Consider a PSPACE-complete problem (e.g., TQBF).
Is there a sub-problem in BPP, that is not known to be in P?
Is there a general technique of finding such sub-problems? Are any of them "natural" (i.e., not completely contrived languages)?
By "sub-problem" I mean a promise problem whose yes/no instances are respectively contained in those of the original language.
For example, asking the same question with NP instead of BPP, we can take the problem $\exists$TQBF (where the promise is that all the quantifiers are existential), namely SAT.
CLARIFICATION: I'm interested in finding some PSPACE-complete problem that has such a BPP sub-problem. It would be good if the PSPACE-complete problem is a "natural" one, such as TQBF, but I'm open to suggestions.