Hacker News new | comments | show | ask | jobs | submit login
The Scientific Case for P≠NP (scottaaronson.com)
125 points by apsec112 7 months ago | hide | past | web | favorite | 100 comments



This is from 2014. In this post, Scott describes various reasonings why one might consider that lead one to suspect that P!=NP.

A surprising amount of verbiage of this post is devoted towards responding to/shaming a particular person who used to comment frequently on Scott's blog, and who thinks it is not safe to assume that P!=NP. I find this off-putting. But sometimes people engage in mini-crusades on the internet... the interested reader can read the prequel and subsequent exchanges between Scott and Lubos throughout comment sections of a number of blogs.


I can personally attest to the truth of Scott's characterization of Lubos Motl's personality. Lubos once publicly called me a "category 5 loon" [1] for something I said in a presentation on quantum mechanics [2]. What he didn't seem to realize was that the thing he was attacking me for was presented as a straw man. There's a lengthy introduction where I said, "The thing I am about to tell you is clearly false, but it seems to follow logically from the things they tell you in quantum mechanics 101. The trick is to figure out where the flaw in the reasoning lies." (And BTW, I can still fool card-carrying physicists with this puzzle even today.)

AFAIK Lubos has never apologized (not that I ever really gave a lot of weight to his opinion).

[1] https://motls.blogspot.com/2012/10/evading-quantum-mechanics...

[2] https://www.youtube.com/watch?v=dEaecUuEqfc


> What he didn't seem to realize was that the thing he was attacking me for was presented as a straw man. There's a lengthy introduction where I said, "The thing I am about to tell you is clearly false, but it seems to follow logically from the things they tell you in quantum mechanics 101.

Could you give me a time stamp for that? When I originally watched that video I didn't catch that, and I can't find it now going back through. I thought you were just describing the "rabbithole" and what going down and coming up taught you. So I thought you clearly understand much more about QM than most software people, but also said blatantly wrong and unsubstantiated things.

And I think you are also missing Motl's final point. He gets the core idea of what you were trying to do, but thought that it made a poor presentation.

> At that moment, I became totally unable to say which of his statements he made are meant seriously and which of them are ironic or statements he wants to disprove. I suppose he essentially understands those things, he just encapsulates them in a completely confused story.


> Could you give me a time stamp for that?

5:20 "I will show you how that story can't possibly be true."

25:10 "So that is the end of step 1, I am now going to go on to step 2 and show you why the story that I have just told you can't possibly be true."

28:13 "The story I have told you up until now is in fact wrong." (And you should listen for another minute to the answer to the subsequent question."

I honestly can't imagine how I could possibly have made it any clearer. If you have a constructive suggestion I'm all ears.


I thought your original quote was direct quote which is why I was confused. If it was a direct quote I likely would have had a very different interpretation of what the presentation.

I would like to give constructive criticism, but feel I still don't understand exactly what what you are trying to say in the presentation.

As it stands the message you seem to be trying to convey in the presentation is that the Copenhagen interpretation leads to a contradiction. This comes from having the "clearly false result" followed by the "Copenhagen is untenable" later. But it seems like from "The thing I am about to tell you is clearly false, but it seems to follow logically from the things they tell you in quantum mechanics 101. The trick is to figure out where the flaw in the reasoning lies" that your true belief is that there is a flaw in the logic instead which changes the meaning of the presentation dramatically.

So is your intent that everything after 28:00ish is completely accurate? Because that is the sense I get from it, and I think there are issues after that point.

At 43:19, you say that the Copenhagen interpretation is scientifically untenable. My understanding is that all the different accepted interpretations are mathematically equivalent so if you are trying to say something true "scientifically untenable" is a very poor choice of words.


> I thought your original quote was direct quote

Actually, what I actually said is more accurate than my off-the-cuff paraphrase in the above comment. There is no flaw in the reasoning. The reasoning is correct, and I say that in the presentation too. It is the premises that are false.

> your true belief is that there is a flaw in the logic

No. See above.

> My understanding is that all the different accepted interpretations are mathematically equivalent

Yes, that's true. The part of the Copenhagen interpretation that is scientifically untenable is the non-mathematical part, the part that says that there is this physical phenomenon called the collapse of the wave function which is non-unitary and outside the scope of the theory, but which causes real measurable effects (i.e. the destruction of interference). That part is demonstrably false, notwithstanding that it is a fair approximation to the truth in many interesting cases, which is the reason that it has survived as long as it has despite being demonstrably false.

What is also false (and the reason I have such a bee in my bonnet about all this) is all the pedagogical baggage associated with the Copenhagen interpretation, how it's intractably weird, and no one understands it, and you shouldn't even try to understand it because you'll "go down the drain" (Feynman's words). That's all bullshit. QM is not hard to understand. The key is that measurement is just large-scale entanglement, and there are no particles. That's it. That's the big reveal. This has been know for decades. I learned it 17 years ago and it was considered old news even then. There's some evidence that Heisenberg knew it but didn't want to advance the hypothesis because there was no way to settle the question experimentally at the time. But since Bell and Aspect and Zurek (and Cerf and Adami) there is no question that it is true. It is as well established a result as any in science, and has been for decades. But for some unfathomable reason it's still controversial.


> Actually, what I actually said is more accurate than my off-the-cuff paraphrase in the above comment. There is no flaw in the reasoning. The reasoning is correct, and I say that in the presentation too. It is the premises that are false.

