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1 TheSymbolicDataBenchmarkProblems CollectionofPolynomialSystems DepartmentofComputerScience UniversityofLeipzig,Germany Hans-GertGrabe Authorsofsoftwareorevenpackagesormodulescapableofspecialsymbolic 1.1Motivation 1 Introduction duringearlydevelopmentphases,reliabletestsandscreeningswithlarge it.whereasthe'copyandpaste'methodisusuallysucientforthisaim setsofdatarequirebatchprocessingandspecialtestbeds. computationsaresoonorlaterfacedwiththeproblemtotestandevaluate software,startandmonitoritsrun,andstoreandevaluatetheoutputof thecomputation. SuchtestbedenvironmentshouldpreparetestdataforinputtotheCA software(e.g.,amaplepackage){andoftenuseevendierenttechnologies specialinputformatsfortheexamples)and,oftenenough,developedanew (shellscriptsandleredirection){theyareusuallyself-made(including Althoughsuchtestbedsaremostlynotrelatedtothetestedsymbolic foreachproject. commonprojectthatcollectsexperiencewithtestbedsandallowsforeasy reuseandmodicationofalreadyexistingcode.thesymbolicdataproject wassetouttomeetthesedemands. Toavoidthisoverheaditwouldbeinterestingtounifysucheortsintoa priatetools,wefocussedonfreesoftwaretools.atthemomentourtestbed ditionsandtestbedsareusuallynotwrittenfromscratchbutreuseappro- SinceeasyreusabilityisbestachievedbycodeunderFreeSoftwarecon-

2 reliesonperl5andthegnutimefunction.arstrelease(mainlydevelopedbyo.bachmann,kaiserslautern,andtheauthor)containsmorethan ofdata.thesepackages(withmorethan15000linesofcode)aredrivenby 40Perlmodulestohandle,translate,sort,validate,andstoredierentkinds astandardinterfaceprogramsymbolicdata.theyareavailablefromour preparationofinputdataandstartingandmonitoringprocesses.thegnu scriptingandpatternmatchingfacilitiesturnedouttobebestsuitedforthe WebsiteunderthetermsoftheGNUPublicLicense(GPL).Perlwithits timefunctionprovidesanindependenttimingtoolwiththepossibilityto DataProjectarosefromquestionsrelatedtocomparisonandbenchmarking timeoutandinterruptprocesses. aboutowntestcomputationsthatoftenenoughcouldnotberepeatedby ofsymbolicsoftware.typicalbenchmarkpaperspublishedsofarreport Asecond(andhistoricallyeventherst)motivationfortheSymbolicexamplesarenotsuppliedorsuppliedonlyinprinted(andoftenmisprinted) interestedpartiesduetodierentreasons:thesoftwareisnotavailable,huge equivalent)formsetc. form,theauthorsrefertoexamplesgivenintheliteratureindierent(non itorystoringbenchmarkexamplesinreliableformatsthatcouldeasilybe accessedbyinterestedparties.ofcourse,thisrequirestogiveawayown testmaterialunderfreesoftwareconditionstothecommunity.thesymbolicdataprojectstarted(withkindlyacknowledgedsupportbythefrench 'UMSMedicis'andtheGerman'FachgruppeComputeralgebra')suchdata Suchproblemscouldbeavoidediftherewasacentralelectronicrepos- 1.2TheSymbolicDataProject{AimsandCurrentState andgeometrytheoremproving. collectionsintheareasofpolynomialsystemsolving(tobereportedbelow) generalbenchmarkdataandtomakethemelectronicallyavailableinamore orlessuniformway. atestbedforsymbolicsoftware,tosystematicallycollectexistingspecialand ThemaintrackfollowedsofarwiththeSymbolicDataprojectwastodevelop requiresalsotodevelopconceptsandtoolstogenerate,store,manipulate, put,outputorintermediateresults.therefore,tocollectbenchmarkdata presentandmaintainit. Notethatsymboliccomputationsoftenleadtovoluminousdataasin-

