Semantics for Web-Based Mathematical Education Systems

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Semantics for Web-Based Mathematical Education Systems

The

^ActiveMath

group:

Erica Melis, Jochen B ¨udenbender, Georgi Goguadze, Paul Libbrecht, Carsten Ullrich melis, jochen, george, paul, cullrich @activemath.org

Universit ¨at des Saarlandes, D-66123 Saarbr ¨ucken, Germany

ABSTRACT

Web-based user-adaptive learning environments suggest a

semanticknowledgerepresentationthatcanbereusedindif-

ferentcontexts. Moreover,iftheseeducationalsystemsem-

ployexternalservicesystemsforsupportorforexploratory

activities,thesemanticrepresentationisabasisfortheinter-

operability of the service systems and for machine-under-

standabledata. ActiveMath is such a learning environ-

mentformathematics. Weshowwhatitsannotatedseman-

tic knowledge representation, extended OMDoc, is like and

how it is used. We also discuss the current bottleneck of

authoring. Since mathematicians are mostlyfamiliar with

authoringL A

T

E

XratherthansemanticXML,ActiveMathof-

fersaL A

T

E

X2OMDoctool. AscomparedwiththedirectOMDoc

authoringwhichisnotyetvisuallysupportedthishaspros

andcons.

1. INTRODUCTION

Many educational systems and on-line documents have

been produced in recent years. Since the encoding of the

domainknowledgeforalearningenvironmentis averyex-

pensiveandtime-consumingtask,reusabilityoftheencoded

knowledge in dierent contexts and for dierent function-

alities is desirable. Therefore, the representation needs to

incorporate an ontology of the domain or, even better, a

uniqueandextensiblesemanticsofthedomainconceptsand

their variousrelationships. Similarly, inter-operability is a

prerequisiteformultipleservicesusedineducationsystems

thatcanaccessandworkwithcommonknowledgesources.

For Semantic Web applications, mathematics is a good

eld to experiment with because it is largely formalized

andhasaclearfundamentalsemanticsindependentofpre-

sentationalissues and because mathematics is a relatively

well-structured eld. For mathematics, anontology needs

to be enhanced by real semantics because mathematical

knowledgeisinherentlydierentfromitspresentation(e.g.,

itsprintedversion). Dierent presentations canmean the

same thing, e.g., 1

2

or 1/2. Conversely, the same presen-

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Semantic Web Workshop 2002 Hawai, USA Copyright by the authors.

tation canmeandierent thingsindierent contexts, e.g.,

( ab

p )=(

a

p )(

b

p

)isfalseinelementaryalgebrabuttrueinthe

theoryofquadraticresidues.

Now,ourlearning environmentActiveMath[8] isaSe-

manticWebapplicationformathematicslearning. Itsknowl-

edgerepresentationisseparatedfromitsfunctionalities. Its

knowledgerepresentationmeetstheaboverequirementsfor

mathematical contentrepresentations andthosefor educa-

tionalapplicationsthatincludetheabovementionedreusabil-

ityandinter-operabilityaswellastherepresentationofped-

agogicalinformation.

ActiveMath' knowledge representation is based on

OpenMath[3],ageneral,standardized,semanticXML-represen-

tation for mathematics. ActiveMath' functionalities re-

quire additionalinformationto beencodedintotheknowl-

edgerepresentation,e.g.,structuralinformationsuchasis-a-

denitionandpedagogicalinformationsuchasthediÆculty

ofanexercise.

ThisarticleshowshowSemanticWebissuessuchasmachine-

understandablerepresentation,reusability,extensibility,and

migration of otherrepresentations are tackled in Active-

Math. Itfocusesonthe knowledgerepresentation andits

currentauthoring. Itsummarizeswhichinformationisrep-

resented in the OMDoc-language which is an extension of

OpenMath. It describesandsubstantiatestheextensionswe

have added for theeducational andother purposesofAc-

tiveMath. It discusseshow content isauthoredpresently

inasituation,wheretoolsfor semanticrepresentationsare

emergingonlyand wherethehabitsofauthorsstill oppose

suchanencoding.

