Logical Characterizations of Bounded Query Classes II: Polynomial-Time Oracle Machines

We have shown that the logics (±HP)*[FOS] and (±HP)1[FOS] are of the same expressibility, and both capture P∥ NP . This result gives us the weakest possible hint that it might be wiser to try and show that (±STC*[FOS] collapses to (±STC)1[FOS] as opposed to trying to show that STC1[FOS] is closed under complementation: an attempt to use the methods of [Imm88] to achieve this latter result has failed (see [BCD89]).

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