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Size, Depth and Energy of Threshold Circuits Computing Parity Function

Author Kei Uchizawa

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Author Details

Kei Uchizawa
  • Graduate School of Science and Engineering, Yamagata University, Japan

Funding

  • Uchizawa, Kei: This work was supported by JSPS KAKENHI Grant Number JP19K11817.

Acknowledgements

We thank the reviewers for their constructive criticisms and suggestions.

Cite As Get BibTex

Kei Uchizawa. Size, Depth and Energy of Threshold Circuits Computing Parity Function. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 54:1-54:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.54

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Circuit complexity
  • neural networks
  • threshold circuits
  • sprase activity
  • tradeoffs

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