How the Web Rallied To Review the P != NP Claim 160
An anonymous reader writes "Remember, about a month ago, when a researcher claimed he had a proof that P != NP? Well, the proof hasn't held up. But blogs and news sites helped spur a massive, open, collaborative effort on the Internet to understand the paper and to see if its ideas could be extended. This article explains what happened, how the proof was supposed to work, and why it failed."
Nerd Superbowl (Score:4, Interesting)
Re:Nerd Superbowl (Score:4, Funny)
I think Mr. Venkatasubramanian is just overgeneralizing from his personal experience. No one interacts with him at conferences because they can't pronounce his name.
Re:Nerd Superbowl (Score:4, Funny)
Yeah. It'd be easier and cooler if he shortened it to Venkman [wikipedia.org].
Re:Nerd Superbowl (Score:5, Funny)
Yeah, but nobody scored...
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There's an extraneous word in your post.
“It was like the Nerd Superbowl."
Yeah, nobody scored.
QED
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Yeah, but nobody scored...
>sniff<... wasn't it beautiful?
The greatest gift (Score:5, Insightful)
No matter the flaws with his paper, this guy has certainly managed to inspire a whole lot of people to delve into a subject and collaborate on it.
Those who think deep thoughts are precious. Those who manage to inspire thousands of others to do so...
OT: sig comment (Score:5, Funny)
People replying to my sig annoy me. That's why I change it all the time.
Time to change again.
Re:The greatest gift (Score:5, Informative)
A simpler proof? Please? (Score:2)
Many of the fundamental proofs in this area aren't so difficult to understand. Certainly in computing theory classes, proofs were generally a page or two and didn't involve (much) advanced math.
Maybe it's just me, but it "feels" like there should be a simpler way to go about showing that P != NP.
Re:A simpler proof? Please? (Score:5, Insightful)
there should be a simpler way to go about showing that P != NP
that simpler way would only exist if P = NP
Re:A simpler proof? Please? (Score:5, Informative)
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This is definitely the kind of thing complexity theorists say. The argument fails entirely because it relies on our intuition of what easy means, and easy is not the same as polynomial, so transferring the intuition is almost intentionally misleading (not that I'm blaming you). It is basing a serious argument on an non-serious characterization of polynomial=easy that it used to help out-siders who don't know what polynomial means to never the less appreciate somewhat what complexity theory is about. Even i
Re:A simpler proof? Please? (Score:4, Interesting)
This is definitely the kind of thing complexity theorists say. The argument fails entirely because it relies on our intuition of what easy means, and easy is not the same as polynomial, so transferring the intuition is almost intentionally misleading (not that I'm blaming you). It is basing a serious argument on an non-serious characterization of polynomial=easy that it used to help out-siders who don't know what polynomial means to never the less appreciate somewhat what complexity theory is about.
I think it's the opposite.
It's basing a non-serious argument on a serious characterization of the mathematical notion of complexity.
It's the original question, "why isn't there a simple proof of P != NP?" that is based on the layman's notion of "easy".
That the answer replaces the vague notion of "easy", with the accurately defined term "polynomial", and replaces the specific "is there an easy answer for this proof?" with the more general "proofs are NP-complete, and so we can expect it to be more complex than polynomial, assuming the thing we're trying prove is true", is not a failing of the answerer.
It's also not the failing of the questioner for being a layman. The point is, sometimes the correct answer to a question can't be put in the terms you want it to be and must, in essence, answer a different question. There are only two correct answers to the original query: "The proof of P!=NP is in the class of NP-complete problems", and "We won't know until we find it (or find the proof that P=NP)".
I personally feel one of the two conveys more useful information.
If someone asked you "How long will it take me to solve a specific but unknown instance of the Traveling Salesman problem?", you could either say: "The Traveling Salesman problem is in general NP-complete, so probably a long time", or you could say "Give me the problem and I'll let you know when I've solved it."
Since the search to find the actual solution, and thus as a side effect figure out how complex it is, is currently underway and in fact the topic of this discussion, that in the interim leaves only one useful answer.
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given a well-behaved axiomatic system A, questions of the form "Is there a proof of statement s from axioms in A with the proof length at most k?