3 1.TounifyeortsofseveralpeopletodevelopPerltoolsforthemanagementofdigitalsymbolicdatafromdierentareasofComputer HencetheSymbolicDataprojectstartedwithtwomaingoals: forspecialtestandbenchmarkpurposesatalocalsite.theyareready Algebra. Thesetools,althoughnotyetperfect,areusefulandcanbeadapted 2.ToprovideacentralrepositoryofdigitalbenchmarkdatafromdierentareasofComputerAlgebra. fordownloadandimprovement. Thisrepositoryathttp:// bythegerman'fachgruppecomputeralgebra')containsthedatacollectedsofarandalsoprovidesaccesstothetoolsanddocumentation. isequallyopentopeoplejoiningthesymbolicdataprojectgroup,andwe Theprojectisorganizedasafreesoftwareproject.TheCVSrepository (viahtmldownloadfromourwebsite)underthetermsofthegnupublic enjoyyourcooperation.toolsanddataarefreelyavailablealsoastar-les License.Thealpharelease0.4,availablesinceMarch2001,contains Toolstomaintaindigitalsymbolicdata(belowyouwillndashort awellelaboratedhtmldocumentation, Digitaldatacollectionsfromtheareasofpolynomialsystemsolving andgeometrytheoremproving, overview), managementofsymbolicnoncomputationalinformation,therelease0.5, DuetothefactthattheSymbolicDatatoolswereusedsofarmainlyforthe andasmallnumberofpublicationsandpresentations. availablesincejanuary2002,oersseparately thesymbolicdatatoolswithaminimaldatacollection(requiredto Thisnewoershouldbeconsideredalsobypeoplewhoareinterestedtouse andthefullsymbolicdatadatacollection. buildthedocumentation) ourtoolsforlocaltestorbenchmarkcomputationsontheirowndataonly.

4 TheSymbolicDatatoolsdevelopedsofararedesignedtomeetthreedierent goals: 1.3TheSymbolicDataTools 1.Tosystematicallycollectandmaintaindigitalbenchmarkdataarising ThedataisstoredinadatabasecomplyingaXML-likesyntaxthat easilymaybeextendedandadapted.thesymbolicdatatoolstogetherwiththeexibilityoftheperllanguageallowtostore,extract, uniedway. Thestandardinterfaceprogramsymbolicdatacanbeusedforthe canbeextendedwithlittleexperienceinperlprogramming.some mostcommonoperations(insertion,validation,extension,update) withoutperlknowledge.duetotheelaborated'actions'conceptit invariousareasofcomputeralgebra. combine,select,modify,presentetc.datawithvariousobjectivesina 2.Tofacilitatetestoreventrustedbenchmarkcomputationsonthecollecteddata. TheSymbolicDataprojectprovidesconceptsandtoolstoextractdata Software,tosetup,start,time,interrupt,andmonitorcomputations fromthesecomputations. ontheseinputdata,andtocollect,analyze,andevaluateoutputdata fromthedatabaseinaformreadablebydierentcomputeralgebra examplesaregivenbelow. Thisrequiresmoreexibilityandhenceadditionalprogrammingeffortsbytheuser.Wealreadydesignedseveraltoolsforabenchmark 3.Toprovidetoolstoaccess,select,translateandpresentdataindierentformats. Thispartoftheprojectisrudimentary.SymbolicDatatemporarypro- Computeenvironment,butthispartoftheprojectisyetunderdevelopment. TocontributedatatotherepositoryortojointheSymbolicDatagroup SQL-compliantdatabases. videsasmallhtmlinterfacefortestpurposesandaninterfaceto pleaseconsultourwebsiteformoreinformation.

