RDF Semantics by Patrick Hayes W3C Recommendation w3/TR/rdf-mt/

RDF Semanticsby Patrick HayesW3C Recommendationhttp://www.w3.org/TR/rdf-mt/ Presented by Jie Bao RPI Sept 4, 2008 Part 1 of RDF/OWL Semantics Tutorial http://tw.rpi.edu/wiki/index.php/RDF_and_OWL_Semantics

A Layer Cake of Languages You Are Here

Outline • What is Semantics? • RDF: Syntax • RDF Graph and Simple Entailment • RDF Interpretation • RDFS Interpretation

What is Semantics Merriam-Webster: the study of meanings Wikipedia: the study of meaning in communication.

What is Semantics? • Intensional Meaning • TW Students are Students with affiliation to the Tetherless World Group • Extensional Meaning • TW Students are the set {Jiao, Ankesh, Jesse,…}

Model Theory Used to link intensional meaning and extensional meaning “Model theory assumes that the language refers to a 'world', and describes the minimal conditions that a world must satisfy in order to assign an appropriate meaning for every expression in the language.” --RDF Semantics Alfred Tarski 1901-1983 Picure source: wikipedia

Model: an Example Expression: A Model: • TW Students are Students with affiliation to the Tetherless World Group …

A Few Jargons

RDF Family RDFS Interpretation RDF Interpretation Simple Interpretation Semantics Syntax

Not Covered in the Talk • Blank Node (b-Node) • Literals (Datatypes) • Containers • Collections • Reification • Annotation • Entailment rules (rule inference)

RDF: Triple and Graph • Triple: (subject, property, object) • UB × U × UBL (Url, Blank node, Literal) • e.g., (Jim, is-a, Professor) • e.g., (Jim, has-surname, “Hendler”) – not covered • e.g.,(Jim, has-pet, _:x) – not covered • Graph: A set of triples Professor is-a Jim “Hendler” has-surname has-pet

Simple Interpretation A simple interpretation I of a vocabulary V is defined by: 1. A non-empty set IR of resources, called the domain or universe of I. 2. A set IP, called the set of properties of I. 3. A mapping IEXT from IP into the powerset of IR x IR i.e. the set of sets of pairs <x,y> with x and y in IR . 4. A mapping IS from URI references in V into (IR union IP) 5. A mapping IL from typed literals in V into IR. 6. A distinguished subset LV of IR, called the set of literal values, which contains all the plain literals in V We do not consider RDF vocabulary (e.g., rdf:type), yet.

Simple Interpretation V IS IR IP IEXT

Simple Interpretation Example V={a, b, c} Picture courtesy of “RDF Semantics”(Figure 1)

Simple Semantic Conditions • if E is a URI reference in V then I(E) = IS(E) • if E is a ground triple s p o. then I(E) = true if s, p and o are in V, I(p) is in IP and <I(s),I(o)> is in IEXT(I(p)) otherwise I(E)= false. • if E is a ground RDF graph then I(E) = false if I(E') = false for some triple E' in E, otherwise I(E) =true • if E is a plain literal "aaa" in V then I(E) = aaa • if E is a plain literal "aaa"@ttt in V then I(E) = <aaa, ttt> • if E is a typed literal in V then I(E) = IL(E) • If E is a blank node and A(E) is defined then [I+A](E) = A(E) • If E is an RDF graph then I(E) = true if [I+A'](E) = true for some mapping A' from blank(E) to IR, otherwise I(E)= false

Note to Simple Interpreation • IP may not be in IR • A property (an element in IP) and its extension (mapping by IEXT) are separated. • Thus avoids paradox like the barber paradox (A barber shaves only those men who do not shave themselves.)

