# Is anything known about NC$^1$ with NP oracle

A few things are known about the class $$\textsf{L}$$ provided with an $$\textsf{NP}$$ oracle ($$\textsf{L}^\textsf{NP} = \Theta_2^\textsf{P}$$ has attracted a bit of attention, for instance [1]) On the other hand, I can't find much about the class $$\textsf{NC}^1$$ with access to an $$\textsf{NP}$$ oracle. Is it because the use of an oracle doesn't play well with the definition of $$\textsf{NC}^1$$? Which I doubt. Most likely, it's because I haven't looked properly.

Except for the direct $$\textsf{NP} \cup \textsf{coNP} \subseteq {\textsf{NC}^1}^{\textsf{NP}} \subseteq \Theta_2^{\textsf{P}}$$, What is known about $${\textsf{NC}^1}^{\textsf{NP}}$$?

• It’s quite nontrivial to decide what is the right way to relativize $\mathrm{NC}^1$ in the first place. See in particular arxiv.org/abs/1204.5508. But any sensible definition should make $\mathrm{NC^{1\,NP}}$ the same as $\Theta^P_2$, as already $\mathrm{AC^{0\,NP}}$ does that (this follows from the representation of $\Theta^P_2$ as in Theorem 4 of Buss & Hay). – Emil Jeřábek Feb 7 at 14:30
• These are not quite research-level questions. I think you should slow down on question asking, and instead study the basic literature on $\Theta^P_2$ first. – Emil Jeřábek Feb 7 at 14:41
• Thanks for the clarification and references. – Abdallah Feb 8 at 2:45
• Apologies that I asked too many basic questions in a row. I'll refrain from doing so in the future. – Abdallah Feb 8 at 2:46