5 TheSymbolicDatadatacollectionisdesignedusingarelationaldatabase 2.1HowDataareOrganized 2 TheSymbolicDataINTPSCollection modelandstoredinaxml-likeasciiformat.thisallowsforeasymanipulationandtranslationofthisdataindierentformats. oneofthevariousdatabaseprogramsasmainenginebutimplementeda les(sd-les)andattachedtotheperlinterfaceinatransparentwayas Perlinterfacetoaccessandmanipulatedata.Datarecordsarestoredas Duetoexibilityreasonswedecidednottouse(atleastatthemoment) recordsoftag/valuepairs(sd-records)usingperl5modulartechnology. informationaboutbenchmarkcollectionsofpolynomialsystemsiscontained intheintpstable.accordingtotherelationaldatabasemodelsecondary TablescorrespondtosubdirectoriesoftheDatadirectorytree.Themain Similarrecordsshareacommonstructureandaregroupedintotables. informationabouttheserecordsisscatteredoverseveralothertables(bib scriptions,geoforgeometrytheoremprovingbackgroundofrelevantintps forbibliographicalreferences,problemsforproblemorproblemclassde- recordsetc.)andlinkedwiththemainrecordthroughacrefattribute. nextpage.foradescriptionofthedierentattributesseebelow. 2.2CrossReferences AtypicalINTPSsd-le,Trinks'example,see[3],isreproducedonthe easilybesearchedfordierentkeys.wedecidedtoputthiscrossreference tionaldatabasemodel,itisusuallystoredinspecialrelationtablesthatcan references.suchrelationalinformationcombinestworecordsand,inarela- Youmayaskformoreinformationaboutthisexample,e.g.,bibliographical extractitassecondarydatainsqlcompliantformforimportintorelation informationintooneofthemain(primary)recordsandtoprovidetoolsto tablesofaclassicaldatabaseenginewithsearchandselectfacilities.this avoidstodevelopanewelaboratedsearchandselectfacilitiesfortheprimary (XMLbased)data. BIBtablerecordsandusestheTrinks'example'sIdasforeignkey.Below thebibrecordof[3]isreproduced. ForTrinks'example,relationalbibliographicalinformationisstoredin isdeclaredasforeigninacrefentryinsomeintpsrecordsthatpointto eachtimeanewpublicationreferstoit.foranalogousreasonsthebibtable ispersistenceinthesensethatwedonotneedtochangeanintpsrecord ThemainreasonforthedecisiontodeclaretheINTPStableasforeign

6 #Record'INTPS/Trinks' <Id> ################################################################# <basis> [35*p+40*z+25*t-27*s, <Key> <Type> Trinks </Type> </Key> </Id> 45*p+35*s-165*b-36, ]</basis> 15*p*t+20*z*s-9*w, -11*b^3+w*p+2*z*t -11*s*b+3*b^2+99*w, 25*p*s-165*b^2+15*w+30*z-18*t, <vars> <dlist> <llist> <ishomog> <degree> [w,p,z,t,s,b] [1,1,2,2,2,3] 0 [4,4,3,5,3,3] 10 </vars> </dlist> </llist> </ishomog> <Date> <PERSON> <Comment> <Version> Mar graebe diff=easy... </Date> </PERSON> </Comment> </Version> </degree> #Endofrecord'INTPS/Trinks' theprimarysourcewherethepolynomialsystemwasmentionedrsttime. ################################################################ SecondarydatamaybesearchedwithanSQLcompliantdatabaseengine forboththeprimaryandtheforeignkeys. Notethatitisnotalwaysaseasyasheretomakesuchajudiciousdecision. FollowingtheXMLphilosophytheattributesofrecords(i.e.,theXMLtag 2.3TheStructureofINTPSRecords butcollectedthisinformationinspecialmetarecordsanddevelopedperl names)andtheirdescriptionsarenotxedwithinthesymbolicdatatools toolstoextractthedescriptionsfromthesemetatables. andxslstylesheetsatthemomenttodescribetagsyntaxandsemantics butarepartofthedata.duetolackingexperiencewedidnotusedtd

7 ... #Record'BIB/Boege_86a' <Id> ################################################################# author= {Boege,W.andGebauer,R.andKredel,H.}, </Id> title= journal= volume= year= {Someexamplesforsolvingsystemsofalgebraic pages= {2}, {83-98}, {J.Symb.Comp.}, {1986}, equationsbycalculating{gr\"obner}bases}, }</bibentry>... <CRef> [INTPS/Hairer_1=>Hairer1,... INTPS/Rose=>Rose, INTPS/Trinks=>Trinks, INTPS/Trinks_1=>SmallTrinks INTPS/Hairer_2=>Hairer2, ]</CRef>... #Endofrecord'BIB/Boege_86a' ################################################################# Templatesareeasilycreated,extendedorchangedvaryingthecorresponding METAtableswithyourfavoritetexteditor. Thisallowsforgreatexibilityandcarefuldesignofdatatablesbyusers. as,e.g.,[2,3,4,7,8,9].eachsuchsystemofpolynomialsisdenedthrough anitebasisinacertainpolynomialringr[x]inalistofvariablesxovera thatuniesthedierentbenchmarkcollectionsofsystemsofpolynomials DesigningthestructureofINTPSrecordswetriedtospecifyaframework basedomainr.itoccursthatmostexamplesmaybereducedtosystemsof polynomialswithintegercoecientsorwithcoecientsinr=z[p]where pisalistofparameters.wedecidedtofocusonsuchsystems. Foruniformityreasonsandtoeasecomparison,werequireofavalid

