What P vs. NP is about

https://vasekrozhon.wordpress.com/2024/08/18/what-p-vs-np-is-actually-about/

by signa11 • 2 years ago

172 87 2 years ago

vasekrozhon.wordpress.com

We recently made a Polylog video about the P vs NP problem. As usual, our goal was to present an underrated topic in a broadly understandable way

Interesting post but not sure why the author calls it underrated. It seems to get a fare share of attention and love amongst those that care about such things.

karmakurtisaani2• 2 years ago

It's hardly an underrated topic. In fact, probably the most well-known from all the millennium prize problems.

Edit: presenting the problem with inverting functions is an unusual approach tho.

theturtletalks• 2 years ago

I was introduced to the P vs. NP problem via the traveling salesman problem and honestly seems like easiest way to explain it to people.

KPGv2• 2 years ago

Personally, I think it's fair to say all Millennium Prize problems are underrated because 99.9% of humans alive have no idea what that is.

gosub1003• 2 years ago

I think that word is in the process of being re-defined. I see it used a lot in the context of older pop music or singers/bands that are no longer at their peak.

My own pet theory is that it's an evolution of speech towards "hardened" claims that you can't easily disagree with. You cannot disagree with "over/under-rated" because there's no official "rating" method. You cannot disagree with "vibes" because the source of a vibration is impossible to determine. They both allow a speaker to make a claim without evidence.

tgv2• 2 years ago

Might be the clickbait treadmill, indeed, or algospeak. But it doesn't need an objective meaning to be treated that way. The word "literally" can no longer be trusted to mean "in a literal sense". It's frequently used as an intensifier.

magicalhippo• 2 years ago

Yeah was gonna say the same. As seen on a recent video: "Blondie is so underrated!"...

Though I guess for a lot of young people it feels like that, with none of their friends talking about such bands.

genewitch• 2 years ago

sounds similar to weasel words https://en.wikipedia.org/wiki/Weasel_word

There's another term, i'm sure, because politicians use it all the time; "I never said X" when it was very heavily implied, and the listener was led down the garden path to the conclusion, only to find out the conclusion is bitter and unpleasant. The speaker can say "oh, that's your own biases/misconceptions/dogma, i never said <something extremely specific>"

I'm in the weeds here, but a simple analogy would be: "I don't like sunny days without clouds or fog." and i say "why don't you like blue skies?" and they say "i never said that" when the salient points of a sunny day with no clouds or fog are 1) there's a sun, and 2) the sky is blue.

thaumasiotes6• 2 years ago

You cannot disagree with "over/under-rated" because there's no official "rating" method.

(A) That's not going to stop anyone; (B) the claim can easily be obviously false. For example, Taylor Swift is underrated.

We have here an example of problem (B); P versus NP is possibly the single most famous problem in computer science. It isn't underrated.

edanm2• 2 years ago

I'm pretty sure the author is specifically referring to the framing of P vs NP used in the video, not to the problem itself, since a bit into the article they write this:

"I think that the framing with inverting a function f is quite underrated."

vasekrozhon9• 2 years ago

Yes! We are not trying to claim that P vs NP is underrated in the video, just the bit with interpreting it as a question about inverting functions. I think using "underrated" is justified, I talked with a bunch of people about this topic and it was a new way of looking at it for a lot of them.

Of course, using the title "What P vs NP is actually about" is a bit of a stretch -- "Interesting take on P vs NP" would be more honest.

munchler• 2 years ago

Framing this as inverting a function is interesting, but raises a question, since any function that maps multiple inputs to the same output is not invertible.

For example, let's say we invert a checker algorithm and "run it backward from YES". Does that find an arbitrary single solution? Does it find all solutions? What if there are an infinite number of solutions?

LegionMammal978• 2 years ago

"An arbitrary single solution" would be the proper answer for NP. The goal is to find some solution that the verifier accepts, like how in SAT the goal is to find some satisfying assignment for all clauses.

delusional• 2 years ago

Given that there is potentially an exponential number of solutions, it would be impossible to enumerate them all in anything smaller than exponential time.

chupasaurus• 2 years ago

There is potentially a factorial number of solutions for NP-complete problems.

cvoss• 2 years ago

In some contexts, the inverse of a function f : A -> B is defined as a function g : B -> A such that g(f(x)) = x and f(g(y)) = y for all x,y.

