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^{NP[n c ]} 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 n²-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(n²) both hold.