Talk:P versus NP problem/Archive 1

message: P = NP problem - Wikipedia, the free encyclopedia
1 Context of the problem; 2 Formal definitions for P and NP ... and tM(w) = number of steps M takes to halt on input w. ..... in polynomial time is b bits long, the above algorithm will try 2b-1 other programs first. ... The Journal of the Operational Research Society 34 (10): 927–934. doi:10.2307/2580891. ...
http://en.wikipedia.org/wiki/P_%3D_NP_problem
The above is the 4th search result here:
link  —Preceding unsigned comment added by 64.183.31.242 (talk) 00:28, 14 July 2009 (UTC)
thread_title:  New discussion 

message: I'm reluctant to contribute to the Wikipedia and hack someone else's text, but I'd like
to point out that prime factorization and NP-completeness don't have anything to do with
each other (or at least, nobody to my knowledge has shown that they do).  The article gives the misleading impression that if someone were to prove P=NP, cryto-algorithms that rely on the hardness of factorizing prime numbers would be in jeapordy.  It would be better, IMO, to stick to propositional SAT to illustrate the difference between P and NP. (as per the article on NP-completeness, which, it seems to me could well be subsumed in the discussion of the two complexity classes.
BTW, I rather disagree that a proof that P = NP would have no practical consequences, but that's another discussion.
- Andre Vellino (vellino@sympatico.ca)
:The article mentions factorization only as a simple and hopefully intuitive example of a problem in NP; it also says explicitly that it is unknown whether the problem is in P. The concept of NP-completeness is not introduced until several paragraphs later. So the article does not suggest a connection between factorization and NP-completeness. AxelBoldt
At the risk of reviving old discussion, I'd like to set the record straight here: If someone proved P=NP then those crypto algorithms WOULD be in jeopardy. Factorization is in NP, even if it's not proven to be NP-Complete, if P=NP then everything in NP, NP-Complete or not, is poly-time solvable. A poly-time algorithm to factor would destroy the assumption that factoring is effectively a one-way problem that is made by public key cryptosystems. 74.12.143.44 (talk) 08:14, 26 November 2007 (UTC)
thread_title:  Old discussion 

message: This question was asked on http://www.slashdot.org a little while ago, and it seems relevant here: What, if any, would be the effects of a proof that P=NP, or that P≠NP, or indeed that the question is undecidable?
- Stuart
The latter two wouldn't have any practical consequences, but it would be a big deal in the world of theoretical computer science (and the last also one in logic and mathematics). If somebody proves P=NP, then the practical consequences depend on the way they prove it. If they explicitly exhibit a polynomial algorithm for an NP complete problem, then we would know polynomial algorithms for all NP problems, and if the degrees of the involved polynomials are reasonable, that would be huge news in many fields, but is considered extremely unlikely. It's much more likely that either the polynomial has a ridiculously high degree, or that the proof wouldn't be constructive and just generally concludes P=NP without exhibiting any algorithm altogether. In those two cases, there wouldn't be any immediate practical consequences either. User:AxelBoldt
I agree with your conclusion.  However, it isn't really possible for there to be a nonconstructive proof of P=NP with no known algorithm.  That's because the construction has already been done.  I could easily give you an algorithm for, say, Travelling Salesman, whose correctness is obvious, but whose asymptotic running time is unknown.  It's easy to prove that if P=NP, then my algorithm is polytime.  If anyone ever publishes a "nonconstructive" proof of P=NP, the construction will already exist.  This is all academic, of course.  The huge multiplicative constant means none of this would have any practical value.  --LC
That's very interesting. How does the construction work? User:AxelBoldt
Assume the goal is a program that, given a TSP instance, returns the certificate (if "yes") or fails to halt (if "no").  Assume you already have a program to check certificates.  Let X=1. Consider the list of all possible programs, sorted first by size, then lexicographically.  Run the first X programs for X steps each, with the TSP instance as an input.  If any of them halts and returns a certificate, check it.  If it's right, halt and return it.  If not, continue.  If none of them succeed, then increment X, and repeat.  Clearly, this algorithm is correct, and its theta running time will be the optimal running time, plus the time to check a certificate.  The constant hidden by the theta is exponential in the length of the first program that solves the problem.  Totally impractical.  I don't remember offhand which books cover this, though I could look it up if needed.  It's been a while since I've visited Wikipedia; it's nice to see that it's still going strong.  --LC
The description is clear, I don't think we need a reference, but I think it's nice enough to put it in the main article. User:AxelBoldt
I've added it, in a form that's hopefully accessible to a wider audience.  --LC
Can you also concoct an algorithm which always gives the correct answer in polynomial time? User:AxelBoldt
Yes, if you can tell me the running time of the first algorithm (or give a polynomial upper bound for it).  Otherwise, I doubt it's possible now.  It's too bad that we can construct this algorithm for NP-complete problems, but not for co-NP-complete problems.  --LC
But then my original statement stands: if we find a non-constructive proof of P=NP, then we still wouldn't have a polynomial algorithm for NP problems, since a polynomial algorithm has to stop after p(n) steps. User:AxelBoldt
I should have said "accept" rather than "solve".  Sorry for the confusion.  If we could construct a particular algorithm, and prove that it is a polytime algorithm accepting a particular language in NP-complete, then we would have proved P=NP.  That would be a constructive proof of P=NP.  If we could prove P=NP without knowing any polytime algorithm accepting an NP-complete language, then that would be a nonconstructive proof.  The latter is impossible.  As soon as someone proves P=NP, we'll immediately know a polytime algorithm that accepts an NP-complete language.  It's the algorithm I gave.  Note that by the standard definition, an algorithm is a polytime algorithm accepting a language if it returns "YES" in polytime on strings in the language, and never halts on strings outside the language.  I'll change the page to say it "accepts the language SUBSET-SUM", rather than "solves the problem SUBSET-SUM".  Actually, I think I'll add a second algorithm too.  This algorithm (under stronger conditions) yields a stronger result.  If there are any algorithms that can be proved to solve (not accept) an NP-complete problem, then this is one of them.  I think that's an interesting result too, so I'll write it up later tonight or tomorrow.  --LC
----
From the article:
:The integer factorization problem is this: given two natural numbers x and y, with n digits each, decide whether x can be evenly divided by
 an integer between 1 and y.
I think the previous example of simply deciding whether a given number is prime was simpler and more intuitive. The main point here is to get the point across that "solving is harder than checking". Was there a reason for changing it? (Also, not both x and y have to have n digits.) AxelBoldt
----
The problem of finding the best move in Chess or Go (on an n by n board) is EXPTIME-Complete
Is this right?  Go has a strictly higher complexity class than Chess.  Isn't Go PSPACE-complete and chess is EXPTIME-complete? - anonymous
:Go with Ko (Japanese rules) is EXPTIME-complete.  Go with other rules might also be EXPTIME-complete, but is only known to be PSPACE-hard.  See this, which was also in the references at the bottom of the article. --LC 18:04 Sep 13, 2002 (UTC)
----
I've also removed this:
*It ignores problems that can be well-approximated.  There are some NP-complete problems where good approximations can be found quickly.
P and NP and NP-complete only contain decision problems, and you can't approximate a decision problem.  Approximations are only possible for problems in classes such as NP-equivalent.  --LC 18:11 Sep 13, 2002 (UTC)
----
What is non-P? This web site says non-P is different from NP. -Jeff
Non-P consists of all those decision problems that cannot be solved in polynomial time. That is indeed very different from NP. AxelBoldt 00:42 Jan 24, 2003 (UTC)
----
I'm a bit confused regarding NP-complete...is it purely a matter of probability? If P intersected with NPC yields the empty set (for certain) but it is uncertain whether P=NP, either NPC=empty set or P does not equal NP. Am i misunderstanding this?
:I'm not sure whether you misunderstand or not.  NPC is not the empty set (satisifiability and many other problems are known to be in NPC).  So if the intersection of P and NPC is empty, problems in NPC are in NP but not in P.   This would imply P!=NP.  However, the great unsolved question is whether the intersection of P and NPC is empty or not.  If you can prove one way or the other, you'd have solved a Millennium Prize challenge and thus be up for a $1 million USD prize, in due course receive a Turing award and maybe even a Fields medal, get tenure at any university computer science department in the world, possibly have the National Security Agency rather eager to talk to you, and be batting off Hollywood starlets of the appropriate gender wherever you travelled.  Well, maybe not the Hollywood starlet part.  --Robert Merkel 04:42, 19 Feb 2005 (UTC)
thread_title:  Effects of P=NP 

