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This chapter presents a powerful technique for proving the existence of certain types of “diagonal” recursive sets: the Uniform Diagonalization Theorem. It allows one to prove the existence of non-complete sets in NP − P, provided that P ≠ NP. We will show also, using this theorem, that under the same hypothesis infinite hierarchies of incomparable sets (with respect to polynomial time reducibilities) exist in NP − P. This theorem allows the original proofs of these results to be considerably simplified, and we will use it later to translate the results to other complexity classes.
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