On Boolean vs. Modular arithmetic for circuits and communication protocols
Identifieur interne : 001296 ( Istex/Corpus ); précédent : 001295; suivant : 001297On Boolean vs. Modular arithmetic for circuits and communication protocols
Auteurs : Carsten DammSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 1998.
Abstract
Abstract: We compare two computational models that appeared in the literature in a Boolean setting and in an analog setting based on modular arithmetic. We prove that in both cases the arithmetic version can to some extend simulate the Boolean version. Although the models are very different, the proofs rely on the same idea based on the Schwartz-Zippel-Theorem. In the first part we prove that depth d semi-unbounded Boolean circuits can be simulated by depth 2d + O(log d + log n) semi-unbounded arithmetic circuits, regardless of the size. This is an improvement on a similar construction in [3] that achieves depth 3d + O(log s + log n), where s is the size of the original circuit. Our construction is simpler and uses fewer random bits. In the second part we prove, that two-party parity communication protocols can approximate nondeterministic communication protocols. A strict simulation of one by the other is impossible as was shown in [2].
Url:
DOI: 10.1007/BFb0055829
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ISTEX:8B735DF5093FB7A183856D1B9A7B094101A52447Le document en format XML
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We prove that in both cases the arithmetic version can to some extend simulate the Boolean version. Although the models are very different, the proofs rely on the same idea based on the Schwartz-Zippel-Theorem. In the first part we prove that depth d semi-unbounded Boolean circuits can be simulated by depth 2d + O(log d + log n) semi-unbounded arithmetic circuits, regardless of the size. This is an improvement on a similar construction in [3] that achieves depth 3d + O(log s + log n), where s is the size of the original circuit. Our construction is simpler and uses fewer random bits. In the second part we prove, that two-party parity communication protocols can approximate nondeterministic communication protocols. 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Computable Real Numbers</ChapterSubCategory><ChapterFirstPage>780</ChapterFirstPage><ChapterLastPage>788</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1998</CopyrightYear></ChapterCopyright><ChapterHistory><OnlineDate><Year>2006</Year><Month>5</Month><Day>28</Day></OnlineDate></ChapterHistory><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><BookID>3-540-64827-5</BookID><BookTitle>Mathematical Foundations of Computer Science 1998</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>Carsten</GivenName><FamilyName>Damm</FamilyName></AuthorName><Contact><Email>damm@uni-trier.de</Email><URL>http://www.informatik.uni-trier.de/~damm/</URL></Contact></Author><Affiliation ID="Aff1"><OrgDivision>Department of Computer Science</OrgDivision><OrgName>University of Trier</OrgName><OrgAddress><Postcode>D-54296</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We compare two computational models that appeared in the literature in a Boolean setting and in an analog setting based on modular arithmetic. We prove that in both cases the arithmetic version can to some extend simulate the Boolean version. Although the models are very different, the proofs rely on the same idea based on the Schwartz-Zippel-Theorem.</Para><Para>In the first part we prove that depth <Emphasis Type="Italic">d</Emphasis> semi-unbounded Boolean circuits can be simulated by depth 2<Emphasis Type="Italic">d</Emphasis> + <Emphasis Type="Italic">O</Emphasis>(log <Emphasis Type="Italic">d</Emphasis> + log <Emphasis Type="Italic">n</Emphasis>) semi-unbounded arithmetic circuits, regardless of the size. This is an improvement on a similar construction in [3] that achieves depth 3<Emphasis Type="Italic">d</Emphasis> + <Emphasis Type="Italic">O</Emphasis>(log <Emphasis Type="Italic">s</Emphasis> + log <Emphasis Type="Italic">n</Emphasis>), where <Emphasis Type="Italic">s</Emphasis> is the size of the original circuit. Our construction is simpler and uses fewer random bits. In the second part we prove, that two-party parity communication protocols can approximate nondeterministic communication protocols. A strict simulation of one by the other is impossible as was shown in [2].</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On Boolean vs. Modular arithmetic for circuits and communication protocols</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>On Boolean vs. Modular arithmetic for circuits and communication protocols</title></titleInfo><name type="personal"><namePart type="given">Carsten</namePart><namePart type="family">Damm</namePart><affiliation>Department of Computer Science, University of Trier, D-54296, Trier, Germany</affiliation><affiliation>E-mail: damm@uni-trier.de</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="ReviewPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2006-05-28</dateIssued><copyrightDate encoding="w3cdtf">1998</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: We compare two computational models that appeared in the literature in a Boolean setting and in an analog setting based on modular arithmetic. We prove that in both cases the arithmetic version can to some extend simulate the Boolean version. Although the models are very different, the proofs rely on the same idea based on the Schwartz-Zippel-Theorem. In the first part we prove that depth d semi-unbounded Boolean circuits can be simulated by depth 2d + O(log d + log n) semi-unbounded arithmetic circuits, regardless of the size. This is an improvement on a similar construction in [3] that achieves depth 3d + O(log s + log n), where s is the size of the original circuit. Our construction is simpler and uses fewer random bits. In the second part we prove, that two-party parity communication protocols can approximate nondeterministic communication protocols. A strict simulation of one by the other is impossible as was shown in [2].</abstract><relatedItem type="host"><titleInfo><title>Mathematical Foundations of Computer Science 1998</title><subTitle>23rd International Symposium, MFCS'98 Brno, Czech Republic, August 24–28, 1998 Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Luboš</namePart><namePart type="family">Brim</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jozef</namePart><namePart type="family">Gruska</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jiří</namePart><namePart type="family">Zlatuška</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Proceedings"></genre><originInfo><copyrightDate encoding="w3cdtf">1998</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book-Subject-Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book-Subject-Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I16005">Theory of Computation</topic><topic authority="SpringerSubjectCodes" authorityURI="I13022">Computer Communication Networks</topic><topic authority="SpringerSubjectCodes" authorityURI="I17028">Discrete Mathematics in Computer Science</topic></subject><identifier type="DOI">10.1007/BFb0055753</identifier><identifier type="ISBN">978-3-540-64827-7</identifier><identifier type="eISBN">978-3-540-68532-6</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">55367</identifier><identifier type="BookID">3-540-64827-5</identifier><identifier type="BookChapterCount">82</identifier><identifier type="BookVolumeNumber">1450</identifier><part><date>1998</date><detail type="volume"><number>1450</number><caption>vol.</caption></detail><extent unit="pages"><start>780</start><end>788</end></extent></part><recordInfo><recordOrigin>Springer-Verlag, 1998</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Lecture Notes in Computer Science</title></titleInfo><name type="personal"><namePart type="given">Gerhard</namePart><namePart type="family">Goos</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Juris</namePart><namePart type="family">Hartmanis</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jan</namePart><namePart type="family">van Leeuwen</namePart><role><roleTerm type="text">editor</roleTerm></role></name><originInfo><copyrightDate encoding="w3cdtf">1998</copyrightDate><issuance>serial</issuance></originInfo><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="SeriesID">558</identifier><recordInfo><recordOrigin>Springer-Verlag, 1998</recordOrigin></recordInfo></relatedItem><identifier type="istex">8B735DF5093FB7A183856D1B9A7B094101A52447</identifier><identifier type="DOI">10.1007/BFb0055829</identifier><identifier type="ChapterID">76</identifier><identifier type="ChapterID">Chap76</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag, 1998</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Springer-Verlag, 1998</recordOrigin></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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