message: This statement has recently been added regarding quantum computers: ...none of the problems they are known to be able to solve are NP-hard.
Is this true?
* I thought factoring was NP-hard.
* Shor's algorithm enables factoring in polynomial time on a quantum computer.
* It's even been implemented on a system of (I think) 7 qubits, where it successfully factored the number 15.
So wouldn't that mean that quantum computers are known to be able to solve an NP-hard problem?
RSpeer 22:52, May 6, 2005 (UTC)
:Factoring probably isn't NP hard. It's in NP intersect co-NP, which is strictly outside NP-hard provided that NP ≠ coNP, which most people believe is true. Deco 23:58, 6 May 2005 (UTC)
thread_title:  Quantum computers and NP-hard problems 

message: Before the article stated that EXPTIME-complete problems require exponential time. This is as incorrect as saying that NP-complete problems require nondeterminism to be solved in polynomial time; while this is believed to be true, it need not be (for example, it may be the case that all problems in EXPTIME can actually be solved in O(nlog n) time). The reason we know that EXPTIME-complete has no intersection with P is because P is not equal to EXP, which requires a proof by diagonalization and is not at all obvious. Deco 00:32, 8 Jun 2005 (UTC)
thread_title:  Small EXPTIME-complete accuracy fix 

message: I added a brief discussion of the oracle result regarding the problem. I tried to keep it simple, but if anyone thinks it can be made easier to understand, I welcome your changes. Deco 07:52, 15 Jun 2005 (UTC)
thread_title:  Oracle discussion 

message: The inclusion of the "......" is interesting.172.191.9.162 (talk) 23:12, 4 May 2009 (UTC)
thread_title:  Why don't we do checks like this to confirm disputes in the article? 

message: Please let me add I simply ask this comment remain here for discussion purposes ONLY for the improvement of this article. You see I really want this to be a great article and I see how complex and abstract the article is it is a bit dense and I would like this here just to show people how some non-traditional content for this article may look. There are so many cool ideas in P Versus NP like how it would revolutionize music potentially and I think someone should mention THIS at least in the article! Things like how it would automate poetry because of the potential mathematical origins of prose to Haiku's may apply to such abstract things as even paintings or scientific discoveries or maybe even movies! Imagine, you've heard the expression, "I thought the movie was a bit formulaic. Imagine if there was actually a FORMULA for movies and we could write it in an equation and plug it into an engine and have it spit out screenplays! This is the type of positive forward thinking content that SHOULD BE   
 > Sum 1. say:    
 > Truth: what becomes true in its construction to be construction it is    
 > a fact that constitutes. We may say it is a (fact) and the proof    
 > constitutes truth.    
 > Sum 1. I am, exist.    
 > Sum sine regno: "I am without a kingdom."    
 > Sic sum ut vides. "Thus I am as you see."    
 > Cogito, ergo sum. "I think, therefore I am."    
 > "Dixit duas res ei rubori fuisse."    
 > He said that two things had abashed him    
 > not equal (≠) (=) equal    
 > Civis Romanus sum: "I am a Roman citizen."    
 > Sum: present active sum, present infinitive esse, perfect active    
 > fuī    
 > "I am as you see."    
 > Sum. (arithmetic) 1. A quantity obtained by addition or aggregation.    
 > I am, exists    
 > The sum of 3 and 4 is 7    
 > ≠    
 > I am, exists    
 > 3+4=7 as you see."    
 > I am, exists    
 > 3+4=7 "I am as you see."    
 > I    
 > Sum (arithmetic) An arithmetic computation, especially one posed to a    
 > student as an exercise (not necessarily limited to addition.) I am,    
 > exists ≠    
 > 2. A quantity of money.    
 > 3. A summary.    
 > 4. A central idea or point.    
 > 5. The utmost degree.    
 > Cum veritate sum    
 > "I am with truth."    
 > Sum 1. Cum veritate sum    
 > I am, exists with truth    
 > = paene (not comparable) ≠    
 > verto: Latin    
 > (vartayati), “‘he turns’”).    
 > present active vertō, present infinitive vertere, perfect active    
 > vertī, supine versum.    
 > I turn, revolve.    
 > I turn around.    
 > I exchange.    
 > I translate.    
 > prīmō 1. at first    
 > satis: 1. enough, filled, plenty    
 > 1. adequately, sufficiently    
 > Cum Veritate Sum paene primo satis    
 > 1. "I am with truth (not comparable) at first to satisfy"    
 > satis prīmō verto    
 > 1. But I satisfy it sufficiently, because at first I turn, revolve.    
 > prīmō 1. at first    
 > Sum venip verto    
 > 1. "Because in essence I begin in reverse"    
 > P==NP    
 > Q.E.D    
 > Martin Musatov. (CC BY-NC)—Preceding unsigned comment added by 71.129.172.196 (talk) 00:34, 23 July 2009 (UTC)
thread_title:  Room for Creativity and Imagination in this Article? 

message: The Caption for the pic at the top of the page says:"Diagram of complexity classes provided that P ≠ NP. If P = NP, then all three classes are equal." which is incorrect.  If P=NP there will still be problems which are not NP-complete.--SurrealWarrior 02:19, 19 Jun 2005 (UTC)
replies: text: No, that's not true. NP-completeness is defined with respect to polynomial-time many-one reductions. Every problem in P is trivially complete under this sort of reduction (simply solve the source problem and produce a yes/no instance of the target problem). Consequently, if P=NP, every problem in P=NP is NP-complete. Deco 08:26, 19 Jun 2005 (UTC)
replies: text: This discussion is over a year inactive, but I want to point out that the caption IS wrong because a decision problem where the answer is always "no" or always "yes" cannot possibly be NP-hard, yet is obviously in P. Eric119 06:01, 23 August 2006 (UTC)
replies: text: You're right. These two problems (and only these two) would fail to be NP-complete. So NP-complete is equal to P=NP minus two problems. I'll revise the caption. Deco 16:37, 23 August 2006 (UTC)
replies: text: Is there a source for your conclusion, Eric and Deco? It doesn't sound right. Always accepting and always rejecting aren't in a special class that acts differently from all other problems. rspeer / ɹəədsɹ  23:25, 23 August 2006 (UTC)
replies: text: Hmmm. At first I thought these two problems were special, since they don't have both yes and no instances for many-one reductions to target (so it would seem that they could only be reduced to themselves). But Papadimitriou asserts: "All languages in L are L-complete." (Computational Complexity, section 16.1, pg. 398). If this is true it seems like the analogous property must hold for P and polynomial-time reductions. These two problems are a special case in Rice's theorem - I'm really unsure about this. Deco 03:43, 24 August 2006 (UTC)

text: Did some more research. In chapter 14 Papadimitriou asserts "For all we know P could be equal to NP [...] Everything in NP would be both polynomial-time computable and NP-complete." Now that I think more about this, I think this has to do with vacuousness in the definition of a many-one reduction. If there is no yes/no instance to target, there is no need for the reduction to translate such instances. I will add back the statement and add an appropriate citation. Deco 03:48, 24 August 2006 (UTC)

text: Your math sounds right, and this document supports you explicitly. The only possible explanation I can find is either that Papadimitriou glossed over these special cases, or he's using a special definition of reduction/completeness (yet his original explanation of reductions is informal and does not reveal such a special case). If he really did miss these cases, then the diagram on page 330, showing P=NP=NP-complete as an explicit scenario, is an even more explicit error. Deco 08:46, 24 August 2006 (UTC)
replies: text: I think it depends on the definition of reduction. Specifically, if you use Turing reduction, a non-empty language can be reduced to an empty one by not using the oracle. But it's a long time ago that I was taught theoretical CS. -- Jitse Niesen (talk) 09:44, 24 August 2006 (UTC)
thread_title:  All three classes are equal 