2. SEMANTIC REPRESENTATION

Althoughtoday'smostcommonrepresentationforknowl-

edgeofweb-basedsystemsisthesyntacticmarkuplanguage

HTML,forameaningfulreuseindierentcontextsandknowl-

edgesharingtheXMLrepresentationisessentialandtheRDF 1

frameworkwithdatarepresentingrelationsbetweenelements

canserveasabasisforbuildinganontology.

2.1 Semantics in Mathematical Knowledge

OMDoc has evolvedhistorically as a standard for mathe-

matical knowledgerepresentation,whichwedecidedto use

for oureducationalsystem. Becauseof thishistoryseveral

features havestilltobeadaptedtosemanticWebdevelop-

ments,e.g. RDF.However, abig advantageofusingOMDoc

isitstrulysemanticavour.

1

http://www.w3.org/RDF/

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ematics? To ensurethe inter-operability of mathematical

systems,akeyword-annotationthatmightsuÆceforsimple

searchfunctionalitiesisnotsuÆcientanymore. Amachine-

readableinputformathematicalsystemssuchasComputer

Algebra Systems (CASs) and theorem provers requires a

mathematical semantics. That is, to provide a basis for

multiplesystems,theactualmathematicalobjects/formulas

have to berepresented. Candidates for the representation

ofmathematicalobjectsandconceptsaretheXML-languages

OpenMath 2

andcontent-MathML.

3

OpenMath isa(European)

standard for the semantic representation of mathematical

formula expressions. It semantically denesasetofmath-

ematicalsymbolsintheOpenMath content dictionariesand

canimportcontentdictionariesintoothers.

However,anextensionofOpenMathisneededbecause(1)

anontologybasedonOpenMathlacksmostrelationsexceptof

theory-inclusion, (2)theOpenMath contentdictionaries are

incomplete and a simple extension mechanism is needed,

(3) OpenMath has no means to structure the content of a

mathematicaldocumentbydividingitintoitslogicalunits

suchas\denition",\theorem",and\proof".

Forthesereasons,OpenMath hasbeenextendedtoOMDoc

[6]whichincludes structuremarkupfor mathematicalcon-

ceptssuchasdefinitionsandtheoremsandforotheritems

suchasexamples, exercises, and elaborative texts. It

alsoallowstodenenewsymbols. OMDocitemsmaycontain

metadata,formalelements,textualelements,andreferences.

Referencescanbeconceptidentiers,URLs,andadditional

codeelements. MetadatainOMDoc represent legalinforma-

tioncomplianttoDublinCoremetadata[11]andextradata

elementformetadataextensions. OMDocallowstorepresent

somemathematicaldependencies: morphismsbetweenthe-

ories,equivalenceofdenitions,proof-for. Implicitlyitcon-

tainsadependencyofsymbolsbytheoccurrenceofsymbols

inanothersymboldenition.

ForourapplicationtheOMDocmetadataandrelationsbe-

tweenelementsareinsuÆcient,because, ononehand,Ac-

tiveMath needs a pedagogical ontology. For its use in

learning environments, we have extended OMDoc. On the

otherhand,relations betweenelementssuchasmathemat-

icaldependency, corollary of a theorem, similar examples,

counterexamplefor aconcept, whichwe introducearealso

generalfortheeldratherthanduetotheeducationalappli-

cation,butarenotpresentinOMDoc. Someofourextensions

are not specic for tutorial applications, such as technical

requirements 4

andbibliographicalreferences.

2.2 Pedagogical Knowledge

Theextensionsdescribedinthefollowingaremotivatedby

the tutorialapplication and generallyapplicable for learn-

ingsystemsratherthanspecicforamathematicssystem.

5

Theyinclude,amongothers

pedagogicalpropertiessuchasdiÆculty. Theyarerel-

evantforActiveMath'user-adaptivitybecausethese

2

http://www.openmath.org/

3

http://www.w3.org/Math/

4

Forclient-serverapplications, theannotationshave toin-

cludethe technical requirementsto display or to invokea

material. Theyneedto beknown, inparticular,to ensure

thatmultimediamaterialisonlyoered,ifthecomputeron

thelearnersclientcanhandleit.

