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How do you know that a decision problem $X$ is NP-complete?, if all other NP-problems polynomially transform to $X$ or if all other NP-problems polynomially reduces (there exist a polynomial time oracle for any problem in NP using an oracle for $X$).

Definitions seem to differ all over the web. Thanks!

David Richerby
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    Hi, Doc. I took the liberty of renaming your problem from $\mathcal{P}$ to $X$. I think things will get super-confusing if we're simultaneously using $\mathcal{P}$ to be a problem and P to be a complexity class! – David Richerby Aug 23 '16 at 15:49
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    "Definitions seem to differ all over the web" -- err, I hope not. Can you quote three different definitions? – Raphael Aug 23 '16 at 16:50

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