message: In the section "Polynomial-time algorithms", it writes:
"No one knows whether polynomial-time algorithms exist for NP-complete languages. But if such algorithms do exist, we already know some of them! For example, the following algorithm correctly accepts an NP-complete language, but no one knows how long it takes in general. This is a polynomial-time algorithm if and only if P = NP."
Then it gives a algorithm that accepts the SUBSET-SUM and claims that the algorithm is a polynomial-time algorithm if and only if P=NP.
But I think this conclusion is not very clear since the number of the program P which produces the suitable subset of S may depend on the size of S.
Furthermore, if we change the IF statement in the algorithm with
           IF the program outputs a complete math proof
               AND each step of the proof is legal
               AND the conclusion is that S does (or does not) have a subset summing to 0
           THEN
               OUTPUT "YES" (or "NO" if that was proved) and HALT
there is another problem: the length of the proof and the number of the program which produces the proof may also depend on the size of S.
----
ok, I have understood. The algorithm is correct.
If there is a polynomial time program to solve the decision problme SUBSET-SUM,
let P_0 be the number of such program. Note that P_0 is independent of the size of input for the 
SUBSET-SUM problem.
We can use the program numbered P_0 to construct a polynomial time program which produce 
a subset of S which adds up to 0. Let P_1 be the number of this new program. Note that P_1 is 
also independent of the size of input.
Assume the input S does has a subset which adds up to 0. Let N_1 be the number of steps
need for program numbered P_1 to produce the correct subset of S. Since program numbered P_1 
is a polynomial-time program, N_1 is a polynomial of the size of S.  Therefore the algorithm 
will halt when N = max(N_1, P_1), P = P_1, and clearly it runs in polynomial time.
The second algorithm is similar.
---
For the above algorithm, we really do two things:  1) Search for P_0, and 2) Use P_0 on an input
string.  We say the overall algorithm runs in polynomial time with respect to the length of the
input because task 1 is a fixed cost, and we assume P_0 is polynomial.  That said, it is infeasible
to search for P_0, so the algorithm is useless to us.
It is interesting to note that for short inputs, the algorithm will likely find a machine that just
outputs the answer rather than solving the problem.  But because such machines are themselves
polynomial on the length of the solution, for large problems, we will eventually find the
correct algorithm before we encounter any of these degenerate programs.
---
"But because such machines are themselves polynomial on the length of the solution, for large problems, we will eventually find the correct algorithm before we encounter any of these degenerate programs."
Let's say S has size 2^forty-bajillion but contains the element '0'.  Then won't the program that simply outputs '0' surely turn up before the "correct" algorithm does?  —Preceding unsigned comment added by 99.88.110.127 (talk) 20:30, 7 November 2009 (UTC)
thread_title:  About the algorithm that accepts the NP-complete language SUBSET-SUM. 

message: I think the example definitely shouldn't be PRIMES. It is quite confusing there. How hard would it be refer to the SAT problem there? --Exa 00:05, 23 January 2006 (UTC)
replies: text: Quite difficult. Really, just try to explain SAT in 50 words or less to someone who's never heard of it - perhaps never even heard of logical propositions. I like the composite problem because it's very simple, very obviously in NP, and familiar even to people with only a middle-school math education (although an NP-complete problem would be nicer). Deco 01:27, 23 January 2006 (UTC)
replies: text: How about TSP or Hamiltonian cycle?  The idea of finding the shortest circuit which travels through a bunch of towns before arriving back at the starting point shouldn't be too hard Keithdunwoody 03:27, 18 February 2006 (UTC)

text: subset sum problem. --Robert Merkel 05:02, 29 March 2006 (UTC)
thread_title:  Better example for a problem in NP 

message: Commonly stated, this is P = NP, which is an assignment statement in most programming languages.
The result of the condition is true if NP is true. So:
  if (P = NP)
    print TRUE
  else
    print FALSe
prints true if NP is true.
:This passes for humour on your planet? --Surturz 04:08, 6 June 2007 (UTC)
thread_title:  Stupid Joke about P = NP 

message: The article describes an algorithm that takes an enumeration of all programs, then runs programs 1..N for N steps each, for N = 1 to infinity.  Then it
says:
:This is a polynomial-time algorithm if and only if P = NP... If P = NP, then this is a polynomial-time algorithm accepting an NP-Complete language. "Accepting" means it gives "YES" answers in polynomial time, but is allowed to run forever when the answer is "NO".
Perhaps I am mistaken, but I think this is complete nonsense.  The "algorithm" is not an algorithm at all, since it is not guaranteed to halt, and in fact won't halt if there is no solution for the problem it is trying to solve.
I suggest that the entire section be deleted, and unless there is a serious objection in a few days, that is what I am going to do. -- Dominus 20:22, 5 April 2006 (UTC)
replies: text: An algorithm does not need to halt in this context. -- Jitse Niesen (talk) 04:49, 6 April 2006 (UTC)
thread_title:  "Polynomial-time algorithms" 

message: on the margins of 'concrete mathematics' by knuth, there is a 'proof' that P=NP; it says simply N=1 => P=NP!  :)--Aryah 18:02, 16 April 2006 (UTC)
thread_title:  trivia 

message: The algorithm that decides subset-sum seems to rely on an unstated assumption: if P=NP, then there exists an algorithm that decides subset-sum and outputs a complete mathematical proof in polynomial time.  This wasn't at all obvious to me, and it was fairly non-trivial to prove.  Is it worth including a proof, or at least a pointer to one? -- Nate Biggs 21:07, 6 June 2006 (UTC)
replies: text: Pointer to one would be good. I wouldn't put a proof here personally, but that's just because I don't really like that section. Deco 23:11, 6 June 2006 (UTC)
replies: text: Any idea where one might find a pointer to one?  Personally, my preference would be to delete the first algorithm entirely (who cares if there's a program that accepts subset-sum?  That has nothing to do with P=NP) and include a short sketch of the proof that the second algorithm is correct. -- Nate Biggs 20:23, 8 June 2006 (UTC)
thread_title:  Another question about the second algorithm 

message: This article should have a more easily-understandable pratical example for the less tech/math-savvy ones, such as this one.
I can't write one myself, if that's what you're thinking. --Procrastinator 17:11, 31 July 2006 (UTC)
thread_title:  Article needs easily-understandable practical example 

message: Hello all,
I'm trying to understand the "Polynomial-time algorithms" section of the article.  I'm not convinced that if P=NP, that in fact an algorithm for an NP problem must be polytime.  I was under the impression that a proof that P=NP would imply that there existed an equivalent polytime algorithm, not that a particular algorithm must be polytime.  Can someone explain this to me? Arkleseizure 21:48, 27 January 2007 (UTC)
replies: text: Your question is a bit vague. Which part of the argument do you not follow? Do you not believe the given algorithm is correct, or do you not believe it runs in polynomial time (given P=NP)? --Mellum 22:02, 27 January 2007 (UTC)
replies: text: I think Arkleseizure is making the point that the article claims that "if P=NP then any algorithm for an NP problem runs in polynomial time", which is not true. However, I don't see where the article makes this claim. -- Jitse Niesen (talk) 07:55, 28 January 2007 (UTC)
thread_title:  Confusion 

message: Hello,
I just read the entire article and I want to say good job to the people who wrote it.  This is a very complex subject but you people have found a way to explain it in a simple, clear, easy to understand way.  Good Job. 70.49.201.164 03:12, 4 February 2007 (UTC)
thread_title:  Good job 

message: The article vividly explains the evidence, theoretical arguments, potential consequences, and history surrounding the statements P≠NP and P=NP.  But Gasarch's poll found that some researches believe that the statements are independent of the accepted axioms.  I feel that this article should give that viewpoint equal exposition as it gives the other two.
Otherwise, terrific article.  Thanks!
--131.193.178.113 20:53, 13 February 2007 (UTC)
replies: text: A good point. We really ought to discuss this viewpoint in more detail, but frankly it involves a deeper background in logic, and I don't think I could explain it myself. I might be able to find some references or hopefully some contributors who know better than I. Deco 21:42, 13 February 2007 (UTC)

text: Usually ZFC. We have a reference for some of them at least, that are quoted in the poll paper. Deco 02:56, 14 February 2007 (UTC)
replies: text: Hm, I took a look at the poll paper, and the author guesses they were talking about ZFC, but we have nothing from the respondents. Except that three other respondents said it was not independent of PRA! So I don't think we have much to go on here. Independence from ZFC, as opposed to other stronger or weaker theories, is nothing special, and no particular reason to single it out. --Trovatore 05:15, 14 February 2007 (UTC)