5

rentlearningsituation;

pedagogicallymotivatedrelationsamongthedierent

piecesofknowledgesuchasis-prerequisite. Thisinfor-

mationcanbeused,e.g.,topresentthelearnerprereq-

uisitesfor understandingaconcept,togeneratelinks,

and togenerate anadaptively chosenand structured

learningcontent.

Ontheonehand,metadatastandardsforlearningresources

6

containtoomanymetadatawhicharenotrelevantforour

purpose. Ontheotherhand,adaptivelypresentingcontent

requiressomemetadatawhicharenotyetspeciedinLOM.

Thisisnotsurprising,sinceIMS 7

willbesoonextendedby

EducationalModellingLanguage(EML) 8

metadata. There-

fore, the ActiveMath metadata extensions of the OMDoc

DTD include some metadata from LOM as well as some

others. Forexample,weextendthelistofpossiblevaluesof

thetypeofattributeofLOMmetadataelementrelationas

describedbelow. Moredetailed, the pedagogicalmetadata

denedintheActiveMath-DTDarefield,abstractness,

difficulty, learning-context whichbelongtothe OMDoc

inFigure1.

Definition of the order of a group element

</Title>

<learning-context use="univ_first_cycle"/>

</relation>

</relation>

</extradata></metadata>

... </CMP>

</definition>

Figure1: ExcerptfromanOMDocfor adenition

fielddescribestheeldtowhichthecontentoftheitem

belongs. ItenablesActiveMathtopresentitemsfrompar-

ticularelds(suchasstatistics,physics,oreconomy),ifthis

is requiredby apedagogical strategy. For instance,if stu-

dentsfromdierentgroupslearnstatistics,e.g.,technicians,

mathematicians,psychologystudents,theyobtaindierent

(motivating) examples and exercises from the appropriate

eld. The meaning of abstractness and difficulty is

self-explaining. They serve to adapt the document to the

learner'scognitivecapabilities andlearning progress. Cur-

rently,these metadatacanhaveoneof threedierent val-

ues. learning-contextspeciesforwhichlearningcontext

thematerialwasintendedoriginally. Thepossiblevaluesof

6

suchastheLearningObjectMetadata(LOM)standardized

byIEEE-LTSC(http://ltsc.ieee.org/)

7

http://www.imsglobal.org/

8

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learning-context are those dened in the IMS-metadata

standards. This information is important incase material

fromdierentsourcesismergedforanewcourse. Further-

more,theOMDocinFigure1hasaverbosity-attribute. The

information aboutthe verbosityof the textualpartsallow

thegenerationofdierentkindsofdocumentsuchasslides

andmoreverbosescripts.

ThefollowingmetadatadenedintheActiveMath-DTD

characterizeexercisesmoreprecisely

the targeted mastery-level of the exercise. Its val-

uescanbeknowledge,comprehension,application,or

transfer

the taskofthelearner whichcanbecalculate,check,

explore,giveexample,model,orprove

levelofinteractivityandaveragelearningtime

thetechnicaltypeoftheexercise. TheActiveMath-

DTDallowsforthevaluesprovidegap,mupad,maple,

omega,multiple choice, andllincurrently.

ActiveMathemploysthesemetadatatouser-adaptivelyse-

lectexercisesfor adocumentandinthesuggestion mecha-

nismaccordingtoaparticularteachingstrategythattargets

aparticularmastery-levelandadaptivelysupportsskillac-

quisition.

