Let us define NP.NP as a class of problems that non-deterministically reduces to 3SAT.
Within the scope of this post, assume non-deterministic reduction is a non-deterministic Turing machine (or algorithm) that computes (in polytime) the function associated with the reduction.
This is inspired by BP.NP; a class of problems with polytime probabilistic reduction to 3SAT.
I see $NP.NP$ is contained in polynomial hierarchy $\Sigma_2 = NP^{NP}$.
(I) Can this upper bound be improved?
(II) Is NP.NP comparable to $P^{NP}$? Or,
(III) NP.NP = NP?
My approach: I am trying to see if I can rule out the containment of co-NP in NP.NP. I don't have a rigorous way to resolve this. Any help or pointer would be much appreciated.
The idea of a non-deterministic reduction in this post is inspired by the np-reduction on page 10 (first paragraph) of the reference:
https://arxiv.org/pdf/quant-ph/0308021
It is similar to many-one reduction (like Karp), but the reduction function is polytime computable by a Non-deterministic machine.
