Is it known that $\textbf{BPP} \subseteq \textbf{P}^{\textbf{NP}}$?

  • 16
    $\begingroup$ No, it’s not known. However, $\mathrm{BPP\subseteq MA\subseteq S_2P\subseteq ZPP^{NP}}$. $\endgroup$ Aug 8 '18 at 11:31
  • 2
    $\begingroup$ Looks like an answer to me. $\endgroup$
    – Auberon
    Aug 10 '18 at 8:39

Heller gives a relativized world where $BPP = EXP^{NP}$ which sits far outside of $P^{NP}$. Showing $BPP \neq EXP^{NP}$ unconditionally would in itself be considered a major breakthrough in derandomization.

  • $\begingroup$ What would $BPP\neq EXP^{NP}$ or $BPP\subseteq P^{NP}$ give in terms of lower bounds? $\endgroup$
    – Turbo
    Feb 23 '19 at 22:03

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