In other contexts, what is meant is g : B -> 2^A such that x is in g(f(x)) for all x and f(x') = f(x) for all x' in g(f(x)). Here, g is said to produce the pre-image under f, and is a generalization of the above definition of inverse for non-injective functions.

In other contexts still, what is meant is g : B -> A such that f(g(y)) = y for all y such that there exists some x with f(x) = y. This is a different generalization called a one-sided inverse. (People probably call it a left inverse or right inverse, but I can never remember which is which because it emerges from an arbitrary notational convention.)

The article uses inverse in this third sense. "Find some input to f which would yield a given output y, if one exists."

analog31• 2 years ago

Informally, an example of this problem is calculus class. A kid who's interested in programming, and takes calculus, can dream up a program that computes the derivative of any expression. But now try doing that with the anti-derivative. And to be sure you can write an expression in multiple ways, but the core problem remains.

Disclosure: I was that kid. My program was a mess, but good enough for a proof of concept.

afiodorov• 2 years ago

Idea is to add enough constraints to the function so that inverting it is non trivial (where inverts finds ANY input). Like the video said inverting multiplication with the output of 15 could yield 15 = 15 * 1, so just make the function f(a, b) = a * b where a > 1 and b > 1 for a more interesting RSA-breaking case.

vasekrozhon9• 2 years ago

Hi, a few commenters on Youtube also asked us this question. I just now added the last section to the blog post that discusses it. Basically, although we only show how to find one arbitrary solution, you can use this trick to also find all of them efficiently in terms of the size of the input and the number of solutions.

nick_g• 2 years ago

Despite being mentioned throughout the post, I don't see a link to the video in question. I saw it when it was posted a few weeks back and really enjoyed it. I would highly recommend giving it a watch, especially if the ramifications of P vs NP are new to you [1]

[1] https://www.youtube.com/watch?v=6OPsH8PK7xM

vasekrozhon9• 2 years ago

Thanks. I also added the link to the post now. Frankly, I did not count on people getting to the blog post in any other way than being curious after watching the video. :D

Pikamander2• 2 years ago

I've never understood why this problem is considered so difficult. I'm not a fancy schmancy computer scientist but even I can see that N = 1 or P = 0.

HappMacDonald• 2 years ago

even I can see that N = 1 or P = 0

wat?

Dylan16807• 2 years ago

It's a joke, they're doing algebra on the letters.

acchow• 2 years ago

I would recommend reading the article.

P vs NP is one of the foundational open problems in computer science.

dehrmann• 2 years ago

They're almost certainly not equal because solving NP in P time requires magically making exponential decisions in P time. What's interesting that it's been so hard to prove one way or the other.

seanhunter4• 2 years ago

I realise you're probably joking but on the off-chance you are not, P and NP are sets containing classes of algorithms. They aren't numbers. See https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pd... for a formal description of the problem.

kens2• 2 years ago

I wasn't expecting to see my (uncredited) die photo of the 8008 processor in the middle of an article about P vs NP!

y42• 2 years ago

just in case someone is wondering.... I assume it's this little beauty

https://www.righto.com/2016/12/die-photos-and-analysis-of_24...

vasekrozhon9• 2 years ago

Sorry, I added the credit now. Hope you don't mind it now? :)

I have to say that I find these pictures extremely aesthetical. It's crazy how the brutally powerful CPUs we have today are successors to processors created not so long ago that also, though very intricate, fit into one picture...

kens2• 2 years ago

Thanks! No worries, I just found it amusing to randomly encounter one of my die photos. I agree, it's remarkable how much complexity there is in a modern CPU compared to the early ones.

pipes2• 2 years ago

BBC "in our time" radio show did an outstanding episode on the topic. As usual it had an amazing calibre of guests but a totally understated style. Also no long pointless intro full of terrible banter or lengthy bullet list of what will be discussed. I wish podcasts could follow this example:

https://www.bbc.co.uk/programmes/b06mtms8

seanhunter4• 2 years ago

I love "In our time". When you said this I thought "I'll know if they're really serious if Tim Gowers[1] is one of the guests". Sure enough, he is.

