The problem of whether P=NP is very closely related to the question of whether there is any problem in NP that cannot be solved with a (family of) polynomial-sized circuit(s). A circuit is a network of AND, OR and NOT gates that process an input (a string of n 0s and 1s) and produce an output (either 0 or 1). It is fairly easy to prove that if a function from to $\{0,1\}$ can be computed in polynomial time, then there is a circuit of polynomial size that will compute that function as well. This means that a potential way to prove that P does not equal NP is to prove that some function in NP cannot be computed by a polynomial-sized circuit. And this is useful because circuits are easier to contemplate from a purely combinatorial point of view than algorithms or Turing machines.

Now one might think that if the function f is monotone, which means that the set of input strings x such that f(x)=1 is closed under changing coordinates from 0 to 1, then there would be no advantage in using NOT gates (though in fact this supposition is known to be wrong). At any rate, it is interesting to try to obtain circuit lower bounds when NOT gates are not allowed. This was for a long time as wide open as the general problem, until it was spectacularly solved by Razborov. It is quite easy to reformulate the problem as the question about unions and intersections that I put at the beginning of the document.

So the short answer to why the result is interesting is that it is a very natural weakening of a statement that would imply that P does not equal NP.

Unfortunately, the insights of Razborov and Rudich and others have told us that, impressive as the result is, it is unlikely to represent a significant step towards a proof that P does not equal NP. Nevertheless, it does hugely improve our understanding of the problem.

But it was only after following the lectures that I realised that monotone circuit complexity had something to do with these problems. I would have been much more excited about the lectures had I been inducted into the significance of the result – perhaps a few comments on the blog as a reminder to amateurs like me as to what it is all about??

Thank you for posting video of your computational complexity course. I am a graduate student in the US, and while I have never had the opportunity to sit in on your courses, I have studied your notes on Combinatorial Number Theory, as well as several of the other expository notes on your website. I find your courses (and notes) to be very interesting and comprehensible. As my graduate program offers very few topics courses, I was hopeful that I could persuade you to consider posting video of future courses on the web as well. While I can understand why some faculty members wouldn’t want to do this, I would also encourage anyone else reading this with the resources to to do so to consider posting video of topics courses!

As an aside, I also need to say that the video production was particularly well done in this case. Most taped math lectures I have seen suffer from bad audio and an inability to read what the lecturer is writing on the board. These were very well done!

I think that, while getting to the bottom of X might look like a purely expository task, it is probably a very important thing to do. This is because new theorems usually come from understanding basic things very well, and not the other way around. In my case, I owe my whole research program to a few months spent trying (successfully!) to get to the bottom of something important and basic but fundamentally simple and often passed over.

The reason why I mention this here is that I think open exposition problems and things we need to get to the bottom of are often indistinguishable in the beginning. So, even if our measure of value is only progress in research, I would still put a very high value on solving open exposition problems.

For this particular problem, I have always been very fond of the Berg-Ulfberg proof: pdf.

My current pet peeve in this regard is the definition of the determinant. I don’t know of a good reference that thoroughly explains how the permutation expansion of the determinant could be written down by someone who’s never seen it before. And while we’re on the subject, I also don’t know a satisfying theoretical reason why determinants should be so absurdly useful in algebraic combinatorics (Gessel-Viennot, the method of Pfaffians).

Incidentally, the idea of “giving a plausible account of how a proof could have been discovered” reminds me of a famous story of Borges, whose title I forget, in which his character decides to take it upon himself to independently write the text of Don Quixote. Fortunately, the exposition problem seems nowhere near that hard. Maybe once I’ve digested the meta-problem of how to write an expository paper of this nature, I’ll give it a try myself.

In the meantime, I vaguely remember liking the proof Berg and Ulfberg gave:

http://scholar.google.com/scholar?q=berg+ulfberg+monotone&hl=en&lr=&btnG=Search