Let be a Boolean function of arguments. What is the smallest circuit that can compute if the set of allowed gates consists of all the unary and binary functions? The size of a circuit is simply the number of gates in it.
It’s easy to show that a random function requires circuits of size (this is tight in the worst case), but a big question in computational complexity is to provide a simple enough function that requires large enough circuits to compute.
Basically, if one comes up with a function that lies in and requires circuits of size super-polynomial in , then , and one of the Millenium Problems is solved!
How far are we from accomplishing this? Well, until recently, the best lower bound for a function from NP was [Blum 1984] (remember that eventually we are aiming at super-polynomial lower bounds!).
I’m very excited to report that very recently a better lower bound has been proved by Magnus Gausdal Find, Sasha Golovnev, Edward Hirsch and Sasha Kulikov! Is it super-polynomial in ? Is it at least super-linear in ?
Well, not really. The new lower bound is ! You may ask: why one should be excited about such a seemingly weak improvement (which, nevertheless, took the authors several years of hard work, as far as I know)?
One reason is that this is the first progress on one of the central questions in computational complexity in more than 30 years. In general, researchers like to invent new models and problems and tend to forget good old “core” questions very easily. I’m happy to see this not being the case here.
Besides, the new result is, in fact, much more general than say the Blum’s. The authors prove that any function that is not constant on every sufficiently high-dimensional affine subspace requires large circuits. Then, they simply invoke the recent result of Ben-Sasson and Kopparty, who construct an explicit function with this property. That is, the paper shows a certain pseudo-randomness property to be enough for better lower bounds. Hopefully, there will be more developments in this direction.
Is this result one more step towards proving ? Time will tell.