8 INTPSrecord,thatitsbasispolynomialsarestoredinexpandedstandard mialandthepolynomialsofthebasisareorderedw.r.t.thedegreereverse formusingthe+,*,and^operators,andthatthemonomialsofapolyno- lexicographicalordering.thesymbolicdatavalidateactioncanxthese propertiesofanintpsrecordifyouhavesingular[6]installedonyour dierentpolynomialsystems.backgroundinformationmaybeofstructural computer1. cernsinvariantpropertiesofthebasisandtheidealgeneratedbyit,e.g., orrelationaltype.structuralinformationaboutapolynomialsystemcon- Furthertagsaredenedtocollectbackgroundinformationaboutthe dim,degree,isoprimes,isoprimedims,etc.,andperlroutinesaredened degreeoftheideal,aprimeorprimarydecompositionoftheideal,orcertain parametersofsuchadescription.severaloptionaltags,likellist,dlist, listsofthelengthsanddegreesofthebasispolynomials,thedimensionor tocollectorevengeneratesuchinformation. tableonthenextpage.theirstructureandsemanticsisstoredinaspecial METAtableMETA/INTPSinthesameXML-formatastherecordsthemselves andthusmayeasilybeextendedormodiedifnecessary.themetatables ThemandatoryandoptionalattributesofINTPSrecordsarelistedinthe initialization. 3arepartoftheDatadirectorytreeandreadinbythePerltoolsduring SymbolicDataprovidesagreatvarietyoftoolstoperformoperationsonthe collecteddata.thesetoolsareofverydierentnatureandrequirements: TheSymbolicDataPerlTools listsofrecords,uptothetransformationofpartsortheentiredatabase theyrangefromtheinsertionandvalidationofsinglerecords,overtheinitiation,controlandevaluationoftestorbenchmarkcomputationsonselected beaccessedinauniquewayasactionsinvokedthroughthesymbolicdata programthatprovidesastandardinterfaceandrealizescommand-lineparsing,initializationofglobalvariablesandrequiredmodules,andexecutionof TheoperationsareimplementedinahierarchyofPerlmodulesandcan intootherrepresentationslikehtmlorsql. theactionsinheritedfromthecommandline. othersystems. storedasaglobalhash$actionsthatmaybeextendedbynewentries. 1ThereisastubtousealsootherCASforthispurpose,butnoimplementationyetfor Itiseasytoaddnewfunctionalitytotheprogramsinceactionsare

9 basis(m) Id,TypeandKey(m) +Keyisgeneratedautomatically). Alistofpolynomialsinexpandedstandardformwithintegercoecients,deninganidealI. Stringsthatidentifytherecordwithinthedatabase(Id=Type vars(m)andparameters(o) basedomain(o) R=k(param)[vars]wherekdenotesthebasiccoecienteld. Thebasiccoecienteld(default:Q). Listsofvariables.Iisconsideredasanidealinthepolynomialring dlist,llist(o) attributes,dim,ishomog,isoprimes,... dierentvariablenotations. roughinvariantstoidentifyrecordscontainingthesamebasisin Listsoftotaldegreesandlengthsofthebasiselements.Thisgives ChangeLog,Version,PERSON,Date suppliedtheinformation.thereisaspecialtablepersonthat MoreinformationaboutIifavailable. licdataandkeepshistoricaltrackoftheiractivities. collectsmoreinformationaboutthepeopleinvolvedwithsymbo- Informationaboutthehistoryoftherecordandthepersonwho CRefAlistofcrossreferencestorelatedrecordsinothertables. Mandatory(m)andoptional(o)attributesofINTPSrecords licdatadocumentationandthesourceofthemoduleactionsspec.pmfor Belowwegivesomeexamplesofuserdenedactions.ConsulttheSymbo- moredetails. Theoverallsyntaxofasymbolicdatacallis lineargumentsuptothemandatoryactionargument(s),andloadstheglobal Onstart-upsymbolicdataloadsallbasicmodules,parsesthecommand- symbolicdata[-rfile]actions[options][args] properties,e.g.,theperlmodulesrequiredfortheaction,adescriptionof theactionetc.theactionhashcanbeextendedusingtherst(optional)-r actionhashwhichspeciesallknown(or,\registered")actionsandtheir

