On P versus NP∩co-NP for decision trees and read-once branching programs
Identifieur interne : 001463 ( Istex/Corpus ); précédent : 001462; suivant : 001464On P versus NP∩co-NP for decision trees and read-once branching programs
Auteurs : S. Jukna ; A. Razborov ; P. Savick ; I. WegenerSource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 1997.
Abstract
Abstract: It is known that if a Boolean function f in n variables has a DNF and a CNF of size ≤ N then f also has a (deterministic) decision tree of size exp(O(log n log2 N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp (Ω(log2 N)) where N is the total number of monomials in minimal DNFs for f and -f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Whereas other examples have the additional property that f is in AC°.
Url:
DOI: 10.1007/BFb0029975
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We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp (Ω(log2 N)) where N is the total number of monomials in minimal DNFs for f and -f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Whereas other examples have the additional property that f is in AC°.</div></front></TEI><istex><corpusName>springer</corpusName><author><json:item><name>S. Jukna</name><affiliations><json:string>Dept. of Computer Science, University of Trier, D-54286, Trier, Germany</json:string><json:string>E-mail: jukna@ti.uni-trier.de</json:string></affiliations></json:item><json:item><name>A. Razborov</name><affiliations><json:string>Steklov Mathematical Institute, Gubkina 8, 117966, Moscow, Russia</json:string><json:string>E-mail: razborov@class.mi.ras.cu</json:string></affiliations></json:item><json:item><name>P. Savický</name><affiliations><json:string>Acad. of Sci. of Czech Republic, Institute of Computer Science, Pod vodárenskou věží 2, 182 07, Praha 8, Czech Republic</json:string><json:string>E-mail: savicky@uivt.cas.sz</json:string></affiliations></json:item><json:item><name>I. 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Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. 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Gombinatorica</title><publicationDate>1989</publicationDate></host></json:item><json:item><host><author><json:item><name>I Wegener</name></json:item></author><title>The Complexity of Boolean Function~</title><publicationDate>1987</publicationDate></host></json:item><json:item><author><json:item><name>A Yao</name></json:item></author><host><pages><last>10</last><first>1</first></pages><author></author><title>Proc. of 26th IEEE FOGS</title><publicationDate>1985</publicationDate></host><title>Separating the polynomial-time hierarchy by oracles</title><publicationDate>1985</publicationDate></json:item></refBibs><genre><json:string>conference</json:string></genre><serie><editor><json:item><name>Gerhard Goos</name></json:item><json:item><name>Juris Hartmanis</name></json:item><json:item><name>Jan van 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We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp (Ω(log2 N)) where N is the total number of monomials in minimal DNFs for f and -f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. 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Proceedings</BookSubTitle><BookVolumeNumber>1295</BookVolumeNumber><BookDOI>10.1007/BFb0029943</BookDOI><BookTitleID>46525</BookTitleID><BookPrintISBN>978-3-540-63437-9</BookPrintISBN><BookElectronicISBN>978-3-540-69547-9</BookElectronicISBN><BookChapterCount>51</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1997</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I16005" Priority="1" Type="Secondary">Theory of Computation</BookSubject><BookSubject Code="I14029" Priority="2" Type="Secondary">Software Engineering</BookSubject><BookSubject Code="I14037" Priority="3" Type="Secondary">Programming Languages, Compilers, Interpreters</BookSubject><BookSubject Code="I17028" Priority="4" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup></BookInfo><BookHeader><EditorGroup><Editor><EditorName DisplayOrder="Western"><GivenName>Igor</GivenName><FamilyName>Prívara</FamilyName></EditorName></Editor><Editor><EditorName