In that case, I think your presentation was accurate to what you wanted to convey, but I disagree that the Copenhagen interpretation has a major problem.

It is perhaps harder to get a good grasp of than another interpretation, but the really hard parts of QM is in the math.

>> My understanding is that all the different accepted interpretations are mathematically equivalent

> Yes, that's true. The part of the Copenhagen interpretation that is scientifically untenable is the non-mathematical part, the part that says that there is this physical phenomenon called the collapse of the wave function which is non-unitary and outside the scope of the theory, but which causes real measurable effects (i.e. the destruction of interference).

I was not quite clear. I meant that all of the predictions are identical (thus leading to the math calculating them being the same). If that is not true, either I have been terribly mislead or you have a groundbreaking result in QM.

> That's all bullshit. QM is not hard to understand.

The really hard part about understanding quantum mechanics is dealing with the underlying math. It is not intuitive. Complex numbers tend to make an intuitive understanding hard. Properly understanding the Quantum Fourier Transform is not easy, and that doesn't even use measurement. I never touched the Copenhagen interpretation in my study of the QFT and it was still hard.


There's a difference between understanding QM and doing QM. (See the preface of Griffith's book [1] for an explanation of what I mean by that distinction.) Doing QM is indeed hard. But it has been claimed by many (with again Feynman being the canonical example) that understanding QM is hard too, so hard in fact that one should not even try. That is false. Anyone who understands high school algebra and what a complex number is can understand QM.

> I meant that all of the predictions are identical

That depends on what you mean. Formally, yes, all the predictions are identical. But informally, the Copenhagen interpretation predicts that the EPRG experiment should yield FTL communications.

> If that is not true, either I have been terribly mislead or you have a groundbreaking result in QM.

I don't know how groundbreaking they are, but QIT has led me to profound insights that Copenhagen could not, e.g. [2] [3].

It also leads to interesting questions, for example: on QIT it is immediately obvious that it is not possible to undo an entanglement without bringing the entangled particles physically back together again (because if this were possible that would lead to FTL). So it is a deep mystery why the two-slit experiment works at all, because every photon must be entangled at a minimum with the particle that emitted it. I don't know, but I suspect that if you pursued this line of thought it would lead to some insight about how lasers work.

[1] http://www.fisica.net/mecanica-quantica/Griffiths%20-%20Intr...

[2] http://blog.rongarret.info/2014/12/quantum-teleportation-dem...

[3] http://blog.rongarret.info/2014/10/parallel-universes-and-ar...


Who is opposed to the idea that measurement is entanglement?


Anyone who defends the Copenhagen interpretation. To be more precise, anyone who thinks that the "measurement problem" is still an unresolved problem, or anyone who defends the concept of "wave function collapse" as having any scientific or pedagogical merit whatsoever, except as a historical example of how science can go badly wrong.

Richard Feynman would be the poster child for this point of view were he alive today. Freeman Dyson defended it to me the last time I spoke to him, but that was 20 years ago so maybe he's come around.

According to a poll taken in 2013 [1] Copenhagen was the most popular interpretation among physicists.

Sean Carroll apparently thinks Copenhagen is tenable [2] along with many of his commenters.

[1] https://arxiv.org/abs/1301.1069

[2] http://www.preposterousuniverse.com/blog/2013/01/17/the-most...


You have linked to the same thing twice. I would like to hear the puzzle.


Oops, sorry about that. Turns out that YouTube hijacks cmd-C on a Mac. :-( I fixed the link.

Here's the corresponding paper if you don't want to sit through the video. The puzzle is presented in section 3:

http://www.flownet.com/ron/QM.pdf


He does no shaming. If you claim that, you have to quote his words with which he does that, and I believe it will be obvious that your statement is biased. He responds, and explains why:

"Anyway, my goal here was never to convince Luboš. I was writing, not for him, but for my other readers: especially for those genuinely unfamiliar with these interesting issues, or intimidated by Luboš’s air of certainty. I felt like I owed it to them to set out, clearly and forcefully, certain facts that all complexity theorists have encountered in their research, but that we hardly ever bother to articulate."


You can sympathize with the sentiment of the article, but there is a preoccupied ("shaming"?) quality in places that moves this from an article about P/NP to an article about showing-up a person who believes something about P/NP.

Perhaps they needed to be shamed? Who knows. But it is transparent...

Quoting:

At one point in his post, Luboš actually compares computer scientists who find P≠NP a plausible working hypothesis to his even greater nemesis: the “climate cataclysmic crackpots.” (Strangely, he forgot to compare us to feminists, Communists, Muslim terrorists, or loop quantum gravity theorists.) Even though the P versus NP and global warming issues might not seem closely linked, part of me is thrilled that Luboš has connected them as he has. If, after seeing this ex-physicist’s “thought process” laid bare on the P versus NP problem—how his arrogance and incuriosity lead him to stake out a laughably-absurd position; how his vanity then causes him to double down after his errors are exposed—if, after seeing this, a single person is led to question Lubošian epistemology more generally, then my efforts will not have been in vain.

Anyway, now that I’ve finally unmasked Luboš


Preoccupied ≠ shaming. If you look at the whole debate, you will see that Aaronson is merely responding in the style established by Luboš, and that style uses ridicule (which is also not shaming, though it may be used in it) quite freely.


