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 1) The following are equivalent: -NP is in SIZE(n
^k
) (has O(n
^k
)-size circuit families) for some k -For each c, P
<sup>NP[n

c

]</sup>
is in SIZE(n
^k
) for some k -ONP/1 is in SIZE(n
^k
) for some k, where ONP is the class of languages accepted obliviously by NP machines, with witnesses for "yes" instances depending only on the input length. 2) For a large number of natural classes C and all k ges C is in SIZE(n
^k
) if and only if C/1 cap P/poly is in SIZE(n
^k
). 3) If there is a d such that MATIME(n) sube NTIME(n
^d
), then P
^NP
does not have O(n
^k
) size circuits for any k > 0. 4) One cannot show n
²
-size circuit lower bounds for oplusP without new nonrelativizing techniques. In particular, the proof that PP nsube SIZE(n
^k
) for all k relies on the (relativizing) result that P
^PP
sube MA rArr PP nsube SIZE(n
^k
), and we give an oracle relative to which P
^oplusP
sube MA and oplusP sube SIZE(n
²
) both hold.