Do you think P=NP? Or were you just curious what these letter combinations could possibly mean in an otherwise well-constructed sentence?

Whether or not P=NP is actually one of the "big" unsolved problems in mathematics. You can be a million bucks richer by proving it one way or the other. (It's one of the seven math problems deemed worthy of a million bucks by the Clay Mathematics Institute.)

So this P=NP (or not) thing is a pretty big deal. What's it all about?

A lot has been written about this problem, and if you want to get up to speed about it, the Wiki entry is not a bad place to start. But I'll try my best to give you a non-technical introduction that will make you care. Because the question that was first asked by the mathematician and computer scientist Stephen Cook is profound. I'm not kidding here: Mathematics, beauty, and Rachmaninov's second piano concerto may be involved. Well, at least, in this post.

I want you to think about two different things. Two seemingly completely unrelated things.

First thing: How long does it take to find the answer to difficult problems? Of course, putting it like this is not precise enough. So let's be more specific. How hard is it to find one rare thing in the midst of $N$ not so rare things?

"Well, it depends", you would answer, and you would be right. It depends on a lot of things. For example, suppose I want you to find a red dot in the middle of very many black dots. That's easy, you can do this visually in no time flat, because the red dot sticks out (left panel below). If the dot you are supposed to find is not so easy to distinguish, then it might take you a while longer (blue dot in the right panel below). If I asked you instead to tell me whether or not any of the dots are different at all (and I don't tell you what the difference is) you may have to look at every dot individually. And if there are $N$ dots, that's how long it would take you on average.

Easy Search Less Easy Search |

But problems of size $N$ (problems that take $N$ steps to solve) are still in the class of easy problems. A little harder are problems that take polynomial time to solve (like $N^2$ steps, or even $N^{10,000}$).

Problems of this type are said to belong to the class "P", where P stands for polynomial. They are (for the most part) easy for a computer. Things get problematic when the $N$ is not in the base, but in the exponent. Like, $2^N$ or $e^N$. The exponential function rises sooo much faster than any polynomial, that the difference between the two (polynomial versus exponential) is not one of degrees. It is one of substance.

Now, the Second thing: Suppose somebody hands you a sheet of paper and claims: "THIS is the solution to this extremely hard problem you have been talking about".

How long will it take you to confirm (or ridicule) said claim? You would think: "Pretty fast I gather", and you would be right (most of the time). In the example of the search above, if I tell you that the red dot is THAT ONE in the upper left corner, all you would have to say is: "OK, right, I see it". For the second panel, it's in the lower right corner (the blue dot). See, it's so much easier if I show you the solution!

There is a name for problems that are really hard, but for which any proposed solution can be checked really quickly. This class of problems is called "NP": the class of problems for which we can verify a solution in polynomial time, but may not be able to

*find*it in polynomial time.
<nerdy comment> Strictly speaking, NP stands for "non-deterministic polynomial". It means that a computer very unlike the one you read this on (a non-deterministic one) could solve the problem in polynomial time. An unpredictable computer, so to speak. Maybe a quantum one. Maybe that has to remain the subject of a different post. One that speculates about whether it is obvious that there must be a classical algorithm that factorizes primes in polynomial time. </end nerdy comment>

Let's just think of the problems in the class NP as those that we have no idea how to solve in polynomial time. I'm oversimplifying here, but this is a blog, not a textbook.

So, I can imagine that you would immediately agree that checking a solution is a hell of a lot faster than finding a solution. Because that makes sense.

We indeed believe that's true. But we do not know.

Let me give you an example hard problem to illustrate all this. It's kind of the grand-daddy of hard problems, perhaps in part because it is the one that Stephen Cook used to formulate the problem. In fact, he showed that if you could solve

*that*problem, then you could solve many many others right away, because this grand-daddy is part of a class of problems that can all be related to each other (called the NP-complete problems). This is cool because it means that if you've solved*any*of the problems in the set in polynomial time, then you've solved them all in polynomial time.
This grand-daddy of problems is the 3-SAT problem, and I urge you to be awake through my description of it. (All other descriptions would put you to sleep much faster anyway.) If you can get through this, I promise that there are symphonies ahead.

"Do I really have to endure a 30 seconds lecture about 3-SAT?"

You brush your teeth, don't you? That's 60 seconds right there! (If not, you should reconsider your brushing routine. Just sayin'.)