Ridiculing is a form of shaming. It might even be the purest form of shaming.

Shaming is to make one "embarrassed or guilty because of one's actions, characteristics, or associations."

He's literally holding up this guy's character, "vanity, arrogance, incuriosity, laughably-abusrd..."

I'm not sure what those characteristics have to do with the P=NP problem.

I'm not saying, perhaps, this guy ought not to feel shame for his prior comments. Possibly he should. But this article is a shaming article. It isnt absolved of that because someone else started it.


"Shaming" denotes ridicule, sure, but is generally used to connote iniquity, i.e. that the one doing the "shaming" is wrong so to do. I think you're getting the pushback you are because you're using this word, with this connotation, to describe a deployment of ridicule whose degree of virtue seems less to you than to others, e.g. https://news.ycombinator.com/item?id=15578837.

Perhaps using other words might improve the light:heat ratio here. And even then the fact merits consideration that, in some cases, ridicule is earned.


Well I've said several times that he may deserved to be shamed.

> ridicule is earned

Your comment only emphasises part of what I've already said.

Of course however, he may not. A person who deploys the ridicule of a person's character (over said, the ridiculing of the idea that P=NP) is one who should be aware that this itself has moral downsides and is open to fair criticism.


Pretty sure Motl is being ridiculed for reasons other than not thinking it very unlikely that P=NP. The man seems to have earned a considerable degree of infamy through exactly the kind of behavior Aaronson calls out.

And I think you may too casually gloss over the point around connotations; a cursory review of the history of privilege rhetoric would suggest there's more substance there than might initially seem to be, and I think this case is parallel.


But isn't Scott's point that "he's an asshole but hear him out, he might have something interesting to say" isn't true because his arguments are intellectually dishonest? He's rejecting Luboš's claims based on a factual argument, garnished with a side of "that dude has issues", and not because he's an ass.


True. "Also he's an ass" is severable.


You may have a point about many applications of ridicule, especially when it is unjustified (i.e. the position being ridiculed is not actually ridiculous), but I think that if you had seen the whole exchange in this case, you would realize that there is little chance that Motl could be coerced into feeling shame by anything, and certainly not this - it's just banter. As there is clearly no possibility of inducing shame, then it seems unlikely that this was intended to do so, and so is not shaming (and if it was, then frankly, its use in this case does not bother me, though I don't think it will be successful.)

But discussing whether this is shaming has no more to do with P=NP than the phrases you quote. On that, I think Aaronson has a point, but I don't think there is much to be gained through speculating about the possibilities. If you work in security, you should be aware that at some undetermined point in the future, anything dependent on P≠NP may no longer be dependable.


> But discussing whether this is shaming has no more to do with P=NP than the phrases you quote.

Yes, and? I'm writing comments about the article, not about P=NP. Since the article contains ridicule, comments about that seem well justified.

I'm not a expert on complexity theory, nor have I claimed to be. The context of discussion in a comment section on an article is the article, not it's purported title.

I haven't criticized the article, and yet people seem to be acting as-if I have. I've only described what it contains. I have made no judgement about it.

> As there is clearly no possibility of inducing shame, then it seems unlikely that this was intended to do so, and so is not shaming

I think you know how bad that argument is. The article contains what it contains.


I am aware that you are not discussing the substantive issue, and I have not mistaken your comments as such - I merely disagree with your characterization of the article's language.

> I think you know how bad that argument is.

Intent is an essential element of shaming. If you have an argument to the contrary, you would be more persuasive if you were to state it, and not simply insinuate that there is one.


Merely because it isnt likely that a person will feel shame does not mean that this article isnt (1) written with that intention; (2) written with that effect; and most importantly, (3) written with that technique.

See, for example, however every major comedian speaks about Trump. That is ridicule and shaming of him, of his personality and character. Yet, he's mostly shameless.

I would guess that actually the people who display shame the least (or seem least self-aware about when it is appropriate to display shame, etc.) are those most likely to incite shaming in others.

Don't you get angry at someone laughing (in ridicule) at you? Doesn't this provoke, sometimes, a brooding preoccupation with ridiculing them back? Is it likely that would be effective?

When people are preoccupied in this way, they are dealing with their own internal emotional state (anger, resentment, etc.) -- they arent thinking, "well this will be effective".

By way of example: all of late-night TV.


> Merely because it isnt likely that a person will feel shame does not mean that this article isnt (1) written with that intention...

If you think intent matters, then this is the wrong way round; whether it is a credible attempt to induce shame is a valid consideration. If you don't think intent matters, then it is beside the point. As you know, I do think intent matters, and when you take the exchange as a whole, I don't see it.

> ... (2) written with that effect...

It seems highly unlikely to have had that effect.

> ...and most importantly, (3) written with that technique.

This is where we differ, as I consider this to be the least important concern, well after intent and effect. If shame was neither intended or resulted, where is the shame to justify calling it shaming? Why should one be concerned about it?

The rest of your post does not appear to be applicable to this case, which, in my opinion, is comprised of the whole exchange, and not just TFA.


It seems ridiculing in the context of a mutually ridiculing discussion.

If you're saying it isnt a mutually ridiculing discussion, rather, it's taking place in a context of affected ridicule for the amusement of its interlocutors -- fine.

I don't know the context, if that is it, then I am wrong.

