There’s a new paper on the arXiv that claims to solve a hard problem:
• Norbert Blum, A solution of the P versus NP problem.
Most papers that claim to solve hard math problems are wrong: that’s why these problems are considered hard. But these papers can still be fun to look at, at least if they’re not obviously wrong. It’s fun to hope that maybe today humanity has found another beautiful grain of truth.
I’m not an expert on the P versus NP problem, so I have no opinion on this paper. So don’t get excited: wait calmly by your radio until you hear from someone who actually works on this stuff.
I found the first paragraph interesting, though. Here it is, together with some highly non-expert commentary. Beware: everything I say could be wrong!
Understanding the power of negations is one of the most challenging problems in complexity theory. With respect to monotone Boolean functions, Razborov  was the first who could shown that the gain, if using negations, can be super-polynomial in comparision to monotone Boolean networks. Tardos  has improved this to exponential.
I guess a ‘Boolean network’ is like a machine where you feed in a string of bits and it computes new bits using the logical operations ‘and’, ‘or’ and ‘not’. If you leave out ‘not’ the Boolean network is monotone, since then making more inputs equal to 1, or ‘true’, is bound to make more of the output bits 1 as well. Blum is saying that including ‘not’ makes some computations vastly more efficient… but that this stuff is hard to understand.
For the characteristic function of an NP-complete problem like the clique function, it is widely believed that negations cannot help enough to improve the Boolean complexity from exponential to polynomial.
A bunch of nodes in a graph are a clique if each of these nodes is connected by an edge to every other. Determining whether a graph with vertices has a clique with more than nodes is a famous problem: the clique decision problem.
For example, here’s a brute-force search for a clique with at least 4 nodes:
The clique decision problem is NP-complete. This means that if you can solve it with a Boolean network whose complexity grows like some polynomial in n, then P = NP. But if you can’t, then P ≠ NP.
(Don’t ask me what the complexity of a Boolean network is; I can guess but I could get it wrong.)
I guess Blum is hinting that the best monotone Boolean network for solving the clique decision problem has a complexity that’s exponential in And then he’s saying it’s widely believed that not gates can’t reduce the complexity to a polynomial.
Since the computation of an one-tape Turing machine can be simulated by a non-monotone Boolean network of size at most the square of the number of steps [15, Ch. 3.9], a superpolynomial lower bound for the non-monotone network complexity of such a function would imply P ≠ NP.
Now he’s saying what I said earlier: if you show it’s impossible to solve the clique decision problem with any Boolean network whose complexity grows like some polynomial in n, then you’ve shown P ≠ NP. This is how Blum intends to prove P ≠ NP.
For the monotone complexity of such a function, exponential lower bounds are known [11, 3, 1, 10, 6, 8, 4, 2, 7].
Should you trust someone who claims they’ve proved P ≠ NP, but can’t manage to get their references listed in increasing order?
But until now, no one could prove a non-linear lower bound for the nonmonotone complexity of any Boolean function in NP.
That’s a great example of how helpless we are: we’ve got all these problems whose complexity should grow faster than any polynomial, and we can’t even prove their complexity grows faster than linear. Sad!
An obvious attempt to get a super-polynomial lower bound for the non-monotone complexity of the clique function could be the extension of the method which has led to the proof of an exponential lower bound of its monotone complexity. This is the so-called “method of approximation” developed by Razborov .
I don’t know about this. All I know is that Razborov and Rudich proved a whole bunch of strategies for proving P ≠ NP can’t possibly work. These strategies are called ‘natural proofs’. Here are some friendly blog articles on their result:
• Timothy Gowers, How not to prove that P is not equal to NP, 3 October 2013.
• Timothy Gowers, Razborov and Rudich’s natural proofs argument, 7 October 2013.
From these I get the impression that what Blum calls ‘Boolean networks’ may be what other people call ‘Boolean circuits’. But I could be wrong!
Razborov  has shown that his approximation method cannot be used to prove better than quadratic lower bounds for the non-monotone complexity of a Boolean function.
So, this method is unable to prove some NP problem can’t be solved in polynomial time and thus prove P ≠ NP. Bummer!
But Razborov uses a very strong distance measure in his proof for the inability of the approximation method. As elaborated in , one can use the approximation method with a weaker distance measure to prove a super-polynomial lower bound for the non-monotone complexity of a Boolean function.
This reference  is to another paper by Blum. And in the end, he claims to use similar methods to prove that the complexity of any Boolean network that solves the clique decision problem must grow faster than a polynomial.
So, if you’re trying to check his proof that P ≠ NP, you should probably start by checking that other paper!
The picture below, by Behnam Esfahbod on Wikicommons, shows the two possible scenarios. The one at left is the one Norbert Blum claims to have shown we’re in.