DisplayOrder="Western"><GivenName>Peter</GivenName><FamilyName>Ružička</FamilyName></EditorName></Editor></EditorGroup></BookHeader><Chapter ID="Chap32" Language="En"><ChapterInfo ChapterType="ReviewPaper" NumberingStyle="Unnumbered" TocLevels="0" ContainsESM="No"><ChapterID>32</ChapterID><ChapterDOI>10.1007/BFb0029975</ChapterDOI><ChapterSequenceNumber>32</ChapterSequenceNumber><ChapterTitle Language="En">On P versus NP∩co-NP for decision trees and read-once branching programs</ChapterTitle><ChapterCategory>Contributed Papers</ChapterCategory><ChapterFirstPage>319</ChapterFirstPage><ChapterLastPage>326</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag</CopyrightHolderName><CopyrightYear>1997</CopyrightYear></ChapterCopyright><ChapterHistory><OnlineDate><Year>2005</Year><Month>6</Month><Day>17</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>3540634371</BookID><BookTitle>Mathematical Foundations of Computer Science 1997</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff1"><AuthorName DisplayOrder="Western"><GivenName>S.</GivenName><FamilyName>Jukna</FamilyName></AuthorName><Contact><Email>jukna@ti.uni-trier.de</Email></Contact></Author><Author AffiliationIDS="Aff2"><AuthorName DisplayOrder="Western"><GivenName>A.</GivenName><FamilyName>Razborov</FamilyName></AuthorName><Contact><Email>razborov@class.mi.ras.cu</Email></Contact></Author><Author AffiliationIDS="Aff3"><AuthorName DisplayOrder="Western"><GivenName>P.</GivenName><FamilyName>Savický</FamilyName></AuthorName><Contact><Email>savicky@uivt.cas.sz</Email></Contact></Author><Author AffiliationIDS="Aff4"><AuthorName DisplayOrder="Western"><GivenName>I.</GivenName><FamilyName>Wegener</FamilyName></AuthorName><Contact><Email>wegener@ls2.informatik.uni-dortmund.de</Email></Contact></Author><Affiliation ID="Aff1"><OrgDivision>Dept. of Computer Science</OrgDivision><OrgName>University of Trier</OrgName><OrgAddress><Postcode>D-54286</Postcode><City>Trier</City><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff2"><OrgName>Steklov Mathematical Institute</OrgName><OrgAddress><Street>Gubkina 8</Street><Postcode>117966</Postcode><City>Moscow</City><Country>Russia</Country></OrgAddress></Affiliation><Affiliation ID="Aff3"><OrgDivision>Acad. of Sci. of Czech Republic</OrgDivision><OrgName>Institute of Computer Science</OrgName><OrgAddress><Street>Pod vodárenskou věží 2</Street><Postcode>182 07</Postcode><City>Praha 8</City><Country>Czech Republic</Country></OrgAddress></Affiliation><Affiliation ID="Aff4"><OrgDivision>Dept. of Computer Science</OrgDivision><OrgName>University of Dortmund</OrgName><OrgAddress><Postcode>D-44221</Postcode><City>Dortmund</City><Country>Germany</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>It is known that if a Boolean function <Emphasis Type="Italic">f</Emphasis> in <Emphasis Type="Italic">n</Emphasis> variables has a DNF and a CNF of size ≤ <Emphasis Type="Italic">N</Emphasis> then <Emphasis Type="Italic">f</Emphasis> also has a (deterministic) decision tree of size exp(<Emphasis Type="Italic">O</Emphasis>(log <Emphasis Type="Italic">n</Emphasis> log<Superscript>2</Superscript><Emphasis Type="Italic">N</Emphasis>)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions <Emphasis Type="Italic">f</Emphasis> that require deterministic trees of size exp (<Emphasis Type="Italic">Ω</Emphasis>(log<Superscript>2</Superscript><Emphasis Type="Italic">N</Emphasis>)) where N is the total number of monomials in minimal DNFs for <Emphasis Type="Italic">f</Emphasis> and -<Emphasis Type="Italic">f</Emphasis>. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Whereas other examples have the additional property that <Emphasis Type="Italic">f</Emphasis> is in AC°.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>Supported by DFG grant Me 1077/101. On leave from Institute of Mathematics, Vilnius, Lithuania.</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>Supported by RBRF grant #96-01-01222.</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>Supported by grant of GA the Czech Republic No. 201/95/0976.</SimplePara></ArticleNote><ArticleNote Type="Misc"><SimplePara>Supported by DFG grant We 1066/8-1.