Abstract. We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos.

FormalConceptAnalysisandResolutioninAlgebraicDomains—PreliminaryReport

PascalHitzlerandMatthiasWendt

Arti cialIntelligenceInstitute,DepartmentofComputerScience

DresdenUniversityofTechnology,Dresden,Germany

{phitzler,mw177754}@inf.tu-dresden.de

Abstract.Werelatetwoformerlyindependentareas:Formalconceptanalysisandlogicofdomains.Wewillestablishacorrespondenebetweencontextualattributelogiconformalcontextsresp.conceptlatticesandaclausallogiconcoherentalgebraiccpos.Weshowhowtoidentifythenotionofformalconceptinthedomaintheoreticsetting.Inparticular,weshowthataspecialinstanceoftheresolutionrulefromthedomainlogiccoincideswiththeconceptclosureoperatorfromformalconceptanalysis.Theresultsshedlightontheuseofcontextsanddomainsforknowledgerepresentationandreasoningpurposes.

1Introduction

Domaintheorywasintroducedinthe1970sbyScottasafoundationforpro-grammingsemantics.Itprovidesanabstractmodelofcomputationusingorderstructuresandtopology,andhasgrownintoarespected eldontheborderlinebetweenMathematicsandComputerScience[1].RelationshipsbetweendomaintheoryandlogicwerenotedearlyonbyScott[2],andsubsequentlydevelopedbymanyauthors,includingSmyth[3],Abramsky[4],andZhang[5].Therehasbeenmuchworkontheuseofdomainlogicsaslogicsoftypesandofprogramcorrectness,withafocusonfunctionalandimperativelanguages.

However,therehasbeenonlylittleworkrelatingdomaintheorytologicalaspectsofknowledgerepresentationandreasoninginarti cialintelligence.TwoexceptionsweretheapplicationofmethodsfromquantitativedomaintheorytothesemanticanalysisoflogicprogrammingparadigmsstudiedbyHitzlerandSeda[6,7],andtheworkofRoundsandZhangontheuseofdomainlogicsfordisjunctivelogicprogramminganddefaultreasoning[8–10].Thelatterauthorsdevelopedanotionofclausallogicincoherentalgebraicdomains,forconveniencehenceforthcalledlogicRZ,basedonconsiderationsconcerningtheSmythpow-erdomain,andextendedittoadisjunctivelogicprogrammingparadigm[10].Anotionofdefaultnegation,inthespiritofanswersetprogramming[11]andReiter’sdefaultlogic[12],wasalsoadded[13].

Thenotionofformalconceptevolvedoutofthephilosophicaltheoryofcon-cepts.Wille[14]proposedthemainideaswhichleadtothedevelopmentofformalconceptanalysisasamathematical eld[15].Theunderlyingphilosophicalra-tionaleisthataconceptisdeterminedbyitsextent,i.e.thecollectionofobjects