Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates
We study the following computational problem: for which values of k , the majority of n bits MAJ n can be computed with a depth two formula whose each gate computes a majority function of at most k bits? The corresponding computational model is denoted by MAJ k ∘ MAJ k . We observe that the minimum...
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| Veröffentlicht in: | Theory of computing systems 2019-07, Vol.63 (5), p.956-986 |
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| container_title | Theory of computing systems |
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| creator | Kulikov, Alexander S. Podolskii, Vladimir V. |
| description | We study the following computational problem: for which values of
k
, the majority of
n
bits MAJ
n
can be computed with a depth two formula whose each gate computes a majority function of at most
k
bits? The corresponding computational model is denoted by MAJ
k
∘ MAJ
k
. We observe that the minimum value of
k
for which there exists a MAJ
k
∘ MAJ
k
circuit that has high correlation with the majority of
n
bits is equal to Θ(
n
1/2
). We then show that for a randomized MAJ
k
∘ MAJ
k
circuit computing the majority of
n
input bits with high probability for every input, the minimum value of
k
is equal to
n
2/3 +
o
(1)
. We show a worst case lower bound: if a MAJ
k
∘ MAJ
k
circuit computes the majority of
n
bits correctly on all inputs, then
k
≥
n
13/19 +
o
(1)
. |
| doi_str_mv | 10.1007/s00224-018-9900-3 |
| format | Article |
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k
, the majority of
n
bits MAJ
n
can be computed with a depth two formula whose each gate computes a majority function of at most
k
bits? The corresponding computational model is denoted by MAJ
k
∘ MAJ
k
. We observe that the minimum value of
k
for which there exists a MAJ
k
∘ MAJ
k
circuit that has high correlation with the majority of
n
bits is equal to Θ(
n
1/2
). We then show that for a randomized MAJ
k
∘ MAJ
k
circuit computing the majority of
n
input bits with high probability for every input, the minimum value of
k
is equal to
n
2/3 +
o
(1)
. We show a worst case lower bound: if a MAJ
k
∘ MAJ
k
circuit computes the majority of
n
bits correctly on all inputs, then
k
≥
n
13/19 +
o
(1)
.</description><identifier>ISSN: 1432-4350</identifier><identifier>EISSN: 1433-0490</identifier><identifier>DOI: 10.1007/s00224-018-9900-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Circuits ; Computation ; Computer Science ; Lower bounds ; Special Issue on Theoretical Aspects of Computer Science (STACS 2017) ; Theory of Computation</subject><ispartof>Theory of computing systems, 2019-07, Vol.63 (5), p.956-986</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2018</rights><rights>Theory of Computing Systems is a copyright of Springer, (2018). All Rights Reserved.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c311t-5e3e9de1fbd1a182f535988667a132377d44e7f19e7322cd23e9706b2ff459fa3</cites><orcidid>0000-0001-7154-138X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00224-018-9900-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00224-018-9900-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Kulikov, Alexander S.</creatorcontrib><creatorcontrib>Podolskii, Vladimir V.</creatorcontrib><title>Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates</title><title>Theory of computing systems</title><addtitle>Theory Comput Syst</addtitle><description>We study the following computational problem: for which values of
k
, the majority of
n
bits MAJ
n
can be computed with a depth two formula whose each gate computes a majority function of at most
k
bits? The corresponding computational model is denoted by MAJ
k
∘ MAJ
k
. We observe that the minimum value of
k
for which there exists a MAJ
k
∘ MAJ
k
circuit that has high correlation with the majority of
n
bits is equal to Θ(
n
1/2
). We then show that for a randomized MAJ
k
∘ MAJ
k
circuit computing the majority of
n
input bits with high probability for every input, the minimum value of
k
is equal to
n
2/3 +
o
(1)
. We show a worst case lower bound: if a MAJ
k
∘ MAJ
k
circuit computes the majority of
n
bits correctly on all inputs, then
k
≥
n
13/19 +
o
(1)
