Norbert Blum on P versus NP

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 [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. 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 n vertices has a clique with more than k 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 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. 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 [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 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 [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 we’re in.

57 Responses to Norbert Blum on P versus NP

  1. Bruce Smith says:

    I found at least one error in your commentary: you defined the “circuit complexity” of a boolean network, not the “algorithmic complexity”, but P != NP is about algorithmic complexity. (The circuit complexity is when you let the network be constructed arbitrarily as a function of the number of inputs n; the algorithmic complexity is when the network has to follow some pattern that would let a Turing machine construct it “reasonably fast” (I’ll defer to experts for a precise definition of “fast”).)

    So the equivalence you state is only an implication, since algorithmic complexity is greater than or equal to circuit complexity. In other words, being able to get the optimal circuit (for a given number of inputs) “from God”, rather than having to construct it using a fixed algorithm independent of n, can (sometimes) give you an advantage in its size.

    (I haven’t yet read the rest of your post. I haven’t yet noticed an error in the paper itself (but haven’t looked very hard, or beyond its page 10).)

    • Bruce Smith says:

      After reading your next paragraph, I take it back — you don’t claim to have defined the “complexity of a Boolean network”, therefore you haven’t been specific enough to make the error I claimed. Normally that complexity is just defined as the number of nodes in the network (sometimes not counting NOT gates, but that doesn’t matter if you’re only worried about “polynomial time or not”) — I had just assumed you meant that definition when I made my prior remark.

      • Bruce Smith says:

        But maybe it’s still worth being more specific about your not-quite-error: using “number of nodes in the network” for “complexity”, this statement is not correct:

        This means that if you can solve it with a Boolean network whose complexity grows like some polynomial in n, then P = NP.

        The reason is that it’s possible in principle that you can solve it with a small network like that, but there is no fast-enough algorithm for constructing that network (given n), so an algorithm to solve the same problem might still have to be superpolynomial time in n.

        If Blum is correct, then, he has proved something significantly stronger than P != NP (though it’s a well-known target for people interested in trying to prove that).

      • John Baez says:

        Bruce wrote:

        After reading your next paragraph, I take it back — you don’t claim to have defined the “complexity of a Boolean network”, therefore you haven’t been specific enough to make the error I claimed.

        Good! I don’t remember doing anything like defining this. In fact I wrote:

        Don’t ask me what the complexity of a Boolean network is; I can guess but I could get it wrong.

  2. Bruce Smith says:

    I finished your post. (Yes, ‘boolean network’ is just another name for ‘boolean circuit’.)

    Note that Blum does claim (on page 4 of his paper) that his proof method is not ruled out by the “natural proofs” limitation, due to the fact that his method can (in principle) only be applied to monotone functions (even though it can be applied to non-monotone circuits used to compute those functions).

  3. gowers says:

    It looks to me as though the paper gets a very low crackpot-index score (where high => crackpot). Of course, that doesn’t mean that it is correct, but I think it may reach the level where the experts feel that it needs to be checked.

    • John Baez says:

      For what it’s worth, the author is a chair of the computer science department at the University of Bonn, and reference [5], the paper the current one relies on, was published in a somewhat reputable place:

      • Norbert Blum, On negations in Boolean networks, in Efficient Algorithms: Essays Dedicated to Kurt Mehlhorn on the Occasion of His 60th Birthday, Lecture Notes in Computer Science 5760 (2009), Springer, Berlin, pp. 18–29.

      On G+, Boyd Smith wrote:

      You definitely need to start with the earlier paper. If these “approximators” can rigorously establish lower bounds, then even if the bounds mentioned in this paper are subtly wrong, “approximators” would be quite the tool in establishing the separation of NP-Complete and P.

      Unfortunately this earlier paper is not open-access. Springer will sell it to you for €24.95. (That sounds so much more tempting than €25!)

  4. Igel says:

    Should you trust someone who claims they’ve proved P ≠ NP, but can’t manage to get their references listed in increasing order?

    The order of the references is increasing by year of publication.

  5. Graham Jones says:

    I was waiting calmly by my radio, and heard instead that Michael Atiyah has a 12-page proof of Feit and Thompson’s odd order theorem. A 5-minute interview begins at about 2:49:00 here: I can’t find a paper.

  6. […] Si quieres la opinión de otro inexperto en complejidad computacional, puedes leer a John Carlos Baez, “Norbert Blum on P versus NP,” Azimuth, 15 Aug 2017. […]

  7. I haven’t seen any big name in the field dismiss this paper, so chances are good that it’s sound. Also, there’s glory and money on the line for such a publication ($1M if not mistaken with glory being a sufficient motivator); the author probably was in a hurry to get it out being excited about it (I also suspect he believed long ago he could solve it, hence he worked patiently alone without collaborators as he didn’t find that necessary). The last thing that was on his mind is minor language typos (there are some) and precise ordering of references. Note that the list of references is rather short for an academic paper; I interpret that to mean “only what I really needed to get this done will be referenced”.