10 fileargument,wherefileisthenameofaperlmodulewhichisloaded beforetheactionsareparsed. ment,e.g.,thesearchandndfacilitiesofsymbolicdata. Notethatsomepartsevenofthebasicfeaturesareyetunderdevelop- systemandhencesomeperlprogramming.thesymbolicdataactionsconceptisbestsuitedtowritesuchextensionsalmostfromscratchintoale andgetthemrunningwith Seethedirectorybin/scriptsforsampleextensions.Wecomebacktothat questioninthenextsection. symbolicdata-rfile... Forexample,startingtestorbenchmarkcomputationsonaspecialCAS requirestranslationofthedataintothespecialinputformatofthetested Dierentoperationsonthedatarequiredierentdegreesofexibility. aredirectlyavailablethroughthesymbolicdatainterface.hereisa(not newrecordsandextractionofsql-compliantcrossreferenceinformation complete)listofsuchactions: Anumberof\standard"actions,mainlyforinsertionandvalidationof Validate Update Insert Manipulationofdatabaseentries: Unique Updatesd-recordinDataBasefromforeignsource Validatesd-record(s) Insertsd-recordintoDataBase Print CreationofnewINTPSrecords: Out Printeldsofsd-record(s)toSTDOUT Testforuniquenessofsd-record(s)w.r.t.DataBase PrintrecordstoSTDOUT CreateINTPS CreateanewINTPSrecordfromaGEOrecordof Homog Flat GenerateanewINTPSrecordwithatbasisfrom equationaltype EvaluationofinformationinBIBrecords: GenerateanewINTPSrecordwithhomogenizedbasisfromaninhomogeneousINTPSrecord anintpsrecordwithparameters GetAllBibs MakeBib extractbibtexentriesfromallbibrecordstostd- OUT CreatealeA.bibfromA.auxandrelevantBIB records

11 ExtractionofCRefinformation: CreateSQL UpdateSQL PrintupdateinformationforSQLtablestoSTD- extractsqltabledenitionstostdout tationformoredetails.e.g.,togenerate/updatethesqlcrossreference Actionscanbedrivenbyvariousoptions.Pleaseconsultthedocumen- informationtableforsomeoftheintpsrecords(withkeymatchingsym*) issuethecommand(inthesymbolicdatahomedirectory) Data/SQL)tocatchcrossreferenceinformation.Yougetalistinglike CRefTableisapredenedSQLtable(alsostoredassd-leinthedirectory symbolicdataupdatesql-tablecreftabledata/intps/sym*.sd... deletefromcreftablewhereid='intps/sym1_311'; insertintocreftablevalues('intps/sym1_311','problems/sym1',''); deletefromcreftablewhereid='intps/sym1_211'; deletefromcreftablewhereid='intps/sym3_5'; insertintocreftablevalues('intps/sym1_211','problems/sym1',''); Theresultmaybepipedtoadatabaseprogram(weusedPostgres95)to updatethecreftablecreatedearlierwiththecommand insertintocreftablevalues('intps/sym3_5','problems/sym3',''); createtablecreftable( thatyieldsoutput symbolicdatacreatesql-tablecreftable Comment Foreign_Id varchar(100)); varchar(80)notnull, ArstseriesofbenchmarkcomputationsonINTPSrecordswasdesignedand 4executedbyOlafBachmannintheyear2000.HedevelopedtheCompute HowtoRunLocalBenchmarkComputations Perlmodulethatrealizescomputationsasanelaboratedinterplaybetween (tablemachine)andexamples(tableintps).seethepaper[1]formore congurationsofcomputeralgebrasoftware(tablecasconfig),machines detailsaboutthisconcept.

