Fixed-Polynomial Size Circuit Bounds

In 1982, Kannan showed that $\Sigma^\P_2$ does not have $n^k$-sized circuits for any $k$. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan's result, we still cannot prove that $\P^\NP$ does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence -- the ``algebrization'' barrier -- that current techniques have inherent limitations in this respect. We explore questions about fixed-polynomial size circuit lower bounds around and beyond the algebrization barrier. We find several connections, including \begin{itemize} \item The following are equivalent: \begin{itemize} \item $\NP$ is in $\SIZE(n^k)$ (has $O(n^k)$-size circuit families) for some $k$ \item For each $c$, $\P^{\NP[n^c]}$ is in $\SIZE(n^k)$ for some $k$ \item $\ONP/1$ is in $\SIZE(n^k)$ for some $k$, where $\ONP$ is the class of languages accepted {\it obliviously} by $\NP$ machines, with witnesses for ``yes'' instances depending only on the input length. \end{itemize} \item For a large number of natural classes ${\cal C}$ and all $k \geq 1$, ${\cal C}$ is in $\SIZE(n^k)$ if and only if ${\cal C}/1\cap\P/\poly$ is in $\SIZE(n^k)$. \item If there is a $d$ such that $\MATIME(n) \subseteq \NTIME(n^d)$, then $\P^{\NP}$ does not have $O(n^k)$ size circuits for any $k ≫ 0$. \item One cannot show $n^2$-size circuit lower bounds for $\oplus \P$ without new nonrelativizing techniques. In particular, the proof that $\PP \not\subseteq \SIZE(n^k)$ for all $k$ relies on the (relativizing) result that $\P^{\PP} \subseteq \MA \Longrightarrow \PP \not\subseteq \SIZE(n^k)$, and we give an oracle relative to which $\P^{\oplus \P} \subseteq \MA$ and $\oplus \P \subseteq \SIZE(n^2)$ both hold. \end{itemize}