W3C


RDF-Based Semantics

W3C Editor's Draft 2126 November 2008

This version:
http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-based-semantics-20081121/http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-based-semantics-20081126/
Latest editor's draft:
http://www.w3.org/2007/OWL/draft/owl2-rdf-based-semantics/
Previous version:
http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-based-semantics-20081008/http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-based-semantics-20081121/ (color-coded diff)
Editors:
Michael Schneider, FZI Research Center for Information Technology
Contributors:
Note: The complete list of contributors is being compiled and will be included in the next draft.

Copyright © 2008 W3C® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.

Abstract

OWL 2 extends the W3C OWL Web Ontology Language with a small but useful set of features that have been requested by users, for which effective reasoning algorithms are now available, and that OWL tool developers are willing to support. The new features include extra syntactic sugar, additional property and qualified cardinality constructors, extended datatype support, simple metamodeling, and extended annotations.
This document provides the RDF-compatible model-theoretic semantics for OWL 2, called "OWL 2 Full".

Status of this Document

May Be Superseded

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

This document is being published as one of a set of 11 documents:

  1. Structural Specification and Functional-Style Syntax
  2. Direct Semantics
  3. RDF-Based Semantics (this document)
  4. Conformance and Test Cases
  5. Mapping to RDF Graphs
  6. XML Serialization
  7. Profiles
  8. Quick Reference Guide
  9. New Features &and Rationale
  10. Manchester Syntax
  11. rdf:text: A Datatype for Internationalized Text

Please Comment By 2008-11-252008-11-28

The OWL Working Group seeks public feedback on these Working Drafts. Please send your comments to public-owl-comments@w3.org (public archive). If possible, please offer specific changes to the text that would address your concern. You may also wish to check the Wiki Version of this document for internal-review comments and changes being drafted which may address your concerns.

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Contents


Editor's Note: It will be considered to finally add "informative" and "normative" to each section's title, following RDF and OWL 1 traditions. Editor's Note: Open Question: There is an open issue with OWL 2 Full's support of n-ary datatypes. I suppose that we need to introduce a new sort of extension, a "datarange extension of arity n", which associates an individual with a set of n-tuples of individuals. Affected parts of the document would be: Introduction, Interpretations, the Semantic Conditions and the Comprehension Principles for the Datatype Complement, and the Restrictions on N-ary Datatypes ("owl:onProperties"). Should this be the case, then the latter semantic conditions have to be considered broken at the moment, since n-tuples of individuals then cannot be instances of class extensions. However, it has been proposed that there is no need to introduce such an additional sort of extension.1 Introduction (Informative)

This document defines the RDF-compatible model-theoretic semantics of OWL 2, called "OWL 2 Full", whichFull". The semantics given here is the OWL 2 semantic extension of RDFS [RDF Semantics] . OWL 2 Full inherits every aspect of the semantic specification of RDFS, and therefore[RDF Semantics]. Therefore, the semantic meaning given to an RDF graph by OWL 2 Full includes the meaning given to the graph by RDFS. Further,Beyond that, OWL 2 Full definesgives additional semanticmeaning to all the language features of OWL 2, by following the design principles that have been applied to the semantics of RDF.

The content of this document is not meant to be self-contained, but builds on top of the RDF Semantics document by only adding theOWL 2 specific parts of the semantics. Hence, the complete definition ofFull accepts every well-formed RDF graph [RDF] as a syntactically valid OWL 2 Full ontology, and gives a precise semantic meaning to it. The semantic meaning is actually givendetermined by the combinationset of these two documents.OWL 2 Full is specified for the OWL 2 Full vocabulary insemantic conditions, which include and extend all the form ofsemantic conditions for RDF and RDFS specified in [RDF Semantics]. OWL 2 Full acts as a vocabulary interpretation, whichfor the RDF and the RDFS vocabularies, and for the OWL 2 Full vocabulary. The OWL 2 Full vocabulary is a set of URIs that occur in the sets of RDF triples, which define the RDF syntax of OWL 2 [OWL 2 RDF Mapping]. The OWL 2 Full semantic conditions specify exactly which triple sets are assigned a specific meaning, and what this meaning is.

OWL 2 Full interpretations are defined on the OWL 2 Full universe. The OWL 2 Full universe is identified with the RDFS universe, and comprises the set of all individuals. It is further divided into sub parts,"parts", namely the classes, the properties, and the datatype values, whichvalues. Thus, the members of these parts are thusalso individuals. Every class has a set of individuals associated with it, the so called "class extension", which is distinguished from the class itself. Analog, every property is associated with a "property extension", which is a binary relation, i.e. a setconsists of pairs of individuals. The classes subsume the datatypes, and the properties subsume the data properties, the annotation properties, and the ontology properties. Individuals may play different roles at the same time. Theytime in an OWL 2 Full ontology. One individual can, for example, be both a class and a property, or both a data property and an annotation property.

Editor's Note: It has been proposed to have some figure here visualizing the parts hierarchy and possibly the IEXT/ICEXT concept as explained in the text above.