However from the author's comments here about how obnoxious the other guy is, I believe my view is more likely to be correct.

NB.

You can infer intent from style. You can infer intent from anything we all do it constantly as a requisite part of ordinary human interaction, and it is often very successful.

"Intent" isnt some radically internal part of an inaccessible void, it's the motivator for your actions which is often trivial to infer in a given context. However, with anything, inferences may be wrong -- this does not make them ordinarily so, impossible, or generally suspicious.

I think intent matters to (1) and (2), it doesnt matter to (3). The claim I was defending, that this article is (somewhat) shaming and this claim be interpreted as any combination of 1/2/3. Personally my view is that it is all of them (possibly, justifiably so).


It reminds me of disses in old rap songs. Eminem yelling at Lynne Cheney[1]. "Who? Oh yeah, she had some kind of thing about rap lyrics 15 years ago."

The dispute is long since forgotten, but the bickering is still there, and it's confusing and pointless to people who weren't around when it happened.

[1]https://www.youtube.com/watch?v=hsBYblypL38



More generally, we can show that the attitude that regards any unproven statement as having a 50% probability of being true is trivially self-contradictory. Suppose X is an unknown number, and all we know is that it's between 0 and 100. We can make the following statements about X, none of which is provable with the information we currently have:

1. X < 1

2. 49 < X < 51

3. X > 99

If we assume that each of these statements has a 50% probability of being true, then we trivially get a contradiction, because the probability of the union of these 3 options becomes 150%, because mutually exclusive probabilities are additive.

So the idea that a 50% chance of truth represents a state of no knowledge about the truth of a statement is just wrong. If we actually knew that any one of the above statements had a 50% probability of being true, we would clearly have more information about X than we would otherwise. There is no sense in which a 50% probability of truth is a useful default.


You are misapplying probabilities here. If you assume no knowledge, and you have two options for X, true or false, then it's a 50% probability of X being true.

Similarly, if there are N possibilities for X, and no prior knowledge for X, then it's 1/N probability of X being on of those possibilities.

The key is what prior knowledge you bring to the table.


The random variable X has an infinite number of possible values (real number between 0 and 100). But each of the statements I made has two options: true or false. Despite each having two options, there's no way that all 3 statements can have a 50% probability of truth. The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options".


> The whole point of my comment was that "no prior knowledge" cannot possibly be equivalent to "equal probability of all options".

But your scenario isn't "no prior knowledge". In your scenario, you know X is a real number between 0 and 100, which is prior knowledge. And, so the correct thing to do, is use that prior knowledge to assign probabilities, which gives you 1%, 2% and 1% for each of your propositions, not 50%.


You don't have any knowledge of X beyond the fact that it's between 0 and 100. It could be uniformly distributed, it could be the number of heads in 100 coin flips, or it could be the average number of yards that a kickoff was returned for in all NFL games last season. It could even just be 73. Assuming a uniform distribution is an arbitrary prior assumption that you're making with no support from any data, and it's no more or less reasonable than any other prior assumption you could make about the distribution. There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval.


> There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval.

I don't agree. Suppose I carry out this process:

Step 1) Generate a random histogram over the bounded interval

Step 2) Pick a random number according to the distribution that histogram represents

If I repeat that process a large number of times (say a few million), what will be the distribution of the result? It will be approximately uniform, and the more times I repeat the process the closer to uniform I get. So the uniform distribution actually is special, as the average case of randomly chosen distributions, and hence why it should act as the default when the only thing we know about the distribution is an upper and lower bound.


If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. "Generate a random distribution on an interval" is not an obviously well-defined procedure, so I don't think you can say anything about what the distribution produced by "averaging" every possible distribution would look like. Which matches the fact that you have no information on what the distribution of X looks like, other than that it is zero outside of the interval [0,100].

Edit: In any case, the process you've described is just begging the question (i.e. it is circular reasoning). In order to prove that the uniform distribution is a natural choice for X, you assume that choosing uniformly from the set of all possible distributions is natural. Even if there was a finite number of possible distributions to choose from such that choosing randomly from that set according to a uniform distribution was a well-defined operation, you're still assuming that it's natural to assign equal probability to all options in order to prove that it's natural to assign equal probability to all options.


> If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that.

So, is this X actually an arbitrary real number? Because real numbers are not known to exist in the physical world. In the physical world, we will take some measurement to a finite number of digits, and also produce an estimate of the error in that measurement. The measurement and the error bounds are all described by rational numbers, of which there are countably many. We don't know whether the set of possible "real values" (as opposed to measurements – if that concept is even meaningful) is actually uncountable – for all we know, all the "real values" could be rational, or could be drawn from some countable subset of the reals. So, if we are dealing with some kind of measured physical quantity, we actually know that the interval [0,100] is countable. We also know that the value could only possibly be measured to some finite degree of accuracy, so we actually know that the interval [0,100] is a finite set.

> In any case, the process you've described is just begging the question (i.e. it is circular reasoning).

Maybe it is just an axiom? At some point we just have to accept some things as axiomatic, so why not accept that? It can't be used to derive a contradiction, and it feels self-evident to me (even if not to you), so I feel justified in accepting it as such.


I think it's probably reasonable to say it's an axiom. That's compatible with my claim that it's an arbitrary choice that is not inherently more correct nor better supported by any information than any other choice, except perhaps in the sense that it feels better or more intuitive.