text: My impression is that, to experts in this subject, the ideas of relativization and of natural proofs are of much greater significance than the idea of undecidability. It should be mentioned, certainly, but not out of proportion to its significance. Your "this position deserves more prominence than it currently has" line comes perilously close to soapboxing: we should be highlighting subjects in proportion to the prominence they already have, not trying to create additional prominence for them. —David Eppstein (talk) 23:27, 29 August 2009 (UTC)
thread_title:  Independent of axioms 

message: I have taken a lot of the technical details out of the introductory paragraph to make it more accessible to people who aren't theoretical computer scientists.  All the intro needs to do is introduce the flavour; the technical details are presented quite adequately in the body of the article.  --Robert Merkel 12:52, 5 May 2007 (UTC)
thread_title:  Simplification of introductory paragraph 

message: Considering that we already have articles on P and NP, this article appears at first glance to be redundant. Only after reading it is it clear that this article discusses the P = NP problem. I'd suggest we rename it to suggest this; maybe P=NP problem or something like that. What do you think? Dcoetzee 22:24, 24 July 2007 (UTC)
replies: text: Its title at http://www.claymath.org/millennium/ is "P vs NP Problem" - if we're renaming (and I'm not at this point advocating a rename), using this name (or, following Wikipedia capitalization guidelines, "P vs NP problem") seems like a reasonable idea. -- Rick Block (talk) 01:10, 25 July 2007 (UTC)
replies: text: I disagree with "vs" - it's not a contest! I do agree with an article move though, I also found it confusing coming here specifically to look up the equality problem and the title of this doesn't suggest it very well. Remy B 08:01, 25 July 2007 (UTC)
replies: text: I think here "vs" is intended in the sense of "as opposed to", as in, is there a difference? Ideally I'd prefer "P≟NP problem", where "≟" is the equals with a question-mark over it, but that might not have wide browser support. Although it looks odd, I'm currently leaning towards "P=NP problem", as that immediately evokes the topic in question. Dcoetzee 00:17, 26 July 2007 (UTC)
replies: text: I dont really think P≟NP problem would help people trying to search for the article, considering the character isnt on a standard keyboard. P=NP problem looks good to me. Remy B 08:06, 26 July 2007 (UTC)
thread_title:  Suggested move 

message: Writing P=NP with no spaces around "=" is generally regarded as incorrect according to Wikipedia conventions; it should by P = NP instead.  I'd move the article to P = NP problem (now a redirect page) instantly if fixing all the links were not going to take a while. Michael Hardy 03:31, 15 September 2007 (UTC)
replies: text: Good point. I can handle automatically fixing redirects, provided everyone agrees this is a good idea. Dcoetzee 07:52, 17 September 2007 (UTC)
thread_title:  Another suggested move 

message: The formal definitions in this Article are not formal at all.  Take a look at the Official Problem Description of the Clay Mathematics Institute to see how P and NP are defined.  ( The Clay Mathematics Institute Official Problem Description (pdf))
--CBKAtTopsails 19:06, 24 October 2007 (UTC)
replies: text: I fail to see the problem with the formal definitions. Can you be more specific? --Mellum 21:48, 24 October 2007 (UTC)

text: So please be bold and fix it.  --Lambiam 08:46, 26 October 2007 (UTC)
replies: text:  Do we really need formal definitions of P, NP, and completeness? Isn't that what P (complexity), NP (complexity) and Karp reduction are for? I would prefer this article to contain just a brief intuitive description of P and NP (yes, even one that glosses over the finer points), then discussion on why the P vs NP problem is important, why it is hard, possible approaches, and consequences. In fact, I don't think there should be anything here as formal as the stuff you have recently been adding. I don't think we need to go all the way down to first principles for each article, or teach complexity theory from the bottom up each time we want to talk about it. It's wikipedia, so let's only summarize the background info here and just re-use good definitions from elsewhere. If someone is interested in the real definitions, there could be a link to the more formal pages mentioned above. Blokhead (talk) 05:21, 20 November 2007 (UTC)
replies: text:  Yeah, the point of this article was always just to give an overview of the problem, and leave the formal definitions of the classes involved to their own articles. A brief summary with appropriate links might not hurt, but nothing too verbose. Dcoetzee 08:09, 20 November 2007 (UTC)
thread_title:  Formal Definition ? 

message: Although someone pointed out that the total number of divisions to go through to check if a given x is composite is [x1/2]-1 instead of x-2 where [x1/2] denotes the greatest integer less that or equal to x1/2, in the theory of computational complexity, two methods are considered the same if they both run in exponential time.  Even if you can cut down the number of divisions to x1/2, the time is still exponential.
To see why, let's assume that x contains k digits.
We can then write
x=ak-110k-1+ak-210k-2+......a110+a0  ( For example, 323=(3)(102)+(2)(10)+3 )
Therefore, x1/2={ak-110k-1+ak-210k-2+......a110+a0}1/2 which clearly is exponential in k, the size of the input.
Although each division is polynomial-time, the total number of divisions is exponential.  Therefore, the total time is exponential.
The question of P vs NP basically asks: "Is there a way to reduce the computation time from exponential to polynomial for all the NP problems?".  Just reducing it from one exponential form to another is not good enough.
--CBKAtTopsails 19:45, 9 November 2007 (UTC)
thread_title:  Some Additional Comments 

message: One of the quotes has a "[sic]" the way it's written:
"The resolution of Fermat's Last Theorem also shows that very simply [sic] questions may be settled only by very deep theories."
I don't know the quote at all, but it could have been:
"The resolution of Fermat's Last Theorem also shows that very simply—questions may be settled only by very deep theories."
Depending on if the "that" refers to the "space of algorithms" or "questions."
71.177.240.147 18:39, 10 November 2007 (UTC)
replies: text: In either case there's an error - the [sic] discourages people from spuriously correcting the accurate quote. Dcoetzee 23:14, 12 November 2007 (UTC)
thread_title:  sic 

message: First of everything, I beleive that formal definitions are needed because people want to use them to establish theorems using rigorous mathematical arguments.  Formal definitions therefore must be precise and cover all applicable situations.
Since this article is rated to be within the scope of WikiProject Mathematics and since you cannot talk about mathematics without talking about formal definitions, I see no reason why formal definitions have to be left out from this article.
I have checked out the articles on P (complexity), NP (complexity) and Karp reduction.  Here are what I found:
(i)  The articles on P (complexity) and Karp reduction do not themselves contain formal definitions for P and NP-complete.  you have to navigate through many other articles spending a lot of time to sort out what information is needed and what information is not needed.  People who do not want to spend a whole week or even a whole month to understand all these other articles in order to get to their original very simple objective (i.e. to find out how P and NP-complete are defined) would simply give up.
(ii)  The article on NP (complexity) has a different way of defining NP.  This different definition, although mathematically equivalent, originated from the concept of nondeterministic machines and is no longer the popular definition.  Most importantly, it is not in tune with the concepts and and ideas talked about in the article of "P = NP Problem".  For example, the concepts of checking relations, verifiers and certificates.
I believe that there are many people out there who became aware of the "P vs NP Problem" because of the $million prize offered by the Clay Mathematics Institute and I believe that what Clay Mathematics Institute is looking for is a settlement of this problem based on some kind of rigorous mathematical argument.
In summary, including a formal definition in this article will serve two important objetcives:
(a)  Provide the convenience of one-stop shopping without having to navigate to many other articles.
(b)  Provide additional flavors in the article for different people who come to search Wikipedia on the subject of "P vs NP" with different perspectives.
--CBKAtTopsails (talk) 16:47, 21 November 2007 (UTC)
replies: text: Responding: NP (complexity) describes both definitions and summarizes a proof of their equivalence. We use the verification definition here because it's more intuitive and approachable without background about nondeterministic machines. It is true that these articles don't currently contain a detailed formal definition, based on the formal sets in the definition of the Turing machine - they rely on background about other concepts in language such as "P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time". Although it would be somewhat pedantic, I think there's reasonable justification for adding more detailed formal definitions to the articles on specific complexity classes - at best, I think this article would benefit primarily from a brief comparison of the formal definitions of P and NP, with reference to their articles for details. Dcoetzee 23:02, 21 November 2007 (UTC)

