Title :
Fixed-Polynomial Size Circuit Bounds
Author :
Fortnow, Lance ; Santhanam, Rahul ; Williams, Ryan
Author_Institution :
Northwestern Univ., Evanston, IL, USA
Abstract :
In 1982, Kannan showed that SigmaP 2 does not have nk-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 PNP 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(nk) (has O(nk)-size circuit families) for some k -For each c, PNP[n c ] is in SIZE(nk) for some k -ONP/1 is in SIZE(nk) 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(nk) if and only if C/1 cap P/poly is in SIZE(nk). 3) If there is a d such that MATIME(n) sube NTIME(nd), then PNP does not have O(nk) 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(nk) for all k relies on the (relativizing) result that PPP sube MA rArr PP nsube SIZE(nk), and we give an oracle relative to which PoplusP sube MA and oplusP sube SIZE(n2) both hold.
Keywords :
computational complexity; formal languages; process algebra; set theory; theorem proving; NP machine; algebrization barrier; fixed-polynomial size circuit bound; formal language; natural class; proof system; Circuits; Computational complexity; Polynomials; USA Councils; circuit complexity; derandomization; lower bounds;
Conference_Titel :
Computational Complexity, 2009. CCC '09. 24th Annual IEEE Conference on
Conference_Location :
Paris
Print_ISBN :
978-0-7695-3717-7
DOI :
10.1109/CCC.2009.21