Well, what happens if we adopt alternative axioms (e.g. some sort of assumption of a normal distribution instead of a uniform one)? Are those alternative axioms consistent?

Maybe we might be able to show that agents adopting the uniform axiom make better (more rewarding) decisions in practice than those who adopt a competing axiom do?

I don't have a proof, but my gut says yes.


> There's nothing special about the uniform distribution that makes it some kind of natural or default prior for a random variable on a bounded interval.

Okay... what's your basis for this? Uniform prior is how Bayesians do it, even in unbounded intervals. It's especially natural when you have a bounded interval.

Again, the whole point is that you are not making use of any other prior information except for the interval bounds.


Well, you have to make some prior assumption, and the uniform seems intuitively reasonable. But I don't think there's a theoretical justification that says the uniform is an inherently better choice than anything else. It's still an arbitrary choice.


I am sure there has been lots of theoretical justification in favor of the uniform prior in Bayesian statistics, especially for bounded intervals. I am not a stats guy though, so I can't show you the justifications in detail.

The whole argument though was that you are not using prior information. And so you cannot assign prior probabilities in a more intelligent manner than uniform.

If you do use prior information like say, that lots of people have tried to solve for P=NP and have not succeeded, you can assign a higher probability to P/=NP.

But that's only if you assume that people trying to solve for P=NP and not succeeding makes it more likely that P/=NP is true. It is an assumption that you have to make explicit and this is the assumption that is being argued for in the OP article.

The thing is that without having priors, you can't do better than uniform.


What made me realize that P?=NP is not that important for practical issues is the unbelievable effectiveness of heuristics. They seem to get extremely close to the optimum than best approximation algorithm known for the problem.


That's not entirely true. I'm a PhD student in theoretical computer science and some of my colleagues work on (hyper-)graph partitioning, both of which are NP-complete. For most real-world instances, nobody has any clue what the optimum cut/imbalance/... is. It would be very useful to evaluate the quality of the heuristics, though. A 0.5% improvement is suddenly much more impressive if you can show that you're only half as far from the optimum as your closest competitor. Or are you not seeing any improvement on instance X because whatever you're trying isn't working, or is it because you've already got the optimum?, etc


One of the best public examples is LKH solver for travelling salesman. I'm not familiar with problems for which proving optimality of a solution to a specific problem instance is not accomplished.

For example, for travelling salesman and vehicle routing there are many instances with known optimum.

Even for those unknown tight bounds are easily achieved.



Unfortunately complexity theorists have studied heuristics as well. If our heuristics are actually good we effectively solve P=NP, therefore there is a strong upper bound to our heuristics. Though the jury is still out as to whether that bound matters for practical purposes.


Heuristics may only work for practical values of n when the answer to P?NP may not be applicable.


This is false. There are NP-complete problems that, assuming P!=NP, there are provably no fast and good heuristics for these problems.


Really? Could I see some examples? Most results I've seen are on fixed approximation algorithms not on heuristics.


Ok, but how long did it take to find a polynomial algorithm for primality checking (published in 2002)? [1] And NP-complete problems are even harder than that! The history of mathematics is rife with conjectures that would be considered true if viewed "scientifically", but have an extremely large counterexample that disproves the whole thing. [2]

[1] https://en.wikipedia.org/wiki/AKS_primality_test [2] https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture


Nothing comes close to P=NP. Again and again, across thousands of problems, almost everything ends up being P or NP-hard, and never both.

It is certainly possible that P=NP, but it will involve basically starting most of the last 30 years of theoretical computer science again.


I love the “electric fence” examples, but I’m not as skeptical as Scott that perhaps humans simply haven’t invented the mathematical tools required to break this fence. Isn’t that something that has happened plenty of times?

You could pretty easily write a parody of this article focused on taking square roots of negative numbers. For generations it looked obvious that no matter which number we tried to square, it would never result in a negative number. You can see the same sort of electric fence that can’t just be a coincidence: we can find the square root of zero, but even the slightest bit less than zero and suddenly there is no square root!

Obviously this sounds silly, because we know that mathemticians invented a new tool: imaginary numbers (a name apparently given mockingly by critical mathematicians), which can be used not only to find square roots of negative numbers, but also to solve problems that don’t obviously involve square roots of negative numbers.


It sounds silly because the analogy is a bit silly. Mathematicians didn't spend centuries trying and failing to take square roots of negative numbers, endlessly studying this as some sort of important question or goal. Complex numbers popped up fairly quickly after the serious study of the roots of polynomials began. They're intrinsic to it.

https://en.wikipedia.org/wiki/Complex_number#History

P≠NP is not like that.


The analogy is not meant to be applied further than “sometimes we invent new mathematical tools solve problems that were impossible to solve previously.”


Yes, and my point is 'it's not a good analogy (nor analogue) for the thing being discussed'. The nature of the things being compared is too dissimilar. Here's a trivial counter-analogy - angle trisection with compass and straightedge. Known since antiquity, proved impossible by Pierre Wantzel, a piddly 180 years ago (with the requisite invention of new mathematical tools).


It was something like 1500 years between the discovery of squareroot 2 and squareroot -1.

If you view mathematics as the exploration of progressively stranger objects, that there's such a gap between discovering two roots should make us cautious of discounting things after a 100 year search.

We don't know what we don't know.


