> 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.<p>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.
To what degree can you say something like "if the kernel doesn't have metaprogramming [or some other set of features] it's fine". When were the last bugs with a reduced feature set?
The implementation of the kernel is relatively trivial, it's an intentional design choice.
This thread has some context. A proof-system researcher found some proof-system bugs and presented them a funny way:<p><a href="https://leanprover.zulipchat.com/#narrow/channel/270676-lean4/topic/Counterexample.20to.20the.20Lean.20Conjecture.20.28Soundness.20Bug.29/near/613135216" rel="nofollow">https://leanprover.zulipchat.com/#narrow/channel/270676-lean...</a><p>A mathematically-inclined reviewer (or an LLM) can quickly identify that it's an exploit. (Two exploits; it's crafted to hit a bug in another proof checker, too.)<p>The post gestures at this, but a natural follow-up, beyond fixing specific bugs around this exploit, would be to task some security-oriented models with proving False in Lean, or with reviewing the code for potentially unsound steps, missing checks, or even useful 'hardening'. That's happening and bugfixes are landing as a result.
Feels appropriate for bugs in a formal proof system:<p>> Beware of bugs in the above code; I have only proved it correct, not tried it.<p>-Knuth, 1977
Reminds me of this: <a href="https://mathoverflow.net/questions/513742/are-we-stuck-with-lean" rel="nofollow">https://mathoverflow.net/questions/513742/are-we-stuck-with-...</a><p>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?
And everyone who's been into this stuff for a while has had their prediction come through.<p>If AI is water, Lean is the pipe and collatz is a clog on one end, then surely we'll find the cracks.
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?<p>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.
We want Lean4 (or any other deduction system that we use, for that matter) to be <i>correct</i>, i.e. "what is a true statement" and "what is a derivable statement" should be the same.<p>"every statement that can be derived also holds" is the difficult part to show, and something we refer to as <i>soundness</i>. For some fancy logics, it's not even possible to show, hence the discovered Kernel Soundness Bug in Lean!<p>"every statement that holds can also be derived", a notion known as <i>completeness</i>, is often a trivial property; in practice, we use <i>refutation completeness</i> instead, i.e. "every statement that doesn't hold can derive false". A bug that would allow a user to prove/derive a previously unproven statement would fall under this category of "completeness bug".<p>However, such completeness bugs immediately show up in testing. Generally, deduction systems have two kinds of rules: a handful of rules that are enough to establish (refutation) completeness, and then a few extra rules to optimize inference. Because so few rules are needed for completeness, lots of test cases will break if one of the rules break.<p>--<p>I'm not actually sure how the completeness situation looks like for proper provers like Lean. It's my graduate student's hubris to assume completeness remains easy to show for more advanced systems than the Superposition calculus ;)
Don't Godel's incompleteness theorems mean that completeness is a property you don't want in a prover (as it means the prover must then be inconsistent and this unsuable) and consistency is a property of the prover you cannot prove using the prover itself?
>"what is a true statement" and "what is a derivable statement" should be the same.<p>you mention completeness in the rest of your comment, so I'm not sure how you aren't aware of this, but the famous incompleteness theorem says that for a consistent set of axioms there will always be true statements you can't prove.[1]<p>[1] <a href="https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems" rel="nofollow">https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...</a>
A correctness bug in a proof checker by definition means that you can prove false.
Reward hacking will continue to be a problem. For sufficiently astonishing AI-created results, we must remember to ask if AI has not instead found it easier to elaborately fool us.
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.
No, Lean allows non-constructive proofs so a proof could be like "if the Riemann hypothesis is true the counterexample is 42 otherwise it is the first nontrivial zero" or something like this, and then you don't get a fully closed counterexample.
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.<p>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.<p>(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.<p>That said, I'm sure there's <i>also</i> 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 <i>anything</i> 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."
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Lean has Claude contributions, what do you expect!<p>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).<p>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.
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.
Do you have any evidence that the bugs were due to LLM code? Or are you just speculating baselessly