We show that if a complexity class C is closed downward under polynomi
al-time majority truth-table reductions (less than or equal to(mtt)(p)
), then practically every other ''polynomial'' closure property it enj
oys is inherited by the corresponding bounded two-sided error class BP
[C]. For instance, the Arthur-Merlin game class AM [B1] enjoys practic
ally every closure property of NP. Our main lemma shows that, for any
relativizable class D which meets two fairly transparent technical con
ditions, we have D-BP[C] subset of or equal to BP[D-C]. Among our appl
ications, we simplify the proof by Toda [To1], [To2] that the polynomi
al hierarchy PH is contained in BP[+P]. We also show that relative to
a random oracle R,PHR is properly contained in +P-R.