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Complexity zoo https://complexityzoo.uwaterloo.ca/Complexity_Zoo:D#dtime states $DTIME(f(n))$ with $PP$ oracle is not in $P/Poly$ if $f(n)$ is superpolynomial.

We know $SUBEXP=\cap_{\epsilon>0}DTIME(2^{n^\epsilon})\not\subseteq DTIME(p(n))$ for any polynomial $p(n)$.

So do we have $SUBEXP^{PP}\not\subseteq P/Poly$? So can we say either $SUBEXP\not\subseteq P/Poly$ or ${PP}\not\subseteq P/Poly$ holds?

I am trying to understand if $C^O\not\subseteq D$ for classes $C,O,D$ then does it follow either $C\not\subseteq D$ or $O\not\subseteq D$?

In other words if we have two classes $C,O$ with $C\subseteq D$ and $O\subseteq D$ then does it mean $C^O\subseteq D$?

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    $\begingroup$ What do you think? Have you tried proving your claim? $\endgroup$ – Yuval Filmus Dec 30 '15 at 9:19
  • $\begingroup$ @YuvalFilmus I am not very sure. Take $C=P$ and $O=NP$ and clearly $P$ and $NP$ are conjectured to be in $\Sigma_2$ and $P^{NP}$ is also in $\Sigma_2$. So I do not know what to say on how to approach such problem although seems to be related to lowness of classes. $\endgroup$ – T.... Dec 30 '15 at 9:30
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    $\begingroup$ P and NP are in $\Sigma_2^P$ by definition. $\endgroup$ – Yuval Filmus Dec 30 '15 at 10:12
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If $D$ is low for itself ($D^D = D$) then $C,O \subseteq D$ should imply $C^O \subseteq D$, though it might depend on the exact definition of the oracle model (for resource-restricted classes there are sometimes delicate issues there). In contrast, if $D$ isn't low for itself then taking $C=O=D$ we get a counterexample.

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  • $\begingroup$ Thank you. Are you saying $P/poly$ is not low for itself? otherwise we will already have an incredible result of either $SUBEXP$ not in $P/poly$ or $PP$ is not in $P/poly$ (meaning either way $PP$ is not in $P/poly$) since we know $SUBEXP^{PP}$ is not in $P/poly$? $\endgroup$ – T.... Dec 30 '15 at 10:33
  • $\begingroup$ No, I think that P/poly is low for itself, so your result follows. $\endgroup$ – Yuval Filmus Dec 30 '15 at 12:01
  • $\begingroup$ isn't $PP$ not in $P/poly$ a new result (this already gives both $PSPACE$ and $NEXP$ not in $P/Poly$) correct? And possibly $\#P$ as well right? $\endgroup$ – T.... Dec 30 '15 at 21:22
  • $\begingroup$ I'm not sure I follow that part of your argument. If SUBEXP is not in P/poly, why does it follow that PP is not in P/poly? $\endgroup$ – Yuval Filmus Dec 30 '15 at 21:25
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    $\begingroup$ We get exactly what you wrote: either SUBEXP is not in P/poly, or PP is not in P/poly. $\endgroup$ – Yuval Filmus Dec 30 '15 at 21:29

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