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A 5 years old did much much better some centuries ago, his first name was Wolfgang. I really don't think that the Arts can be produced by AIs, simply because art is about human feelings and communication of those subtle feelings.

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This is about the most important question of all mathematical questions!

I personally doubt that this is the answer but it is worth reading.

I personally doubt that this is the answer but it is worth reading.

**P ≠ NP?**

+Alok Tiwari pointed out a new paper by Norbert Blum, which claims to solve a famous math problem. So if this paper is wrong, don't blame me. Blame him. ʘ‿ʘ

• Norbert Blum, A solution of the P versus NP problem, https://arxiv.org/abs/1708.03486.

Just kidding! Most papers that claim to solve hard math problems are wrong: that's why these problems are considered hard. Alok Tiwari knows this.

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 = 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 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 [12] was the first who could shown that the gain, if using negations, can be super-polynomial in comparision to monotone Boolean networks. Tardos [16] 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. The author 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 n vertices has a clique with more than k nodes is a famous problem: the

**clique decision problem**.

The clique decision problem is

**NP-complete**. This means, among other things, that if you can't solve it with any Boolean network whose complexity grows like some polynomial in n, then P ≠ NP.

(Don't ask me what the complexity of a Boolean network is.)

I guess Blum is hinting that the best

*monotone*Boolean network for solving the clique decision problem has a complexity that's exponential in n. 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 [11].*

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. So he's a smart cookie.

*Razborov [13] 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 a problem can't be solved in polynomial time. Bummer!

*But Razborov uses a very strong distance measure in his proof for the inability of the approximation method. As elaborated in [5], 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 [5] 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.

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Awesome and never seen before drone footage of Versailles Palace and gardens. A must see!

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Bubbles & human stupidity! Greed doesn't lead to anything positive.

Beware!

Beware!

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For already many years Carl Cox creates elegant and creative musical textures that are really enjoyable & danceable !

#electro

#carlcox

Global Underground 038: Black Rock Desert https://g.co/kgs/JJg4N6

#electro

#carlcox

Global Underground 038: Black Rock Desert https://g.co/kgs/JJg4N6

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Jesus Christ was an ultra pacifist so please stop this insanity to pretend that God would in any manner give a green light or whatsoever message to this mister Donald. This is truly unbearable and totally shameful to pretend this. Please leave JC out of all this. This adviser must be a complete fake evangelist, literally!

‘God has given Trump authority to take out Kim Jong Un,’ evangelical adviser says

http://wapo.st/2umZe47

#shameonyou

#fakegodmessages

‘God has given Trump authority to take out Kim Jong Un,’ evangelical adviser says

http://wapo.st/2umZe47

#shameonyou

#fakegodmessages

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This vision of the brilliant Prokofiev's piano concertos is simply the best I ever listen to. Those recordings give the right level of musical attention that this masterpieces compositions deserve.

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Very nice!

Miniature Scenes Set Amongst Office Supplies by Derrick Lin

http://www.thisiscolossal.com/2017/08/miniature-scenes-set-amongst-office-supplies-by-derrick-lin/

http://www.thisiscolossal.com/2017/08/miniature-scenes-set-amongst-office-supplies-by-derrick-lin/

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