1Chapter 3RDF and RDFS SemanticsIntroductionz RDF has a very simple data modelz But it is quite liberal in what you can sayz Semantics can be given using axiomatically– relating it to another representation, e.g., first order logic, for which a semantic model exists – May result in an executable semanticsz Semantics can be given by RDF Model Theory (MT)RDF/RDFS “Liberality”z No distinction between classes and instances (individuals)<Species, type, Class><Lion, type, Species><Leo, type, Lion>z Properties can themselves have properties<hasDaughter, subPropertyOf, hasChild><hasDaughter, type, familyProperty>z No distinction between language constructors and ontology vocabulary, so constructors can be applied to themselves/each other<type, range, Class><Property, type, Class><type, subPropertyOf, subClassOf>Semantics and model theoriesz Ontology/KR languages aim to model (part of) worldz Terms in language correspond to entities in worldz MT defines relationship between syntax and interpretations– Can be many interpretations (models) of one piece of syntax– Models supposed to be analogue of (part of) worldz e.g., elements of model correspond to objects in world– Formal relationship between syntax and modelsz structure of models reflect relationships specified in syntax– Inference (e.g., subsumption) defined in terms of MTz e.g., TA B iff in every model of T, ext(A) ⊆ ext(B)2Set Based Model Theoryz Many logics (including standard FOL) use a model theory based on Zermelo-Frankel set theoryz The domain of discourse (i.e., the part of the world being modelled) is represented as a set (often referred as Δ)z Objects in the world are interpreted as elements of Δ– Classes/concepts (unary predicates) are subsets of Δ– Properties/roles (binary predicates) are subsets of ΔΔ (i.e., Δ2)– Ternary predicates are subsets of Δ3etc.z The sub-class relationship between classes can be interpreted as set inclusionz Doesn’t work for RDF, because in RDF a class (set) can be a member (element) of another class (set)– In Z-F set theory, elements of classes are atomic (no structure)Set Based Model Theory ExampleWorldInterpretationDaisy isA CowCow kindOf AnimalMary isA PersonPerson kindOf AnimalZ123ABC isA CarΔ{⇔ ,…} ⊆ Δ × ΔabModelMary drives Z123ABCSet Based Model Theory Examplez Formally, the vocabulary is the set of names we use in our model of (part of) the world{Daisy, Cow, Animal, Mary, Person, Z123ABC, Car, drives, …}z An interpretation I is a tuple Δ, ·I– Δ is the domain (a set)– ·Iis a mapping that mapsz Names of objects to elements of Δz Names of unary predicates (classes/concepts) to subsets of Δz Names of binary predicates (properties/roles) to subsets of Δ × Δz And so on for higher arity predicates (if any)z RDF has “Non-standard” semantics in order to deal with thisz Semantics given by RDF Model Theory (MT)z In RDF MT, an interpretation I of a vocabulary V consists of: – IR, a non-empty set of resources (corresponds to Δ)– IS, a mapping from V into IR (corresponds to ·I)– IP, a distinguished subset of IR (the properties)z A vocabulary element v ∈ V is a property iff IS(v) ∈ IP– IEXT, a mapping from IP into the powerset of IR × IRz I.e., property elements mapped to subsets of IR × IR– IL, a mapping from typed literals into IRRDF Semantics3Example RDF Simple Interpretationz RDF Imposes semantic conditions on interpretations, e.g.:– x is in IP if and only if <x, IS(rdf:Property)> is in IEXT(I(rdf:type))z All RDF interpretations must satisfy certain axiomatic triples, e.g.:– rdf:type rdf:type rdf:Property– rdf:subject rdf:type rdf:Property– rdf:predicate rdf:type rdf:Property– rdf:object rdf:type rdf:Property– rdf:first rdf:type rdf:Property– rdf:rest rdf:type rdf:Property– rdf:value rdf:type rdf:Property– …RDF Semantic ConditionsExample RDF Interpretationz RDFS simply adds semantic conditions and axiomatic triples thatgive meaning to schema vocabularyz Class interpretation ICEXT simply induced by rdf:type, i.e.:– x is in ICEXT(y) if and only if <x,y> is in IEXT(IS(rdf:type))z Other semantic conditions include:– If <x,y> is in IEXT(IS(rdfs:domain)) and <u,v> is in IEXT(x) then u is in ICEXT(y)– If <x,y> is in IEXT(IS(rdfs:subClassOf)) then x and y are in IC and ICEXT(x) is a subset of ICEXT(y)– IEXT(IS(rdfs:subClassOf)) is transitive and reflexive on ICz Axiomatic triples include:– rdf:type rdfs:domain rdfs:Resource– rdfs:domain rdfs:domain rdf:PropertyRDFS Semantics4RDFS Interpretation Examplez If RDFS graph includes triples<Species, type, Class> <Lion, type, Species> <Leo, type, Lion><Lion, subClassOf, Mammal ><Mammal, subClassOf, Animal>z Interpretation conditions imply existence of triples<Lion, subClassOf, Animal> <Leo, type, Mammal> <Leo, type, Animal>…RDFS Axiomsz Another way to define the semantics of RDF and RDFS is to give axioms that relate it to well understood representation, such as FOL, that has a formal semantics.z A benefit of this approach is that the axioms may provide the basis of an “executable semantics”z For a list of FOL axioms (in N3) defining RDFS vocabulary, see– http://www.csee.umbc.edu/691m/n3/rdfs-rules.n3RDFS Inference Rules{?S ?P ?O} => {?P a rdf:Property}.{?P rdfs:domain ?C. ?S ?P ?O} => {?S a ?C}.{?P rdfs:range ?C. ?S ?P ?O} => {?O a ?C}.{?S ?P ?O} => {?S a rdfs:Resource. ?O a rdfs:Resource}.{?Q rdfs:subPropertyOf ?R. ?P rdfs:subPropertyOf ?Q} => {?P rdfs:subPropertyOf ?R}.{?P @has rdfs:subPropertyOf ?R. ?S ?P ?O} => {?S ?R ?O}.{?C a rdfs:Class} => {?C rdfs:subClassOf rdfs:Resource}.{?A rdfs:subClassOf ?B. ?S a ?A} => {?S a ?B}.{?B rdfs:subClassOf ?C. ?A rdfs:subClassOf ?B} => {?A rdfs:subClassOf ?C}.{?X a rdfs:ContainerMembershipProperty} => {?X rdfs:subPropertyOf rdfs:member}.{?X a rdfs:Datatype} => {?X rdfs:subClassOf rdfs:Literal}.RDFS Classesrdf:Alt rdfs:subClassOf rdfs:Container.rdf:Bag rdfs:subClassOf rdfs:Container.rdfs:ContainerMembershipProperty rdfs:subClassOf rdf:Property.rdfs:Datatype rdfs:subClassOf rdfs:Class.rdf:Seq rdfs:subClassOf rdfs:Container.rdf:XMLLiteral rdfs:subClassOf rdfs:Literal; a rdfs:Datatype.5RDFS Propertiesrdfs:label rdfs:domain rdfs:Resource; rdfs:range rdfs:Literal.rdfs:comment rdfs:domain rdfs:Resource; rdfs:range rdfs:Literal.rdfs:seeAlso rdfs:domain rdfs:Resource; rdfs:range rdfs:Resource.rdfs:isDefinedBy rdfs:domain rdfs:Resource; rdfs:range rdfs:Resource;
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