An Analog Characterization of Elementarily Computable Functions over the Real Numbers
Identifieur interne : 000772 ( Istex/Corpus ); précédent : 000771; suivant : 000773An Analog Characterization of Elementarily Computable Functions over the Real Numbers
Auteurs : Olivier Bournez ; Emmanuel HainrySource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.
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DOI: 10.1007/978-3-540-27836-8_25
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We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. 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We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. 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type="first">Friedemann</forename><surname>Mattern</surname></persName><affiliation><orgName type="institution">ETH Zurich</orgName><address><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">John</forename><forename type="first">C.</forename><surname>Mitchell</surname></persName><affiliation><orgName type="institution">Stanford University</orgName><address><settlement>CA</settlement><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moni</forename><surname>Naor</surname></persName><affiliation><orgName type="institution">Weizmann Institute of Science</orgName><address><settlement>Rehovot</settlement><country key="IL">ISRAEL</country></address></affiliation></editor><editor><persName><forename type="first">Oscar</forename><surname>Nierstrasz</surname></persName><affiliation><orgName type="institution">University of Bern</orgName><address><country key="CH">SWITZERLAND</country></address></affiliation></editor><editor><persName><forename type="first">C.</forename><surname>Pandu Rangan</surname></persName><affiliation><orgName type="institution">Indian Institute of Technology</orgName><address><settlement>Madras</settlement><country key="IN">INDIA</country></address></affiliation></editor><editor><persName><forename type="first">Bernhard</forename><surname>Steffen</surname></persName><affiliation><orgName type="institution">University of Dortmund</orgName><address><country key="DE">GERMANY</country></address></affiliation></editor><editor><persName><forename type="first">Madhu</forename><surname>Sudan</surname></persName><affiliation><orgName type="institution">Massachusetts Institute of Technology</orgName><address><settlement>MA</settlement><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Demetri</forename><surname>Terzopoulos</surname></persName><affiliation><orgName type="institution">New York University</orgName><address><settlement>NY</settlement><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Dough</forename><surname>Tygar</surname></persName><affiliation><orgName type="institution">University of California</orgName><address><settlement>Berkeley</settlement><region>CA</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Moshe</forename><forename type="first">Y.</forename><surname>Vardi</surname></persName><affiliation><orgName type="institution">Rice University</orgName><address><settlement>Houston</settlement><region>TX</region><country key="US">UNITED STATES</country></address></affiliation></editor><editor><persName><forename type="first">Gerhard</forename><surname>Weikum</surname></persName><affiliation><orgName type="institution">Max-Planck Institute of Computer Science</orgName><address><settlement>Saarbruecken</settlement><country key="DE">GERMANY</country></address></affiliation></editor><idno type="pISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="seriesID">558</idno></series></biblStruct></sourceDesc></fileDesc><profileDesc><abstract xml:lang="en"><head>Abstract</head><p>We present an <hi rend="italic">analog</hi> and <hi rend="italic">machine-independent</hi> algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning <hi rend="italic">recursive analysis</hi>, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning <hi rend="italic">analog models</hi>, our results provide a characterization of the power of a natural class of analog models over the real numbers.</p></abstract><textClass ana="subject"><keywords scheme="book-subject-collection"><list><label>SUCO11645</label><item><term>Computer Science</term></item></list></keywords></textClass><textClass ana="subject"><keywords scheme="book-subject"><list><label>I</label><item><term type="Primary">Computer Science</term></item><label>I16005</label><item><term type="Secondary" subtype="priority-1">Theory of Computation</term></item><label>I14002</label><item><term type="Secondary" subtype="priority-2">Software Engineering/Programming and Operating Systems</term></item><label>I17028</label><item><term type="Secondary" subtype="priority-3">Discrete Mathematics in Computer Science</term></item><label>I1701X</label><item><term type="Secondary" subtype="priority-4">Numeric Computing</term></item><label>I15017</label><item><term 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ID="Aff9"><OrgName>Indian Institute of Technology</OrgName><OrgAddress><City>Madras</City><Country>India</Country></OrgAddress></Affiliation><Affiliation ID="Aff10"><OrgName>University of Dortmund</OrgName><OrgAddress><Country>Germany</Country></OrgAddress></Affiliation><Affiliation ID="Aff11"><OrgName>Massachusetts Institute of Technology</OrgName><OrgAddress><City>MA</City><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff12"><OrgName>New York University</OrgName><OrgAddress><City>NY</City><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff13"><OrgName>University of California</OrgName><OrgAddress><City>Berkeley</City><State>CA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff14"><OrgName>Rice University</OrgName><OrgAddress><City>Houston</City><State>TX</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff15"><OrgName>Max-Planck Institute of Computer Science</OrgName><OrgAddress><City>Saarbruecken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SeriesHeader><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-540-27836-8</BookID><BookTitle>Automata, Languages and