Didn't Gödel prove that you can't prove this or any statement like this?
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He proved there are statements you can't prove (or even express). He didn't prove that this was one of them.
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Didn't Gödel prove that you can't prove this or any statement like this?
No, the statement given looks at bounded proof lengths (that is proofs of at most some length). Those can be listed completely up to any given bound. What Godel's shows you is that you can't in general ask "is there a proof of statement s from axioms in A" but the class here is "Is there a proof of statement s from axioms in A with the proof length at most k?" which is much easier to answer.
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No wonder Gödel killed himself.
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that simpler way would only exist if P = NP
Why? I can only guess your reasoning. Correct me if I am wrong:
"Proving P=NP can be accomplished by finding a polynomial time algorithm to the NP-hard problem of your choice. You give the problem, the algorithm, you prove is correct and that is poly-time. Success.
Proving P!=NP is the same (by definition) that proving that there is a problem in NP for wich there is no poly-time algorithm that solves it. Infinite problems and infinite algorithms... looks that proving no matching exists is necessarily hard."
I
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But if you found it, this experience shows it would take thousands of people to figure out if it was correct, so P != NP.
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So.....the simpler way doesn't exist, and therefore P != NP
DONE and DONE
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Or when P = 0.
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Considering that Wiles's proof for Fermat's Last Theorem, which is a number theory problem that can be trivially stated, was ridiculously complex and used some crazy maths that weren't even discovered in Fermat's time, I don't think you can really estimate the size of a proof by the complexity of the problem stated.
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But we don't know that the current proof is the *only* proof. There may very well be a simpler one out there.
As for the problem simplicity vs. the proof simplicity, that's not what I said. I stated that related problems (in the same field) have simple proofs.
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He made it up.
Or made a misstake and realized it later.
Fact is that _after_ the famous "not enough space here for the solution" stuff he still did quite some work so proof a _very_ limited subset of the last theorem.
Which does not make any sense if he had had a proof for the superset long before.
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It could make sense because the method of the proof in the limited case is in fact very interesting (the infinite descent). Yours is not a sufficient argument.
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some proofs are simple, sure, but look up busy beaver stuff to see some loooong proofs.
In cs courses they tend to stick to the most short and sweet proofs simply because the long ones would be unintelligible to almost everyone.
Re:A simpler proof? Please? (Score:5, Informative)
I don't think you can really estimate the size of a proof by the complexity of the problem stated.
You are correct that you cannot. In fact, this is a consequence of Godel's theorem. Proof sketch: Assume we have some nice axiomatic system A, that can model the arithmetic of the natural numbers (say Peano arithmetic), and assume that this system is not stupid (axioms are recursively enumerable, valid proofs are recursively enumerable, system is consistent. I think that's all I need but there may be some other silly issues). Assume that there is a computable function f, such that any true statement in A of length n has a proof of length at most f(n). Then I claim that we can use this to resolve whether any given statement has a proof in A by looking at all proofs of length up to f(n). This contradicts standard corollaries of Godel's theorem. So no such f can exist. Thus, minimum proof length for some statements must be much longer than the length of the statements.
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Many of the fundamental proofs in this area aren't so difficult to understand. Certainly in computing theory classes, proofs were generally a page or two and didn't involve (much) advanced math.
Maybe it's just me, but it "feels" like there should be a simpler way to go about showing that P != NP.
You "feel"? If there has even been a most unsubstantiated and unscientific subjective expression of feelings over fact, this is it.
Re:A simpler proof? Please? (Score:5, Insightful)
Science may lead to facts, but it's not an automated process. Believe it or not, human emotions and intuition are involved with every scientific discovery!
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Yes, but they needed to be removed from any rigorous study.
Nature doesn't care how you feel..ever.
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Science may lead to facts, but it's not an automated process. Believe it or not, human emotions and intuition are involved with every scientific discovery!
Perhaps so, but human emotions and intuitions behind great discoveries or at least serious attempts at scientific discoveries are based on evidence that suggest the belief is in the right track.
When it comes to your belief that a proof of NP != P should be simple, what do you base it on? You would have done a much better service to your hypothesis by giving concrete examples of this instead of mentioning the existence of beliefs and emotions in the scientific process and thus feel scientific by proxy.
