On closure properties of bounded two-sided error complexity classes

We show that if a complexity class C is closed downward under polynomial-time majority truth-table reductions (≤_mtt^p), then practically every other "polynomial" closure property it enjoys is inherited by the corresponding bounded two-sided error class BP[C]. For instance, the Arthur-Merlin game class AM [B1] enjoys practically every closure property of NP. Our main lemma shows that, for any relativizable class D which meets two fairly transparent technical conditions, we have C^BP[C] {Mathematical expression}BP[D^C]. Among our applications, we simplify the proof by Toda [Tol], [To2] that the polynomial hierarchy PH is contained in BP[⊕P]. We also show that relative to a random oracle R, PH^R is properly contained in ⊕P^R.