Agents Learning in a Three-Valued Logical Setting

AgentsLearninginaThree-ValuedLogicalSettingEvelinaLammaFabrizioRiguzziDEIS,UniversitadiBologna,VialeRisorgimento240136Bologna,Italy,felamma,friguzzig@deis.unibo . itLusMonizPereiraCentrodeIntelig^enciaArti cial(CENTRIA),DepartamendeInformatica,UniversidadeNovadeLisb oa,2825MontedaCaparica,Portugallmp@di.fct. unl.ptAbstractWeshowthattheadoptionofathree-valuedsettingforinductiveconceptlearningisparticularlyusefullearninginsingleandmultipleagentsystems.Distinguishingb etweenwhatistrue,falseandwhatisunknowncanb eusefulinsituationswheredecisionshavetob etakenonthebasisofscarceinfor-mation.Suchsituationo ccurs,forexample,whenanagentincrementallygathersinformationfromthesur-roundingworldandhastoselectitsownactionsonthebasisofsuchacquiredknowledge.Inathree-valuedsetting,welearnde nitionforb oththetargetconceptanditsopp osite,consider-ingp ositiveandnegativexamplesasinstancesoftwodisjointclasses.Tothispurp ose,weadoptExtendedLogicPrograms(ELP)underaWell-FoundedSeman-ticswithexplicitnegation(WFSX)astherepresenta-tionformalismforlearning.StandardInductiveLogicProgrammingtechniquesarethenemployedtolearntheconceptanditsopp osite.Thelearntde nitionsofthep ositiveandnegativconceptsmayoverlap,b othwhenlearningconictingrulesforapredicateanditsexplicitnegationbysin-gleagentorwhencombiningtheknowledgelearnedbymultipleagents.Inthepap er,wehandleissueofstrategiccombinationofp ossiblycontradictorylearntde nitions.1Intro ductionMostworkoninductiveconceptlearningconsidersatwo-valuedsetting.Insuchasetting,whatisnoten-tailedbythelearnedtheoryisconsideredfalse,onbasisoftheClosedWorldAssumption(CWA)[25]. However,inpractice,itismoreoftenthecasethatearecon dentab outthetruthorfalsity ofonlyalimitednumb eroffacts,andarenotabletodrawanyconclu-sionab outtheremainingones,b ecauseavailableinformationisto oscarce.Likeithasb eenp ointedoutin[9,8],thisistypicallythecaseofanautonomousagentthat,inanincrementalway,gathersinforma-tionfromitssurroundingworld.Suchanagentneedstodistinguishb etweenwhatistrue,falseandwhatisunknown,andthereforeneedstolearnwithinaricherthree-valuedsetting.Forthispurp ose,weadopttheclassofExtendedLogicPrgrams(ELP,forshort,inthesequel)asrepresentationlanguageforlearninginathree-valuedsetting.ELPcontaintwokindsofnegation:defaultnegationplusasecondformofnegation,calledex-plicit,whosecombinationhasb eenrecognizedasaveryexpressivemeansofknowledgerepresentation.TheadoptionofELPallowsonetodealdirectlyinthelanguagewithincompleteknowledge,exceptionsthroughdefaultnegation,aswellwithtrulynega-tiveinformationthroughexplicitnegation[22,34]. Forinstance,in[3,5,10718]itisshownhowELPareapplicabletosuchdiversedomainsofknowledgerepresentationasconcepthierarchies,reasoningab outactions,b eliefrevision,counterfactuals,diagnosis,up-datesanddebugging.Inthiswork,e rstdiscussvariousapproachesandstrategiesthatcanb eadoptedinInductiveLogicPro-gramming(ILP,henceforth)forlearningwithathree-valuedsettingsbyasingleagent.Then,wediscusshowtheknowledgelearnedbyseparateagentscanb ecombinedtoobtainacommonknowledgebase.Asin[14,13],thelearningpro cessasingleagentstartsfromasetofp ositiveandnegativexamplesplussomebackgroundknowledgeintheformofanex-1