Themetadatarelationisusedtorepresentseveralrela-

tionshipsbetweenOMDocitems. Thetypeofthisrelationis

speciedinthetype-attributewhichcanhavethefollowing

meanings:

the previous knowledge required to understand the

item. Forinstance,theelementinFigure1depends-on

establishesdependenciesonpreviousconcepts.

thesimilaritybetweentheitemandanotherone,e.g.,

for two examples, denitions, exercises etc. that are

similaras,e.g.,showninFigure2.

afor-relationshipswhichcanbeusedinordertochar-

acterizeanitemwithanadditionalfunctionality. For

instance,anitemthatis aproofforatheoremcould

aswellbeanexampleforamethodapplicationoran

examplecouldbeacounterexampleforaconcept.

acitation-relationshipreferringtoanadditionalbib-extra

elementwhichisdenedinActiveMath-DTDtoen-

able afullfeatured citationmechanism.

OnecanarguethatourXMLspecicationsare\heavyonat-

tributes". There are some pros and cons for this. Pros:

when using attributes one can x the default values for

them,whereastheforbodyofanelementwecanonlyspec-

ify the type of data that canbe placed inside (e.g. child

elements,PCDATAetc.). Cons: nodirectstandardscom-

pliance (translation needed). Also note that, when using

attributes,theneedtointerpretethelabelsisnotintroduc-

inganyadditionaleortforXMLdatamanipulationengines.

ThedevelopementofActiveMathmetadatawillbecon-

tinuedby: separatingthemetadataandthecontentdatabases;

IMS content packaging (as soon as EML is integrated in

Definition of left cosets

</Title>

</relation>

...

</relation>

</extradata></metadata>

</definition>

Figure2: Therelation toasimilar denition

3. SUPPORT FOR AUTHORING

AuthoringontologicalXMLisinvestigatedinseveralprojects,

e.g., the SemanticWeb project 9

and the Ontology Editor

Protege[9].

10

Authoringtoolsfor the describedtrulysemantic(math-

ematical) XML are still insuÆcientlydeveloped. Suchtools

willnotonlyhavetosupporttheauthorinemployingorex-

tendinga(mathematical)ontologybutalsoinauthoringor

choosingpedagogicalmetadata,inauthoringexerciseswith

external service systems, and support her in writing (ab-

stracted)mathematicalformulasthatlatercanbepresented

viastylesheets. Thisisaseriousbottleneckcurrently.

So,whataretherealisticalternativescurrently?First,use

astillpreliminarytool,QMath,brieydescribedinx3.1that

supports the authoring of mathematical formulas. QMath

provides atleastsome supportbutisnot suÆcientlycom-

fortable for the average author. Second, translate from a

syntactically oriented macro-based encoding and heuristi-

cally addsemanticsas describedinx3.2. Third, wait until

the open-source community including our group has pro-

ducedanicevisualtool(someofitexists already,e.g., our

visualeditorfortablesofcontent).

3.1 QMath

QMath is a tool and a migration format for producing

OMDoc documents. It was developed by Alberto Gonzales

Palomo and is currentlyused by ActiveMath authors to

writeOMDocs. AQMath documentis easytoproduce. This

ispartiallyduetothefactthatQMathsupportsunicodeand

theauthorisfreetowriteherformulasdirectlybyusinguni-

codesymbols. Look,for instanceatthefollowing example,

correspondingtotheL A

T

E

XsourcefromtheFigure4:

Definition:[<-df1]

:"The definition of cartesian product"

:for:[cartesian_product]

:depends_on:[def_pair]

...

$MN Df suchthat(pair(x,y),x 2M&y2N)$.

Figure3: Excerpt fromaQMathdocument

Theauthorcandeneacontextforadocument. A con-

text containsuser-denedshortcuts forOMDoc elements. It

alsocontainsthereferencesofsymbolsusedinthedocument

9

http://www.semanticweb.org/

10

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theauthorusesasymbol,QMathsuggeststhesymboldecla-

rationprocedurewhichassignsameaningtothesymbolby

anexplicitreferenceto asymbolinacontentdictionaryor

viaapre-recordedcontext.

QMath supports metadata elements of OMDoc as well as

someadditional metadataofActiveMath. Thismetadata

supportcanbe extended. Assoonasanewelementis de-

claredinadocumentorinacontextQMathperformsatrans-

formationtoXMLmarkup.

