Tableau Algorithm. Presentation Outline Description Logics Reasoning Tasks Structural Subsumption Tableau Algorithm Examples Q & A

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Tableau Algorithm1Presentation OutlineDescription LogicsReasoning TasksStructural SubsumptionTableau AlgorithmExamplesQ & A2DL BasicsConcepts (unary predicates/formulae with one free variable)E.g., Person, Doctor, HappyParentRoles (binary predicates/formulae with two free variables)E.g., hasChild, loves, hasBrother, hasDaughterIndividuals (constants)E.g., John, Mary, ItalyOperators (for forming concepts and roles) restricted so that:Satisfiability/subsumption is decidable and, if possible, of low complexity3DL SemanticsInterpretation domain IInterpretation function IIndividuals iI 2 IJohnMaryConcepts CI ILawyerDoctorVehicleRoles rI I IhasChildowns(Lawyer u Doctor)4A TBox is a set of schema axioms (sentences), e.g.:

{Doctor v Person, HappyParent Person u 8hasChild.(Doctor t 9hasChild.Doctor)}

An ABox is a set of data axioms (ground facts), e.g.:

{John:HappyParent, John hasChild Mary}

A Knowledge Base (KB) is just a TBox plus an ABox

DL Knowledge Base5Example of TBoxWoman Person Female Man Person Woman Mother Woman hasChild.Person Father Man hasChild.Person Parent Father Mother Grandmother Mother hasChild.Parent MotherWithManyChildren Mother 3 hasChild MotherWithoutDaughter Mother hasChild.Woman Wife Woman hasHusband.Man Example of ABoxMotherWithoutDaughter(mary)Father(peter) hasChild(mary, peter)hasChild(peter, harry) hasChild(mary, paul) Reasoning TasksWhether a TBox description is satisable (i.e., non-contradictory)Whether one description subsumes another one in a TBox - organize the concepts of a terminology into a hierarchy according to their generalityFind out whether the set of assertions in a ABox is consistent (has a model)Whether the assertions in the ABox entail that a particular individual is an instance of a given concept descriptionA concept description can also be conceived as a query - retrieve the individuals that satisfy the query. Types of ReasoningThe simplest form of reasoning involves computing the subsumption relation between two concept expressions, i.e., verifying whether one expression always denotes a subset of the objects denoted by another expression.Parent is a specialization of Person, i.e., Person subsumes Parent

ABTypes of ReasoningA more complex reasoning task consists in checking whether a certain assertion is logically implied by a knowledge base.For example, Bill is an instance of ParentStructural SubsumptionNormalize descriptionsCompare syntactical structure of normal formsNormal form of C: A1 Am R1.C1Rn.CnNormal form of D: B1 Bk S1.D1Sl.DlC D iff:For all i, 1