My question is the following:
Can a (probabilistic, deterministic, ndtm) oracle turing machine $A$ calling an oracle residing in a superior (more difficult) complexity class $B$, have less power then the oracle called ($A^B < B$).
My believe is that this new class should be at least as powerful as $B$?
For example, assuming that i've $P^X$, where $X$ resides in a superior complexity class.
Could $P^X$ be weaker than $X$ in some cases and, if yes, is a meaningful complexity class?