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

  • 15
    No, it’s not known. However, $\mathrm{BPP\subseteq MA\subseteq S_2P\subseteq ZPP^{NP}}$. – Emil Jeřábek Aug 8 at 11:31
  • 1
    Looks like an answer to me. – Auberon Aug 10 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.

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