Every RDF graph is a syntactically validIn OWL 2 Full ontology, which receives its semantic meaning by applyingontologies, usually no care is needed to ensure that URI references are actually in the setappropriate part of the OWL universe. These "localizing" assumptions will typically follow from applying the OWL 2 Full semantic conditions . For ontologies importing other ontologies, the whole imports closure of that ontology will generally have to be taken into account.conditions.

A strong relationship holds between the RDF-Based Semantics ofOWL 2 Full and the Direct Semantics of OWL 2 DL, in that[OWL 2 Direct Semantics]. OWL 2 Full is, in somea certain sense, able to reflect all logical conclusions of the Direct Semantics, when applied to an OWL 2 DL.DL ontology [OWL 2 Structural Specification] in RDF graph form. The precise relationship is stated by the OWL 2 correspondence theorem.

The italicized keywords MUST , MUST NOT , SHOULD , SHOULDcontent of this document is not , and MAY specify certain aspectsmeant to be self-contained, but builds on top of the normative behavior ofRDF Semantics document [RDF Semantics] by only adding the OWL 2 specific aspects of the semantics. Hence, the complete definition of OWL 2 Full is actually given by the combination of these two documents.

The italicized keywords MUST, MUST NOT, SHOULD, SHOULD NOT, and MAY specify certain aspects of the normative behavior of OWL 2 tools, and are interpreted as specified in RFC 2119 [RFC 2119].

2 Vocabulary

The OWL 2 Full vocabulary is a set of URI references in the OWL namespace, owl:, which is given by the URI reference

http://www.w3.org/2002/07/owl#

Table 2.1 lists the OWL 2 Full vocabulary, which extends the RDF and RDFS vocabulary as specified by Sections 3.1 and 4.1 of the[RDF Semantics .]. Excluded are those URI references from the OWL namespace, which are mentioned in one of the other tables in this section.

Table 2.1: OWL 2 Full Vocabulary
owl:AllDifferent owl:AllDisjointClasses owl:AllDisjointProperties owl:allValuesFrom owl:Annotation owl:AnnotationProperty owl:assertionProperty owl:AsymmetricProperty owl:Axiom owl:backwardCompatibleWith owl:bottomDataProperty owl:bottomObjectProperty owl:cardinality owl:Class owl:complementOf owl:DataRange owl:datatypeComplementOf owl:DatatypeProperty owl:deprecated owl:DeprecatedClass owl:DeprecatedProperty owl:differentFrom owl:disjointUnionOf owl:disjointWith owl:distinctMembers owl:equivalentClass owl:equivalentProperty owl:FunctionalProperty owl:hasKey owl:hasSelf owl:hasValue owl:imports owl:incompatibleWith owl:intersectionOf owl:InverseFunctionalProperty owl:inverseOf owl:IrreflexiveProperty owl:maxCardinality owl:maxQualifiedCardinality owl:members owl:minCardinality owl:minQualifiedCardinality owl:NamedIndividual owl:NegativePropertyAssertion owl:Nothing owl:object owl:ObjectProperty owl:onClass owl:onDataRange owl:onDatatype owl:oneOf owl:onProperty owl:onProperties owl:Ontology owl:OntologyProperty owl:predicate owl:priorVersion owl:propertyChain owl:propertyDisjointWith owl:qualifiedCardinality owl:ReflexiveProperty owl:Restriction owl:sameAs owl:someValuesFrom owl:sourceIndividual owl:subject owl:SymmetricProperty owl:targetIndividual owl:targetValue owl:Thing owl:topDataProperty owl:topObjectProperty owl:TransitiveProperty owl:unionOf owl:versionInfo owl:withRestrictions

Note: The use of the URI reference owl:DataRange has been deprecated as of OWL 2. The URI reference rdfs:Datatype SHOULD be used instead.

Table 2.2 lists the set of datatypes supported byof OWL 2 Full. The datatyperdf:XMLLiteral is described in Section 3.1 of the[RDF Semantics .]. rdf:text is described in [RDF:TEXT]. All other datatypes are described in Section 4 of the[OWL 2 Structural Specification].

Table 2.2: Datatypes of OWL 2 Full
xsd:anyURI xsd:base64Binary xsd:boolean xsd:byte owl:dateTime xsd:decimal xsd:double xsd:float xsd:hexBinary xsd:int xsd:integer xsd:language xsd:long xsd:Name xsd:NCName xsd:negativeInteger xsd:NMTOKEN xsd:nonNegativeInteger xsd:nonPositiveInteger xsd:normalizedString xsd:positiveInteger owl:rational owl:real owl:realPlus xsd:short xsd:string rdf:text xsd:token xsd:unsignedByte xsd:unsignedInt xsd:unsignedLong xsd:unsignedShort rdf:XMLLiteral

Feature At Risk #1: owl:rational support

Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to public-owl-comments@w3.org.

Editor'sThe owl:rational datatype might be removed from OWL 2 if implementation experience reveals problems with supporting this datatype.

