Learning Tree Languages from Positive Examples and Membership Queries
Identifieur interne : 002536 ( Istex/Corpus ); précédent : 002535; suivant : 002537Learning Tree Languages from Positive Examples and Membership Queries
Auteurs : Jérôme Besombes ; Jean-Yves MarionSource :
- Lecture Notes in Computer Science [ 0302-9743 ]
Abstract
Abstract: We investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm. We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset $\mathcal{E}$ of a regular tree language $\mathcal{L}$ is representative for $\mathcal{L}$ if every transition of the minimal tree automaton for $\mathcal{L}$ is used at least once for the derivation of an element of the set $\mathcal{E}$ .
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DOI: 10.1007/978-3-540-30215-5_33
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Institute of Science</OrgName><OrgAddress><City>Rehovot</City><Country>Israel</Country></OrgAddress></Affiliation><Affiliation ID="Aff8"><OrgName>University of Bern</OrgName><OrgAddress><Country>Switzerland</Country></OrgAddress></Affiliation><Affiliation 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><SubSeries><SubSeriesInfo><SubSeriesID>1244</SubSeriesID><SubSeriesPrintISSN>0302-9743</SubSeriesPrintISSN><SubSeriesElectronicISSN>1611-3349</SubSeriesElectronicISSN><SubSeriesTitle Language="En">Lecture Notes in Artificial Intelligence</SubSeriesTitle></SubSeriesInfo><SubSeriesHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Jaime</GivenName><GivenName>G.</GivenName><FamilyName>Carbonell</FamilyName></EditorName></Editor><Editor AffiliationIDS="Aff17"><EditorName DisplayOrder="Western"><GivenName>Jörg</GivenName><FamilyName>Siekmann</FamilyName></EditorName></Editor><Affiliation ID="Aff16"><OrgName>Carnegie Mellon University</OrgName><OrgAddress><City>Pittsburgh</City><State>PA</State><Country>USA</Country></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgName>University of Saarland</OrgName><OrgAddress><City>Saarbrücken</City><Country>Germany</Country></OrgAddress></Affiliation></EditorGroup></SubSeriesHeader></SubSeries><Book Language="En"><BookInfo BookProductType="Proceedings" ContainsESM="No" Language="En" MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-540-30215-5</BookID><BookTitle>Algorithmic Learning Theory</BookTitle><BookSubTitle>15th International Conference, ALT 2004, Padova, Italy, October 2-5, 2004. Proceedings</BookSubTitle><BookVolumeNumber>3244</BookVolumeNumber><BookSequenceNumber>3244</BookSequenceNumber><BookDOI>10.1007/b100989</BookDOI><BookTitleID>112794</BookTitleID><BookPrintISBN>978-3-540-23356-5</BookPrintISBN><BookElectronicISBN>978-3-540-30215-5</BookElectronicISBN><BookChapterCount>37</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2004</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I21017" Priority="1" Type="Secondary">Artificial Intelligence (incl. Robotics)</BookSubject><BookSubject Code="I16013" Priority="2" Type="Secondary">Computation by Abstract Devices</BookSubject><BookSubject Code="I16021" Priority="3" Type="Secondary">Algorithm Analysis and Problem Complexity</BookSubject><BookSubject Code="I16048" Priority="4" Type="Secondary">Mathematical Logic and Formal Languages</BookSubject><BookSubject Code="I21033" Priority="5" Type="Secondary">Document Preparation and Text Processing</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID><SubSeriesID>1244</SubSeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff18"><EditorName DisplayOrder="Western"><GivenName>Shoham</GivenName><FamilyName>Ben-David</FamilyName></EditorName><Contact><Email>shoham@il.ibm.com</Email></Contact></Editor><Editor AffiliationIDS="Aff19"><EditorName DisplayOrder="Western"><GivenName>John</GivenName><FamilyName>Case</FamilyName></EditorName><Contact><Email>case@cis.udel.edu</Email></Contact></Editor><Editor AffiliationIDS="Aff20"><EditorName DisplayOrder="Western"><GivenName>Akira</GivenName><FamilyName>Maruoka</FamilyName></EditorName><Contact><Email>amaruoka@isenshu-u.ac.jp</Email></Contact></Editor><Affiliation ID="Aff18"><OrgName>David R. Cheriton School of Computer Science University of Waterloo</OrgName><OrgAddress><Country> </Country></OrgAddress></Affiliation><Affiliation ID="Aff19"><OrgDivision>Department of Computer & Information Sciences</OrgDivision><OrgName>University of Delaware</OrgName><OrgAddress><Street>103 Smith Hall</Street><Postcode>DE 19716</Postcode><City>Newark</City></OrgAddress></Affiliation><Affiliation ID="Aff20"><OrgDivision>Dept. of Information Technology and Electronics</OrgDivision><OrgName>Ishinomaki Senshu University</OrgName><OrgAddress><Country> </Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Part ID="Part2"><PartInfo TocLevels="0"><PartID>2</PartID><PartSequenceNumber>2</PartSequenceNumber><PartTitle>Regular Contributions</PartTitle><PartChapterCount>29</PartChapterCount><PartContext><SeriesID>558</SeriesID><BookTitle>Algorithmic Learning Theory</BookTitle></PartContext></PartInfo><SubPart ID="SubPart8"><SubPartInfo><SubPartID>8</SubPartID><SubPartSequenceNumber>8</SubPartSequenceNumber><SubPartTitle>Query and Reinforcement Learning</SubPartTitle><SubPartChapterCount>3</SubPartChapterCount></SubPartInfo><Chapter ID="Chap33" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>33</ChapterID><ChapterDOI>10.1007/978-3-540-30215-5_33</ChapterDOI><ChapterSequenceNumber>33</ChapterSequenceNumber><ChapterTitle Language="En">Learning Tree Languages from Positive Examples and Membership Queries</ChapterTitle><ChapterFirstPage>440</ChapterFirstPage><ChapterLastPage>453</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>978-3-540-30215-5</BookID><BookTitle>Algorithmic Learning Theory</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff21"><AuthorName DisplayOrder="Western"><GivenName>Jérôme</GivenName><FamilyName>Besombes</FamilyName></AuthorName><Contact><Email>Jerome.Besombes@loria.fr</Email></Contact></Author><Author AffiliationIDS="Aff21"><AuthorName DisplayOrder="Western"><GivenName>Jean-Yves</GivenName><FamilyName>Marion</FamilyName></AuthorName><Contact><Email>Jean-Yves.Marion@loria.fr</Email></Contact></Author><Affiliation ID="Aff21"><OrgDivision>LORIA – INPL</OrgDivision><OrgName>Ecole Nationale Supérieure des Mines de Nancy</OrgName><OrgAddress><Street>615, rue du jardin botanique</Street><Postcode>54602</Postcode><City>Villers-lès-Nancy</City><Country>France</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>We investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm.</Para><Para>We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset <InlineEquation ID="IEq1"><InlineMediaObject><ImageObject FileRef="978-3-540-30215-5_33_Chapter_TeX2GIFIEq1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{E}$</EquationSource></InlineEquation> of a regular tree language <InlineEquation ID="IEq2"><InlineMediaObject><ImageObject FileRef="978-3-540-30215-5_33_Chapter_TeX2GIFIEq2.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{L}$</EquationSource></InlineEquation> is representative for <InlineEquation ID="IEq3"><InlineMediaObject><ImageObject FileRef="978-3-540-30215-5_33_Chapter_TeX2GIFIEq3.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{L}$</EquationSource></InlineEquation> if every transition of the minimal tree automaton for <InlineEquation ID="IEq4"><InlineMediaObject><ImageObject FileRef="978-3-540-30215-5_33_Chapter_TeX2GIFIEq4.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{L}$</EquationSource></InlineEquation> is used at least once for the derivation of an element of the set <InlineEquation ID="IEq5"><InlineMediaObject><ImageObject FileRef="978-3-540-30215-5_33_Chapter_TeX2GIFIEq5.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"></ImageObject></InlineMediaObject><EquationSource Format="TEX">$\mathcal{E}$</EquationSource></InlineEquation>.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></SubPart></Part></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Learning Tree Languages from Positive Examples and Membership Queries</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>Learning Tree Languages from Positive Examples and Membership Queries</title></titleInfo><name type="personal"><namePart type="given">Jérôme</namePart><namePart type="family">Besombes</namePart><affiliation>LORIA – INPL, Ecole Nationale Supérieure des Mines de Nancy, 615, rue du jardin botanique, 54602, Villers-lès-Nancy, France</affiliation><affiliation>E-mail: Jerome.Besombes@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Jean-Yves</namePart><namePart type="family">Marion</namePart><affiliation>LORIA – INPL, Ecole Nationale Supérieure des Mines de Nancy, 615, rue du jardin botanique, 54602, Villers-lès-Nancy, France</affiliation><affiliation>E-mail: Jean-Yves.Marion@loria.fr</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference" displayLabel="proceedings" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-BFHXPBJJ-3"></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 investigate regular tree languages exact learning from positive examples and membership queries. Input data are trees of the language to infer. The learner computes new trees from the inputs and asks to the oracle whether or not they belong to the language. From the answers, the learner may ask further membership queries until he finds the correct grammar that generates the target language. This paradigm was introduced by Angluin in the seminal work [1] for the case of regular word language. Neither negative examples, equivalence queries nor counter examples are allowed in this paradigm. We describe an efficient algorithm which is polynomial in the size of the examples for learning the whole class of regular tree languages. The convergence is insured when the set of examples contains a representative sample of the language to guess. A finite subset $\mathcal{E}$ of a regular tree language $\mathcal{L}$ is representative for $\mathcal{L}$ if every transition of the minimal tree automaton for $\mathcal{L}$ is used at least once for the derivation of an element of the set $\mathcal{E}$ .</abstract><relatedItem type="host"><titleInfo><title>Algorithmic Learning Theory</title><subTitle>15th International Conference, ALT 2004, Padova, Italy, October 2-5, 2004. Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Shoham</namePart><namePart type="family">Ben-David</namePart><affiliation>David R. Cheriton School of Computer Science University of Waterloo, </affiliation><affiliation>E-mail: shoham@il.ibm.com</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">John</namePart><namePart type="family">Case</namePart><affiliation>Department of Computer & Information Sciences, University of Delaware, 103 Smith Hall, DE 19716, Newark</affiliation><affiliation>E-mail: case@cis.udel.edu</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Akira</namePart><namePart type="family">Maruoka</namePart><affiliation>Dept. of Information Technology and Electronics, Ishinomaki Senshu University, </affiliation><affiliation>E-mail: amaruoka@isenshu-u.ac.jp</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="book-series" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0G6R5W5T-Z">book-series</genre><originInfo><publisher>Springer</publisher><copyrightDate encoding="w3cdtf">2004</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="I21017">Artificial Intelligence (incl. 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