Rule-Based Abduction for Logic Programming

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Rule-BasedAbductionforLogicProgrammingDecember1994KouichiHirataJ?ED9L0lAbstractInordertocapturethenatureofinference,aphilosopherPeirceclassiedinferenceintothreefundamentalkinds:deduction,induction,andabduction.Inthisclassica-tion,whichbasedontheformofsyllogisms,abductionischaracterizedastheinferenceofacaseAfromaruleA!CandaresultC.Furthermore,healsoplacedthesethreekindsofinferenceateachstageofscienticinquiry.Accordingtohim,everyscienticinquirybeginswithanobservationofasurprisingfact.Therststage,abduction,ofscienticinquiryproposesahypothesistoexplainwhythefactarises.Thesecondstage,deduction,derivesnewconclusionsfromthehypothesis.Thethirdstage,in-duction,testsempiricallyorcorroboratesthehypothesisandtheconclusions.Hence,abductionisnotonlyakindofinference,butalsoamethodofscienticdiscovery.Theinferenceschemaofabductionastherststageofscienticinquiryisdescribedinthefollowingthreesteps:1.AsurprisingfactCisobserved.2.IfAweretrue,thenCwouldbeamatterofcourse.3.Hence,thereisreasontosuspectthatAistrue.Incomputerscience,thesecondstage,deduction,hasbeendevelopedfromview-pointsofautomatedtheoremprovingandlogicprogramming.Thethirdstage,induc-tion,hasbeenstudiedfromviewpointsofinductiveinferenceandmachinelearning.Fortherststage,abduction,therearealsomanyresearchesinvariouselds.Inordertosystematicallyunderstandthemandclearlydiscussabduction,rstweclassifyab-ductionintovetypes:rule-selectingabduction,rule-ndingabduction,rule-generatingabduction,theory-selectingabduction,andtheory-generatingabduction.Inthisthesis,weexaminesuchvariousresearchesonabductionsofardeveloped,andshowthatmostofthemcanbeplacedinourclassication.Furthermore,weinvestigatetherstthreetypesofabduction,whichwecalltogetherrule-basedabduction,forlogicprogramming.Therule-selectingabductionforlogicprogrammingisabductionwhichselectsaruleinaprogramandproposesahypothesistoexplainasurprisingfact.Fromthephilosophicalviewpointwementionedabove,weshouldconsidertheprocessofabduc-tionwhichterminates.Hence,itisamainpurposeinthisthesistoidentifytheclassioflogicprogramsforwhichtheprocessofabductionterminates.Werstintroducetheconceptofhead-reducingprograms.Then,weshowthatallthederivationsforahead-reducingprogramandasurprisingfactarenite.Hence,alltheprocessesofrule-selectingabductionforahead-reducingprogramarenite.Ingeneral,abductioniscloselyrelatedtononmonotonicreasoning.Thus,inthisthesis,wecomparerule-selectingabductionwithdefaultlogic.Inordertoformulatetherule-selectingabductionfordefaultlogic,wedeneasurprisingfactandahypothesisinthedefaultlogic.Weshowthat,ifthereexistsahypothesiswhichexplainsasurprisingfact,thentherealsoexistsanextensionofagivendefaulttheory,whichincludesthesurprisingfact.ThisextensioniscorrespondingtotheleastHerbrandmodelofthedeniteprogramobtainingfromthedefaulttheory.Furthermore,weextendtheconceptofhead-reducingnesstothatofbreadth-rsthead-reducingprograms,andtherule-selectingabductiontothebreadth-rstrule-selectingabduction.Wealsoshowthatthereexistsanitederivationforabreadth-rsthead-reducingprogramandasurprisingfact.Hence,theprocessofbreadth-rstrule-selectingabductionforabreadth-rsthead-reducingprogramisnite.Therule-ndingabductionforlogicprogrammingisabductionwhichndsaruleinaprograminthesetofprogramsandproposesahypothesistoexplainasurprisingfact.Inrule-ndingabduction,weareinterestedinhowtochooseprogramsfromthesetofprograms.Then,wepayourattentiontochoosingprogramsforwhichtheprocessofrule-ndingabductionterminates.Weintroducetwoconceptsofloop-pairandloop-elimination.Theloop-pairsyn-tacticallydetermineswhetherornotthereexistsaninniteprocessofrule-ndingabductionforthechoiceofprograms.Weshowthat,ifaloop-pairappearsinaderiva-tion,thenthederivationbecomesinnite.Ontheotherhand,theloop-eliminationisatransformationofprograms.Byusingloop-elimination,wecanchoosetheprogramsforwhichrule-ndingabductionterminates.Wealsoshowthat,forgiventwoprograms,ifwetransformoneprogrambyloop-elimination,thenallthederivationsforunionofthetransformedprogramandtherestarenite.Inotherwords,byloop-elimination,wecanchoosetheprogramswhoseprooftreeshavenoinnitebranches.Inthisthesis,wealsodiscussanalogicalreasoningfromtheviewpointofabduc-tion.Inthisthesis,weadopttheformulationofanalogicalreasoningbyHaraguchiiiandArikawa.Intheirformulation,themainproblemishowtodetectananalogy.Inordertosolvethisproblem,wealsoadopttheconceptofpartialisomorphicgeneraliza-tions.Byusingtheseconcepts,weintroducetheconceptofdeduciblehypotheses,andformulaterule-ndingabductionwithanalogy.Weshowthatadeduciblehypothesisiscorrectinthesenseofanalogicalreasoning,andshowthatitispolynomialtimecom-putablewithrespecttothelengthofasurprisingfactandthesizeofaprooftree.Wedesignanalgorithmofrule-ndingabductionwithanalogy,andrealizeitasaPrologprogram.Therule-generatingabductionforlogicprogrammingisabductionwhichgeneratesaruleandproposesahypothesistoexplainasurprisingfact.Inrule-generatingab-duction,onlyonesurprisingfactisgiven.Inordertogeneratearuleandproposeahypothesis,weneedtogeneralizethesurprising

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