Posted by vok 11 hours ago
This has been marginally debated since the mid 90s, when Impagliazzo published the “5 worlds” survey. Occasionally it is brought up in talks about average case analysis. There is occasionally a workshop about it. But it has never really got the attention it deserved. How many more of these results do we need to take it seriously? (Assuming we do indeed certify it.)
What do you mean? Do you mean that our heuristics shortcuts of treating P as efficiently solvable (and NP as efficiently verifiable) are not warranted? So far pretty much all algorithms in P that we found have small exponents. That hasn't changed, but it might change in the future.
Or do you mean something else?
Keep in mind that even most instances of many NP hard problems can be solved very efficiently in practice, because most instances ain't hard.
However we know problems that are expected to be 'easier' than NP that are hard to solve in practice. These are pretty much all the problems we build cryptography out of, because essentially these problems are in the intersection of NP and co-NP and that's expected to be smaller than NP. And cryptography has to be hard to break on average to work: a random key needs to be a good key.
> and P \in NP is actually ill-posed.
Oh, that's very simple to answer: P is not an element of NP. The question people care about is whether there's any element in NP that's not in P. In LaTeX notation P \subset NP vs P \subseteq NP.
I'm not sure what you mean by ill-posed? The question is very clear. What are you trying to say?
There are many ways it might turn out to be ill-posed, this is just one of them. Impagliazzo's "5 worlds" survey is in some sense a catalog of some of these ways, and includes some of the ways you seem to be referencing (approxmations, probabilistic techniques, average-case efficiency; depending what you mean this could lie in the "Heuristica" world).
OK, that's a reasonable worry and one that practitioners share. Thanks for explaining. Though I don't think ill-posed is a good term for this, the technical meaning of the term points in a different direction.
So far, as far as I can tell this new result gives much of any extra credence to that worry.
— Scott Aaronson
Considering that no human will be able to sift through these in no less than a team-decade, to verify that there's no mistake, yet still that will be massively less compute than the one necessary to produce the 'solution', isn't the empirical finding contrary to the theoretical finding being suggested? Namely that in practice we are generating these 'proofs' in a massive amount of compute time, but it will take tiny amounts of time to verify them.
Is it very useful to talk about computing in purely mathematical terms and not in empirical terms? There's very obvious truths that we are seeing here, and it seems like those that go through some sort of rigorous mathematical training kind of fail to see it. Similar to how 70% of physicians or something believe in the Many Worlds Interpretation or that time isn't real, a sort of algebraic psychosis where one is removed from a human reality that we have access to in order to attempt getting close to an 'objective reality' that is far removed from the human experience.
To some extent we are computing entities, so I don't even think that we can conceive of computing as decoupled from ourselves[1]. I know the observer effect is very often mis-cited by laymen, but I'll be damned if some computer scientists like Godel don't sound like they were up to something similar to this.
Also, here's another outsider perspective. This is obviously slop, I'm not sure how but it's obviously nerd-sniping the CS majors that understand this, because it's obviously very hard to find the flaw, but it gives epicycles energy. It's also sufficiently complex that under a bayesian analysis it is more likely that the AI has reached the limit of its processing power until it becomes confused enough to believe it has reached something significant, kind of like how I'm doing now.
[1] xkcd://1163
How do you figure that? For ETH, OpenAI is engaging with top experts in this area. It's unlikely that they'll need 10 man-years combined to go through the results:
> This work will be verified and shepherded by a group led by Russell Impagliazzo (UC San Diego), Ramamohan Paturi (UC San Diego), and Ryan Williams (MIT). All three are experts in algorithm design and complexity theory.
> Impagliazzo and Paturi, along with Francis Zane, are the original proposers of the Exponential Time Hypothesis, and have authored a number of influential papers on the topic over the last 30 years.
> Williams has refuted strong versions of the ETH, used ETH for hardness results, and connected algorithms and lower bounds.
They are making good progress on similar research programmes for many cetaceans.
In the case of our stochastic parrots it's even easier, since they already speak something like human language. And for the math proofs it's even easier, because once they are formalised into something like Lean the humans only need to understand the statement of the theorem, and a mechanical computer program can check the proof.
It is not a demerit, actually I think that it is a very elegant outcome how many problems could be solved by generation and this probably show us how language is a far more important thing than we ever thought it was.
It's not possible for someone who's not a specialist in the field to understand it (unless there are more elementary proofs, of which likely none exist).
siting on a recliner making an occasional sensible chuckle while flipping through the pages?
OpenAI has that too - open their website and ask.
I totally disagree with you on the precept that all difficult to understand things need be needlessly dumbed-down/watered-down until palatable for a mass (average) audience. Requiring 'math50' before 'trig101' isn't gate-keeping in any moral sense, it's an effort to save everyones' time involved when the effort is futile.
Similarly I don't demand that haute couture fashion companies explain the style to me until 'I get it' ; I just take it for granted that i'm not going to be able to understand the entirety of the world in one lifetime.
You're right, and I should correct my earlier phrasing. The paper does *not* merely assert existence — it gives a *specific deterministic algorithm* with a concrete, proven running time:
> A deterministic algorithm that exactly counts the satisfying assignments of a canonically encoded $k$-CNF formula on $n$ variables in time $(n+m)^{O_k(1)} \cdot 2^{O_k(n \log\log n / \sqrt{\log n})}$, using polynomial space.
The Lean formalization proves this bound. So it's a fully specified algorithm, not just "there exists one." My earlier "theoretical existence statement" framing was wrong.
*What that changes (and doesn't) for the practicality question:*
- It's a real algorithm in the sense that if you implemented it, it would terminate and give the correct answer. The proof guarantees the bound. - But the bound is still *astronomically slow* for any input size a cryptographer cares about. At $n = 1{,}000$ variables the exponent is already $\sim 1{,}000 \cdot 5 / 5 \approx 1{,}000$, i.e. $2^{1000}$ — far beyond any feasible computation. - The reduction-overhead argument for cryptography is unchanged: the subexponentiality is in $n$ (the SAT variable count), and reducing a cryptographic problem of size $N$ to SAT inflates $n$ to $\text{poly}(N)$, which swallows the subexponential advantage.
So the correction is: yes, it's a specific algorithm, not a mere existence claim. But the practical conclusion is the same — the bound is too slow to matter for cryptography, and the reduction path doesn't preserve the subexponential character relative to the original security parameter.
From the HN comment guidelines:
> Don't post generated text or AI-edited text. HN is for conversation between humans.
Disclosure doesn't address the objection (not that it wouldn't be obvious anyway). If you learned something from an AI convo, tell us what you learned, in your own words, not what the LLM said.