I wonder, what would happen, if in the definition of $PH$ (Polynomial Hierarchy, see, e.g., here), the role of $NP$ would be replaced by $RP$?
It seems, we could still build a hierarchy, the same way as $PH$ is built, just using $RP$ everywhere instead of $NP$, and $coRP$ instead of $coNP$. Let us call it Randomized Polynomial Hierarchy ($RPH$).
My first guess is that $RPH\subseteq BPP$, or maybe $RPH=BPP$. It is based on the known fact that $NP=RP$ implies $PH=BPP$. Nevertheless, if $P\neq RP$, then $RPH$ could still be a proper, infinite hierarchy within $BPP$.
Of course, the edge of the issue is blunted by the fact that $P=RP$ is conjectured (even $P=BPP$), which would flatten $RPH$ into $P$. However, $P=RP$ is not known at this time, and it has resisted all proof attempts so far. Therefore, $RPH$ still has at least some chance to be a proper hierarchy.
While $RPH$, admittedly, has a good chance to be "flat," could the concept still be useful for something nontrivial? Here is an example: if one can prove $RPH=BPP$, then it would yield that $P=RP$ implies $P=BPP$, which, I think, would be an interesting result.
Is anything known about this?