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And you know this from the experience of all the scientific discoveries you made? Come on. Read any biography or autobiography about scientists or mathematicians, and you'll see they're not robots devoid of emotions.
Damn... (Score:5, Funny)
Step #1: Wait for him to prove and confirm P!=NP
Step #2: Solve for N:
So P!=NP,
therefore P!/P=N,
thus the Ps cancel and we are left with N=!.
Step #3: ???
Step #4: Profit!
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Step #3: ???
Step #4: Profit!
Boy, that joke just never gets old, does it?
Ah, but what if it had held up??? (Score:4, Insightful)
We would be reading this instead:
"Remember, about a month ago, when a researcher claimed he had a proof that P != NP? Well, after a month of vigorous examination by ordinary netizens and Nobel-prize-winning mathematicians, it looks like it's going to hold up. Blogs and news sites helped spur a massive, open, collaborative effort on the Internet to understand the paper and to see if its ideas could be extended. This article explains what happened, how the proof works, and the holes experts and laymen attempted to punch in it and why the proof is still standing."
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Nobody who can add to the P != NP discussion intelligently is an ordinary netizens*.
Look, no one without an interest in math gave a hoot. In fact pretty much any advanced math problem is going to be self selected to be pretty much all people who can contribute.
Had it been about Vitamin D research, then having it open would have been a nightmare.
*And that is the stupidest word to come out of the internet, far far worse the blogosphere.
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Nobody who can add to the P != NP discussion intelligently is an ordinary netizens*.
Look, no one without an interest in math gave a hoot. In fact pretty much any advanced math problem is going to be self selected to be pretty much all people who can contribute.
Had it been about Vitamin D research, then having it open would have been a nightmare.
*And that is the stupidest word to come out of the internet, far far worse the blogosphere.
I didn't follow this as closely as I should have, but I do recall that the wiki for this proof was actually edited by many non-mathematicians. People corrected the English in the proof, fixed various grammatical mistakes, and clarified ideas to make them more precise or simple.
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If it had held up, someone would have already set about producing a computing system that was capable of constructing all proofs and all complex structures of everything, and formatting and submitting them as patents.
Many of these would be business models and means of winning elections regardless of public opinion.
Within a few years, our legislative and economic systems would be taken over by the people operating the machine, and they would change the law and, legally, make us their slaves.
You might say I'm
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Yeah, instead what we read is:
"I have fixed all the issues that were raised about the preliminary version in a revised manuscript; clarified some concepts; and obtained simpler proofs of several claims. Once I hear back from the journal as part of due process, I will put up the final version on this website." - http://www.hpl.hp.com/personal/Vinay_Deolalikar/
Oh hey, that's practically the same thing.
Great story (Score:4, Insightful)
This has been one of the best slashdot posts in a long, long while.
I'm gonna have to renew my subscription to Science News. Kudos to Ms. Rehmeyer.
Not so great article (Score:2)
The first paragraph of the article is just nonsense. It claims that if we knew P=NP "computers would acquire mind-boggling powers such as near-perfect translation, ...". Wow! Imagine that! All we have to do is prove a theorem and suddenly we can write amazingly fast programs. But of course, we could just *assume* that P=NP and write the same programs. All a proof would do is give us some hope (or fear) that various problems would turn out to be more tractable than otherwise.
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She elaborates later on that, if P=NP, the proof will provide general hint in transforming NP into P whose solution can be more easily found, hence the "mind-boggling" possibilities of solving many problems currently thought infeasible to solve.
"Assuming" P=NP doesn't help because we don't know in general how to transform NP into P or if it's even possible. In fact, it's suspected it's impossible so one will never likely even to try.
Um... pretty old news, folks. (Score:2)
I mean really. Nothing new here, folks. Move along now.
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"It's new to me, and ti's new to a lot of other people. so STFU or get out."
That's pretty funny. Did I stop you from reading TFA? Or any of the other comments? I think it's pretty amusing to draw this kind of reaction for simply saying that this happened a month ago.
But you are free to skip my comments if you want to. Nobody is making you read them. So if you don't like them, it's you who can GTFO.