11

3.2 L

^A

TEX2

^OMDoc

Forhandlingmacro-basedencodings,wefacetwodemands:

(1)the migration of existing mathematical learning docu-

mentsourcesencodedinpresentationallanguages,sayL A

T

E X,

and(2)thenewencodingbyauthors/mathsprofessorswho

areusedtowritingL A

T

E

Xandopposeauthoringatrulyse-

mantic representation. In the following, we describe our

eortsinbothdirections.

The migration ofan existing large L A

T

E

X documentwas

thegoal ofacase studywe conductedin2001. TheL A

T

E X

sources of the document [4] were not intended to be mi-

gratedtoasemanticrepresentationoriginally. Itunrestrict-

edlyusesauthor-specicmacros.AlthoughtheL A

T

E

Xsource

was alreadysplitinto 'slices'andprovidedsomedependen-

cies,thedocumentwasdesignedprettylinearlyratherthan

appropriateforahypertextpresentationanditwasdiÆcult

toprepareforareuse. Forinstance,itincludedtextsuchas

"Aswehaveseeninthepreviousexample...".

The logical structureof the in the existing L A

T

E X docu-

ment hadto be carefully redesignedin orderto obtain in-

dependentlyreusable items related via dependencies. For

instance, many basic pieces in the text still included an-

otherone. E.g., introductionsor elaborations containeda

denition.

Another problem was the use of not-represented abbre-

viations. Thetext contained elements like ... the correct

notation for this should be

@f

@x

(a) but we shall write only

@f

@x

sinceitis clearthat weare talkingaboutthederivative

inthepointa.Asemanticrepresentationwouldneedtore-

fertothesamemathematicalobjectthatmayhavedierent

annotations,e.g.,abbrevandadefault.

Thepurely syntacticuseofnotationis oneofthe major

problems. InasentencelikeBecarefullwithournotations!

In some cases (a;b) will mean an open interval and oth-

erwise just an ordered pair. (a;b) purely syntactical and

itssemantics iscontext-dependent. Anautomatictransla-

tionis almost impossibleor at least has ahighly context-

dependent heuristics. The presentation-oriented represen-

tation and mis-use of L A

T

E

X is another problem. This is

obviouswhentheL A

T

E

X encoding$\{x: x\in A$ und $x$

ist rational $\}$) of the formula fx: x 2 A und x ist

rationalg is analyzed and shows that a mathematical for-

mulaisscatteredintopiecesandcombinedagainbynatural

languagetext.

Apart the context-dependent and presentation-oriented

representation, presentationaland semantic information is

mixedespecially in the mathematical formulas and there-

foreheuristics for correctlyparsing all formulas cannotbe

11

FormoreinformationonQMathseehttp://www.matracas.

callyimpossible.

Ourexperiencesuggeststhatitisimpossibletotranslate

a L A

T

E

X source automatically that has been writtenwith-

outthegoalofasemanticrepresentationinmind. Thisnot

onlyrequirestoimplementmanydocument-specicheuris-

tics, it boilsdowntoabout50% manual translation. Even

worse,often theoriginal encoding doesnotallowa unique

translationtosemanticallyrepresentedformulasandmaybe

author'sworkagain.

A solutioncanonlyrely onaquasi-semanticmarkupin

L A

T

E

X that usesmacrosand environmentsto encodeinfor-

mationneededforsemanticknowledgerepresentationandis

extensible. Thishasbeenattemptedinanothercase study,

where weinstructed'conservative'authorsto writestrictly

denedL A

T

E

Xsourcesandprovidedatoolforanautomatic

conversiontoOMDocviaQMath.

WedenedL A

T

E

Xmacrosandenvironmentsfortherepre-

sentationofsemanticinformationandmeta-datatosupport

the automaticconversiontoOMDoc. Ifanelementhasnon-

emptychildren,itisencodedbyanenvironment,otherwise

a macro is used 12

. We speciedrestrictions { inparticu-

larforwritingformulasinthatL A

T

E

X. Themostimportant

restrictionsare

use thesymbolsalready denedinOpenMath content

dictionariesinordertoensurereusability,

specifytheinterpretationofsourceformulaswrittenin

L A

T

E

X,e.g. inxorprexnotation

use the pre-dened L A

T

E

X environments and macros

for dening OMDoc elements, i.e., structure elements

andmetadata.