    I am not from this field but came close to joining in (Ph.D. drop out from U. of Toronto 30 years ago, where Cook worked and Razborov and Rudich came to visit) and have no understanding of the technical arguments presented.

    • John Baez says:

      Thanks! By the way, my remark about the ordering of references was a joke: I don’t really care much about the ordering of references, and I would never use that as a way to judge the mathematical content of a paper.

  8. Gustav Nordh says:

    I’m confused by the following point in the paper:
    On p. 27 there is a description of a DNF’ representation of a standard network (De Morgan circuit).
    On p. 29 Thm. 5, it is claimed that if C is standard network computing a monotone function f, then all monomials contained in DNF'(C) are
    implicants of f.

    I fail to see this. For example, let f be the monotone function computed by the circuit C = (x or y) and (not x or y) and (x or not y).
    If I’ve understood the construction correctly, DNF'(C) = x or y or xy, but obviously neither x nor y is an implicant of f.

    • Karl Wimmer says:

      To add to this, I don’t see why, from the first two paragraphs of the proof of Theorem 5, we can conclude that DNF'(g_0) computes f. I see only some sort of one-sided-ness that DNF'(g_0) computes a function such that f = 1 implies that this function is also 1.

      The third paragraph doesn’t help me either: surely DNF'(g_0) and its DNF/CNF-switch compute the same function, but it does not immediately follow that the DNF/CNF-switch computes f (because DNF'(g_0) might not), so we can’t make any conclusions about f-clauses.

      (Aside: this one-sided-ness is consistent with Gustav’s example above.)

      From a different viewpoint, surely a standard network computing a monotone function could compute non-monotone functions at internal nodes. Theorem 5 doesn’t apply to non-monotone functions, so DNF'(g) might not correctly compute the sub-function in the network whose output node is g (which will happen for many non-monotone functions). Because of this, I’m not convinced that this inductive construction of DNF'(g_0) will necessarily be correct in the end.

      If I’m totally off-base here, please let me know!

    • Todd Trimble says:

      Others have picked up on this comment: see here and here.

  9. […] Baez has very quickly put together a post explaining the very basics of Blum’s argument.  Even more briefly, Blum claims to have shown that the best-case complexity of a function solving […]

  10. @whut says:

    This is one of those Millennium Problems, right?

    Just blogged a post this morning on another Millennium Problem – this regarding Navier-Stokes:

    The key is to constrain the premise and then solve an applied science problem

  11. wb says:

    Scott Aaronson makes an argument on his latest blog post that this proof is most probably wrong.

    • John Baez says:

      Thanks for pointing that out. Here is Aaronson’s argument:

      Unrelated Update: To everyone who keeps asking me about the “new” P≠NP proof: I’d again bet $200,000 that the proof won’t stand, except that the last time I tried that, it didn’t achieve its purpose, which was to get people to stop asking me about it. So: the paper claims to give an exponential circuit lower bound for Andreev’s function, but that function as defined in Section 7 of the paper seems clearly to have a polynomial-size circuit, based on polynomial interpolation (thanks to Luca Trevisan for this observation). So I don’t see how this can possibly stand. Please just stop asking, and if the thing hasn’t been fully refuted by the end of the week, you can come back and tell me I was a closed-minded fool.

      Over on Facebook, Luca Trevisan wrote:

      Andreev’s function, which is claimed to have superpolynomial circuit complexity (abstract, then section 7), is just univariate polynomial interpolation in a finite field, which, if I am not missing something, is solvable by Gaussian elimination.

      Alas, I have no understanding of this issue.

      • Todd Trimble says:

        Luca seems to have retracted this particular counterargument in the comment section of the FB post.

      • John Baez says:

        Yes, Luca Trevisan wrote:

        You are right, guys, I misunderstood the definition of Andreev’s function: it’s not clear that it reduces to polynomial interpolation.

        Scott Aaronson has removed his mention of Luca Trevisan’s objection from his blog.

  12. Bruce Bartlett says:

    Thanks for this.

  13. Stefan P. says:

    Further discussions are at . They are currently a better typesetted version of the comments here, but that might still be useful.

  14. Bruce Smith says:

    If anyone is looking for “realtime” informed discussion of this paper — the main place I can find is the stackexchange post referenced by Todd Trimble above: .

    There is also an older opinion on Quora at , and a discussion at . These are a few hours less up to date, but might continue to receive new comments.

    Does anyone know of other places?

  15. MWL says:

    I would like to point out that one of Norbert Blum’s co-workers in Bonn, Mathias Hauptmann, has published a preprint on the arxiv that separates P from a class in the polynomial hierarchy. In other words, the polynomial hierarchy does not collapse.

    • Mathias Hauptmann, On alternation and the union theorem.

    I can by no means judge this proof. But since Norbert Blum’s proof raises so much scrutiny, his co-worker Hauptmann’s proof is probably now of broader interest to you.

    • John Baez says:

      Wow, I would never have understood the (potential) importance of this paper from its abstract! Thanks!

