero, in fact.
This was never about the Collatz conjecture itself. If I understand the original discussion correctly (as of a few days ago, not sure if new stuff has come to light), everybody agreed that that framing was just a flashy gimmick. And some Lean maintainers were unhappy about it, since this framing just added noise to the reproducer; they would have preferred a simple proof of False. Nobody ever thought that the disproof might be real.
This had nothing to do with Collatz and everything to do with a Lean bug.
The proof was not actually a proof at all, because it was unsound (despite Lean admitting the proof).
So essentially: One cannot trust the code produced by an LLM, even if the code is a formal proof passing the verifier.
Trust in formal reasoning is necessarily always conditional. You have to start somewhere. The good thing about verified formal proofs is that the only way they can be in error is if the verifier is faulty. This drastically limits the possible reasons for error.
(In practice, there’s also the possible error that the proved formal statement means something different than what you thought it meant.)
One can only trust a verifier as far as they can trust anything made out of software.
I would expect proofs that exploit kernel bugs to look fishy, so someone reading the proof could catch the smell.
That said, I'm sure there's also room for underhanded Lean programming as well, which would be even more interesting.
It has nothing to do with LLMs. One cannot trust the proof of anything if one cannot trust the verifier.
"A soundness bug in the Lean kernel (#14576) was reported and fixed during the week of July 27. [...] On July 25, Ramana Kumar published a repository containing a sorry-free "disproof" of the Collatz conjecture, produced with AI assistance. It is not a valid proof because it exploits a bug in the kernel's handling of nested inductive types. On July 28, Kiran Gopinathan reduced it to a small proof of False and opened issue #14576."
Lean has Claude contributions, what do you expect!
Use Coq or Isabelle or any other decent theorem prover. Lean is just hyped.
The relevant kernel code isn't written by Claude though. According to git blame, 95% of the code in inductive.cpp is 5+ years old, with only ~3 hunks (totaling less than 30 lines) coming within the last 12 months including this fix. The Lean kernel in general does not seem to change very much, e.g. the last two years are very sparse in terms of activity with relatively contained changes when I examine the history (especially so when contrasted with the pace of the rest of the project).
Beyond that it looks like a pretty simple oversight. Coq and Isabelle have also had 'prove False' bugs, it isn't the end of the world. Stuff like this happens.
Yeah people don't seem to get that the whole point of having a tightly checked kernel is so you don't have to care so much about the rest of it. Tactic heavy proofs have been "slop" long before LLMs got involved, and they lean heavily on the kernel rejecting nonsense.
What is the error rate of human programmers? Anyone who tells you that either human or A.I. code is magically exempt from issues is selling you something.
There are humans who make very few mistakes. The presence of Claude in a project however tells you something about the attitude of the project:
- Probably too close to corporations.
- Does not care about slop.
We have seen many projects that adopted AI under corporate pressure circle the drain. Often the corporations themselves backpedaled after some months.
> We have seen many projects that adopted AI under corporate pressure circle the drain.
Which?
> Often the corporations themselves backpedaled after some months.
Which
> There are humans who make very few mistakes
Mmm there are humans who think they make very few mistakes. Ones who actually make few mistakes, not sure about that one. Could be a mistake that they catch themselves very quickly, but I just don’t think humans are very good at generating 100% reliable output on the first try anywhere close to most of the time.
Why can’t your two criteria be applied to projects with human contributors without further due diligence?
Claude can produce code that is professionally reviewed and found to be correct.
Do you have any evidence that the bugs were due to LLM code? Or are you just speculating baselessly