text: I'm not suggesting that - just suggesting that formal definitions could be useful in the articles on specific classes, and once they're added it'd be redundant to provide complete formal definitions of every class here; it'd suffice to describe relevant differences. I recognise that the "formal definition" in this section wasn't formal, it was just more detailed.  And no, I'm certainly not referring to section 2.1 of NP (complexity) - the definition is in the introduction: it's sufficient to say that it's the set of languages recognizable in polynomial time by a nondeterministic Turing machine, or equivalently the set of languages verifiable by a deterministic Turing machine in polynomial time, given a valid proof string. This is not a detailed formal definition, but it is rigorous and correct. Also keep in mind that formal definitions of related classes are very similar (e.g. NP and NEXP) and I would prefer to centralize this type of information in articles like NTIME. Dcoetzee 14:06, 25 November 2007 (UTC)

text: The articles should discuss both definitions, and their equivalence. Many sources use one as the definition and prove the other - that's always the way with equivalent definitions. I'm not saying we shouldn't define them, just that it's redundant to completely formally define the classes both here and in the more detailed articles on the classes themselves. It's important to avoid duplicating content. This is an appropriate place for comparing/contrasting formal definitions of P and NP. Dcoetzee 23:11, 27 November 2007 (UTC)

text: I just took a look at the additions to the article and I'm okay with them, at least after deleting the fluffy "intuitive explanation" paragraph that didn't add anything and contained lots of errors. The section was a good idea. :-) Dcoetzee 23:55, 27 November 2007 (UTC)
thread_title:  Some more comments on formal definition 

message: There have been some talks about the practicality of the polynomial time definition on this page.  The section "Is P really practical?" in this article states the following:
"All of the above discussion has assumed that P means "easy" and "not in P" means "hard".  While this is a common and reasonably accurate assumption in complexity theory, it is not always true in practice, for several reasons:
*It ignores constant factors.  A problem that takes time 10100n is in P (it is linear time), but is completely impractical.  A problem that takes time (1+10-100)n is not in P (it is exponential time), but for n a googol takes time essentially e = 2.718… and so is very practical for all feasible values of n.
*It ignores the size of the exponents.  A problem with time n100 is in P, yet impractical. Problems have been proven to exist in P that require arbitrarily large exponents (see time hierarchy theorem).  A problem with time 2n/100 is not in P, yet is practical for n up into the low thousands.".
According to the opinion of at least one expert in this field, all the known polynomial-time algorithms for solving any practical problems in the real world run in time O(n5) or less with reasonable constants.  The following is a direct quote from Section 1.5.1 of Chapter 1 in the the textbook, "Computational Complexity: A Modern Approach" written by Sanjeev Arora and Boaz Barak of Princeton University (See http://www.cs.princeton.edu/theory/complexity/modelchap.pdf)
"1.5.1 On the philosophical importance of P
The class P is felt to capture the notion of decision problems with “feasible” decision procedures.
Of course, one may argue whether DTIME(n100) really represents “feasible” computation in the
real world. However, in practice, whenever we show that a problem is in P, we usually find an n3
or n5 time algorithm (with reasonable constants), and not an n100 algorithm. (It has also happened
a few times that the first polynomial-time algorithm for a problem had high complexity, say n20,
but soon somebody simplified it to say an n5 algorithm.)"
In all fairness, I think the burden of proof lies on those who made the criticism to come up with a specific example to support their argument.  In other words, you need to show that there is a practical problem which requires an n100 time algorithm to solve it and otherwise cannot be solved in any other more efficient way.  
If you don't have a specific example, all you talk about is still speculation.  It is not about practicality.
--CBKAtTopsails 20:37, 4 December 2007 (UTC)
replies: text: There are many fixed-parameter tractable algorithms and approximation algorithms which can have arbitrarily large polynomial exponent (based on the parameter values) and which are potentially useful in practice. Dcoetzee 00:07, 8 December 2007 (UTC)
replies: text: See AKS primality test for an n6+ε primality testing algorithm, the best yet.  --Vaughan Pratt (talk) 05:00, 13 March 2008 (UTC)

text: Just as a note, "fixed-parameter tractable" does not mean "polynomial-time for any fixed parameter value k"; it also demands that the exponent is a constant independent of k. --Mellum (talk) 11:40, 8 December 2007 (UTC)
replies: text: See for example Parameterized Intractability of Distinguishing Substring Selection, which describes a problem that has a PTAS but has no EPTAS, under reasonable assumptions; that is, if you wish to solve it within a small ratio, the exponent of the resulting algorithm will necessarily be large. Dcoetzee 06:28, 12 December 2007 (UTC)
thread_title:  About the practicality of the polynomial time definition 

message: {{RFCsci | section=Request for comments: "the" verifier or "a" verifier? !! reason=Should the definition of NP-complete use "the" verifier or "a" verifier? !! time=08:38, 22 December 2007 (UTC)
replies: text: * As explained in my comment above, I think both wordings are valid, although they are interpreted differently. One is a free-standing valid statement in its own right ("a"), while the other refers to a previously-defined object ("the"). As such I have no stake in the outcome of this discussion. Dcoetzee 07:11, 25 December 2007 (UTC)
thread_title:  Request for comments: "the" verifier or "a" verifier? 

message: There's a section that refers to a trivial problem DUH but does not provide any description of it. Can someone look into it? Thanks. --Uchchwhash (talk) 14:23, 14 February 2008 (UTC)
replies: text: It is actually described: given a description of a Turing machine M guaranteed to halt in polynomial time, does there exists a polynomial-size input that M will accept? However, I find this very unclear and I don't see the relevance of mentioning this contrived problem; a reference to the list of NP-complete problems, most of which are much more natural, would seem to serve the purpose better of conveying the fact of the existence of NP-complete problems.  --Lambiam 04:57, 15 February 2008 (UTC)
replies: text: The importance of this problem from a pedogogical perspective is that it's straightforward to prove that the problem is in NP and NP-hard. I revised to briefen this paragraph, and remove the name DUH, which I think is too informal. Dcoetzee 08:02, 15 February 2008 (UTC)
thread_title:  DUH 

message: I had started a discussion above, titled "Suggested addition to "Is P really practical"". I'd like to continue the discussion that discussion here, because it seems like people pay more attention to sections at the bottom of the talk page than sections in the middle of it. First, let me repost my last post from above:
:I apologize, Dcoetzee. I thought I had addressed your first objection. You said my original proposal was much too long, and I thought you were just saying that it should be about the same length as the first point and the second point in the list. The version of my paragraph that I added to the article wasn't the same as my original proposal. It did have about the same number of words as the first and second points. Are you saying that your version is superior to my version, in and of itself? Or are you just saying that, in general, a shorter version is better? I can understand that, because, as you said, this discussion is somewhat off-topic from the P=NP problem, so I guess you might be saying that you just want to have less off-topic text in this article. I saw that CRGreatHouse added a link to "Cobham's thesis." What do you think about moving this whole section out of the P=NP article and into the Cobham's thesis article? And I think that if it gets moved, then, after we move it, it should have a discussion of hardware, including some examples with numbers. I think this is the reason for Nicolaennio's confusion in his posts on this talk page. See my response to him below. Also, I wasn't citing the talk page. I was just trying to put a link similar to the links that already exist in wikipedia that say things like "where?--see talk page". For example, see the first paragraph of this article: Tying (commerce). Sorry for the confusion. --Navigatr85 20:56, 2 December 2007 (UTC)
replies: text: I think moving this out is a great idea and Cobham's thesis is a great place for it to go. It can still be briefly summarized here and linked using , which I think is a good idea. I have a lot more to write about Cobham's thesis too that would go quite astray of this topic. :-) Regarding your original contributions, I don't honestly remember them, and I will evaluate them when you add them in that article. Dcoetzee 19:53, 10 March 2008 (UTC)
thread_title:  Moving "Is P really practical" 

message: Is this known to be trivial or should it be cited in this Article?
(* complete dictionary of words, one per row, number of rows is (alpha^word), using words of length "word" and an alphabet of "alpha" number of characters *)
alpha=4;word=3;dict=.;
dict=Partition[Flatten[Tuples[Reverse[IdentityMatrix[alpha]],word]],(alpha*word)]
PseudoInverse[dict]==((Transpose[dict])*((alpha)^-(word -1)))-((word - 1)/(alpha^word)/word)
True
There is nothing special for alpha=4;word=3 - The method works for all cases where word < alpha, but that is a more verbose loop, and the Mathematica PseudoInverse function takes too long to calculate for values of alpha and word that are large, whereas the transpose side of the equation is fast.
100TWdoug (talk) 15:11, 27 March 2008 (UTC)
replies: text: It can only be cited if you have a reference for this, published by a reliable source. However, it is not clear what this has to do with the problem whether .  --Lambiam 19:43, 27 March 2008 (UTC)
thread_title:  Should this be Cited? 

