A Weight-Size Trade-Off for Circuits with MOD m Gates (1994)
| Venue: | In Proc. 26th Ann. ACM Symp. Theor. Comput |
| Citations: | 10 - 1 self |
BibTeX
@INPROCEEDINGS{Grolmusz94aweight-size,
author = {Vince Grolmusz},
title = {A Weight-Size Trade-Off for Circuits with MOD m Gates},
booktitle = {In Proc. 26th Ann. ACM Symp. Theor. Comput},
year = {1994},
pages = {68--74}
}
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Abstract
: We prove that any depth--3 circuit with MOD m gates of unbounded fan-in on the lowest level, AND gates on the second, and a weighted threshold gate on the top needs either exponential size or exponential weights to compute the inner product of two vectors of length n over GF(2). More exactly we prove that log(wM ) = \Omega\Gamma n), where w is the sum of the absolute values of the weights, and M is the maximum fan--in of the AND gates on level 2. Setting all weights to 1, we have got a trade--off between the numbers of the MOD m gates and the AND gates. By our knowledge, this is the first trade--off result involving hard--to--handle MOD m gates. In contrast, with n AND gates at the bottom and a single MOD 2 gate at the top one can compute the inner product function. The lower--bound proof does not use any monotonicity or uniformity assumptions, and all of our gates have unbounded fan--in. The key step in the proof is a random evaluation protocol of a circuit with MOD m gates. ...