The irrationality of the square root of 2 was known long (longer than 1500 years) before the invention of complex numbers, never mind root 2 itself. But whatever the duration, there wasn't relatively uninterrupted, continuous study of mathematics over it - that's a fairly recent thing. Mathematical (and other) knowledge was routinely lost or not seriously pursued at all. This supposed 'gap' between two fairly arbitrary discoveries doesn't inform our understanding of the development of mathematics itself as much as you seem to think.


> It was something like 1500 years between the discovery of squareroot 2 and squareroot -1.

It was something like 200 years between the discovery of sqrt(-1) and more-or-less universal acceptance of -1 as a number in the first place (yes, imaginary numbers predate the universal acceptance of negative numbers). European mathematics in particular favored geometric interpretation of numbers over algebraic interpretations until relatively recently, and while irrational numbers come about very easily in geometry, negative numbers don't have an immediately obvious interpretation, particularly if you exclude it is a convention of directionality.


Introducing imaginary numbers doesn’t make it untrue that negative numbers don’t have a square root in the reals. Your analogy is more akin to maybe physicists will find something that can do nondeterministic computation in polynomial time—interesting in its own right but irrelevant to the mathematical problem as stated.


Of course not. The new tool doesn’t change what is true, but it certainly let us prove things that we couldn’t possibly prove without the tool.


But this is another area where if P=NP you might expect hints to leak through. As Aaronson points out elsewhere, there's a number of problems where people had found practically efficient algorithms, or polynomial-time randomized algorithms, or something, long before they found an actual polynomial-time worst case algorithm. (In addition to primality, linear programming is another example.) For NP-complete problems there's none of that, no such hints leaking through.


We had plenty of evidence that a deterministic primality testing algorithm exists, though. The Miller-Rabin test was known to be randomized polynomial time since the 70s, and assuming the generalized Riemann hypothesis it can be made deterministic polynomial.

And while there was some nuemerical evidence that the Polya conjecture was false, there really wasnt all that much evidence that the conjecture was false. The first counterexample is around 9 billion-- you could find it yourself if you wanted.


> Ok, but how long did it take to find a polynomial algorithm for primality checking

Proving P=NP doesn't require finding a polynomial time algorithm to solve NP-complete problems. Finding such an algorithm is a much greater challenge than proving that one can exist.


Nice post, but he loses sympathy with me for shaming that guy.


Normally I would agree, but it should be noted that Lubos is a major asshole, and deserves every bit of shaming he gets. He's easily in the top 3 biggest assholes I've had to deal with in my academic career, and that's saying something.

He has it all: extreme levels of arrogance, narcissism, pedantry, absolute lack of empathy, extreme aggressivity. He is also a notable science denialist: climate change denier, evolution denier...


Could be. But how difficult is it to just ignore him, or if there is such an urge, write in a way that subtly addresses him while other readers have no clue of what is going on, or use phrases such as "some people say ..."? Plenty of good options, if you ask me.


You're missing the history where Lubos was spamming Aaron's other articles on these topics with all of the claims that Aaron deconstructs in this post. Readers of Aaron's blog were rightly confused about what to think. This post is the proper response to help your readers make sense of this.


"Scott", or "Aaronson", but not "Aaron".


Nothing wrong with politely pushing back against assholes. We should not tolerate behavior like Lubos' in civilized society.



"Telling truth to competence."

Brilliant.


Are there other theories that rely on the assumption that p≠np?

Like how there are theories that begin "assuming the reimann hypothesis|goldbach conjecture is correct.."

Disclaimer.. it is my intended inference to show that p=np


The existence of one-way functions is one of those things that requires P≠NP. So all cryptographic hash functions would be broken.


I was thinking along the lines of a practical and novel solution to another problem built on top of the assumption p≠np

But fair point, that is usually why I say such confidence in those technologies is disconcerting


(2014)


> Because, like any other successful scientific hypothesis, the P≠NP hypothesis has passed severe tests that it had no good reason to pass were it false.

Is this the crux of the post?

That seems like an extremely crap reason to believe a mathematical theorem. What am I missing?

I also don't find his heuristic convincing -- to the point his description is making me question if P=NP is actually possible after all: a close complex border that seems to yield increasing numbers of "close calls" sounds exactly like the situation where there's a lurking, sparse class of "touches" that we haven't thought of, particularly when under 100 years of research has been put into the topic.


That seems like an extremely crap reason to believe a mathematical theorem. What am I missing?

What's the alternative?

As I see it, the possible ways to think about it are as follows:

1) You must think of it as 50/50 possibility of being true until an absolute proof is found.

2) You look at places where it could fail, and if it passes then you update the likelihood based on that.

To me (and as this post argues) it makes sense to follow (2). Not only is it more useful practically (you can use that assumption to solve other things, conditional on it being true) but also it reflects the reality of the world. Even if it turns out not to be true in every case it is clear that in the majority of cases it is true.

Maybe it is untrue but it is possible to define the set of cases where it is untrue. If this is the case, it seems likely that class is very small, so it is mostly true. That is the opposite of a proof of course, but it does seem reasonable to think about your confidence of it being true as being somewhat proportional to the likely size of the class.

close complex border that seems to yield increasing numbers of "close calls" sounds exactly like the situation where there's a lurking, sparse class of "touches" that we haven't thought of

That's not really how it works though. If it keeps happening and those classes don't touch, then it typically means they aren't as close as they seem (ie, there is an extra dimension in which they are a long way apart). In this case it seems like the P/NP divide is actually a thing which means things which appear close actually aren't, so if you analyze them in that space they don't appear close together at all.


