Generalized lowness and highness and probabilistic complexity classes

Abstract

We introduce generalized notions of low and high complexity classes and study their relation to structural questions concerning bounded probabilistic polynomial-time complexity classes. We show, for example, that for a bounded probabilistic polynomial-time complexity class$C$ =BPΣ ^P _k ,L $C$=H $C$ implies that the polynomial hierarchy collapses to$C$. This extends Schöning's result for$C$=Σ ^P _k (L $C$ andH $C$ are the low and high sets defined by$C$). We also show, with one exception, that containment relations between the bounded probabilistic classes and the polynomial hierarchy are preserved by their low and high counterparts.LBPP andLBPNP are characterized asNP ∩BPP andNP ∩ co-BPNP, respectively. These characterizations are then used to recover Boppana, Hastad, and Zachos's result that if co-NP ⊂BPNP, then the polynomial hierarchy collapses toBPNP, and Ko's result that ifNP ⊂BPP, then the polynomial hierarchy collapses toBPP.