12 teamandthispartremainsexperimentalstillnow. elaboratedenvironmentifyouhavedataavailableinelectronicformandthe TherewasnocontinuationoftheseeortswhenOlaflefttheproject lesȧsanexampleweconsiderbenchmarkcomputationstotestthesolve PerlscriptingfacilitiesathandtocreateCASinputlesandanalyzeoutput Butitiseasytosetuplocalbenchmarkcomputationsalsowithoutan facilityofmupadonzerodimensionalidealsasdescribedin[5].toset programas newactionsolvetest.thisactioniscalledviathesymbolicdatainterface upsuchcomputationswecreatealescripts/compute.plthatdenesa symbolicdataparsestheinputline,expandsthesd-lenamesandcallsthe actiononeachofthesd-records.hencethemostdicultpartofanaction denitionisthecallslotthatcontainsaperlfunctiontobeexecutedon symbolicdata-r"scripts/compute.pl"solvetest[sd-les] thecorrespondingsd-record. containsthemupadcodeoftheexampleandstartsasystemcall InourexamplethisPerlfunctioncreatesaninputle/tmp/Key.inthat viatimedsystem.timedsystemisaspecialsymbolicdataperlfunctionde- nedinthemoduletimedsystem.pmthatallowstotimeandtrapacomputation.itisbasedonthegnutimefunction.werefertothesymbolicdata mupad<$infile>$outfile2>&1 eachexamplethecomeuptimeofthecasoftware.analternativesolution usesthe(systemdependent)innertimefunction(e.g.,mupad'straperror onlinedocumentationformoredetails. function)totimecomputationsandisdescribedbelow. Suchanapproachpossiblydoesnotmeetyourneedssinceitincludesfor GNUtime.Notethatdierentkeysofanactionshashentrymaycarryalso detailedhtmldescription(key'description')abouttheaction. verbose(key'verbose')andusageinformation(key'example')andevena OnthenextpageyoundalistingofCompute.plforthesolutionwith recordandarrangesthemasmupadinputlines(thecodebetweenthe ale/tmp/key.in.thenmupadisstartedwithatimeboundof100s.to EOT's).Duringexecutionoftheactiononthatrecordthiscodeiswrittento Thefunctionthecomputationextractstherelevantvaluesfromthesd- solvetheproblem.uponsuccesstheoutputofthecomputationiswrittento ale/tmp/key.outthatcanbeanalyzedeitherbyhandorwithadditional Perlfunctions.

13 {verbose=>"benchmarkcomputationswithmupadandtimedsystem", $ACTIONS->{SolveTest}= req=>['timedsystem.pm'], call=>sub {my$r=shift; #createtheinfile open(fh,">$infile")or close(fh); printfhthecomputation($r); my$infile="/tmp/$r->{key}.in"; #setupthecomputation my$outfile="/tmp/$r->{key}.out"; die"can'topen$infileforwriting:$!\n"; my$maxtime=100; my$syscall="mupad<$infile>$outfile2>&1"; #startthecomputation #evaluatethecomputation if($l[0]<0) },return$r; {print("$r->{key}notfinishedwithin$maxtimesec.\n");} else example=>'symbolicdata-r"$sd_home/bin/scripts/compute.pl"' {printf("$r->{key}:usertime%1.2f,systemtime%1.2f.\n",.'solvetest$sd_home/data/intps/sym1_211.sd', $l[1],$l[2]);} }; PRETTYPRINT:=FALSE; subthecomputation vars:=$r->{vars}; {my$r=shift; polys:=$r->{basis}; tt:=time((sol:=solve(polys,vars)));sol;nops(sol); my$s=<<eot; quit; EOTreturn$s; tt:=time((sol1:=numeric::solve(polys,vars)));sol1;nops(sol1); }

14 mersandwillnotbediscussedhere. ThecodeitselfisstraightforwardforslightlyexperiencedPerlprogram- separatelywithmupadandinspecttheresults.hereistherelevantperl usethesymbolicdatatoolstogenerateanappropriateinputle,runit codeforanewactiontraptest: ForasolutionusingMuPAD'straperrorfunctioninsteadofGNUtime {verbose=>"benchmarkcomputationswithmupadandtraperror", $ACTIONS->{TrapTest}= argvcall=>sub {shift;my$arg=expandargv(shift); #createtheinfile my$l; open(fh,">$infile")or my$infile="/tmp/mupad.in"; printfhinittext(); printfhexittext(); close(fh); die"can'topen$infileforwriting:$!\n"; },print"inputfilewrittento$infile\n"; example=>'symbolicdata-r"$sd_home/bin/scripts/compute.pl"' lenames.thenwecreatethe(single)inputle/tmp/mupad.incontaining };Therstlinescollectthesd-recordstobetestedfromtheir(expanded).'TrapTest$SD_HOME/Data/INTPS/S*.sd', thedierentexamples.thisrequiressomeadditionalcode,mainlyforthe functiontrapcomputation,reproducedonthenextpage. Nowstartthetestcomputationas 5 ExtendingtheDataBase mupad</tmp/mupad.in>/tmp/mupad.out2>&1 Inasimilarfashionthedatabasemaybeextendedtoincorporatenewexamplesevenfromnewapplicationareas.Wedocumentarstscratchextension RaymondHemmecke,whorunstheWebsitehttp:// toexamplesfromintegerprogrammingthatarosefromaconversationwith

15 subinittext{return"prettyprint:=false;\n";} subexittext{return"quit;\n";} my$time=10; //Example$r->{Key} subtrapcomputation {my$r=shift; deletesol,sol1,sol2; vars:=$r->{vars}; polys:=$r->{basis}; my$s=<<eot; EOTreturn$s; traperror((sol1:=numeric::solve(polys,vars)),$time);sol1; traperror((sol:=solve(polys,vars)),$time);sol; } Givenaninteger-valuedmatrixAwithncolumnsonemayaskthechallengingquestionstocomputetheHilbertbasisortheextremalraysofthe conefx2zn:ax=0;x0g. exampleawithlistsofinteger-valuedvectors,oneperline.theinteger thematrix. valuesareseparatedbywhitespaces.arstlinegivesthedimensionsof Hemmecke'sdatacollectioncontainslesA.mat,A.hil,A.rayforeach AuxiliaryPerlcodefortheTrapTestaction wedeneanewtabletestsets,i.e.,createsuchsubdirectoriesofdataand alreadydenedinthedata/metarootdirectoryeachnewrecordshouldhave Data/META.Besidestandardattributes(Key,Type,PERSON,Date,...) ToinsertrecordsofthenewapplicationintotheSymbolicDatadatabase rayforthehilbertbasisandthelistofextremalrays.valuesforthelatter amandatoryattributematforthematrixaandoptionalattributeshiland attributesareoptionalsinceeithertheircomputationmaybetochallenging ortheoutputtoheavy.forthelattercasewecreatean(optional)attribute filetostorethelocationofthecorrespondingleatwww.testsets.de.for themomentmatriceswillbestoredaslistsofvectorsinhemmecke'sformat skippingthe(redundant)dimensioninformation.insteadwedeneanother mandatoryattributedimwiththeambientspacedimensionasvalue.

16 sd-recordsdim,mat,hil,ray,andfile,i.e.,lesdim.sd,mat.sd,hil.sd, ray.sd,andfile.sdinthedata/meta/testsetsdirectory.atypicalsuch METAsd-leisreproducedbelow. TogenerateanewtablewiththeseattributesonehastocreateMETA <Type> <Id> #Record'META/TESTSETS/mat' ############################################################### </Id> <level> <description>generatingsetofvectors <Syntax> <Key> 1 (- \d \s)* TESTSETS/mat </description> </level> </Syntax> </Key> </Type> <Date> <PERSON> <Version> #Endofrecord'META/TESTSETS/mat' Jan graebe... </PERSON> </Date> </Version> ################################################################ withacopyofametasd-lefromanotherdirectoryastemplate.level oneindicatesmandatorytags,levelgreateroneoptionaltags(defaultis3). TheSyntaxMetaattributethatdenesavalidsyntaxofmatvaluesisgiven METAsd-lescanbecreatedwithyourfavoritetexteditorstarting inperlregexpnotation.take.*toposenorestrictions. $ACTIONS->{Create}= {verbose=>"createtestsetsfrom*.matfiles", byanotheractionwrittenfromscratch: NowyoucanaddnewrecordstotheTESTSETStable.Thiscanberealized argvcall=>sub {shift;my$arg=shift; },mapcreatenewrecord($_),grep(/\.mat$/,@{$arg}); example=>'symbolicdata-r"$sd_home/bin/scripts/testsets.pl"'.'create<files>', #getremainingargs andstorestheminsd-lesinthetemporarydirectory/tmp.valuesforthe };Itispartofaletestsets.plandcreatesnewrecordsfromles*.mat otherattributescanbeaddedinlatersteps(actually,aslightextensionof