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"Nothing here... move along" is a stock humorous/sarcastic phrase, in case you didn't know. I was not really telling anybody to do anything.
But even if I were, it was not intended to be either literal or nasty. People are completely free to read whatever they want. They certainly aren't obligated to take my advice. On the other hand, also in case you did not know, "STFU" is hardly helpful or humorous. It's generally cons
Obvious or oblivious? (Score:3, Funny)
It is the greatest question in computer science. A negative answer would likely give a fundamentally deeper understanding of the nature of computation. And a positive answer would transform our world: Computers would acquire mind-boggling powers such as near-perfect translation, speech recognition and object identification; the hardest questions in mathematics would melt like butter under computation’s power; and current computer security methods would be as easy to crack as a TSA-approved suitcase lock.
Proof that P!=NP: We haven't made any really hard problems really easy. If P=NP, then computers automatically acquire mind-boggling powers and the ability to crack encryption. Presumably that would have already happened if P=NP, therefor P!=NP. QED.
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It is the greatest question in computer science. A negative answer would likely give a fundamentally deeper understanding of the nature of computation. And a positive answer would transform our world: Computers would acquire mind-boggling powers such as near-perfect translation, speech recognition and object identification; the hardest questions in mathematics would melt like butter under computation’s power; and current computer security methods would be as easy to crack as a TSA-approved suitcase lock.
Proof that P!=NP: We haven't made any really hard problems really easy. If P=NP, then computers automatically acquire mind-boggling powers and the ability to crack encryption. Presumably that would have already happened if P=NP, therefor P!=NP. QED.
Following that logic, the world was flat until Galileo stated otherwise.
Really.... we humans are ignorant enough that we could be footling around with a limited set of algorithms for aeons and not stumble on any that solve NP problems. This isn't QED, it just shows that our set of known algorithms is painfully limited.
Think about Quicksort: that's an algorithm that really opened my eyes to P=?NP, as it showed how you could really optimize a sort by limiting the scope of the elements being sorted. I had ne
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Following that logic, the world was flat until Galileo stated otherwise.
While I agree with the rest your post, the middle ages were not quite as dark as you think. It was pretty well known at the time of Galileo, that he earth was not flat. Actually that was known since the time of the greeks (and Eratosthenes even calculated the diameter pretty accurately in about 300 BC). Aristoteles was a proponent of a spherical earth, and Aristoteles was *the* most known philosopher during the middle ages, and much of his philosophy blended into christianity so a spherical earth was never
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Yeah... or there is something else we haven't figured out.
They are looking for mathematical proof.
If you have 3 oranges, and one goes away you will have 2 oranges QED. That does NOT answer why one orange went away.
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This is not a perfect comparison, but consider that before Calculus, or rather the fundamental theorem, integrating was inexact, clunky, and was a very difficult problem that was impossible to give an exact answer to. But with the FTC it became possible to
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Dude, do you have any idea what a "proof" is?
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And that's interesting. There's a disconnect of sorts between formal logic and reason, tied in to the difference between deductive and inductive methods. Of course a proof is nice, and a necessary part of advancing mathematics; but a person would not be in error for believing N != NP. At some point a preponderance of evidence, or probability, or reasoning by analog
To summarize where the proof went wrong... (Score:5, Informative)
To summarize what difficulty the proof ran into: There's a general class of NP-complete problems known as SAT. SAT is essentially given a collection of Boolean variables (so can have values "yes" or "no") and given some logical statement of those variables is there an assignment to those variables that makes the statement true? So for example, for A ^ ~ A, there isn't one, but for say A v B there are satisfactory solutions. This problem is the canonical NP-complete problem. Now, the attempted proof examined k-SAT, which is a subset of SAT known to also be NP-complete. k-SAT is the same thing as SAT but each statement must be a sequence of ands containing k inputs into set of ors. So for example if one was looking at 3-SAT "(A v B v ~ C) ^ (A v A v ~D)" would be a valid example. Now, it happens that for k>2, k-SAT is NP-complete. Deolalikar tried to examine the statistical properties of k-SAT and derive a contradiction from the assumption that k-SAT was polynomial time solvable. However, this runs into issues because from a statistical perspective 2-SAT is known to look statistically more or less the same as k-SAT, and 2-SAT is polynomial time solvable. This is a deep barrier which the proof did not overcome.