ThefollowingisanexampleofusingspecialOMDoc-oriented

L A

T

E

Xenvironments:

\begin{definition}{df1}{carte sian _pro duct}

{The definition of cartesian product}

\depends-on{def_pair} ...

$M \times N \Df \suchthat{\pair{x}{y}}

{x \in M \and y \in N }$.

\end{definition}

Figure 4: Example ofwriting restricted L A

T

E X

create aseparate leto dene newsymbols and add

XSLpresentationalinformationtothedenedsymbols,

also dene own DTD extensionif necessary and XSL

presentationforit.

Iftheserequirementaremet,ourtoolautomatically con-

vertstherestrictedL A

T

E

XsourcesviaQMathtoOMDoc.

To summarize: as comparedwith a directauthoring of

OMDoc in QMath, authoring in a restricted and augmented

L A

T

E

Xismorefamiliartomathematiciansevenifnotstrictly

simpler. Althoughthedirectcontrol ofthelayoutofadoc-

ument by editing the generated PDF-document is very at-

tractivetoauthors,itkeepsthepresentationandloosesthe

representationandthusdestroysthesemanticandmetadata

informationneededfortheSemanticWebapplication.

12

For more details see http://www.activemath.org/~ilo/

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4. USAGE IN

^ActiveMath

Insteadofaconclusion,wewanttosummarizewhatAc-

tiveMathisable todowiththeknowledgerepresentation

andwhatthefutureactivitieswillbeinthisdirection.

FortheActiveMathsystem,thereuseofcontentindif-

ferent contexts is particularly important because its user-

adaptivity implies thatthe samecontentcanbe presented

indierentwaysdependingontheuserandinthelearning

situation.

ActiveMath' user-adaptive functionalities such as the

presentationofthecontentandthedynamicsuggestiongen-

eration use the structureinformation and metadata anno-

tatingtheunitsofthecontent.

ActiveMath has the following components: a session

manager,coursegenerator,themathematicalknowledgebase,

a presentation planner, a user model, a pedagogical mod-

uleandexternalmathematicalsystems(ActiveMathinte-

gratesseveralservicesystemsforcalculation,proof,andex-

plorationsuchastheproofplannermega[7]andtheCom-

puterAlgebraSystemsMuPAD[10]andMaple). Here,the

usermodelis acomponentto store, readandupdate data

aboutthelearner'sprole. Itcontainshistoryoftheactions,

alistofpreferencesoftheuserandalistofcompetenceas-

sessments. The user's actions are analyzed by evaluators

thatcalculateupdatesoftheusermodel.

ThecoursegenerationinActiveMathisrealizedasfol-

lows: requests of the user are sent from the browser via

a web-server to the session manager. When the user has

chosenhergoalconceptsandscenario,thesessionmanager

sendsthisrequesttothecoursegenerator. Thecoursegen-

eratorcontactsthemathematicalknowledgebaseinorderto

calculatewhichmathematicalconceptsarerequiredforun-

derstandingthegoalconcepts. Thentheinformationabout

theuser'sknowledgeis requestedfrom theusermodeland

thecollected IDsofOMDoc itemsannotatedwiththeuser's

knowledgemastery-levelsareenteredasfactsintotheknowl-

edgebase of theexpertsystem. Thenthe rules are evalu-

ated and generate aninstructional list of XML items to be

presented. Here, the metadata such as diÆculty are not

onlyusedtoselectappropriateexercisesandexamplesfora

learnerbutalsofortheevaluationoftheuser'sactivities.

The XML content is transformedto aformat suitable for

presentationviaXSL.AnXSLstylesheetspeciesthepresen-

tationofourXMLdocuments,bydescribinghowaninstance

istransformedintoHTMLortoL A

T

E

XorFlash.

