Published by MIT Press. Copyright 1999 Massachusetts Institute of Technology.
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We show that the permanent cannot be computed by uniform constant-depth threshold circuits of size T(n) for any function T such that for all k, T^(k)(n) = o(2^n). 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, Therien, and Vollmer,
showing that there are problems in
the counting hierarchy that require superpolynomial-size uniform TC0
proof uses ``leaf languages'' as a tool in obtaining their
separations. 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 fixed level of the counting
hierarchy. (It only shows that hard problems must exist at
We also present related and somewhat weaker lower bounds, extending the theorem of Caussinus et al. showing that ACC^0 is properly contained in ModPH.