<P> Donald Knuth has stated that he has come to believe that P = NP, but is reserved about the impact of a possible proof: </P> <P> (...) I don't believe that the equality P = NP will turn out to be helpful even if it is proved, because such a proof will almost surely be nonconstructive . </P> <P> A proof that showed that P ≠ NP would lack the practical computational benefits of a proof that P = NP, but would nevertheless represent a very significant advance in computational complexity theory and provide guidance for future research . It would allow one to show in a formal way that many common problems cannot be solved efficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems . Due to widespread belief in P ≠ NP, much of this focusing of research has already taken place . </P> <P> Also P ≠ NP still leaves open the average - case complexity of hard problems in NP . For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable . Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average - case complexity question . These range from "Algorithmica", where P = NP and problems like SAT can be solved efficiently in all instances, to "Cryptomania", where P ≠ NP and generating hard instances of problems outside P is easy, with three intermediate possibilities reflecting different possible distributions of difficulty over instances of NP - hard problems . The "world" where P ≠ NP but all problems in NP are tractable in the average case is called "Heuristica" in the paper . A Princeton University workshop in 2009 studied the status of the five worlds . </P>

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