RDF Semanticsby Patrick Hayes

W3C Recommendation

http://www.w3.org/TR/rdf-mt/

Presented by Jie BaoRPI

Sept 4, 2008

Part 1 of RDF/OWL Semantics Tutorialhttp://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 Tarski1901-1983Picure source: wikipedia

Model: an Example

Expression:

A Model:

TW Students are Students with affiliation to the Tetherless World Group

…

A Few Jargons

Outline

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

RDF Family

RDFS Interpretation

RDF Interpretation

Simple Interpretation

Syntax Semantics

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

Jim

Professor

“Hendler”is-a

has-surname

has-pet

Outline

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

Simple InterpretationA 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

IP IR

IEXT

V

IS

Simple Interpretation ExampleV={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.)

Outline

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

RDF Vocabulary (rdfV)

• rdf:type rdf:Property • rdf:XMLLiteral rdf:nil rdf:List rdf:Statement

rdf:subject rdf:predicate rdf:object rdf:first rdf:rest rdf:Seq rdf:Bag rdf: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

Picture courtesy of “RDF Semantics”(Figure 2)

Vocabulary: rdfV + V={a,b,c}

Outline

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

RDFS Vocabulary (rdfsV)

• rdfs:domain rdfs:range rdfs:Resource • rdfs:Class rdfs:subClassOf rdfs:subPropertyOf• rdfs:Literal rdfs:Datatype • rdfs:member rdfs:Container

rdfs:ContainerMembershipProperty • rdfs:comment rdfs:seeAlso rdfs:isDefinedBy

rdfs: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.