Chicago Journal of Theoretical Computer Science

Volume 1999

Article 7

Published by MIT Press. Copyright 1999 Massachusetts Institute of Technology.

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The Permanent Requires Large Uniform Threshold Circuits

Eric Allender (Rutgers University)
6 August 1999

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 circuits. Their 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 some level of the counting hierarchy.)

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.

DOI: 10.4086/cjtcs.1999.007
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Last modified: Sat Mar 20 09:32:31 CST 1999