Characterizations of some complexity classes between Θ2p and Δ2p
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We give some characterizations of the classes P NP [O(log k n)]. First, we show that these classes are equal to classes ACk−1(NP). Second, we prove that they are also equivalent to some classes defined in the Extended Boolean hierarchy. As a last characterization, we show that there exists a strong connection between classes defined by polynomial time Turing machines with few queries to an NP oracle and classes defined by small size circuits with NP oracle gates. With these results we solve open questions that arose in [Wa-90] and [AW-90]. Finally, we give an oracle relative to which classes P NP [O(log k n)] and P NP [O(logk+1n)] are different.
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