I am interested in complexity classes of the form $A^{B}$, where $A$ and $B$ are complexity classes such that $A \subsetneq \mathsf{P}$ and $\mathsf{NP} \subsetneq B$ are (believed to be) true.First, is this a well-defined question? Are there any known results about the structure of such classes?Second, is there an explicit problem in one of $\mathsf{NC}^{\mathsf{NP}}$, $\mathsf{NL}^{\mathsf{NP}}$, $\mathsf{FNC}^{\# \mathsf{P} }$, or $\mathsf{FNL}^{\# \mathsf{P}}$ and which is not in $\mathsf{P}$ under some assumption? After some research, I have not been able to find such a problem.
How powerful are weak complexity classes with powerful oracles?
cc.complexity theory;complexity classes