Programming</BookTitle><BookSubTitle>31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings</BookSubTitle><BookVolumeNumber>3142</BookVolumeNumber><BookSequenceNumber>3142</BookSequenceNumber><BookDOI>10.1007/b99859</BookDOI><BookTitleID>111983</BookTitleID><BookPrintISBN>978-3-540-22849-3</BookPrintISBN><BookElectronicISBN>978-3-540-27836-8</BookElectronicISBN><BookChapterCount>102</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2004</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I16005" Priority="1" Type="Secondary">Theory of Computation</BookSubject><BookSubject Code="I14002" Priority="2" Type="Secondary">Software Engineering/Programming and Operating Systems</BookSubject><BookSubject Code="I17028" Priority="3" Type="Secondary">Discrete Mathematics in Computer Science</BookSubject><BookSubject Code="I1701X" Priority="4" Type="Secondary">Numeric Computing</BookSubject><BookSubject Code="I15017" Priority="5" Type="Secondary">Data Structures</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Josep</GivenName><FamilyName>Díaz</FamilyName></EditorName><Contact><Email>diaz@lsi.upc.edu</Email></Contact></Editor><Editor AffiliationIDS="Aff17"><EditorName DisplayOrder="Western"><GivenName>Juhani</GivenName><FamilyName>Karhumäki</FamilyName></EditorName><Contact><Email>karhumak@utu.fi</Email></Contact></Editor><Editor AffiliationIDS="Aff18"><EditorName DisplayOrder="Western"><GivenName>Arto</GivenName><FamilyName>Lepistö</FamilyName></EditorName><Contact><Email>alepisto@utu.fi</Email></Contact></Editor><Editor AffiliationIDS="Aff19"><EditorName DisplayOrder="Western"><GivenName>Donald</GivenName><FamilyName>Sannella</FamilyName></EditorName><Contact><Email>dts@inf.ed.ac.uk</Email></Contact></Editor><Affiliation ID="Aff16"><OrgDivision>Departament de Llenguatges i Sistemes Informatics</OrgDivision><OrgName>Universitat Politecnica de Catalunya, Campus Nord - Ed. Omega</OrgName><OrgAddress><Street>240 Jordi Girona Salgado</Street><Postcode>1-3 E-08034</Postcode><City>Barcelona</City></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgDivision>Department of Mathematics and Turku Centre for Computer Science TUCS</OrgDivision><OrgName>University of Turku</OrgName><OrgAddress><Postcode>20014</Postcode><City>Turku</City><Country>Finland</Country></OrgAddress></Affiliation><Affiliation ID="Aff18"><OrgDivision>Department of Mathematics</OrgDivision><OrgName>University of Turku</OrgName><OrgAddress><City>Turku</City><Country>Finland</Country></OrgAddress></Affiliation><Affiliation ID="Aff19"><OrgDivision>Laboratory for Foundations of Computer Science</OrgDivision><OrgName>University of Edinburgh</OrgName><OrgAddress><Country> </Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Part ID="Part2"><PartInfo TocLevels="0"><PartID>2</PartID><PartSequenceNumber>2</PartSequenceNumber><PartTitle>Contributed Papers</PartTitle><PartChapterCount>96</PartChapterCount><PartContext><SeriesID>558</SeriesID><BookTitle>Automata, Languages and Programming</BookTitle></PartContext></PartInfo><Chapter ID="Chap25" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>25</ChapterID><ChapterDOI>10.1007/978-3-540-27836-8_25</ChapterDOI><ChapterSequenceNumber>25</ChapterSequenceNumber><ChapterTitle Language="En">An Analog Characterization of Elementarily Computable Functions over the Real Numbers</ChapterTitle><ChapterFirstPage>269</ChapterFirstPage><ChapterLastPage>280</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2004</CopyrightYear></ChapterCopyright><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><PartID>2</PartID><BookID>978-3-540-27836-8</BookID><BookTitle>Automata, Languages and Programming</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff20"><AuthorName DisplayOrder="Western"><GivenName>Olivier</GivenName><FamilyName>Bournez</FamilyName></AuthorName><Contact><Email>Olivier.Bournez@loria.fr</Email></Contact></Author><Author AffiliationIDS="Aff20"><AuthorName DisplayOrder="Western"><GivenName>Emmanuel</GivenName><FamilyName>Hainry</FamilyName></AuthorName><Contact><Email>Emmanuel.Hainry@loria.fr</Email></Contact></Author><Affiliation ID="Aff20"><OrgName>LORIA/INRIA</OrgName><OrgAddress><Street>615 Rue du Jardin Botanique</Street><Postbox>BP101</Postbox><Postcode>54602</Postcode><City>Villers lès Nancy</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We present an <Emphasis Type="Italic">analog</Emphasis> and <Emphasis Type="Italic">machine-independent</Emphasis> algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning <Emphasis Type="Italic">recursive analysis</Emphasis>, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning <Emphasis Type="Italic">analog models</Emphasis>, our results provide a characterization of the power of a natural class of analog models over the real numbers.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Part></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>An Analog Characterization of Elementarily Computable Functions over the Real Numbers</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>An Analog Characterization of Elementarily Computable Functions over the Real Numbers</title></titleInfo><name type="personal"><namePart type="given">Olivier</namePart><namePart type="family">Bournez</namePart><affiliation>LORIA/INRIA, 615 Rue du Jardin Botanique, BP101, 54602, Villers lès Nancy, France</affiliation><affiliation>E-mail: Olivier.Bournez@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Emmanuel</namePart><namePart type="family">Hainry</namePart><affiliation>LORIA/INRIA, 615 Rue du Jardin Botanique, BP101, 54602, Villers lès Nancy, France</affiliation><affiliation>E-mail: Emmanuel.Hainry@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" type="conference" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-BFHXPBJJ-3">conference</genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2004</dateIssued><copyrightDate encoding="w3cdtf">2004</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.</abstract><relatedItem type="host"><titleInfo><title>Automata, Languages and Programming</title><subTitle>31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Josep</namePart><namePart type="family">Díaz</namePart><affiliation>Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, Campus Nord - Ed. 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