The Permanent Requires Large Uniform Threshold Circuits

We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n), for any function T such that for all k, T (n) = o(2). 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 TC 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 hierarchy.) We also present related and somewhat weaker lower bounds, extending the theorem of [CMTV96] showing that ACC is properly contained in ModPH. A preliminary version of this work appeared in Proc. 2nd International Computing and Combinatorics Conference (COCOON '96). Supported in part by NSF grant CCR-9509603.