Thesemanticrepresentationisabasisformergingcontent

from dierent sources and presenting the merged content

consistently.

TheActiveMathknowledgerepresentation isproviding

twoontologies: themathematicalandthe educationalone.

Mathematicalontologyis alsousefulfor other mathappli-

cations. Themathematicalconcepts(elementsofontology)

aretheskeleton(macrolevel)ofadocument,andtheedu-

cationalonesprovidetheinformationforbuildingthemicro

levelstructure.

Alternative Usages

ActiveMath'supportforexploratoryandinteractivelearn-

ing will be improved. This includes the investigation of

elaborate search functions based on the the semantic and

partiallyformalrepresentation.

Anextstepisthemachine-understandabledescriptionof

many situations, includingthe automatedadviseto auser

forchoosinganappropriatesystemtoperformataskorfor

agent-basedcomputations(see[5]).

Semantically represented repositories will be useful not

justfor learningenvironmentsbutalsofor working mathe-

maticians. For instance, OMDoc-structured repositories can

improve the organization and searchability of mathemati-

cal knowledge. Today the digitallibraries,haveto face the

manuallycontrolledentryofauthorandclassicationinfor-

mationandoftenmakeuseofkeywordsauthoredbyreview-

ers. Todaythesearchcapabilitiesarelimitedtotextualand

keywordsearch.

13

WeunderstandourresearchaspartofthelargerEuropean

initiativefor web-based mathematical knowledgerepresen-

tationandmanagement. Itsrstworkshop[1] coveredtop-

icsrangingfrompublishingoflargecollectionsofelectronic

preprints to tools for managing mathematical documents.

Thereishopeforacriticalmassofcontentencodedinase-

manticXMLsincetheinitiativewillworkonthisaswellason

toolstomaintainandusethecontentdataandmetadata.

5. REFERENCES

[1] ElectronicProceedingsoftheFirst International

WorkshoponMathematicalKnowledgeManagement,

September2001.

[2] J.Budenbender,G.Goguadze, P.Libbrecht,E.Melis,

andC.Ullrich.MetadataforActiveMath.Seki

ReportSR-02-01,Fachbereich14Informatik,

UniversitatdesSaarlandes,2002.

[3] O.CaprottiandA.M. Cohen.Draftoftheopenmath

standard.OpenMathConsortium,1998,

http://www.nag.co.uk/projects /Open Math /omst d/.

[4] B.I.DahnandH.Wolters.AnalysisIndividuell.

Springer-Verlag,2000.

[5] M.DewarandD.Carlisle.Mathematicalsoftware: the

nextgeneration? In[1]

[6] M.Kohlhase.OMDoc: Towards aninternetstandardfor

theadministration,distributionandteachingof

mathematicalknowledge.InAISC'2000,2000.

[7] E.MelisandJ.H.Siekmann.Knowledge-basedproof

planning.ArticialIntelligence,115(1):65{105,1999.

[8] E.Melis, E.Andres,G.Goguadze,P.Libbrecht,M.

Pollet,andC.Ullrich.ActiveMath: System

description.InJohannaD.Moore,CarolRedeld,and

W.LewisJohnson,eds,ArticialIntelligencein

Education,pages580{582.IOSPress,2001.

[9] N.F.Noy,R.W.Fergerson,andM.A.Musen.The

knowledgemodelofProtege-2000: Combining

interoperabilityandexibility.In2thInternational

Conference onKnowledgeEngineeringandKnowledge

Management(EKAW'2000),2000.

[10] A.SorgatzandR.Hillebrand.MuPAD-Ein

ComputeralgebraSystemI.Linux Magazin,(12),1995.

[11] TheDublinCoreMetadataInitiative,1998,

http://purl.org/DC/.

13

TheAmericanMathematicalSociety'sE-Printservice,e.g.,

allowsthesearchthroughthereviewsonlywhicharetradi-

tionally encodedin the T

E

X language. Searching through

formulasinthis languageis oneof themost unpredictable