So here it goes. 30 seconds.

You know about logic already. If A and B are two variables that can be either TRUE or FALSE, then you know whether A$\vee B$ is TRUE or FALSE, depending on the truth value of A and B. Now let's get C into it. I can now ask, for example, whether A$\vee$B$\vee\neg$ C is true. This $\neg$ symbol, by the way, is just the logical NOT. So, if A is TRUE, then $\neg$A is FALSE. And $\vee$ stands for OR, of course, while $\wedge$ simply means AND.

Well, you say, obviously, this clause A$\vee$B$\vee\neg$ C is TRUE if any of A or B is true, or C is false. Indeed, to check a single clause, that's easy. But what if we take two clauses (each made from three variables (three, get it, 3-SAT?) and ask whether the AND of these two clauses is true. Like, for example, (A$\vee$B$\vee\neg$ C)$\wedge (\neg$ B$\vee$C$\vee\neg$ D).

"Well OK", you say, now I have to determine whether each of the clauses is TRUE to figure out whether the AND of the two clauses is true. And I see what you did there: in any of the clauses, variables can reoccur. "And you're down 15 seconds, by the way. "

All right. Now imagine I have a given number of variables A,B,C, to whatever. Like N of them. And I make tons of clauses from them (each of them ORs of three variables), ANDed together. And I call the resulting expression a "statement". And I ask you the all-important question:

"Could the resulting statement ever be TRUE?"

It could look like this:

(R$\vee$T$\vee\neg$ C)$\wedge (\neg$ B$\vee$C$\vee\neg$ D)$\wedge(\neg$ W$\vee$B$\vee\neg$ M)$\wedge$(F$\vee\neg$ G$\vee$ D)$\wedge$(W$\vee$T$\vee\neg $R)

Or it could be a lot longer. Asking whether the statement can ever be TRUE is asking whether the statement can be "satisfied". And because the statement is composed of clauses with three variables, it is the "3-Satisfiability" problem: 3-SAT.

Now, I don't expect you to be able to answer this question just by looking at the statement. And, I also realize that my 30 seconds are up. But you could of course just test it. If there are $N$ variables, you just have to test $2^N$ possible cominations of TRUE and FALSE for these variables, plug them into the statement, and check if it ever returns 1. But you understand immediately that this will take a long time if $N$ is large. An exponential amount of time, because $2^N$ depends on $N$ exponentially.

(Really it does. Because $2^N=e^{\ln(2) N}$. If you are more comfortable with exponentials to the base e.)

Now, let me imagine for a moment that you have been thinking about this for longer than the 30 seconds it took you to read through this simple example. Please, let me indulge in that thought just for the sake of it.

You may have come up with this simple thought:

"What if..... every variable appears only once in any clause? Shouldn't it be obvious that some combination will result in a TRUE statement?"

Yes. Yes indeed, that is quite right. I'm impressed.

"But wait, wait. What if there are only, like, two variables A and B, but a gazillion clauses. Isn't it obvious that there are bound to be contradictions between clauses so that not all clauses could possibly be right at the same time (kind of like A and $\neg A$ cannot be both be TRUE at the same time! Am I right? Am I?"

Gosh, you. You're like, totally right. If that happens (a small number of variables, a large number of clauses) you can fairly quickly rule out that the statement can ever be TRUE. Indeed, figuring out whether a statement of many clauses but a small number of variables, or a large number of variables and just a few clauses, is usually quick and easy.

For anything inbetween, however, we think it'll take checking essentially all $2^N$ to figure out satisfiability. Sorry. Unless, of course, there is a trick nobody has yet thought of. An algorithm of unheard of power.

Remember when I wrote that I want you to think about two things? Two very different things?

The first was: How long does it take to solve a really hard problem? In the case of 3-SAT, we think it'll take an "NP" amount of time, on average. NP: a very very long time. Possibly as long as the universe has been around (for big $N$.). But we can check any proposed solution quickly.

Just imagine: someone gives you a sequence of variables A,B,C, etc. and claims that

"A=TRUE, B=FALSE, C=FALSE.... will satisfy the statement!"

You can check that in no time at all, by just plugging it into the formula. VoilĂ ! Either that someone was right, or not. The time to check any proposed solution is polynomial in the problem size, that is, it is P.

