The Permanent Requires Large Uniform Threshold Circuits

We show that thepermanent cannot be computed by uniform constantdepth threshold circuits of size Tn, for any function T such that for all k, Tk n = o2n. More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size on the uniform constant-depth threshold circuit model. In particular, this lower bound applies to any problem that is hard for the complexity classes PP or P. This extends a recent result by Caussinus, McKenzie, Th erien, and Vollmer CMTV96 , showing that there are problems in the counting hierarchy that require superpolynomial-size uniform TC0 circuits. The proof in CMTV96 uses leaf languages" as a tool in obtaining their separations, and their proof does not immediately yield larger lower bounds for the complexity of these problems, and it also does not yield a lower bound for any particular problem at any xed level of the counting hierarchy. It only shows that hard problems must exist at some level of the counting ...

• Type: Journal
• Year: 1999
• Where: CJTCS
• Authors: Eric Allender
• Keywords: CJTCS 1999, Constantdepth Threshold Circuits, Counting Hierarchy, Threshold Circuit