

Proving P!=NP: "...Ryan has taken the first real baby step in decades." - amichail
http://blog.computationalcomplexity.org/2010/11/breakthrough-circuit-lower-bound.html

======
chriseidhof
Could anybody with a more mathematical background explain what this means?

~~~
randomwalker
First of all this has no relevance if you're not a complexity theorist. No
"practical" impact, not even remotely. A real downer, I know, but I thought
that had to be stated up front.

The question is why are complexity theorists excited about this.

When a problem is too hard to solve, you look for easier versions to solve
first. Here the easier version is to prove that the complexity class NEXP
(which is intuitively vastly more powerful than even NP) is bigger than a
certain class of circuits of constant depth (circuits with polynomial depth
are known to be roughly equivalent to P, and constant-depth circuits are
intuitively vastly _less_ powerful).

ETA: the circuits considered here are "non-uniform," which makes the previous
paragraph slightly inaccurate. Nevertheless, the point stands that the goal
here is to separate two complexity classes that are intuitively very different
in their power.

Circuit lower bounds -- proving that a restricted class of circuits, as above,
is in fact limited in its computational power -- have been notorious for their
hardness. The reason one would want to prove this kind of statement is not
because these circuits are objects of practical interest, but because of the
_proof techniques_ that would come out as a side-effect, in the hope that
these techniques would be applicable in solving other, harder problems.

So that's the gist of it: the 3 reasons this theorem is exciting are: proof
techniques, proof techniques and proof techniques.

Now some of the statements from the post should hopefully make a lot more
sense:

 _This approach converts weak algorithms for solving circuit satisfiability
questions into circuit lower bounds. Ryan's proof doesn't use deep
mathematical techniques but rather puts together a series of known tools in
amazingly clever ways._

 _Ryan breaks through the natural proofs barrier in an interesting way. ... he
avoids the issue by using diagonalization and so his proof does not fulfill
the constructivity requirement of natural proofs._

What's going on here is that there are often meta-proofs in complexity theory
showing that a certain approach to proofs won't work. The "natural proofs
barrier" being referred to is (apparently) a limit on what you can achieve by
a certain type of "natural" construction that is explained in this post by
Lipton: [http://rjlipton.wordpress.com/2009/03/25/whos-afraid-of-
natu...](http://rjlipton.wordpress.com/2009/03/25/whos-afraid-of-natural-
proofs/)

The one thing I haven't touched upon is why there are "mod m" gates in the
circuit class under consideration. That is also (surprise, surprise) related
to proof techniques. As Luca Trevisan explains, using mod m gates instead of
binary gates or mod-prime gates disables two well-known classes of proofs
("fixing variables to random values" and "low-degree polynomial
approximation"), ensuring that some heavy artillery will need to be developed
in order to prove statements about them.
[http://lucatrevisan.wordpress.com/2010/11/08/a-circuit-
lower...](http://lucatrevisan.wordpress.com/2010/11/08/a-circuit-lower-bound-
breakthrough/)

------
foobarbazoo
"Diagonalization" strikes again.

There's a lesson here kids: when you can't prove something, "diagonilization"
is the proof-tool of choice. Sure, if you include it, you won't have proven
anything either, or useful, or correct, but you might just get a PhD.

(For those who are curious: diagonilization is a proof that takes the form:
"Imagine you did this impossible thing. With the result of doing that
impossible thing, now try and do this other, reasonable thing. You can't! So
whatever properties I can derive from not being able to do this other,
reasonable thing are true. (Please ignore that the first step is impossible,
and focus on my brilliance in discussing the second, reasonable step.)"

That is, in a nutshell, the diagonilization "proof". The rest of the details
are just there to make it look like it's actually logic or math. Best not to
look too closely.

Also, you'll be shocked, SHOCKED to learn that not a single result in the real
world relies on the diagonilization proof (though a lot of real-world PhDs
would no doubt be affected by it's loss). _sigh_ Mathematicians are idiots
when they try and do logic. Stick with numbers, boys.

