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Abductive concept learning

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Abductive concept learning
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  AbductiveConceptLearning  A.C.KakasandF.RiguzziJanuary31,1997  Abstract Weinvestigatehowabductionandinductioncanbeintegratedinordertoobtainamorepowerfullearningframework.Inparticular,wediscussthepossibleapplicationsoftheAbductiveConceptLearningframework,anextensionoftheInductiveLogicProgramminglearningparadigmtothecaseinwhichboththebackgroundandthetargettheoryareabduc-tivelogicprograms.Inthisframework,wecanlearninthepresenceofincompleteinformationinthebackgroundknowledgeand/orinthetrain-ingsetbyexploitingthehypotheticalreasoningofabduction.WerststatetherequirementsforasystemthatperformsACL,andthenillus-tratehowsuchasystemcouldbeusedtosolvesomeoftheproblemsofextensionalandintensionalILPsystems.Abductivelogicprogramsareapowerfulmeansofrepresentingconcepts:weinvestigatethedierentusesofintegrityconstraintsinthetargettheory.Finally,wepresentan algorithmforACL,whichperformsadepth-rstsearchinthespaceofclauseorderingsandabest-rstsearchinthespaceofclauserenements,togetherwithanappropriateheuristicfunction. 1Introduction  Inthispaperweinvestigatehowabductionandinductioncanbeintegratedin ordertoobtainamorepowerfullearningframework.Inparticular,wediscusshowtheInductiveLogicProgramming(ILP)learningparadigm12,2]canbeextendedinordertolearnabductivelogicprogramsinsteadofdeniteornormallogicprograms.Theextendedinductiveproblemresultingfromthisintegrationwasintro-ducedin6]andwascalledAbductiveConceptLearning. Denition1.1  AbductiveConceptLearning(ACL) Given   asetofpositiveexamples E  +  ,  asetofnegativeexamples E    , 1   anabductivetheory  AT  =  h T;A;IC  i asbackgroundtheory. Find  Anewabductivetheory  AT  0 =  h T  0 ;A;IC  0 i suchthat  foreach  e +  2  E  +  , AT  0 j =  A  e +  ,  foreach  e   2  E    , AT  0 6j =  A  e   . Therefore,ACLdiersfromILPbecauseboththebackgroundknowledgeandthelearnedprogramareabductivelogicprograms.Asaconsequence,thenotionofentailmentofILPmustbesubstitutedwiththenotionofabductiveentailment( j =  A  ).ACLisanewlearningparadigmthatcontainsILPasaspecialcase.Inthefollowing,wewillconsideramodiedversionofthisdenitionofACL,inwhichthelastconditionissubstitutedby   foreach  e   2  E    , AT  0 j =  A  note   . WewillcalltherstdenitionACL1,whilethelatterACL2.Whenmentioning simplyACLwewillbereferringtoACL2.Insection2wewilldiscussthedierencesbetweenthetwodenitions.ACLcanbeusedwhenthebackgroundtheoryisavailableintheformofan abductivelogicprogram.Thisisusuallythecasewhenthebackgroundknowl-edgeisincomplete,becausewearelackingsomeinformationonthedomain.In thiscase,asystemforACLcanbeusedtolearndespitetheincompleteness.Someofthebackgroundpredicateswillbeconsideredasabducible:thesearepredicatesforwhichwesuspectorweknowthattheyhaveanincompletede-nition.Duringthelearningprocess,wecanmakeassumptions,bymeansofabduction,aboutthesepredicates,providedthattheseassumptionsareconsis-tentwiththeavailablepartialdenitiongivenintermsofrulesandintegrity constraints.Theseassumptionsaremadeinordertoprovidesupportforthetheorythatwearelearning.WewillshowthatACLisabletodealalsowithadierenttypeofincom-pleteness:sparsenessofthetrainingset.Inthiscasethetargetpredicatesareconsideredabducibleand,duringthelearningprocess,asystemforACLtriestocompletethetrainingdatabyusingabduction.Inthiscase,theassumptionshavetobeconsistentwiththetrainingdataandwiththepartialdenition