The BBC also had him on (the Today programme iirc but it was a while ago) the one time to comment when someone claimed to have proved it.

[1] https://en.wikipedia.org/wiki/Timothy_Gowers

pipes2• 2 years ago

Thanks, it's been years since I listened to this. Now after reading about Timothy gowers I want to listen to it again.

dleink• 2 years ago

The only banter is when the producer offers tea at the end, which seems incredibly necessary to me.

antoine-levitt• 2 years ago

I just skimmed the article and didn't watch the video, but the bit about backpropagation is just wrong. Backpropagation doesn't compute an inverse of the jacobian, it computes its transpose. (although a similar idea to backpropagation could possibly be used to compute an inverse of several reversible layers, but that's not typically how neural networks work)

bigallen• 2 years ago

This is one of the many comments I see on HN where I genuinely have no idea whether it’s satire or just high-level technical talk. I assume the latter, but I don’t know enough to disprove the former

seanhunter4• 2 years ago

It's not satire. If you have some function with several inputs, the Jacobian is a matrix of all the partial derivatives of this function with respect to each input variable[1]. Since derivatives give the slope of a function, if you think about your function as being like a bumpy surface with the height at each point being the output, this matrix tells you which way (and how far) to change any input if you want to go "up" or "down" in the output.

Backpropogation is a way to optimise a neural network. You want to know how best to nudge the weights of the network to optimise some loss function, so what you do is compute the gradient (ie partial derivative) of that function with respect to each of the weights. This allows you to then tweak the weights of the function so your network gets better at whatever task you're trying to get it to learn. See [2] to understand how this works and and [3] to understand how this relates to the Jacobian, but generally if you're trying to go "downhill" in your loss function it's easy to see intuitively that knowing which way the function slopes (ie the effect of tweaking each of the weights) is important and that's what the Jacobian tells you.

The inverse of a matrix[4] and its transpose[5] are two different operations in linear algebra. Transpose turns rows into columns and columns into rows and the inverse of a matrix is a little harder to grasp maybe without background, but you could think of multiplying one matrix by the inverse of another as a little like division (since you can't actually divide matrices).[6]

[1] https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinan...

[2] https://www.youtube.com/watch?v=Ilg3gGewQ5U

[3] https://www.youtube.com/watch?v=tIeHLnjs5U8

[4] https://math.libretexts.org/Workbench/1250_Draft_4/07%3A_Mat...

[5] https://math.libretexts.org/Bookshelves/Linear_Algebra/Funda...

[6] algebraists please don't shoot me for that.

moralestapia• 2 years ago

"So, for example, if you can solve some problem \Pi by running a SAT solver ten times, this doesn’t mean that you have reduced that problem to SAT— in reduction, you can only run the SAT solver once."

This is also not true.

davmre• 2 years ago

Backprop itself doesn't invert the computation, but it does give you the direction for an incremental move towards the inverse (a 'nudge' as the article puts it). That is, given a sufficiently nice function f and an appropriate loss ||f(x) - y*||^2, gradient descent wrt x will indeed recover the inverse x* = f^{-1}(y*) since that is what minimizes the loss. I assume this what the article is pointing at.

If you want to be picky, it's true that the direct analogue of continuous optimization would be discrete optimization (integer programming, TSP, etc) rather than decision problems like SAT. But there are straightforward reductions between the two so it's common to speak of optimization problems as being in P or NP even though that's not entirely accurate.

vasekrozhon9• 2 years ago

Hi, the author of the blog post here.

The section was written horribly -- while I was talking about backpropagation, I was thinking about "what can be done in polynomial time" and there's a mismatch as you explained. Thanks and shame on me, I rewrote it.

In any case, I would recommend watching the video first, the article is just accompanied stream of consciousness for those few who really liked the video.

usednoise4sale2• 2 years ago

I know no one will believe this but I'm reasonably sure I proved P!=NP earlier this week.

It was very similar to a previous unsolved problem in a "haha, history repeats itself" sort of way.

This comment might have a lot more comments someday.