      • anonymous 2 says:

        Yes, the abstract clearly implies PH does not collapse and P!=PSPACE, which is a spectacular result in its own right if correct. So that paper’s getting no attention and hasn’t been submitted for publication is a sign that it hasn’t stood up to scrutiny. It’s been up since Feb 2016 or so.

        • Markus Blumenstock says:

          Err…doesn’t Hauptmann’s purported P \neq \Sigma_2^p imply P \neq NP?
          If P \neq \Sigma_2^p, then P \neq PH (as P \subseteq \Sigma_2^p \subseteq PH).
          If P \neq PH, then P \neq NP (see book of Arora & Barak, Theorem 5.4, 2., use contraposition).

  16. vznvzn says:

    strangely Hauptmann and Blum are at the same university. here was some reaction to Hauptmanns proof which experts didnt take seriously at the time, and still feel it needs/ deserves more attn from experts (over a year later). in contrast Blums proof has gone viral at this point. a bit amused/ delighted, my hits for this old Hauptmann claim are burning up lately. a lot from Germany :)

    Blum’s attack is in a very nice area of complexity theory, circuit complexity theory, that i have been studying for over a decade, and have written in my blog years ago that it may be one of the most viable attack directions on P vs NP. would like to see a collaborative attack in this area, polymath style. in any case the review of this Blum proof is already a natural polymath-oriented effort. if only it could be sustained/built on further as world experts on this topic coalesce/huddle. great to see cyberteamwork in action at times. <3 :) :cool: :star:

    am planning to blog on this soon, a bit overwhelmed this moment/morning though.

    ps hi gowers! small world! :) :P

  17. […] Normbert Blum argumenta que la mejor red booleana monotona para resolver el problema del clique está acotada exponenancialmente por abajo y que ese límite también aplica a las redes no monótonas. Eso implicaría que P≠NP  según el. Ya han salido diversos científicos cuestionando su forma de demostrar el problema, como John Baez. […]

  18. […] are following an ongoing discussion on StackExchange and in comments to a post by Luca Trevisan, a post by John Baez, and a Hacker News thread, among several other […]

  19. vloodin says:

    Tardos’ example seems to refute the proof.

    • Todd Trimble says:

      There is some commentary on this at the CS Theory Stackexchange thread, here and here. So there is growing consensus that the proof is cooked; if so, this confirms Scott Aaronson’s prediction that the purported proof would be refuted by the end of the week.

      • Larson says:

        It is now very clear that the proof is wrong. See for an exposition on the fatal error which was exposed.

      • John Baez says:

        Thanks! Yes, Razborov is a god of complexity theory, so this report by Mikhail if true is enough for me:

        I am familiar with Alexander Razborov whose previous work is extremely crucial and serves as a foundation for Blum’s proof. I had the good luck of meeting him today and wasted no time in asking for his opinion on this whole matter, on whether he had even seen the proof or not and what are his thoughts about it if he did.

        To my surprise, he replied that he indeed was aware of Blum’s paper but didn’t care to read it initially. But as more fame was given to it, he did get a chance to read it and detected a flaw immediately: namely that the reasonings given by Berg and Ulfberg hold perfectly for the function of Tardos, and since this is so, Blum’s proof is necessarily incorrect as it contradicts the core of Theorem 6 in his paper.

        Idolvon’s remark pointing out in detail “the single error” is also worth reading, though I’m not technically qualified to evaluate it.

    • John Baez says:

      vloodin wrote:

      Tardos’ example seems to refute the proof.

      By the way, there’s now a Wikipedia article on this graph invariant introduced by Éva Tardos in 1988:

      • Wikipedia, Tardos function.

  20. […] been covered at length elsewhere: see especially Luca’s post, Dick Lipton’s post, John Baez’s post, and the CS Theory StackExchange […]

  21. Bruce Smith says:

    This comment on Scott Aaronson’s blog says that Norbert Blum has retracted his claim:

    • John Baez says:

      Yes, Norbert Blum has added this comment on the arXiv:

      The proof is wrong. I shall elaborate precisely what the mistake is. For doing this, I need some time. I shall put the explanation on my homepage.

      So this story is done… for now! I hope everyone trying to prove P = NP is now aware of the Tardos function, the example which could have let Norbert Blum know his proof could not possibly work. There’s now a Wikipedia entry on it:

      • Wikipeda, Tardos function.

      On G+, Timothy Gowers wrote:

      The difference between Norbert Blum and a crank

      One obvious difference is that Blum has already established himself as an extremely reputable researcher: he has published serious results in the area and is very well acquainted with all the literature about techniques that cannot work to prove that P≠NP, giving careful explanations of why they did not apply to his proof attempt. But even stronger evidence of non-crankdom is that when people pointed out that his proof couldn’t work, instead of clinging to it, he retracted it, and now promises to write a detailed account of what is wrong. Similar classy behaviour was exhibited by Edward Nelson a few years ago when his attempted proof of the inconsistency of ZFC turned out to contain a somewhat subtle error.

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