availableforthetargetpredicate.Theaimofthelearningprocessistoproduceacompletedenitionofthetargetpredicatesdespitetheincompletenessoftheavailableinformation.Ifthisisnotpossible,wecouldconsiderthemasabducible(orincomplete)inthenaltheoryandtrytorestricttheassumptionsthatcanbemadeonthembylearning integrityconstraints.Forabduciblepredicatesinthebackgroundknowledge,wewouldalsoliketondstricterboundariesbyinferringintegrityconstraints.Letusillustratetheseideasbymeansofanexample.2  Example1.2  Supposewewanttolearntheconcept father .Letthebackground theorybe: T  =  f parent ( john;mary  ) ;male ( john  ) ;parent ( david;steve ) ;parent ( katy;ellen  ) ;female ( katy  ) g A  =  f male;female g andletthetrainingdatabe: E  +  =  f father ( john;mary  ) ;father ( david;steve ) g E    =  f father ( katy;ellen  ) g Inthiscase,wewouldlikeanACLsystemtolearntherule  father ( X;Y  )   parent ( X;Y  ) ;male ( X  ) : makingtheassumptions =  f male ( david  ) ;notmale ( katy  ) g .Byconsidering thebackgroundknowledgetogetherwiththeseassumptions,wecouldinferthe integrityconstraint:   male ( X  ) ;female ( X  ) : ThenextexampleshowsthatACLcanbeparticularlyusefulforperform-ingmultiplepredicatelearning,becauseassumptionsmadewhilelearningoneconceptcanbeusedastrainingdataforlearninganotherconcept. Example1.3  Supposewewanttolearntheconcepts grandfather and  father .Letthebackgroundtheorybe: T  =  f parent ( john;mary  ) ;father ( john;mary  ) ;male ( john  ) ;parent ( mary;ellen  ) ;female ( mary  ) ;parent ( ellen;sue ) ;parent ( david;steve ) ;parent ( steve;jim  ) g A  =  f father;male;female g andletthetrainingdatabe: E  +  =  f grandfather ( john;ellen  ) ;grandfather ( david;jim  ) g E    =  f grandfather ( mary;sue ) g AnACLsystemshouldlearntherule: grandfather ( X;Y  )   father ( X;Z  ) ;parent ( Z;Y  ) : abducing    1 =  f father ( david;steve ) ;notfather ( mary;ellen  ) g .Nowwelearn adenitionforfatherusing  T      1 : father ( X;Y  )   parent ( X;Y  ) ;male ( X  ) abducing    2 =  f male ( david  ) ;notmale ( mary  ) g .Moreover,asinexample1.2,wecaninferanintegrityconstraintfrom  T      1     2 :   male ( X  ) ;female ( X  ) : Inthepreviousexamples,wehaveseenthatintegrityconstraintsaregen-eratedforpredicatesofthebackgroundknowledge.However,whenthetargetpredicateisabducibleaswellandwehavemadeassumptionsaboutitinthelearningphase,wemaywanttogenerateconstraintsonit,asitisshowninthenextexample.3  Example1.4  Letthebackgroundtheorybe  T  =  f r ( a  ) ;q ( b;a  ) ;q ( c;d  ) ;s ( e ) g A  =  f p  g andthetrainingdatabe  E  +  =  f p  ( a  ) ;p  ( b ) ;p  ( c ) g E    =  f p  ( e ) g Inthiscase,therules p  ( X  )   r ( X  ) :p  ( X  )   q ( X;Y  ) ;p  ( Y  ) : couldbelearned,abducing  =  f p  ( d  ) g .Atthispointwecouldstopthelearning processbecausewehaveacompleteandconsistentprogramw.r.t.tothetraining data.Butthelearnedprogramdoenotgivesacompletedenitionoftheconcept p  ,becausewehavemadeanassumptionaboutitwhichisnotcoveredbyany rule.Therefore  p  willremainabducibleaswellinthelearnedtheory,allowing fornewassumptionson  p  .However,wecouldmaketheassumption  p  ( e ) ,which isfalsebecauseitisanegativeexamplefor  p  .Thereforewehavetogenerate integrityconstraintson  p  ,inordertoavoidtheassumptionoffalsehypothesis.Inthiscase,theintegrityconstraint   p  ( X  ) ;s ( X  ) : couldbegenerated,whichprohibitstheassumptionof  p  ( e )LetusnowsummarizethetasksthatanalgorithmforACLhastoperform,startingfromthesimpleroneandgoingtothemorecomplex.