OrigamiPastrami• 2 years ago

I don't believe you, but you can always prove me wrong and share your proof.

usednoise4sale2• 2 years ago

Well, it was recent, so I'm currently in the 'feedback and validation' stage. I've reached out to experts for feedback, but these things do take time.

If you have a genuine interest, and especially if you could provide feedback or warm introductions to someone who can, I'm happy to share.

You can email me at atmanthedog at Google's free email service domain. Put something HN related in the subject. *If you do, please share your mathematical background so I can better tailor a response.

geysersam• 2 years ago

"I have discovered a truly marvelous demonstration of the proposition that P/=NP but this HN comment is too short to contain it..."

Eagerly awaiting your published proof

seanhunter4• 2 years ago

That's fantastic if true. Make sure you check out the Clay institute's page and collect your prize.[1]

[1] https://www.claymath.org/millennium/p-vs-np/ and especially https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pd...

fbarred• 2 years ago

You might avoid common pitfalls by reading "Eight Signs A Claimed P≠NP Proof Is Wrong": https://scottaaronson.blog/?p=458.

variadix• 2 years ago

Is the interesting part of P vs NP (since it seems very unlikely that P = NP) the mathematical machinery that would be required to prove that P != NP? Presumably doing so would require a way to determine a lower bound for _all_ possible algorithms that solve a particular problem.

vasekrozhon9• 2 years ago

Exactly! In theoretical computer science, we have a lot of experience coming up with new algorithms but no idea how to prove lower bounds. So I would say that new lower-bound technique is perhaps the most interesting thing we would learn from the proof of P!=NP.

But, of course, P vs NP is clearly central to computer science and math for a bunch of other reasons, hopefully, our video provided some intuitions.

bloaf• 2 years ago

For an example of harder-than-np-complete, I was shocked to find out how hard vector reachability is after being presented with a vector reachability problem and assuming I could just look up a reasonable-time algorithm.

I incorrectly assumed it would have some basic linear algebra solution because of how simple the problem seemed.

enugu• 2 years ago

This is a good insight.

To me, the essence of deep learning has nothing to do with trying to mimick biological systems or something in that sense; it’s the observation that if your circuits are continuous, there’s a clear algorithmic way of inverting/optimizing them using backpropagation.
From the perspective of someone used to algorithms like Dijkstra’s algorithm, quicksort, and so on, this declarative approach of thinking in terms of loss functions and architectures, rather than how the net is actually optimized, sounds very alien. But this is how the whole algorithmic world would look like if P equaled NP! In that world, we’d all program declaratively in Prolog and use some kind of .solve() function at the end that would internally run a fast SAT solver to solve the problem defined by our declarations.

yldedly2• 2 years ago

I think a much closer analogy to function inversion is MCMC (or Bayesian inference in general), where we can easily compute the density p(x) of any point x, but finding the x given a p(x) is intractable. Strictly speaking it's about finding a set of x's that are distributed as p(X), not finding the x given any single density p(x), but it's close.

Relatedly, probabilistic programming was originally imagined pretty much like your second quote: you define a model, get some data, run them both through the built-in inference engine, and you get the parameters of the model likely to have produced the data. In practice though, there's no universal inference engine that works for everything (some people disagree, but they're NUTS ;) I guess pretty much for the same reason P is probably not equal to NP.

vasekrozhon9• 2 years ago

I guess pretty much for the same reason P is probably not equal to NP.

Yep, in particular there are classes called #P and PP that are closely connected to NP that can capture the hardness of problems like computing partition functions, sampling from posterior distribution and so on.

vasekrozhon9• 2 years ago

Hi, the author of the blog post here.

1) With others, I work on high-effort Youtube videos, our channel's called Polylog. We recently made a video about an underrated take on P vs NP. [1]

2) After working on the high-effort video, I always write a low-effort stream-of-consiousness-style very-much-unedited blog post where I clarify some details or explain some more esoteric stuff that did not make it into the video. The blog post you are discussing is such a blog post. So you should watch the video first and read the blog post only if you really liked the video and want to know more. :)

3) If you have any questions, I'll be happy to answer them!

[1] https://www.youtube.com/watch?v=6OPsH8PK7xM

What P vs. NP is about

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