.</description><subject>Circuits</subject><subject>Computation</subject><subject>Computer Science</subject><subject>Lower bounds</subject><subject>Special Issue on Theoretical Aspects of Computer Science (STACS 2017)</subject><subject>Theory of Computation</subject><issn>1432-4350</issn><issn>1433-0490</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><recordid>eNp1kMFOwzAMhiMEEmPwANwqcQ44cdo0R1TYhlTEBc5R1iajE2tLkmra29NRpJ042bK_35Y-Qm4Z3DMA-RAAOBcUWE6VAqB4RmZMIFIQCs5_e04FpnBJrkLYAgDmADNSFt2uH2LTbpJXs-18Ew_J-pAUXRuiaWPyZPv4eVoVja-GJoZk34zjstsnC9PSpk2WJtpwTS6c-Qr25q_Oycfi-b1Y0fJt-VI8lrRCxiJNLVpVW-bWNTMs5y7FVOV5lknDkKOUtRBWOqasRM6rmo-4hGzNnROpcgbn5G662_vue7Ah6m03-HZ8qTlDhVkqpRgpNlGV70Lw1uneNzvjD5qBPkrTkzQ9StNHaRrHDJ8yYWTbjfWny_-HfgDg4G3R</recordid><startdate>20190701</startdate><enddate>20190701</enddate><creator>Kulikov, Alexander S.</creator><creator>Podolskii, Vladimir V.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2P</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PRINS</scope><scope>PTHSS</scope><scope>PYYUZ</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0001-7154-138X</orcidid></search><sort><creationdate>20190701</creationdate><title>Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates</title><author>Kulikov, Alexander S. ; Podolskii, Vladimir V.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c311t-5e3e9de1fbd1a182f535988667a132377d44e7f19e7322cd23e9706b2ff459fa3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Circuits</topic><topic>Computation</topic><topic>Computer Science</topic><topic>Lower bounds</topic><topic>Special Issue on Theoretical Aspects of Computer Science (STACS 2017)</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Kulikov, Alexander S.</creatorcontrib><creatorcontrib>Podolskii, Vladimir V.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Database (1962 - current)</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection (Proquest) (PQ_SDU_P3)</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ABI/INFORM global</collection><collection>Computing Database</collection><collection>ProQuest Science Journals</collection><collection>Engineering Database</collection><collection>ProQuest advanced technologies & aerospace journals</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>One Business (ProQuest)</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>ProQuest Central China</collection><collection>Engineering collection</collection><collection>ABI/INFORM Collection China</collection><collection>ProQuest Central Basic</collection><jtitle>Theory of computing systems</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Kulikov, Alexander S.</au><au>Podolskii, Vladimir V.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates</atitle><jtitle>Theory of computing systems</jtitle><stitle>Theory Comput Syst</stitle><date>2019-07-01</date><risdate>2019</risdate><volume>63</volume><issue>5</issue><spage>956</spage><epage>986</epage><pages>956-986</pages><issn>1432-4350</issn><eissn>1433-0490</eissn><abstract>We study the following computational problem: for which values of
k
, the majority of
n
bits MAJ
n
can be computed with a depth two formula whose each gate computes a majority function of at most
k
bits? The corresponding computational model is denoted by MAJ
k
∘ MAJ
k
. We observe that the minimum value of
k
for which there exists a MAJ
k
∘ MAJ
k
circuit that has high correlation with the majority of
n
bits is equal to Θ(
n
1/2
). We then show that for a randomized MAJ
k
∘ MAJ
k
circuit computing the majority of
n
input bits with high probability for every input, the minimum value of
k
is equal to
n
2/3 +
o
(1)
. We show a worst case lower bound: if a MAJ
k
∘ MAJ
k
circuit computes the majority of
n
bits correctly on all inputs, then
k
≥
n
13/19 +
o
(1)
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00224-018-9900-3</doi><tpages>31</tpages><orcidid>https://orcid.org/0000-0001-7154-138X</orcidid><oa>free_for_read</oa></addata></record> |
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| source | SpringerLink Journals (MCLS); Business Source Complete |
| subjects | Circuits Computation Computer Science Lower bounds Special Issue on Theoretical Aspects of Computer Science (STACS 2017) Theory of Computation |
| title | Computing Majority by Constant Depth Majority Circuits with Low Fan-in Gates |
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