(1)Makeassumptionsaboutabducibleliteralsfromthebackgroundknowl-edgeinordertocoverpositiveexamplesandruleoutnegativeones.(2)Makeassumptionsontargetliteralsagaininordertocoverpositiveand ruleoutnegativeexamples.Inthiswayweenlargethetrainingsetand thereforenewlearningstepsmayberequired.(3)Inferintegrityconstraintsontheassumptionsmadeonabduciblepredi-catesfromthebackgroundknowledge,thereforereducingtheincomplete-nessoftheirdenition.(4)Inferintegrityconstraintsontargetpredicates.Forthiscasewehaveto distinguishtwodierentusesofconstraints:(i)inthecasewhichtherulesinferredarecorrect(theydonotderivefalsefacts),integrityconstraintscanbeusedtolimittheassumptionsaboutthefactsthatcannotbeprovedusingtherules;(ii)inthecaseinwhichtherulesinferredarenotcorrect,wecanusein-tegrityconstraintstospecializetheminordertoavoidthederivation offalsefacts.Inthiscase,everythingthathasbeenderivedusing therulesmustbeveriedagainstintegrityconstraints.4   Inexamples1.2and1.3thesystemperformstasks(1)and(3),whilein example1.4tasks(2)and(4)(i)areperformed.Uptonow,twoalgorithmsforACLhavebeendened.Theoneproposed in8]extendsanintensionaltop-downILPsystem2]withabduction,whiletheoneproposedin11]extendsFOIL15],anextensionaltop-downsystem2],withabduction.Bothofthemperformonlytask(1)above.Inthispaperwepresentanalgorithmwhichisabletoperformallfourtasks.Insection2weillustratethedierencesbetweenthetwodenitionsofACL.Theproblemsofextensionalsystemsarediscussedinsection3anditisshown howtasks(1)and(2)canbeusedtosolvesomeofthem.Insection4wediscusshowthesametaskscanhelpintensionalsystemstoovercomesomeproblemswhenlearningmultiplepredicatesandnormallogicprograms.Therelation betweenrulesandintegrityconstraintsincase(4)isdiscussedinsection5,whileinsection6weconcentrateoncase(ii).Analgorithmthatperformsallfourtasksaboveispresentedinsection7.Theheuristicsforthisalgorithmarediscussedin8.Insection9,weshowhowthisalgorithmcanbeextendedin ordertocomplywithACL1.Finally,insection10,weconcludesummarizing themainpointsanalyzedinthepaperanddiscussingfutureworks. 2DierencesbetweenthetwodenitionsofACL  ACL1requiresthatthenegativeexamplescannotbeabductivelyderivedfrom thelearnedtheory.Thismeansthattheremustbenowayofmakingassump-tionsthatallowtoderivethenegativeexamples.InACL2instead,wetesteach negativeexample e   bytryingtoabductivelyderive note   .Inordertoderive note   ,wecanmakeassumptionstosupportthegoal.Thereforethiscondition isweakerthanthepreviousoneanditallowsustoexploitabductionnotonly tocoverpositiveexamplesmoreeasilybutalsotoruleoutnegativeexamples.Letusillustratethisdierencebymeansofanexample. Example2.1  Considerexample1.2andsupposethattheintegrityconstraint   male ( X  ) ;female ( X  ) : isinthebackgroundknowledge.Supposealsothatthefact female ( katy  ) isnotinthebackgroundknowledge.Wewanttotestthenegativeexample  father ( katy;ellen  ) accordingtoACL1.Thetestfails,becauseitispossible toderive  father ( katy;ellen  ) byassuming  male ( katy  ) ,sincetheintegritycon-straintisnotviolatedbythisassumption.Ifwetesttheexampleaccordingto ACL2,thetestsucceedsbecausewecanderive  notfather ( katy;ellen  ) byabduc-ing  notmale ( katy  ) .Ifweconsiderthesamelearningproblembutwith  female ( katy  ) inthe backgroundknowledge,nowthederivationof  father ( katy;ellen  ) fails,andthe negativeexampleisnotcoveredalsoaccordingtoACL1.Thederivationof  notfather ( katy;ellen  ) onitsturn,succeedswithouttheabductionofanything. 5
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