Is P (time to check a solution) equal to NP (time to find a solution)? That's preposterous right? "No way!" you exclaim. Yet, nobody has ever been able to prove that this is not so. People have tried, and even claimed they succeeded.

So we have to entertain, at this point in time, the possibility that P=NP, as preposterous as that may sound. Because the people who have been trying to prove the opposite aren't exactly amateurs. There is, after all, a million bucks riding on it. So could it be that P=NP? What would the world be like it if it was true?

First, let us think about this: Is anything like this (P=NP) even possible at all? It would mean that we can create rare things that are "TRUE" (by some measure) as easily as we can check that any thing is TRUE (by some measure). Let's go back to 3-SAT to see what this means. Let's imagine that we deal with a hard case of 3-SAT: that there is actually only a single lonesome set of assignements to the variables that results in the statement to be TRUE. In that case, the statement is satisfiable, but proving it will require finding the proverbial needle in a haystack. We know that we can check every proposition quickly. But if we don't have any way of guesssing what a good solution is, then we have to check all of them. Checking each may be quick. Checking all is not, because there are an exponential number of candidates.

How could we find this one needle in a haystack? Is something like this even thinkable in principle?

It is indeed! Think of a different problem: creating a poignant and meaningful piece of poetry.

"That's ridiculous!", you exclaim, "How can you liken a creative process such as writing poetry to an automated procedure?"

Hold on there for a moment. Take you favorite piece of poetry. Think about it. Recite it in your head.

What, you don't have one? Really, you should spend a bit more time away from the computer, smell the roses, so to speak. What about this one, about the thing with feathers? No? What about Hamlet's soliloquy? Well, you should have a favorite piece of poetry, is all I'm saying. I have more, but I can tell you're getting tired of this discussion already.

Now, given the medium in which those pieces of poetry that I referred to are rendered (sequences of letters drawn from an alphabet of 26 symbols, plus the space), they are extraordinarily rare. I think I can even trust you to calculate how rare, if you imagine the "blind monkey" way of writing a poetry on a type writer. You know, the kind that (theoretically speaking) can reproduce all the works of Shakespeare when randomly hacking away at a keyboard.

The chance of even producing a Shakespearean sonnet by this method before the sun engulfs this planet are pretty much nil, you can imagine. That's if the monkey types completely randomly. Now, what if it's not completely random? What if, say, you write random 9 letter sequences, and keep all those that match any word (or combination of words) that Shakespeare ever wrote?

Well, that would be an algorithm that still runs for a long time, but it can actually produce a Shakespeare sonnet! The one called "A Lover's Complaint", no less. How is this possible?

The answer is that any deviation from a random search is an algorithm, a heuristic. Heuristics can accelerate search at sometimes formidable rates. The one I described is, when you think about it, an evolutionary one. You keep the good stuff (even if it was produced by a random process), and you try, try, again.

Yes it is true, evolution is an algorithm--a heuristic--that can produce complexity from a diet of random chance. I'm pretty sure you're with me on this. But there are plenty of other algorithms. For example (for the problem of reconstituting Shakespeare's writings) you could bias the way letters are produced, for example by imposing blanks between letters at the rate dictated by the average length of words in English.

"This is cheating!", you might exclaim. But that is precisely what an algorithm is: finding ways to cheat, so that you don't have to examine every possible solution.

In the light of this, we can now formulate the central problem of P=NP.

Are there ways to cheat so that we can find the truly hard stuff fairly quickly? For example, as quickly as we can check whether the proposed solution really is a solution?

Let's use the art metaphor one more time. There are "solutions" (time series of notes) that we call music that are generally acknowledged to be "good". I'm sure you can be with me here: this is neither well-defined nor do I think it can be. (This is not stopping us from trying: stay tuned for my lab's efforts in this direction). And not everybody agrees. If there was an obvious algorithm for good music, then we would not be paying successful composers the money they earn. "Not everybody can be a Mozart", is what we say.

Think about it: we (usually) can tell fairly quickly whether a piece of music is any good. Can we write such a piece as quickly?

Now, we don't usually think of our musical appreciation abilities as algorithmic, but perhaps we should. The generation of music can be algorithmic (and in fact the generation of music via evolutionary algorithms was the subject of a recent article within the Proceedings of the National Academy of Sciences). And yours truly wrote a commentary about that article here. But I digress. As is my wont.

