Structure of Complexity Classes: Separations, Collapses, and Completeness

During the last few years, unprecedented progress has been made in structural complexity theory; class inclusions and relativized separations were discovered, and hierarchies collapsed. We survey this progress, highlighting the central role of counting techniques. We also present a new result whose proof demonstrates the power of combinatorial arguments: there is a relativized world in which UP has no Turing complete sets.

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