Constructing Oracles by Lower Bound Techniques for Circuits

Separating or collapsing complexity hierarchies has always been one of the most important problems in complexity theory. For most interesting hierarchies, however, we have been so far unable to either separate them or collapse them. Among these unsolvable questions, whether P equals NP is perhaps the most famous one. In view of the fundamental difficulty of these questions, a less interesting but more realistic alternative is to consider the question in the relativized form. Although a separating or collapsing result in the relativized form does not imply directly any solution to the original unrelativized question, it is hoped that from such results we do gain more insight into the original questions and develop new proof techniques toward their solutions. Recent investigation in the theory of relativization shows some interesting progress in this direction. In particular, some separating results on the relativized polynomial hierarchy have been found using the lower bound results on constant-depth circuits [Yao, 1985; Hastad, 1986, 1987]. This new proof technique turns out to be very powerful, capable of even collapsing the same hierarchy (using, of course, different oracles) [Ko, 1989].