If P=NP, then there must be an algorithm that creates beatutiful music, beautiful poetry, beautiful fiction, you name it. Of course, just because we haven't found it (trust me, we haven't), that doesn't mean it does not exist. Can a machine be programmed to produce these kind of rare pieces?

The answer to the latter question is, of course, "We don't know". All I know is that it is unlikely that a computational algorithm (the way we understand it) can produce Rachmaninov's second piano concerto, for example. I have a personal view about this, because I've worked on his 2nd (and to a lesser extent 3rd) concerto since 1982. But I digress as usual.

I'm not arguing here for a non-algorithmic view of human performance or ingenuity. What I am suggesting is that the origin of these rare achievements is found in the history of the machine that produced it (the personal life experience of the composer, here, Sergej). Mathematical algorithms do not take experiences into account: they produce solutions from given input data. But the human algorithm takes a lot more input into account: this input data could be essentially infinite (our experience of the world up to the present). Even more, the algorithm people use is not deterministic, in the sense that no two people are alike. We get a "solution" very very rarely. Some personal histories, blended with exquisite mental algorithms operating within that person's brain, can produce unmatched beauty, once in a while.

So where does this leave us? NP is probably not P, because if it was then there would be no place for genius in this world. Genius is an algorithm that takes advantage of history: the accumulated experiences (guided by curiosity) that make up a person's baggage. It's good that NP does not equal P. Life, the world, the universe, would be boring if NP=P. It's not a proof, and I don't expect to receive a check in the mail for it. But I'm fairly confident that I'm right about that.

"This is cheating!", you might exclaim. But that is precisely what an algorithm is: finding ways to cheat, so that you don't have to examine every possible solution.

In the light of this, we can now formulate the central problem of P=NP.

Are there ways to cheat so that we can find the truly hard stuff fairly quickly? For example, as quickly as we can check whether the proposed solution really is a solution?

Let's use the art metaphor one more time. There are "solutions" (time series of notes) that we call music that are generally acknowledged to be "good". I'm sure you can be with me here: this is neither well-defined nor do I think it can be. (This is not stopping us from trying: stay tuned for my lab's efforts in this direction). And not everybody agrees. If there was an obvious algorithm for good music, then we would not be paying successful composers the money they earn. "Not everybody can be a Mozart", is what we say.

Think about it: we (usually) can tell fairly quickly whether a piece of music is any good. Can we write such a piece as quickly?

Every fibre in our body says: "Hell no!"

Now, we don't usually think of our musical appreciation abilities as algorithmic, but perhaps we should. The generation of music can be algorithmic (and in fact the generation of music via evolutionary algorithms was the subject of a recent article within the Proceedings of the National Academy of Sciences). And yours truly wrote a commentary about that article here. But I digress. As is my wont.

If P=NP, then there must be an algorithm that creates beatutiful music, beautiful poetry, beautiful fiction, you name it. Of course, just because we haven't found it (trust me, we haven't), that doesn't mean it does not exist. Can a machine be programmed to produce these kind of rare pieces?

The answer to the latter question is, of course, "We don't know". All I know is that it is unlikely that a computational algorithm (the way we understand it) can produce Rachmaninov's second piano concerto, for example. I have a personal view about this, because I've worked on his 2nd (and to a lesser extent 3rd) concerto since 1982. But I digress as usual.

Sergej Rachmaninov (1873-1943)

So where does this leave us? NP is probably not P, because if it was then there would be no place for genius in this world. Genius is an algorithm that takes advantage of history: the accumulated experiences (guided by curiosity) that make up a person's baggage. It's good that NP does not equal P. Life, the world, the universe, would be boring if NP=P. It's not a proof, and I don't expect to receive a check in the mail for it. But I'm fairly confident that I'm right about that.

"We get a 'solution' very very rarely."

ReplyDeleteWell said, though I might put it slightly differently.

We very rarely use algorithms/methods which actually 'solve' problems, at least for the general case of the problem. We (and evolution, and machine learning, and...) generally come up with heuristics and various tricks which get a *good enough answer most of the time*.

The term of art is "probably approximately correct" (PAC), and I think that the PAC framework is actually rather profound.

Anyway, I'm not sure there is a direct implication for P=NP (which I strongly suspect is false). Maybe that, as far as we know, nature has been really good at finding P (or small N) approximations when it is faced with an NP problem. Seems like a rather practical approach for us to try and emulate.

I couldn't agree more.

ReplyDelete