RDF Vocabulary (rdfV) • rdf:typerdf:Property • rdf:XMLLiteralrdf:nilrdf:Listrdf:Statementrdf:subjectrdf:predicaterdf:objectrdf:firstrdf:restrdf:Seqrdf:Bagrdf:Alt rdf:_1 rdf:_2 ... rdf:value

RDF Semantic Conditions • x is in IP if and only if <x, I(rdf:Property)> is in IEXT(I(rdf:type)) • Thus, RDF properties (IP) must be resources (IR) in the universe. • (rdf:type rdf:type rdf:Property ) is always true • More conditions for literals

RDF Interpretation Example Vocabulary: rdfV + V={a,b,c} Picture courtesy of “RDF Semantics”(Figure 2)

RDFS Vocabulary (rdfsV) • rdfs:domainrdfs:rangerdfs:Resource • rdfs:Classrdfs:subClassOfrdfs:subPropertyOf • rdfs:Literalrdfs:Datatype • rdfs:memberrdfs:Containerrdfs:ContainerMembershipProperty • rdfs:commentrdfs:seeAlsordfs:isDefinedByrdfs:label

RDFS Semantic Conditions On classes • x is in ICEXT(y) if and only if <x,y> is in IEXT(I(rdf:type)) • IC = ICEXT(I(rdfs:Class)) • IR = ICEXT(I(rdfs:Resource)) • LV = ICEXT(I(rdfs:Literal)) • If x is in IC then <x, I(rdfs:Resource)> is in IEXT(I(rdfs:subClassOf)) • If <x,y> is in IEXT(I(rdfs:subClassOf)) then x and y are in IC and ICEXT(x) is a subset of ICEXT(y) • IEXT(I(rdfs:subClassOf)) is transitive and reflexive on IC

RDFS Semantic Conditions On properties • If <x,y> is in IEXT(I(rdfs:domain)) and <u,v> is in IEXT(x) then u is in ICEXT(y) • If <x,y> is in IEXT(I(rdfs:range)) and <u,v> is in IEXT(x) then v is in ICEXT(y) • IEXT(I(rdfs:subPropertyOf)) is transitive and reflexive on IP • If <x,y> is in IEXT(I(rdfs:subPropertyOf)) then x and y are in IP and IEXT(x) is a subset of IEXT(y) More for container and literals

RDFS Axiomatic triples Domains • rdf:type rdfs:domain rdfs:Resource .rdfs:domain rdfs:domain rdf:Property .rdfs:range rdfs:domain rdf:Property .rdfs:subPropertyOf rdfs:domain rdf:Property .rdfs:subClassOf rdfs:domain rdfs:Class .

RDFS Axiomatic triples Ranges • rdf:type rdfs:range rdfs:Class .rdfs:domain rdfs:range rdfs:Class .rdfs:range rdfs:range rdfs:Class .rdfs:subPropertyOf rdfs:range rdf:Property .rdfs:subClassOf rdfs:range rdfs:Class . More for container, reification, literal, and annotation…

RDFS-Valid Triples • rdfs:Resource rdf:type rdfs:Class . • rdfs:Class rdf:type rdfs:Class . • rdf:Property rdf:type rdfs:Class . • rdfs:domain rdf:type rdf:Property .rdfs:range rdf:type rdf:Property .rdfs:subPropertyOf rdf:type rdf:Property .rdfs:subClassOf rdf:type rdf:Property.

Conclusions • Model Theory gives semantics to RDF(S) • RDF and RDFS vocabularies pose semantic constraints on interpretations • RDF: type, Property • RDFS: domain, range, Resource, Class, subClassOf subPropertyOf • Will see OWL 1 and OWL 2 extensions to RDF(S) in the future

More on RDF Semantics • Herman J. ter Horst - Completeness, decidability and complexity of entailment for RDF Schema and a semantic extension involving the OWL vocabulary. In J. Web Sem. 3(2-3):79-115, 2005. • Jos de Bruijn, Stijn Heymans - Logical Foundations of (e)RDF(S): Complexity and Reasoning. In ISWC/ASWC pp. 86-99, 2007. • Jeff Z. Pan, Ian Horrocks - RDFS(FA) and RDF MT: Two Semantics for RDFS. In International Semantic Web Conference pp. 30-46, 2003.

RDF Semantics by Patrick Hayes W3C Recommendation w3/TR/rdf-mt/