message: As discussed above, I moved most of the information out of this section and into the Cobham's thesis article. This was done because Dcoetzee and I agreed that the discussion was not very relevant to the P=NP problem. However, I ended up deleting some of the discussion of quantum computers, not moving it, because it was irrelevant to Cobham's thesis. Maybe that information could be added back into the P=NP article. I would consider adding something like this:
:*There is a misconception that quantum computers could help with the P=NP problem. However, quantum algorithms have not to date solved any NP-hard problem in polynomial time. Also, the definition of P and NP are in terms of classical computing models like Turing machines. Therefore, even if a quantum computer algorithm were discovered to efficiently solve an NP-hard problem, we would only have a way of physically solving difficult problems quickly, not a proof that the mathematical classes P and NP are equal.
But then, If we added that, I think we might need a citation for the first sentence of that paragraph. On the other hand, maybe we don't need any discussion of quantum computers in this article. What does everyone think? Navigatr85 04:16, 1 April 2008 (UTC)
thread_title:  Moved "Is P really practical" 

message: This section, to the best of my knowledge, is incorrect.  Whilst I'm not a descriptive complexity expert, to the best of my knowledge, nobody has come up with a logical characterisation of P in the way that Fagin did with NP.  Rather, extensions to FOL, such as adding LFP, IFP, orderings and Peano arithmetic are proposed, then counterexamples are eventually found.  This seems to be supported by a talk I attended recently by Dr. Anuj Danwar, a descriptive complexity theorist.DPMulligan (talk) 20:52, 13 July 2008 (UTC)
replies: text: The reference for this is precisely the one given, Neil Immerman's "Languages Which Capture Complexity Classes," specifically Theorem 4.7. Do you believe the source is being misinterpreted? Dcoetzee 08:42, 14 July 2008 (UTC)

text: You are correct. I have removed the section. Thanks for noticing this. :-) Dcoetzee 18:57, 15 July 2008 (UTC)

text: Thanks - I have limited expertise in this area and appreciate your clarification. Dcoetzee 22:34, 11 September 2008 (UTC)
thread_title:  Logical characterization of problem 

message: I see in the beginning of the section that Clay Mathematics Institute is offering a million dollars prize.  But in the 'Results about difficulty of proof...", it says "A million-dollars prize...".  This generate some confusion.  I'm editing this to say "The Clay Mathematics Institute million-dollars prize..."  This makes it clear to a specific prize and not some generic prize. 24.16.28.198 (talk) 16:39, 2 October 2008 (UTC)
thread_title:  Million dollars prize. 

message: The following was added to the easy-vs-hard section:
thumb|310 px|The graph shows time (average of 100 instances in msec using a 933 MHz Pentium III) vs.problem size for knapsack problems for a state-of-the-art specialized algorithm. Quadratic fit suggests that empirical algorithmic complexity for instances with 50–10,000 variables is O((log n)2).Similarly, the graph on the right shows the empirical complexity for solving the knapsack problem, demonstrating a sub-linear (!) growth. The argument that an NP-Complete problem therefore is solveable only heuristically is false (but quite common)Goulimis C. N. (2007). "Appeal to NP-Completeness Considered Harmful".  Interfaces, Vol. 37, No. 6, November–December 2007, pp. 584–586.
I think this is misleading for two reasons, one theoretical and one practical:
*Theoretically, a program can only be sub-linear if it is not required to read all its input.  Therefore, the behavior shown in the graph cannot be expected to extrapolate to larger sizes.   Since NP-complete is about the limit in large cases, this seems like it should be called out.
*In practice, many NP complete problems have both 'easy' and 'hard' problem instances.  For example, a knapsack problem can work on N numbers of M bits each, for a total size of NxM.  If either N or M is small, very efficient algorithms are possible, and well known.  But if N is roughly equal to M, these can be very hard.  For example, I'd be very surprised if the program in question can solve a problem with 1000 numbers of 1000 bits in the time implied by the graph.  At the very least you need to state how the problem instances were created.
LouScheffer (talk) 04:06, 11 December 2008 (UTC)
replies: text: As I mentioned on my talk page, I have moved this content out to Cobham's thesis. I believe it's appropriate to go into detail about the issue in the article dedicated to the topic, and only summarize it here. Also, the growth here is quadratic in terms of the size of the input in bits, not sublinear (I changed that while I was at it, please visit Cobham's thesis if you disagree). Dcoetzee 20:25, 15 December 2008 (UTC)
thread_title:  Sub-linear growth 

message: Although the article hints to it by starting " It is considered to be the most important problem in the field" would it be possible to have examples of why, if it were to be solved, it is so important?  I'm thinking particularly of the cryptanalysis perspective but there's a lot more to it.  This would be an additional item to the computer science and maths content for less technical readers.  —Preceding unsigned comment added by 81.153.145.240 (talk) 06:06, 24 December 2008 (UTC)
replies: text: Good question. I can think of plenty of computer science possibilities - many improvements could be made to compilers like optimal register allocation, and circuits could be fit into smaller spaces with optimal layout and gate counts. Some types of programs could be generated from high-level specifications. It would facilitate motion planning for robots. Any puzzle could be solved rapidly, which is an approachable example but not a terribly useful one. Realtime rendering of graphics could be made much more efficient. Outside of computer science, I'm not sure - maybe some stages of protein structure prediction? Dcoetzee 11:22, 24 December 2008 (UTC)
replies: text: Lance Fortnow’s blog on what would happen if P=NP. Russel Impagliazzo’s paper of what the world would look like under different complexity-theoretic assumptions. Thore Husfeldt (talk) 15:29, 26 December 2008 (UTC)
thread_title:  Practical applications 

message: Shouldn't this be P vs NP problem (or P ≠ NP problem)? That seems the more common name in the literature. Shreevatsa (talk) 20:25, 21 February 2009 (UTC)
replies: text: There was some debate over this a long time ago. Originally for a long time it was called Complexity classes P and NP, but that was confusing because it's clearly about the problem and the classes have their own separate articles. I notice that Mathworld and the official Millenium Prize Problems description refer to it as "The P versus NP problem", so your first suggestions sounds quite appropriate. Dcoetzee 13:55, 9 March 2009 (UTC)
replies: text: Great, if no one objects we can move it to P versus NP problem. Shreevatsa (talk) 14:27, 9 March 2009 (UTC)

text: Unfortunately, the page is now protected against being moved by normal users. Does anyone oppose the move? Is someone reading this an admin who could make the move? Thanks, Shreevatsa (talk) 04:22, 17 March 2009 (UTC)
replies: text: This isn't an opposition - I don't know enough about the subject to oppose the move; it's just a comment - but I think that "P = NP problem" is clearer: the problem is to prove or disprove P = NP, whereas 'versus' means against - P against NP problem - which I think is a less accurate description. Just my AU$0.02. --Aseld talk 07:37, 17 March 2009 (UTC)
replies: text: Well, "P = NP" is a statement, and one believed to be false... so "P = NP problem" seems odd to me. ("P ≠ NP problem" would be slightly better, but still, appending "problem" to a statement seems odd.) I can't think of anything better though, but "P versus NP" seems the more commonly used name. Anyway, if you find "P versus NP" sounds odd too, I guess there is really no perfect name. :) It's probably best to leave it as it is, considering the many mentions of "P = NP" in the article. Shreevatsa (talk) 08:28, 17 March 2009 (UTC)
thread_title:  Name 

message: User:Shreevatsa changed "algebrization" to "arithmetization." He's correct - I think I accidentally added the word "algebrization." Quote: "In their STOC’08 paper [AW08], Aaronson and Wigderson present the concept of “algebrization barrier”: the arithmetizing techniques cannot resolve the main open questions in complexity."  Dcoetzee 13:53, 9 March 2009 (UTC)
replies: text: I wasn't the one who noticed it, the change was done by 65.96.170.103 here, on the duplicate of this article created by the Musatov character, presumably. I searched for "arithmetization" in the AW paper before inserting it here. :) Shreevatsa (talk) 14:26, 9 March 2009 (UTC)
thread_title:  Verified arithmetization edit 