> Maybe it is untrue but it is possible to define the set of cases where it is untrue. If this is the case, it seems likely that class is very small, so it is mostly true. That is the opposite of a proof of course, but it does seem reasonable to think about your confidence of it being true as being somewhat proportional to the likely size of the class.

Any NP-complete problem reduces to any other, so if P=NP for any of them then it's true in general. Some more efficiently than others, of course.


This is a fair point.

I guess my only response would be that some of the complexity classes in "some more efficiently than others, of course" might be impractical even if they are theoretically possible.


Which might be the saving grace should the equality turn out to be true, given the sheer number of scary-powerful algorithms that happen to be NP-complete. Certainly it'd be convenient if they were solvable, but... probably not safe, you understand?

Not really relevant here, though.


What's the argument here, though?

When we find a single bridge, all the complexity classes change, so we can only really divide the problems here into "ones we've found polynomial time solutions for" and "ones we haven't".

There may be something interesting about the ones we havent -- or we just didn't think of something obvious/a lot of special cases with high degree polynomials.

It's worth mentioning the near collisions happen in the low-dimensionality region of the border -- which doesn't have the same separating forecast, as it becomes easier to touch in higher dimensions. We can think of a sphere and a cube -- in low dimensions, a sphere lies inside of a cube; in higher dimensions, the surface bulges past the sides. You just don't see it until something like the 17th dimension. (Similarly, knot theory explodes with complexity in higher dimensions.)

Taking a "scientific approach" to asymptotic problems just makes the CS people advocating it seem vaguely hacky, since they're not even advancing a proper density argument -- it's just such a low-dimensional, poorly bounded argument that it's clear they didn't make a real calculation, they tried to dress up personal feelings in science.


The argument isn't that P≠NP is true because it has passed tests. The argument is that it seems to be true because it has passed tests. Specifically, it seems much more likely to be true than false.

Consider this. You have to bet on either P=NP or P≠NP. Where do you put your money. That is the question we are trying to answer.


Both, but mostly on the null hypothesis, and on its converse I think I must insist the payout be in gold.


There is no relevant null hypothesis here.


Sloppy use of language. The by far most probable case given all the priors available. And, in the same spirit of clarity, I'd throw a small bet on P/=NP precisely because of the very long odds, and insist on being paid in gold because in the case of a constructive proof I wouldn't expect to find anything in my bank account for long. Or anyone's.


My point is that his heuristic has dropped my expected value for betting on P != NP, because his argument is one that suggests we will find a class, not that we won't.

His description of "passing tests" actually sounds like the buildup of stress before breakage, where many near-faults are generated.

I'm less confident of the resolution post reading that -- so it seems extremely weird to present as evidence that P != NP, as it shifted my beliefs the exact opposite way.


Then perhaps you haven't properly understood the argument he's making. Your "stress before breakage" argument is literally the gambler's fallacy.


> Your "stress before breakage" argument is literally the gambler's fallacy.

It's not even anything close: the gambler's fallacy requires that the events you're trying to relate are uncorrelated.

The logical structure of P vs NP is anything but uncorrelated along its surface, and a rough surface along the low-dimensionality region is a good indication of a rough surface along the higher-dimensionality regions.

So -- the fallacy requires you be trying to correlate uncorrelated things; I am not. I'm correlating things that are going to be correlated because of how the spaces are constructed.


> The logical structure of P vs NP is anything but uncorrelated along its surface, and a rough surface along the low-dimensionality region is a good indication of a rough surface along the higher-dimensionality regions.

Is it? This strikes me as pretty contentious.


How so?

Clearly there's a reason you think that isn't true -- perhaps you could share it instead of hiding behind just calling it "contentious".

It seems the stronger claim to suppose that the lower dimensions are rough and the top are smooth, especially when you think about where that roughness comes from.

Let's imagine a class "UP" which is the useful polynomial algorithms -- things we might actually compute.

The arguments raised here are a heuristic that UP != NP, which it certainly supports. My objection is that not only does that argument for UP != NP not support P != NP, it actually is evidence for P = NP, because UP seems to get asymtotically close to NP, and we know there are portions of P outside of UP -- suggesting we can find a "high" P region that sits atop UP and gets nudged over the line as UP approaches NP.

The suggestion that UP approaches NP but there's no P that crosses bears some supporting. Our disagreement seems to be over the obviousness of that relationship, and how much UP resembles P as a whole.


If all shots from storm troopers seem to barely miss, does that indicate that eventually we a storm trooper will miss? Or does it indicate a conspiracy among storm troopers to miss.

There are (apparantely) so many near misses for which it isn't clear why it should be a near miss. A great explanation for this near missing is the underlying fact that all NP problems contain some 'brute force' core that can never be polynomial.

Also note that the blog post does not at all concern your class UP. None of these are algorithms designed to be run. They are all about exploring abstract complexity classes.


My point is stormtroopers:empire::UP:P

You're correct that stormtroopers will never hit, wrong that stormtroopers are representative of the empire, and wrong about the correlation between stormtrooper hits and empire hits, because you're ignoring higher order effects (eg, while stormtroopers always miss, the empire must hit something to be the "bad guy" -- and they do: Luke's parents, Alderaan, Obi-Wan, Hoth, Luke's arm, etc.