There are other deep barriers that the paper did not obviously overcome, including what is known as the "natural proof" barrier and the "relativization" barrier. The last essentially says that P=NP is true in some other computing models very similar to the standard Turing model (you consider Turing machines with special black boxes called oracles attached which answer specific questions quickly.) Similarly, it turns out that P != NP for some oracles as well. Thus, any valid resolution of P=NP will have to break down in some more or less obvious way when one tries to run the proof through for an oracle machine. If one can't point to where in a proof this would occur, this is a good indication that the proof is not valid.
Overall, I'm highly pessimistic that we are going to resolve P=NP anytime soon although I strongly believe that P != NP. There are currently much weaker claims than P=NP that we still cannot prove. We can't as far as I'm aware even get a strongly non-trivial result of the form for some explicit constant C, "No NP complete problem can be solved in polynomial time with a polynomial of degree at most C." And that's much weaker than showing that P != NP, because P !=NP is essentially that statement made for any value of C. We seem to need serious new insights and possibly lots of new machinery and structures before we can have a really good chance at cracking this nut.
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Re:To summarize where the proof went wrong... (Score:5, Funny)
Math is hard, but some types of math are really hard.
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Re:To summarize where the proof went wrong... (Score:4, Interesting)
The proof tried to show that 3-SAT is not solvable in polynomial time. But the same techniques (if valid) would have also proven that 2-SAT (a simpler version of 3-SAT) is not solvable in polynomial time. That's a problem since we already have techniques for solving 2-SAT in polynomial time. In general if your proof technique can be used to prove something known to not be true, then your proof technique is broken.
The "relativization" barrier is similar. In trying to prove "P!=NP", it is really easy to also end up accidentally "proving" for certain oracles "X" that "P^X!=NP^X" when we already know that for those oracles "P^X=NP^X". The converse is also true: In trying to prove "P=NP" it is really easy to also end up accidentally "proving" for certain oracles "Y" that "P^Y=NP^Y" when we already know that for those oracles "P^Y!=NP^Y".
Re:To summarize where the proof went wrong... (Score:5, Funny)
So for example if one was looking at 3-SAT "(A v B v ~ C) ^ (A v A v ~D)" would be a valid example. Now, it happens that for k>2, k-SAT is NP-complete.
Oh, that explains it.
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Wow, yeah... ASCII doesn't lend itself to symbolic logic at all. I don't suppose you can clarify.
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Could this problem just be the CompSci version of Godel's Theorem, or Heisenberg Uncertainty principle, etc.? I bet Douglas Hofstadter could make a book out of it...
"A self-referential definition, by definition, is not a definition".
This is likely a dumb question but... (Score:2)
... given that SAT already is NP complete, doesn't that already prove P!=NP? For P to be equal to NP, it has to be equal for SAT too, which it isn't.
So isn't it then coming down to: WHICH classifications can be made, so P==NP for class X and P!=NP for class Y?
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NP is a set of problem. The main issue is about two subsets of NP: P and NP-complete. We have lots of examples of problems in each subset. We think that these subsets are disjoint, but no one has proven that fact. We have proven that if they are not disjoint then they are equal. Therefore one way to prove that they are equal is to show a problem is both in P and NP-complete.
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I'm curious to know the best lower bound for any decision problem. I don't even know any good lower bounds for problems where the output must be at most linear in the size of the input.
maybe not proven, but seems obvious (Score:2, Flamebait)
In C code:
choice1="P";
choice2="NP";
if (choice1 != choice2)
yeahbaby++;
Submit that! Science, math, logic... it's just too easy
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C with first-class strings? Yeah, right. ;-)
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You will pay a million bucks for code that doesn't work correctly? shit, if I work for you I would have all the money in the world!
One problem in the article for me... (Score:2)
He translated the problem of P versus NP out of computer science entirely and into the world of formal logic, using an area known as "finite model theory" that has produced remarkable results.
*face hits palm*
Computer science uses formal logic in it's proofs all the time (at least as formal as mathematics).