message: I changed the first of these sentences to the second in this article:
: 
: Most computer scientists believe that P≠NP.
: Most computer scientists believe that P ≠ NP.
: 
This sort of thing is codified in WP:MOSMATH.  There must be some people to whom the first form above does not appear conspicuously jarring. Michael Hardy (talk) 13:20, 13 March 2009 (UTC)
thread_title:  MOSMATH 

message: I noticed this link: Qeden, a wiki that aims to solve the Millennium Prize Problems ("down for maintenance" as of 15th of July 2008) 
on the external links section of the page is dead.  There is no reason for it to be included on the page yet for some reason it has been defended like its a piece of the holy grail.  It could be interferring with Musatov's proof.  And since it's a dead link, it holds no reason or logic for it to be included.  In fact the link points to a page for a domain that is not only NOT under construction, but is in fact EXPIRED.  Can we arrange to have it deleted?  —Preceding unsigned comment added by 172.190.234.231 (talk) 01:54, 22 March 2009 (UTC)
thread_title:  Proof by Musatov 

message: http://www.knowledgerush.com/kr/encyclopedia/Big_O_notation/
Please reference this for some interesting p=np generated content.  —Preceding unsigned comment added by 172.190.234.231 (talk) 02:15, 22 March 2009 (UTC)
replies: text: I'm afraid that isn't nearly relevant enough, and largely duplicates our own article on big O notation. Dcoetzee 04:24, 10 April 2009 (UTC)
thread_title:  Link from Musatov 

message: It has been noted that an attempt has been made to censor a reference to ArXiv because it is  not a refereeable source. However, there exists several other pages across Wikipedia where ArXiv is used as a reference. We cannot pick and choose sources as this is not in equality aligned with the Five Pillars of Wikipedia. Additionally it must be considered and noted that there are several pages which exist as uncontested references on this page that are not refereaable sources.  Specifically the works of Mr. Scott Aaronson on his blog and the other reference which exists under the heading Complexity Zoo. The reference to ArXiv cannot legally be excluded unless it is standardized by the Wikipedia communnity that all references must refereeable sources. If this is indeed the case the source should then rightfully be removed. However this is not the case as I am unaware of any instance where ArXiv requires all references to indeed be refereeable.  In dubiously is the fact that there exists other references uncontested which do not come from referereeable sources that remain and are allowed to attempt to prove counterpoint to the established fact.  I would like to raise this to discussion if need be but I default to the wisdom of the community to support this fair engagement on this page. Anonymousone2 (talk) 04:54, 12 April 2009 (UTC)
replies: text: ArXiv is an indiscriminate paper archive and is not, in general, considered a reliable source, although some publications on it are reliable. I must insist you stop adding your reference to the article, which refers to a paper for a controversial claim that has not been peer-reviewed and is almost certainly false. ArXiv is rife with false papers on this very subject, in no small part because of the Millenium Prize Problem reward. Dcoetzee 05:14, 12 April 2009 (UTC)
thread_title:  Requested Rerence usage 

message: Copyright infringing text deleted.
If the term Succinct Problem must indeed be included in the P=NP problem it should then be allowed to exist on Wikipedia.  For clarification if Succinct Problem is insistently included in the P=NP page it the term Succinct Problem should be allowed existence on Wikipedia.  It seems there may be a system glitch where the page Succinct Problem keeps getting flagged and deleted each time it is created. May we please investigate this.  
I suggest these two alternatives/outcomes:
1) The term Succinct Problem is given a page on Wikipedia.
2) The term Succinct Problem is removed from the P=NP problem page.
Thank you. I look forward to continued cooperation and investigation!  In the spirit of wholehearted academic truth and positivity!!!Anonymousone2 (talk) 05:02, 12 April 2009 (UTC)
replies: text: The two attempted creations of Succinct problem tonight were deleted as copyright infringements. I have deleted the copyvio text from Anonymousone2's comment above. —C.Fred (talk) 05:05, 12 April 2009 (UTC)

text: A page on succinct circuit problems (which I assume is what you're referring to) would be useful, but must be written in your own original language. You cannot copy and paste text from other websites unless those websites explicitly release it under a GFDL-compatible license. Moreover, the account that created those articles has been blocked, and your statements here raise the question of whether you are also another account created by the same person. Please don't give us cause to investigate this. Dcoetzee 05:15, 12 April 2009 (UTC)
thread_title:  Succinct Problem Creation Reference - Please help me fill this out guys![[User:Anonymousone2|Anonymousone2]] ([[User talk:Anonymousone2|talk]]) 05:02, 12 April 2009 (UTC) 

message: Mr. Aaronson's blog is not a refereeable source yet it makes claims with respect to the P=NP Problem. If it is required that all referenced sources be refereeable then Mr. Aaronson's claim must also be removed. We must make specific arguments for our actions and not be vague with respect to such careful complex and sensitive matters. Please reference where in the Wikipedia requirement for References does it say this article is not submissable. If you cannot produce this material then the argument remains valid.  —Preceding unsigned comment added by Anonymousone2 (talk • contribs) 05:23, 12 April 2009 (UTC)
replies: text: The requirement is only that they be reliable. Scott Aaronson is an authority in complexity. The author of the paper you are attempting to cite is not. Your reference will not be included, and your persistence will not be rewarded. Policy does not specifically mandate what type of references are permitted or not, because it requires editorial judgement and discretion to determine this. Dcoetzee 05:26, 12 April 2009 (UTC)

text: First of all, that is an essay and does not have the force of policy. Second, it does not make the claim that arXiv papers are reliable, merely that they undergo some cursory review. Third, Wikipedia is not a strict system of rules that must be followed; it is clear in this case that this paper is inappropriate, regardless of how you might interpret the rules to accomodate it. There is no point in attempting to shift the discussion to Reliable source examples, because I do not disagree with what the page already says. Dcoetzee 05:36, 12 April 2009 (UTC)

text: No, it does not. Some papers in ArXiv meet it. ArXiv does not have the resources to review papers for correctness. They verify only that the paper is "appropriate for the topic area." ArXiv has for many years hosted false proofs and disproofs of P=NP. This one is not special, and will not be included. Do you really think this paper would go unnoticed if it had correctly resolved the biggest open problem in computer science? Wikipedia is not a platform for promotion of dubious research. Also, please stop adding "the claim" above, as that is a copyright violation, entirely irrelevent to discussion of the article, and also clearly meaningless nonsense. Dcoetzee 05:48, 12 April 2009 (UTC)

text: Additionally, you are in violation of the Wikipedia:Three revert rule. I am reporting you for edit warring and you may be temporarily blocked from editing as a consequence. Dcoetzee 05:49, 12 April 2009 (UTC)
replies: text: Please see your edit warring report and let me know if you have any problems. Dcoetzee 06:08, 12 April 2009 (UTC)

text: Additionally, it has come to my attention that you are probably a sock puppet of the banned user User:Martin.musatov, who has a clear conflict of interest in introducing this reference, and additionally should not be evading his block by posting under another account. Dcoetzee 06:15, 12 April 2009 (UTC)
thread_title:  Equality in references 

message: BTW, I rather disagree that a proof that P = NP would have no practical ...... very formal proof that P < NP - it is being under review by Discrete Applied ...... you are probably a sock puppet 
I found this while using a new search engine called MeAmI.
The inclusion of the "......" is interesting.172.191.9.162 (talk) 23:12, 4 May 2009 (UTC)
thread_title:  Musatov "Proof" 

