When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2020.54
URN: urn:nbn:de:0030-drops-133988
URL: https://drops.dagstuhl.de/opus/volltexte/2020/13398/
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### Size, Depth and Energy of Threshold Circuits Computing Parity Function

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### Abstract

We investigate relations among the size, depth and energy of threshold circuits computing the n-variable parity function PAR_n, where the energy is a complexity measure for sparsity on computation of threshold circuits, and is defined to be the maximum number of gates outputting "1" over all the input assignments. We show that PAR_n is hard for threshold circuits of small size, depth and energy:
- If a depth-2 threshold circuit C of size s and energy e computes PAR_n, it holds that 2^{n/(elog ^e n)} ≤ s; and
- if a threshold circuit C of size s, depth d and energy e computes PAR_n, it holds that 2^{n/(e2^{e+d}log ^e n)} ≤ s. We then provide several upper bounds:
- PAR_n is computable by a depth-2 threshold circuit of size O(2^{n-2e}) and energy e;
- PAR_n is computable by a depth-3 threshold circuit of size O(2^{n/(e-1)} + 2^{e-2}) and energy e; and
- PAR_n is computable by a threshold circuit of size O((e+d)2^{n-m}), depth d + O(1) and energy e + O(1), where m = max (((e-1)/(d-1))^{d-1}, ((d-1)/(e-1))^{e-1}). Our lower and upper bounds imply that threshold circuits need exponential size if both depth and energy are constant, which contrasts with the fact that PAR_n is computable by a threshold circuit of size O(n) and depth 2 if there is no restriction on the energy. Our results also suggest that any threshold circuit computing the parity function needs depth to be sparse if its size is bounded.

### BibTeX - Entry

```@InProceedings{uchizawa:LIPIcs:2020:13398,
author =	{Kei Uchizawa},
title =	{{Size, Depth and Energy of Threshold Circuits Computing Parity Function}},
booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
pages =	{54:1--54:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-173-3},
ISSN =	{1868-8969},
year =	{2020},
volume =	{181},
editor =	{Yixin Cao and Siu-Wing Cheng and Minming Li},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},