Fixed-Polynomial Size Circuit Bounds

—In 1982, Kannan showed that ΣP 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 • The following are equivalent: – NP is in SIZE(nk ) (has O(nk )-size circuit families) for some k – For each c, PNP[nc ] 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. • For a large number of natural classes C and all k 1, C is in SIZE(nk ) if and on...

• Type: Conference
• Year: 2009
• Where: COCO
• Authors: Lance Fortnow, Rahul Santhanam, Ryan Williams
• Keywords: Circuit Lower Bounds, COCO 2009, Nk -sized Circuits, Size Circuits, Theoretical Computer Science