Abstract
The best known size lower bounds against unrestricted circuits have remained around 3n for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than 5n. In this work, we propose a nongateelimination approach for obtaining circuit lower bounds, via certain depththree lower bounds. We prove that every (unboundeddepth) circuit of size s can be expressed as an OR of 2^{s/3.9} 16CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around n^{3o(1)} for decades. Under a plausible hypothesis about probabilistic polynomials, we show that n^{4ε}size DeMorgan formulas have 2^{n^{1Ω(ε)}}size depth3 circuits which are approximate sums of n^{1Ω(ε)}degree polynomials over F₂. While these structural results do not immediately lead to new lower bounds, they do suggest new avenues of attack on these longstanding lower bound problems.
Our results complement the classical depth3 reduction results of Valiant, which show that logarithmicdepth circuits of linear size can be computed by an OR of 2^{ε n} n^δCNFs, and slightly stronger results for seriesparallel circuits. It is known that no purely graphtheoretic reduction could yield interesting depth3 circuits from circuits of superlogarithmic depth. We overcome this limitation (for smallsize circuits) by taking into account both the graphtheoretic and functional properties of circuits and formulas.
We show that improvements of the following pseudorandom constructions imply superlinear circuit lower bounds for logdepth circuits via Valiant’s reduction: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth3 circuits with constant bottom fanin. On the other hand, our depth reductions show that even modest improvements of the known constructions give elementary proofs of improved (but still linear) circuit lower bounds.
BibTeX  Entry
@InProceedings{golovnev_et_al:LIPIcs.ITCS.2021.24,
author = {Alexander Golovnev and Alexander S. Kulikov and R. Ryan Williams},
title = {{Circuit Depth Reductions}},
booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
pages = {24:124:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771771},
ISSN = {18688969},
year = {2021},
volume = {185},
editor = {James R. Lee},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13563},
URN = {urn:nbn:de:0030drops135633},
doi = {10.4230/LIPIcs.ITCS.2021.24},
annote = {Keywords: Circuit complexity, formula complexity, pseudorandomness, matrix rigidity}
}
Keywords: 

Circuit complexity, formula complexity, pseudorandomness, matrix rigidity 
Collection: 

12th Innovations in Theoretical Computer Science Conference (ITCS 2021) 
Issue Date: 

2021 
Date of publication: 

04.02.2021 