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TR14-126 | 9th October 2014 20:08
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#### Separating Cook Completeness from Karp-Levin Completeness under a Worst-Case Hardness Hypothesis

**Abstract:**
We show that there is a language that is Turing complete for NP but not many-one complete for NP, under a {\em worst-case} hardness hypothesis. Our hypothesis asserts the existence of a non-deterministic, double-exponential time machine that runs in time $O(2^{2^{n^c}})$ (for some $c > 1$) accepting $\Sigma^*$ whose accepting computations cannot be computed by bounded-error, probabilistic machines running in time $O(2^{2^{\beta 2^{n^c}}})$ (for some $\beta > 0$).

This is the first result that separates completeness notions for NP under a worst-case hardness hypothesis.