</SimplePara></ArticleNote></ChapterHeader><NoBody></NoBody></Chapter></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On P versus NP∩co-NP for decision trees and read-once branching programs</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>On P versus NP∩co-NP for decision trees and read-once branching programs</title></titleInfo><name type="personal"><namePart type="given">S.</namePart><namePart type="family">Jukna</namePart><affiliation>Dept. of Computer Science, University of Trier, D-54286, Trier, Germany</affiliation><affiliation>E-mail: jukna@ti.uni-trier.de</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">A.</namePart><namePart type="family">Razborov</namePart><affiliation>Steklov Mathematical Institute, Gubkina 8, 117966, Moscow, Russia</affiliation><affiliation>E-mail: razborov@class.mi.ras.cu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">P.</namePart><namePart type="family">Savický</namePart><affiliation>Acad. of Sci. of Czech Republic, Institute of Computer Science, Pod vodárenskou věží 2, 182 07, Praha 8, Czech Republic</affiliation><affiliation>E-mail: savicky@uivt.cas.sz</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">I.</namePart><namePart type="family">Wegener</namePart><affiliation>Dept. of Computer Science, University of Dortmund, D-44221, Dortmund, Germany</affiliation><affiliation>E-mail: wegener@ls2.informatik.uni-dortmund.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">2005-06-17</dateIssued><copyrightDate encoding="w3cdtf">1997</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: It is known that if a Boolean function f in n variables has a DNF and a CNF of size ≤ N then f also has a (deterministic) decision tree of size exp(O(log n log2 N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp (Ω(log2 N)) where N is the total number of monomials in minimal DNFs for f and -f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen-Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Whereas other examples have the additional property that f is in AC°.</abstract><relatedItem type="host"><titleInfo><title>Mathematical Foundations of Computer Science 1997</title><subTitle>22nd International Symposium, MFCS '97 Bratislava, Slovakia, August 25–29, 1997 Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Igor</namePart><namePart type="family">Prívara</namePart><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Peter</namePart><namePart type="family">Ružička</namePart><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" displayLabel="Proceedings"></genre><originInfo><copyrightDate encoding="w3cdtf">1997</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="I14029">Software Engineering</topic><topic authority="SpringerSubjectCodes" authorityURI="I14037">Programming Languages, Compilers, Interpreters</topic><topic authority="SpringerSubjectCodes" authorityURI="I17028">Discrete Mathematics in Computer Science</topic></subject><identifier type="DOI">10.1007/BFb0029943</identifier><identifier type="ISBN">978-3-540-63437-9</identifier><identifier type="eISBN">978-3-540-69547-9</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">46525</identifier><identifier type="BookID">3540634371</identifier><identifier type="BookChapterCount">51</identifier><identifier type="BookVolumeNumber">1295</identifier><part><date>1997</date><detail type="volume"><number>1295</number><caption>vol.</caption></detail><extent unit="pages"><start>319</start><end>326</end></extent></part><recordInfo><recordOrigin>Springer-Verlag, 1997</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">1997</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, 1997</recordOrigin></recordInfo></relatedItem><identifier type="istex">278B80A20CF16EC1AF26906B6DE75764480D29EF</identifier><identifier type="DOI">10.1007/BFb0029975</identifier><identifier type="ChapterID">32</identifier><identifier type="ChapterID">Chap32</identifier><accessCondition type="use and reproduction" contentType="copyright">Springer-Verlag, 1997</accessCondition><recordInfo><recordContentSource>SPRINGER</recordContentSource><recordOrigin>Springer-Verlag, 1997</recordOrigin></recordInfo></mods></metadata></istex></record>Pour manipuler ce document sous Unix (Dilib)
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