But purely from the existence of stormtroopers consistently missing, you can infer there's a Darth Vader that must score some hits, or the plot wouldn't work. I think we might be in a similar case for P ?= NP, and you're all way too blasse predicting from stormtroopers.

Very little research has been done on P - UP, where UP is all polynomials with degree less than 10,000 and coefficients less than 10^100.

In fact, I don't know a single P-UP algorithm that wasn't explicitly constructed to be -- do you?

I just think most of us are subtly talking about UP, not P. (Including the blog post.) And that you've incorrectly inferred a "plot" when all you've observed is "redshirt fire".


UP is much harder to deal with for a very simple reason. Composition of polynomials is still polynomial. Composition of polynomials of low degree leads to polynomials with high degree.

You keep assuming we only look at UP, whereas actually our theoretical work explicitly looks at P. Often, we don't even care about actual efficiency of the algorithm. The constants might be astronomical, but we still like the algorithm for its asymptotic properties.

I'm not versed in recent research, but I imagine the composition argument above often leads to polynomial performance with stupid degrees. I expect people only compute the actual degree when they thing it might be low.

Again though, this is not my field of research.


His earlier post (linked from this) has stronger reasons.

I think it’s advisable to keep an open mind on the issue, and not simply on the actual question itself; it would not be surprising to me if the solution comes from something like adopting a different framework.

For example, time bounds and complexity classes smell a lot like conservation laws: “transforming such and such arrangement into such and such arrangement requires at least this much computation.”

That said, it’s possible (a) we’re currently failing to consider some “term”, (b) generally ignoring this term doesn’t cause problems when proving lower bounds but (c) the complex border and your “sparse touches” correspond to situations where the missing term plays a more significant role.

That’s a case where P probably isn’t equal to NP but keeping an open mind at least leads in more interesting directions, imho.

I also think people don’t take seriously the possibility of P being equal to NP but with intrinsically high degree. I say this not to be cute—“what if p is np but still de-facto intractable?”—but because I don’t think anyone has a great intuition for, say, what kinds of algorithms have polynomial solutions of minimum degree, say, 8...at least not in the same way we have good intuition for which algorithms are linear, nlogn, n^2, n^3, and so on.

It’s hard for me, at least, to feel overly confident in the “we’ve been working on finding a fast algorithm for seventy years and gotten nowhere” when our algorithmic intuition vis-a-vis higher polynomial degree seems so under-developed.

Even if you don’t consider it likely that p equals np, you can still follow this line of thought and consider the possibility that these “sparse touches” may be cases where the slippery problems like graph isomorphism (etc) correspond to problems with polynomial running times of (unexpectedly) high degree...and thus we keep finding these sparse touches along a seemingly-complex border because we don’t yet have a solid intuition for the capabilities of polynomial algorithms of high degree.

And so on and so forth. P probably isn’t NP but being dogmatic about it is neither fun nor interesting.


You touch on something I had thought, but not said:

His argument seems to be "well, the border doesnt touch using under degree 100 polynomials, so it must never touch!"

I'm disinclined to believe that order 10^374738393874 polynomials can be accurately forecast by order under 100 polynomials, and suspect that portion of the border simply hasn't been examined at all or only in the most basic cases.


The "idiot's explanation of P=NP" that refers to "easy" or "practical" is at this point fairly well established as disproven. It's possible for P=NP, but for that to happen, the algorithm pretty much has to be completely impractical.

To summarize the argument in more general terms: the class of algorithms in P is those where you can derive some global property in terms of increment local decisions. For example, building the shortest path between two nodes in a graph can be done by always picking the closest node. By contrast, NP-hard algorithms have the problem that incremental local decisions don't let that happen: you might need to recolor an entire optimally-colored subgraph to admit a new node.

Furthermore, we know from research that there tends to be a very sharp transition from P to NP-hard in terms of transformation, where relaxing a single condition goes straight from P to NP-hard without any intermediate "we don't know where there is" space. This tends to suggest that it's not really so much a question of problems being P or NP, but rather of instances being intrinsically easy to hard.

The question of P=NP then is really about whether these hard instances are really exponentially hard, or can we embed them into a simpler instance using some combinatorial construct that merely makes it look exponentially hard. So it's still possible for P=NP, but the practical question of "will we get an efficient, guaranteed for all instances, solution to these problems?" is answered in the negative. When you do the combinatorial embeddings to make the polynomial time algorithms work, you end up getting constants that look like 3^2^2^4, and there's no way to shrink those constants to practical ones.


....Duh?

But it's absolutely insane to take the impossibility of low-constant, low-degree embedding as evidence as a general lack, given the rich structure of "large" numbers.

Which is what it's being used as in this post.


Hmm.. good point. It'd be interesting to think of examples where something with a "complex border" turned out true, and examples where it turned out false.

(On a side not, I love this kind of scientific approach to believing unproved theorems, and Polya has a lot of thoughts on the topic)


Mathematics is full of things which show asymptotic behavior, so I'm not sure why the ability to find points where two things are arbitrarily "close" to each other would be read as indicating the likeliehood of a point where they "touch".




Guidelines | FAQ | Support | API | Security | Lists | Bookmarklet | Legal | Apply to YC | Contact

Search: