Anotherpossibilityistoentersomealreadyknownimplicationsbeforestartingtheexploration.Theseimplications,theuseralreadyknowstobevalid,

Transcription

1 BackgroundImplicationsandExceptions AttributeExplorationwith jectsinaspeciedcontext.thisknowledgerepresentationisespeciallyuseful Summary:Implicationsbetweenattributescanrepresentknowledgeaboutob- TechnischeHochschuleDarmstadt,FachbereichMathematik cspringer-verlagberlin{heidelberg1995 erdstumme offormalconceptanalysisthatsupportstheacquisitionofthisknowledge.fora plicationsbetweenattributesofthiscontexttogetherwithalistofobjectswhich whenitisnotpossibletolistallspeciedobjects.attributeexplorationisatool speciedcontextthisinteractiveproceduredeterminesaminimallistofvalidim- backgroundimplications)andallvalidimplications.thelistofimplicationscan setofimplicationsthatllsthegapbetweenpreviouslygivenimplications(called besimpliedfurtherifexceptionsareallowedfortheimplications. arecounterexamplesforallimplicationsnotvalidinthecontext.thispaperdescribeshowtheexplorationcanbemodiedsuchthatitdeterminesaminimal 1.Introduction speciedcontext.thisknowledgerepresentationisespeciallyusefulwhenit isnotpossibletolistallspeciedobjects.attributeexploration(cf.anter Implicationsbetweenattributescanrepresentknowledgeaboutobjectsina (1987),Wille(1989))isatoolofformalconceptanalysisthatsupportsthe acquisitionofthisknowledge.formalconceptanalysiswasintroducedin Wille(1982)andhasgrownduringthelastfteenyearstoanusefultoolin determinesaminimallistofimplicationsthatissucienttodeduceallvalid dataanalysis. Foraspeciedcontexttheinteractiveprocedureofattributeexploration implicationsbetweentheattributesofthecontexttogetherwithalistof hasalreadysomeideahowtheattributesarerelated.beforestartingthe objectasacounterexample. Usually,theuserdoesnotstartsuchanexplorationfromscratch,buthe objectswhicharecounterexamplesfortheimplicationsthatarenotvalidin usercaneitheracceptasuggestedimplicationorhemustsupplyanew thecontext.asaninteractiveprocedure,theprogramsuggestsimplications thenumberofremainingpossibleimplications. explorationhecanenteralistofobjectswhichmaysignicantlydecrease totheuserwhichdonotcontradicttoalreadygivencounterexamples.the Anotherpossibilityistoentersomealreadyknownimplicationsbeforestartingtheexploration.Theseimplications,theuseralreadyknowstobevalid,

2 InP.Burmeistersimplementation(1987),itisalsopossibletoenterbackgroundimplications.Thisprogramhoweverdeterminesaminimalsetof areherecalledbackgroundimplications.thispaperdescribesageneral- ofimplicationswhich togetherwiththebackgroundimplications is izationofattributeexploration,thatitisabletodetermineaminimallist sucienttodeduceallvalidimplications. thisprogramdoesnotdependinanywayonthebackgroundimplications onestartswith.intheapproachpresentedinthispaperthebackground usedtodecreasethenumberofquestionstotheuser.theresultinglistin implications regardlessofthebackgroundimplications.theyareonly implicationsareusedtominimizeonlythenumberofadditionallyneeded describeacontextbyaminimalnumberofimplications.theremaybemore Inthenextsectionthebasicdenitionsofformalconceptanalysisarerecalledandexplainedbyanexample,beforeimplicationsincontextsand implicationsisdescribed.itisilluminatedbyanexampleinthelastsection, sectiontheinteractiveprocedureofattributeexplorationwithbackground thenotionsofcompletenessandirredundancyareintroduced.inthethird implicationsifsomeofthemareobvious. implications.thisisbasedonthebeliefthatitisnotalwaysthebestto theresultoftheexploration. 2.ImplicationsofContextsandtheL-Duquenne- wherewealsodiscuss,howtheadmissionofexceptionscanfurthersimplify Firstwebrieyrecallthebasicdenitionsofformalconceptanalysis(cf. Denition:Wecall(;M;I)acontext,whereandMaresetsandIis anterandwille(1995))andgiveanexample. arelationbetweenandm(i.e.im).theelementsofandm read:\theobjectghastheattributem". arecalledobjectsandattributes,respectively,andgim(:()(g;m)2i)is uigues-basis ForeverysetAofobjectswedenethesetA0:=fm2MjgIm forallg2agofallattributessharedbyallobjectsina.duallytheset B0:=fg2jgImforallm2Bgisthesetofallobjectshavingall Nowaconceptofthecontext(;M;I)isapair(A;B)withA,BM, attributesinbm. Inmumandsupremumintheconceptlatticearecalculatedasfollows: A0=B,andB0=A.ThesetAiscalledtheextentoftheconcept,theset Btheintent.Thehierarchicalsubconcept-superconcept-relationisgivenby (A1;B1)(A2;B2):()A1A2(()B1B2).Thesetofallconcepts whichiscalledtheconceptlatticeof(;m;i)andisdenotedbyb(;m;i). ofacontext(;m;i)togetherwiththisorderrelationisacompletelattice t2t(at;bt)=(\ ^ t2tat;([ t2tbt)00);_ t2t(at;bt)=(([ t2tat)00;\ t2tbt):

3 malcontextshowninfig.1.itsobjectsarethegraphs1upto18and relationshipscontainedintheunderlyingdatacontext:weconsiderthefor- itsattributesaretenattributesofundirectedgraphs(cf.wilson(1975)): Thefollowingexampleshowshowtheconceptlatticeunfoldstheconceptual connected,disconnected,bipartite,complete,completebipartite,tree,forest, planar,eulerian,hamiltonian.thiscontext(togetherwithalistofimplications)istheresultoftheattributeexplorationthatisdescribedinthe (g),whichisdenedastheconceptwiththesmallestextentcontaining g,islabeledwith\g".dually,foreveryattributemitsattributeconcept extentwhicharelinkedtoitbyadescendingpathanditcontainsallthose (m),whichisdenedastheconceptwiththesmallestintentcontaining bedeterminedinthediagram:aconceptcontainsallthoseobjectsinits m,islabeledwith\m".thentheextentandintentofeveryconceptcan itsextentandtheattributesconnected,complete,eulerian,andhamiltonian attributesinitsintentthatarelinkedtoitbyanascendingpath.the nected),(planar)and(bipartite),andtheotherby(connected),(planar) rightmostconceptinfig.2forexamplehasthegraphs3,14and18in Inthediagramonecanseetwocubesatthetop:oneisspannedby(discon- initsintent. and(bipartite).thisindicatesthatinbothcasesthethreeinvolvedat- Thedominatingpartinthelatticeliesbetween(connected)and(13).It tributesareindependent. isthedirectproductofa6-element\ladder"witha4-element\rectangle", butitcanalsobeseenas4-dimensionalhypercubesthataregluedtogether ateightvertices.theupperoneliesbetween(connected)and(15)and thirdsection. Itissucienttolabelthelinediagramnotwiththecompleteconcepts,but onlywiththeattributesandobjects:foreveryobjectg,itsobjectconcept and18. rian),and(hamiltonian).thisshowsthat,forconnectedgraphs,thefour attributesplanar,bipartite,eulerian,andhamiltonianareindependent. pointismissing itismarkedinthediagramwithalittledotleftof(15). Inthelowerhypercube(between(bipartite)^(connected))and(13)one ThisindicatesthateverycompletebipartiteHamiltonianplanargraphisalso Eulerian.Infactthereexist(uptoisomorphism)onlytwosuchgraphs,13 isspannedby(planar)^(connected),(bipartite)^(connected),(eule- Denition:AnimplicationbetweenattributesinMisapair(X;Y)of implicationx!yisvalidinacontextkifitisrespectedbyeveryobject intent.theimplicationisthencalledanimplicationofthecontextk.an subsetsxandyofm.itisdenotedbyx!yandisread\ximplies MthatrespectsLalsorespectsX!Y. TrespectsasetLofimplicationsifitrespectseveryimplicationinL.An implicationx!yisentailedbyasetlofimplicationsifeverysubsetof Y".AsubsetTofMrespectstheimplicationifX6TorYT.Theset

4 disconnected bipartite completebipartite tree connected planar Eulerian forest 1 Hamiltonian Lemma1AnimplicationX!YisvalidinacontextKifandonlyif 10 Figure1:Contextofgraphs B(K):Inourexamplefcomplete,Euleriang!fHamiltoniangisanimplicationofthecontext,whichcorrespondstotheequality(complete)plicationscanequivalentlybeunderstoodasequalitiesintheV-semilattice betweentheattributesofthecontext,becausethesetofallintentsisexactly thelargestclosuresystemonmthatrespectsalltheseimplications.theim- Thestructureofaconceptlatticeisalreadydescribedbyallimplications YX00.ThenitisalsorespectedbyeveryconceptintentofK lattice. (Eulerian)=(complete)^(Eulerian)^(Hamiltonian)intheconcept 20

5 complete connected disconnected bipartite planar tree forest Hamiltonian Eulerian complete bipartite Figure2:ConceptlatticeofthecontextinFig.1 Someoftheseimplicationsmaybeknowninadvance,andinthispaper theywillbereferredtoasbackgroundimplications.inaformalsenseevery implicationofacontextcanbeabackgroundimplication.thequestionis nowhowtodescribethestructureoftheconceptlatticewithimplications inthemostecientwaywhenbackgroundimplicationsaregiven.weare lookingforaminimallistthatis\llingthegap"betweenthebackground implicationsandallvalidimplications. Denition:LetKbeanitecontextandLasetof(background)implicationsofK.AsetBofimplicationsofKiscalledL-complete,ifevery implicationofkisentailedbyl[b.itiscalledl-irredundantifnoimplicationa!b2bisentailedby(bnfa!bg)[l.al-basisisa L-completeandL-irredundantsetofimplicationsofK.IfLisemptythen Biscalledcomplete,irredundant,andabasis,respectively.AsubsetPof Miscalledpseudo-intentofKifP6=P00andifforeverypseudo-intentQ withqptheinclusionq00pholds.

6 J.-L.uiguesandV.Duquenne(1986)showthatB:=fP!P00jPisa backgroundimplicationsisdenotedbyp7!p:=p[pl[pll[:::with Duquenne-uigues-basis.WeobtainaL-basisbygeneralizingthisdenition. pseudointentgisabasisof(;m;i)ifmisnite.thisbasisiscalled XL:=X[SfBMjAX;A!B2Lg. willbenite. Denition:TheclosureoperatoronthesetMofattributesinducedbythe foreveryl-pseudo-intentqwithqp,theinclusionq00pholds. AsubsetPofMiscalledL-pseudo-intentofKifP=P6=P00andif, Withoutfurthermentioning,allsetsofattributesconsideredinthefollowing ThesetBL:=fP!P00jPisaL-pseudo-intentgofimplicationsiscalled Proof.Obviously,allimplicationsinBLareimplicationsofK.Weprove L-Duquenne-uigues-basis. Theorem2BLisaL-basisofK. LetP!P002BL.WeshowthatP!P00isnotentailedby(BLn fp!p00g)[lbecauseprespectsallimplicationsin(blnfp!p00g)[l ThisisacontradictionbecauseTdoesnotrespectthisimplication. T6=T00.ThenTisaL-pseudo-intentbydenitionandsoT!T002BL. FurthermoreTrespectsQ!Q00foreveryL-pseudo-intentQT.Suppose thatblisl-completebyshowingthateverysubsettmrespectingall (andprovesothatblisl-irredundant):asp=p,itclearlyrespectsall implicationsinl[blisanintent:ast!tisentailedbylwehavet=t. Duquenne-uigues-basisisB=fcd!abcd;b!ab;ad!abcd;ac!abcdg.For A!A.ThecontextinFig.3showsthatingeneralthisisnotthecase.The pseudo-intentswithandthendeletingthe(trivial)implicationsoftheform OnemayaskifitispossibletogettheL-pseudo-intentsbyjustclosingthe implicationsinl.forq!q002blnfp!p00gwithqpwehave Q00P,asPisaL-pseudo-intent.HencePrespectsalsoQ!Q00. thebackgroundimplicationl:=fcd!agwegetthel-duquenne-uiguesbasisbl=fb!ab;ad!abcd;ac!abcdgwhilefp!p00jpispseudo-intent acquisitiontoolthatcanbeusedtodeterminetheduquenne-uigues-basis B.anter(1987)presentsattributeexplorationasaninteractiveknowledge 3.AttributeExplorationwithBackgroundImplications isnotl-irredundant. withp6=p00gadditionallycontainsacd!abcd.ingeneraltheresultingset ofacontextthatiseithertoolargeforacompleteinputintothecomputer orthatisevennotcompletelyknown.itisbasedonhisnext-closure- Algorithmthatecientlycalculatesclosuresystems.

7 1 2 3abcd TheAttributeExplorationprocedurecanbemodiedsuchthatitcanbe 4 setlofbackgroundimplications.thereforeweproceedsimilartoanter closuresystemonthesetmofallattributes: usedtodetermineinteractivelythel-duquenne-uigues-basisforagiven (1987).FirstweshowthatthesetofallintentsandL-pseudo-intentsisa Figure3: c Lemma3Let(;M;I)beacontext,letLbeasetofimplicationsof andq6p.thenp\qisanintent. (;M;I),andletPandQbeintentsorL-pseudo-intentswithP6Q Proof.PasQandthereforealsoP\QrespectallimplicationsinL[BL exceptp!p00andq!q00.becauseofp6p\qandq6p\qthe ofsimplicityweassumethatm:=f1;:::;ng. NextweintroducethelecticalorderonthesetofsubsetsofM.Forthesake setp\qrespectstheseimplications,too.henceitmustbeanintent. Corollary4ThesetofallintentsandL-pseudo-intentsofanitecontext A<B:()(9i2BnA:A\f1;:::;i?1g=B\f1;:::;i?1g)for Denition:ThelecticalorderonP(M)isdenedby (;M;I)isaclosuresystemonM;withtheclosureoperatorX7!X:= A;BM. X[X[X[:::,whereX:=X[SfBMjA!B2BL;AXg. ForA;BMandi2Mwedene A<iB:()(i2BnAandA\f1;:::;i?1g=B\f1;:::;i?1g) subsetbmthelecticallynextintentorl-pseudo-intentisthesetbi, andai:=((a\f1;:::;i?1g)\fig). lastintentorl-pseudo-intentism. whereiisthemaximalelementinmnbwithb<ibi.thelectically Theorem5ThelecticallyrstintentorL-pseudo-intentis;.Foragiven TheNext-Closure-algorithmofB.anter(1987)listsallclosedsetsofa closuresystemonanitesetinthelecticalorder.inthenexttheoremitis appliedtotheclosuresystemofallintentsandalll-pseudo-intents: a 1 b 2 d 3 4

8 ThistheoremprovidesthecentralpartoftheAttributeExplorationwith backgroundimplications,whichwedescribenow:wewanttodetermine thel-duquenne-uigues-basisofacontext(;m;i)(whichisapriorinot Hofobjects.ThesetHmayalsobeempty. completelygiven)forasetlofbackgroundimplicationsofthecontext. Thealgorithmstartswithapartialcontext(H;M;I\(HM)forasubset Algorithm.Setk:=1andBL:=;. (1)DeterminethekthL-pseudo-intentPkof(H;M;I\(HM)byapplyingTheorem5.IfMisreached(asintent)thenSTOP.BListhen thel-duquenne-uigues-basis. (2)Asktheuser:\IstheimplicationPk!P00 (3)otostep(1). {Iftheansweris\No",thenaskforanobjectgthatdoesnot {Iftheansweris\Yes",thenaddPk!P00 respectthisimplication.therowfgg0thenalsohastobeentered bytheuser.addgtoh. kvalid?" ktoblandincreasek. lecticallyrstl-pseudo-intentsof(h;m;j).let(;m;i)beanitecontext Theorem6Let(H;M;J)beanitecontextandletP1,:::,Pkbethek changewhenanobjectisaddedthatrespectsallpreviouslyacceptedimplicationsandallbackgroundimplications: ThealgorithmiscorrectbecausethelecticallyrstL-pseudo-intentsdonot L-pseudo-intent. PJJ L-pseudo-intentsof(;M;I). fggirespectsallpi!pjj withhandj=i\(hm),inwhichalltheimplicationsinl[fpi! Proof.Fori=1;:::;kwehavePII islecticallylessthanpj,theassertionisaconsequenceofthedenitionof iji=1;:::;kgarevalid.thenp1,:::,pkarealsotheklecticallyrst i.aseveryl-pseudo-intentqpjof(h;m;j) i=pjj ibecauseforeveryg2theset setm:=fconnected(conn),disconnected(disc),bipartite(bip),complete InthissectionweseehowthecontextinFig.1isproduced.Westartwiththe Eulerian(eul),Hamiltonian(ham)gofattributesandwanttoknowwhich (comp),completebipartite(cbip),tree(tree),forest(for),planar(plan), 4.AnExplorationofraphs Astheclassofundirectedgraphscontainsinnitelymanyisomorphism implicationsbetweentheseattributesarevalidforallundirectedgraphs. classes,thereisnopossibilitytodeterminetheduquenne-uigues-basisfor

9 Furthermoreweknowthatatreeisjustdenedasaconnectedforestand thatacompletebipartitegraphisalwaysbipartite.thisjustiesthefollowingbackgroundimplications: Whenwelookatthelistofattributesthenweseethatconnectedanddisconnectedarecontradictingeachother(i.e.,nographcanhavebothattributes). graphs.thegraphstheusergivesascounterexamplesduringtheattribute explorationarejustthesetypicalgraphs. theinnitecontext(;m;i)directly.onehastoworkwithsome\typical" anemptyseth.inthefollowingtheattributesappearinginthepremiseof Wedonothaveanyobjectsatthebeginning,sotheexplorationstartswith animplicationwillnotbelistedintheconclusionagain. fconn,discg!m fconn,forg!ftreeg fcbipg!fbipg ftreeg!fconn,forg TherstL-pseudo-intentistheemptyset.Thereforethedialoguestarts NowHcontainstheobject1.InthisenlargedcontexttherstL-pseudointentisstilltheemptyset,butonthisstepwehave;00=fconn,plan, withthequestion: hamg.q:is;!fconn,plan,hamgvalid? Q:Is;!Mvalid? A:No.1hastheattributesconn,plan,ham. 3isaddedtoH.InthecontextwithH=f1;2;3gtheemptysetis 2isaddedtothesetH. Q:Is;!fplangvalid? A:No.2hastheattributesdisc,bip,for,plan. anintent.thenextl-pseudo-intentisfhamg. A:No.3hastheattributesconn,comp,eul,ham. Theimplicationfhamg!fconngisaddedtoBLwhichwasemptyuptonow. Q:Isfhamg!fconngvalid? A:Yes. Q:Isfeulg!fconn,comp,hamgvalid? A:No.4hastheattributesconn,bip,cbip,plan,eul. :::

10 andthefollowingimplicationsareaccepted: Duringtheexplorationthegraphs1to20aregivenascounterexamples fconn,bip,tree,for,plan,hamg!fcomp,cbip,eulg fconn,bip,tree,for,plan,eulg!fcomp,cbip,hamg fconn,comp,eulg!fhamg fdisc,bip,cbipg!ffor,plang fcompg!fconng fhamg!fconng fconn,bip,cbip,plan,hamg!feulg feulg!fconng fconn,bip,compg!fcbip,tree,for,plang fforg!fbip,plang ThesetenimplicationsconstitutetheL-Duquenne-uigues-basisBL.Every implicationthatisvalidinthecontextcanbededucedfromthemandthe fourbackgroundimplications.theduquenne-uigues-basisconsistsofall andthefourthbackgroundimplicationandthetwoimplicationsftreeg! implicationsinthel-duquenne-uigues-basisandadditionallyoftherst fconn,bip,for,plangandfconn,bip,for,plang!ftreeg.inthisexample thecardinalityofthel-duquenne-uigues-basisisjustthedierenceofthe asexceptionsthatcontradictanimplicationandare(uptoisomorphism) andkeepinmindtheexceptions.forexamplewecanregardallgraphs cases,wemayconrmsomeimplicationsthataretrueforalmostallgraphs Duringtheexplorationtherearesomeimplicationsthatcanbedeniedby cardinalitiesoftheduquenne-uigues-basis,butingeneralitmaybelarger. Thebeginningoftheexplorationdialogueremainsunchanged.Therstdifferenceappearswiththequestion:\Isfcompg!fconn,eul,hamgvalid?", because6is(uptoisomorphism)theonlyconnectedcompleteplanarhamiltoniangraphwhichisnoteulerianandisthereforenotallowedascounterexample.howevertheimplicationhastobedenied:raph06infig.4 servesasnewcounterexample.thenextsuggestionfcompg!fconn,hamg fconn,eul,hamg!fcompg,because14isanexceptioninthesensedened of14thegraph014willbeusedasacounterexamplefortheimplication willbeacceptedwiththeexception7,whichistheonlycompletegraph thatisnothamiltonian.inthiswaytheexplorationcontinues.instead above. ThisapproachyieldsthefollowinglistofimplicationsthatisaL-basisfor allgraphsexceptfor6,7,13,14,and18.behindeveryimplication uniqueinhavingexactlytheirattributes. thegeneralstructureofgraphtheorywithoutbotheringwithpathological onlyonecounterexample(uptoisomorphism).ifwewanttodetermine arelisteditsexceptions.

11 disconnected bipartite completebipartite tree connected planar Eulerian forest Figure4:Additionalgraphsfortheexplorationwithexceptions fdisc,bip,cbipg!ffor,plang fcompg!fconn,eul,hamg fhamg!fconng feulg!fconng fforg!fbip,plang (6;7) 14 Hamiltonian completeplanarhamiltoniangraphs:14and18.that18istheonly fconn,bip,tree,for,plan,hamg!m fconn,bip,tree,for,plan,eulg!m fconn,comp,plan,eul,hamg!m (14;18) HamiltoniantreeandtheonlyEuleriantreeisexpressedby7thresp.8th The6thimplicationindicatesthatthereexist(uptoisomorphism)onlytwo fconn,bip,cbip,plan,hamg!feulg fconn,bip,comp,eul,hamg!m (13;18) implication.itisalsotheonlyhamiltonianbipartitecompletegraph(10th implication). (18) TheresultingconceptlatticeisshowninFig.5.Theimplicationsvalidin 7,13,14,and18.Theconceptlatticeoftheseexceptionsisshown thislatticeareexactlythosewhicharevalidforallgraphsexceptfor6, infig.6.inparticularonecanseeinthediagramthatallexceptionsare connectedplanargraphs. atedbysomeconceptsisinteractivelydetermined. AttributeexplorationdeterminestheV-semilatticethatisgeneratedbythe attributeconcepts.inpremiseandconclusiononlyconjunctionsofattributes areallowed.disjunctionsbecomeinvolvedindistributiveconceptexploration (cf.stumme(1995)),wherethecompletedistributivelatticethatisgener- 6

12 disconnected bipartite connected planar 11 9 complete bipartite Eulerian Hamiltonian 10 complete Figure5:Conceptlatticeresultingoftheexplorationwithexceptions 1 forest 4 17 tree 20 planar Figure6:Conceptlatticeoftheexceptions connected complete bipartite Hamiltonian complete bipartite forest Eulerian tree disconnected

13 Kontexte.TechnischeHochschuleDarmstadt(Latestversion1995forAtariST References: Burmeister,P.(1987):ProgrammzurformalenBegrisanalyseeinwertiger anter,b.(1987):algorithmenzurbegrisanalyse.in:b.anter,r.wille,k. informativesresultantd'untableaudedonneesbinaires.math.sci.humaines andmsdos) E.Wol(Eds.):BeitragezurBegrisanalyse.B.I.-Wissenschaftsverlag,Mannheim,Wien,Zurich,241{25grisverbandausgewahlterraphen.Mittelseminar,THDarmstadt Duquenne,V.anduigues,J.-L.(1986):Famillesminimalesd'implications 95,5{18 Ehrenberger,P.,Heiss,R.,Ihringer,Cl.,andVogel,N.(1992):Be- Wille,R.(1982):Restructuringlatticetheory:Anapproachbasedonhierarchiesofconcepts.In:I.Rival(Ed.):Orderedsets.Reidel,Dordrecht{Boston, anter,b.andwille,r.(1995):formalebegrisanalyse:mathematische acquisitioninformalconceptanalysis.(inpreparation) Stumme,.(1995):DistributiveConceptExploration atoolforknowledge rundlagen.springer,berlin,heidelberg(toappear) 445{470 In:E.Diday(Ed.):Dataanalysis,learningsymbolicandnumericknowledge. NovaSciencePublisher,NewYork,Budapest,365{380 Wilson,R.J.(1975):Introductiontographtheory.Longman,London Wille,R.(1989):Knowledgeacquisitionbymethodsofformalconceptanalysis.