message: All of the above discussion has assumed that P means "easy" and "not in P" means "hard". This assumption...
The first line in P = NP problem#Does NP mean "hard" and P mean "easy"? is a little ambiguous. I can't tell if it is inferring NP means "not in P", or is saying problems that are not in the P space?
All of the above discussion has assumed that P means "easy" and NP means "hard". This assumption...
Is this better?
R.Vinson (talk) 15:48, 3 June 2009 (UTC)
replies: text: No, it is worse. NP = not quite as easy as P, but still easier than a lot of other complexity classes. NP doesn't stand for "not P", it stands for "nondeterministic polynomial", i.e. solvable in polynomial time (like P) by a nondeterministic machine (stronger than the machines used to define P). So it makes no sense to imagine that NP stands for hard problems — what then would the many problems that are not even in NP be? —David Eppstein (talk) 16:04, 3 June 2009 (UTC)
replies: text:  I understand the definition of P and NP that is why I brought it up. The section is titled "Does NP mean hard and P mean easy?", then the first sentence states "P means easy and not in P means hard". This draws a correlation that NP is the equivalent of saying not in P. I'm not particular found of calling P easy or NP hard. I thinking stating the discussion above assumes P is fast (efficient) and NP is slower (less efficient). R.Vinson (talk) 17:52, 3 June 2009 (UTC)
replies: text:  This is a misunderstanding in several ways. NP contains P, as well as every trivial problem. It doesn't make sense to call NP hard, except perhaps in a worst-case sort of sense (e.g. the hardest problem in NP is hard). Additionally, whether NP-hard problems are hard depends on whether  P=NP, which we obviously don't know (I certainly hope we're not "assuming" P ≠ NP in this article!). Finally, the Cobham's thesis article which this section summarizes doesn't mention NP, and Cobham's paper predated the invention of NP. For all these reasons, the text should remain as it is. Dcoetzee 03:48, 4 June 2009 (UTC)

text:  Just to summarize: you are saying most of the section is flawed, so it should remain as is? R.Vinson (talk) 04:44, 4 June 2009 (UTC)
replies: text: We are saying that your reading of the section seems to be flawed, so you should read it (and our answers here) again. Specifically, NP cannot mean hard, because NP contains all of the easy problems and does not contain most of the hard ones. That's why it says "not in P" rather than "NP" in that sentence. —David Eppstein (talk) 04:48, 4 June 2009 (UTC)

text: You keep making assumption about my understanding. Let me rephrase my initial question. NP does not mean "not in P", why is the article making a linguistic correlation between the two? It is ambiguous at the least. R.Vinson (talk) 05:36, 4 June 2009 (UTC)
replies: text: Oh, I didn't notice the section title. Looks like David fixed the problem. Dcoetzee 07:44, 4 June 2009 (UTC)
thread_title:  "not in P" 

message: It is said in the text of the article: the computer is ... sequential (it performs actions one after the other). As of 2009, these assumptions are satisfied by all practical computers yet devised.
Strictly speaking multi-CPU/multicore computers (such as Core2Duo based systems) are not sequential. Well, it is true that these can be modeled as sequential. For example a Core2Duo system can be modeled as odd instructions executed by one core and even instructions executed by the other core. (This model could add additional certainty however.)
porton (talk) 16:35, 18 June 2009 (UTC)
replies: text: The discussion is, frankly, rather immaterial here... this article has a bad tendency to collect general statements about complexity theory and Cobham's thesis. Dcoetzee 21:39, 18 June 2009 (UTC)
thread_title:  Sequentiality 

message: Sci.Math on USENET that 2SAT is in P, but Max-2SAT is NP-hard. There exists α s.t. given a 2- 
CNF formula ϕ, deciding if Opt(ϕ) α is NP-hard. ... Proof. Opt 1, m 
sets covering .. 
2SAT is in P, but Max-2SAT is NP-hard => .... Proof. Opt 1, m sets. 
covering everything. ... possible performance guarantee for the Max. p- 
Cover unless P=NP. ... 
Q.E.D. ?
Not the place for 'new solutions, I know, but is USENET Sci.Math citable for computation problems if the proof checks out? Thanks!
:You're asserting essentially that Max-2SAT can be solved by a polytime reduction to 2SAT. Using the trivial reduction you gave, this is untrue, because this only works for formulas that are satisfiable (if the formula is satisfiable, which you can check in polytime, then you know the maximum number of clauses that are satisfiable, namely all clauses). On the other hand, if the formula is not satisfiable, this tells you almost nothing about the maximum number of satisfying clauses, other than that it's not all of them. To put it another way, this only works when the threshold in Max-2SAT is set to the number of clauses, which is a special case. Dcoetzee 05:59, 14 August 2009 (UTC)
thread_title:  2SAT and Max-2SAT 

message: This page is quite long now. Can we archive this talk page? (I'm not too sure how to do that.) --Robin (talk) 13:37, 16 September 2009 (UTC)
replies: text: This page is 250kb. When this page is bombarded by spam, it becomes around 400kb. We really should archive this page (manually or with the bot). --Robin (talk) 14:33, 22 November 2009 (UTC)

text: The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 
The result of the move request was move to P versus NP problem. David Eppstein (talk) 19:12, 24 October 2009 (UTC)
thread_title:  Archive 

message: :Feel free to state your position on the renaming proposal by beginning a new line in this section with *Support or *Oppose, then sign your comment with ~~~~. Since polling is not a substitute for discussion, please explain your reasons, taking into account Wikipedia's naming conventions.
* Support. Move, per nom and "use common name". (Also, odd to call it "P = NP problem" when most scientists don't believe P = NP.) Shreevatsa (talk) 03:37, 20 October 2009 (UTC)
thread_title: Survey

message: If you read # ^ Aaronson, Scott, Is P Versus NP Formally Independent?, http://www.scottaaronson.com/papers/pnp.pdf .
If P=NP was true, then this very problem itself must be easy, right? But it's obviously hard, so P ≠ NP, as it shows the difficulty of its own proof. Should we add this? I find it a bit funny. Money is tight (talk) 04:13, 13 December 2009 (UTC)
replies: text: It's already there. See the last paragraph of P versus NP problem#Results about difficulty of proof. —David Eppstein (talk) 05:02, 13 December 2009 (UTC)

text: The result is: independence from weak set-theoretic axioms => too-good-to-be-true algorithms. Expressed that way, it's not about proofs, it's about whether there are models of those axioms in which P=NP and other models in which P≠NP. —David Eppstein (talk) 16:57, 14 December 2009 (UTC)
replies: text: Maybe something like there is no possibility of this sort of independence would be clearer.  I agree with Pcap that it is still possible to have a notion of the question's independence, just as it's possible to have a notion of 2 being equal to 3.  It just wouldn't actually be independent of the theories in question. --Trovatore (talk) 03:12, 16 December 2009 (UTC)
replies: text: Whoever edits that section, I'd just like to point out that there's a "using currently known techniques" clause. The result follows only if P vs NP is proved independent of (say) ZFC using currently known techniques. It is entirely plausible that a new set theoretic technique is developed which shows P vs NP to be independent of ZFC, and that would not imply efficient algorithms for NP. --Robin (talk) 04:35, 16 December 2009 (UTC)
replies: text: Oh, well that's quite different from what I'd heard.  In that case I think that needs to be made clear, because it certainly wasn't clear to me.  I'll have to take a look at Ben-David's paper. --Trovatore (talk) 04:58, 16 December 2009 (UTC)
thread_title:  New section to add 

message: "Therefore, if one believes (as most complexity theorists do) that problems in NP do not have efficient algorithms, it would follow that such notions of independence cannot be possible." (emphasis mine) "Notions" should be "proofs" there no? I don't see how the notions are threatened. Pcap ping 12:00, 14 December 2009 (UTC)
replies: text: The result is: independence from weak set-theoretic axioms => too-good-to-be-true algorithms. Expressed that way, it's not about proofs, it's about whether there are models of those axioms in which P=NP and other models in which P≠NP. —David Eppstein (talk) 16:57, 14 December 2009 (UTC)
replies: text: Maybe something like there is no possibility of this sort of independence would be clearer.  I agree with Pcap that it is still possible to have a notion of the question's independence, just as it's possible to have a notion of 2 being equal to 3.  It just wouldn't actually be independent of the theories in question. --Trovatore (talk) 03:12, 16 December 2009 (UTC)
replies: text: Whoever edits that section, I'd just like to point out that there's a "using currently known techniques" clause. The result follows only if P vs NP is proved independent of (say) ZFC using currently known techniques. It is entirely plausible that a new set theoretic technique is developed which shows P vs NP to be independent of ZFC, and that would not imply efficient algorithms for NP. --Robin (talk) 04:35, 16 December 2009 (UTC)
replies: text: Oh, well that's quite different from what I'd heard.  In that case I think that needs to be made clear, because it certainly wasn't clear to me.  I'll have to take a look at Ben-David's paper. --Trovatore (talk) 04:58, 16 December 2009 (UTC)
thread_title:  Odd sentence there