For example, choose k logical requirements at random and ask: What is the probability that there is some binary number of n digits that will satisfy them all? If the number of requirements is huge (i.e., k is large) and the number of possible solutions is tiny (i.e., n is small), then of course there will never be correct solutions, no matter how long the problem is calculated.
This too is a facepalming moment. It's akin to saying "If you flip a coin 100 times, of course it will land on heads at least once." Except that a probability of 1/(2^100) != 0.
slashdot has confusing hyperlinks in its summaries (Score:5, Interesting)
This summary had three hyperlinks:
1. "claimed he had a proof"
2. "hasn't held up"
3. "spur a massive effort"
It was missing the only IMPORTANT hyperlink:
4. "this article". --- The slashdot submission was about an article. I'd like to read the article. I'd like a hyperlink which unambiguously takes me to the article. As it was, I didn't know which of the hyperlinks would take me to the article.
1. "claimed he had a proof" -- did this hyperlink take me to his claim? No: it took me to a online collaborative discussion of his claim (i.e. the original slashdot article).
2. "didn't hold up" -- did this hyperlink take me to the announcement that it didn't hold up? No: it took me to a slashdot article that maybe had a link to the statement about how it didn't hold up.
3. "spur a massive effort" -- did this hyperlink take me to that effort? Or did it take me to the spur in question? No: it took me to a REVIEW of that effort.
The hyperlinks in Slashdot summaries are always confusing.
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I too would like the actual article that he seems to refer to.
It refers to the last link [sciencenews.org] in the summary, the only non-Slashdot link. I figured that out by going to the original submission [slashdot.org].
Linking to the original submission is one nice thing that Slashdot does right. I fully agree with the grandparent that Slashdot often has confusing links in their summaries.
Pi, what a waste of time (Score:2, Insightful)
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Take a CAD program and figure the area of a circle of radius one. In some cases you get an interesting value of pi.
--
pi = 22/7. 22/7 repeats. Therefore pi is not transcendental.
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I can't imagine why you'd ever stop thinking about it.
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"For 99.9% of the public, 3.14xxx is good enough."
Yep. That's what I use, although I have trouble finding the 'x' key on most calculators.
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That's why I use "159" in place of those three 'x' characters. It actually seems to help somehow.
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There are infinitely many digits of pi, and they don't repeat, FYI. Therefore, each new digit is new information of a sort, and there are people who think we might learn something from the statistical probabilities of the digits and subsequences.
As far as wasting computer resources, do you know anything we have more of, relative to a few decades ago? There's several desktop, laptop, and netbook computers in this house, not to mention a couple of iPhones. The iPhones have more power than 1980 supercomp
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My Kurt Reply: (Score:2)
You Sir, are a Gödeless troglodyte.
Loutish Word of the Day: "Diaeresis"
Re:So... we disproved P != NP (Score:5, Funny)
P!=NP
(P-1)! * P=NP
N=(P-1)!
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Technically, if P = 1, (P-1)! will equate to 1 (as 0! = 1), and therefore P will be equal to NP.
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No, but it does prove that you've been Whooshed!
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God.
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Well, the author hasn't been convinced by the crowd:
"Deolalikar hasn't withdrawn his article. He claims to have fixed the errors and will be submitting the revised article to a journal to go through an ordinary peer review process."
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If he was really interesting in making it accurate then he'd allow the public to see the revised paper and attempt to pick holes in it again.
unless of course he just cares more about not being proved wrong than being right.
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he's also interested in getting as much of the credit as possible. Clearly he thinks that he is using the best strategy of crowdsourcing vs. proprietary approach.
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It seems that saying it as P!=NP is just too geeky.
There is no such thing as "too geeky". Especially here.
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Except that the NP-Complete portion of NP have been found to be translatable to one another in polynomial time. So a very interesting (likely the most interesting) portion of NP would be known to be P with a basically common solution if a polynomial solution for one of them was found in the general case.
NP-Complete includes, for example, the Traveling Salesman Problem, the Bin-Packing (or Knapsack) Problem, Trench Cutting Problem, multiprocessor scheduling, microcode bit optimization, and job shop schedulin