17 testsets.plgraspsalsothesevalues).createnewrecordisauserdened Perlfunctionthatcreatessd-leswiththedesiredcontentinatemporary directory. withthecommand InasecondsteptheselesareinsertedintotheSymbolicDatadatabase Thiswillvalidatethenewrecords,generate(asfaraspossible)andinsert accordingtotheidtagvalue.hencetheactuallenameofthetemporary missingtagvalues,formattheoutputnicelyandstoreitinthedatabase symbolicdatainsert-fix/tmp/*.sd ledoesnotmatter.weuseincreasingnumbersaslenames: my$i=0; {local$/; subcreatenewrecord my$fn=shift; my($r,$a); #evaluate*.mat #setkeyandtype open(fh,$fn)ordie; $r->{type}="testsets"; ($r->{key}=$fn)=~s/\.mat$//; $_=<FH>; closefh; #outputtheresult $r=record->new($r); desiredlocation.missingattributevalues(ofid,dateandperson)are }Record->newblesses$rtoasd-recordand$r->Outwritesittothe $r->out("/tmp/".$i++.".sd"); generatedduringinsertion.

18 zippedtar-lessd-tools.tgzandsd-data.tgzfromourcentralrepository Youmaydownloadthetools,dataanddocumentationofSymbolicDataas 6 HowtoLocallyInstalltheToolsandData athttp:// (orhigher)installedonyoursystem. ToworkwiththetoolsofSymbolicDatayoumusthavePerlversion5 ToinstalltheSymbolicDatasoftwareanddata,runGNUtar tar-xzfsd-tools.tgz containingtheperltools,data,anddocumentationsourcesofthesymbolic- ThiswillcreateadirectorySymbolicDatawithseveralsubdirectories tar-xzfsd-data.tgz rungnumake Dataproject. SettheenvironmentvariableSD_HOMEtothatdirectory,changetoitand fromtheirsourcesatthislocation.thisisalsoarsttestforthesymbolic- tocreateanewdirectorysdhtmlandgeneratethehtmldocumentation makeall Datatoolstobeproperlyinstalled. documentationforfurtherdetails. References WerefertotheSymbolicData/READMEleandtheSymbolicDataHTML [1]O.BachmannandH.-G.Grabe.TheSymbolicDataProject:Towards Seehttp:// forcomputeralgebra,universityofkaiserslautern. algebrasoftware.reportsoncomputeralgebra27,jan2000.centre anelectronicrepositoryoftoolsanddataforbenchmarksofcomputer [2]D.BiniandB.Mourrain.Polynomialtestsuite,1996. [3]W.Boege,R.Gebauer,andH.Kredel.SomeexamplesforsolvingsystemsofalgebraicequationsbycalculatingGrobnerbases.J.Symb. Seehttp://www-sop.inria.fr/saga/POL. Comp.,2:83{98,1986.

19 [4]S.R.CzaporandK.O.Geddes.OnimplementingBuchberger'salgorithm [5]H.-G.Grabe.AboutthepolynomialsystemsolvefacilityofAxiom, forgrobnerbases.inproc.symsac'86,pages233{238.waterloo, Macsyma,Maple,Mathematica,MuPAD,andReduce.InM.Wester, Canada,1986. [6]G.-M.Greuel,G.Pster,andH.Schonemann.Singular2.0.ACom- editor,computeralgebrasystems:apracticalguide,chapter8,pages 121{151.Wiley,Chichester,1999. puteralgebrasystemforpolynomialcomputations,centreforcom- puteralgebra,universityofkaiserslautern,2001. [7]PoSSo:PolynomialSystemSolving,1993{ [8]D.Wang.IrreducibledecompositionofalgebraicvarietiesviacharacteristicsetsandGrobnerbases.ComputerAidedGeometricDesign,9:471 {484,1992. Seehttp://posso.dm.unipi.it. [9]D.Wang.Solvingpolynomialequations:characteristicsetsandtriangularsystems.Math.andComp.inSimulation,42:339{351,1996.