Computational complexity, theory of computation, polynomial-time hierarchy, randomness, oracles.
We simplify the proof by S. Toda [Tod89] that the polynomial hierarchy PH is contained in BP[ӨP]. Our methods bypass the technical quantifier interchange lemmas in the original proof, and clarify the counting principles on which the result depends. We also show that relative to a random oracle R, PHR is strictly contained in ӨPR.
Regan, Kenneth W. and Royer, James S., "A Simpler Proof of PH C BP[ӨP]" (1990). Electrical Engineering and Computer Science Technical Reports. 96.