Feature At Risk #2: owl:dateTime name

Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to public-owl-comments@w3.org.

The name owl:dateTime is currently a placeholder. XML Schema 1.1 Working Group will introduce a datatype for date-time with required timezone. Once this is done, owl:dateTime will be changed to whatever name XML Schema chooses. If the schedule of the XML Schema 1.1 Working Group slips the OWL 2 Working Group will consider possible alternatives, so the name is potentially at risk . Editor's Note: The datatype owl:rational is at risk , pending implementation experience. Table 2.2: Datatypes of OWL 2 Full xsd:anyURI xsd:base64Binary xsd:boolean xsd:byte owl:dateTime xsd:decimal xsd:double xsd:float xsd:hexBinary xsd:int xsd:integer xsd:language xsd:long xsd:Name xsd:NCName xsd:negativeInteger xsd:NMTOKEN xsd:nonNegativeInteger xsd:nonPositiveInteger xsd:normalizedString xsd:positiveInteger owl:rational owl:real owl:realPlus xsd:short xsd:string rdf:text xsd:token xsd:unsignedByte xsd:unsignedInt xsd:unsignedLong xsd:unsignedShort rdf:XMLLiteralalternatives.

Table 2.3 lists the set of datatype facets supported byof OWL 2 Full. Section 4 of the[OWL 2 Structural Specification] describes the meaning of each facet, andto which datatypes it can be applied, respectively. Editor's Note:and which values it can take for a given datatype. The facet "langPattern" itrdf:langPattern is currently not clear, whether it will eventually have the namespace "rdf:", and whether this facet will be defined by the OWL 2 Structural Specification or elsewhere, for example by a distinguished document describing the datatype rdf:text.further described in [RDF:TEXT].

Table 2.3: Datatype Facets of OWL 2 Full
rdf:langPattern xsd:length xsd:maxExclusive xsd:maxInclusive xsd:maxLength xsd:minExclusive xsd:minInclusive xsd:minLength xsd:pattern

3 Ontologies

Editor's Note: Should this section on ontologies and importing be kept in this semantics document? And if so, what should be said about the semantics of an RDF graph having imports statements? Obviously, such an RDF graph has a clear semantic meaning in OWL 2 Full without regarding the imports clousure. Perhaps we should state that in such a case the whole imports closure SHOULD be regarded as the ontology instead of the single RDF graph, with the semantic meaning being the one of that ontology.

Every well-formed RDF graph [RDF] is a syntactically valid OWL 2 Full ontology. If a OWL 2 Full ontology imports other OWL 2 Full ontologies, then the whole imports closure of that ontology has to be taken into account.

Definition 3.1 (Import Closure): Let K be a collection of RDF graphs. K is imports closed iff for every triple in any element of K of the form x owl:imports u then K contains a graph that is referred to by u. The imports closure of a collection of RDF graphs is the smallest imports closed collection of RDF graphs containing the graphs.

AnA OWL 2 Full ontology MAY contain an ontology header, with optional information about the ontology's version,if the ontology's author wants to explicitly signal that the ontologyan RDF graph is intended as a OWL 2 Full ontology. Such an ontology header MAY additionally contain information about the ontology's version. The OWL 2 Mapping to RDF [OWL 2 RDF Mapping] provides details about the syntax of ontology headers.

4 Interpretations

OWL 2 Full provides a vocabulary interpretation and vocabulary entailment (see Section 2.1 of the[RDF Semantics )]) for the RDF vocabulary, theand RDFS vocabulary,vocabularies, and the OWL 2 Full vocabulary.

From the RDF Semantics ,[RDF Semantics], let V be a set of URI references and literals containing the RDF and RDFS vocabulary, and let D be a datatype map according to Section 5.1 of the[RDF Semantics .]. A D-interpretation I of V is a tuple

I = ⟨ IR, IP, IEXT, IS, IL, LV ⟩.

IR is the domain of discourse or universe, i.e., a nonempty set that contains the denotations of URI references and literals in V. IP is a subset of IR, the properties of I. LV is a subset of IR that covers at least the value spaces of all datatypes in D. IEXT is used to associate properties with their property extension, and is a mapping from IP to P(IR × IR), where P is the powerset. IS is a mapping from URI references in V to their denotations in IR. IL is a mapping from typed literals in V to their denotations in IR, where in particular the denotations ofwhich maps all well-typed literals are membersto instances of LV (Section 5.1 of the[RDF Semantics] explains why the range of IL is actually IR instead of LV).

The set of classes IC is defined as IC = { x ∈ IR | ⟨x,I(rdfs:Class)⟩ ∈ IEXT(I(rdf:type)) }. The mapping ICEXT from IC to P (IR) associates classes with their class extension, and is defined as ICEXT(c) = { x ∈ IR | ⟨x,c⟩ ∈ IEXT(I(rdf:type)) } for c ∈ IC.As detailed in the[RDF Semantics ,], a D-interpretation has to meet additional semantic conditions, which constrain the set of RDF graphs that are true under this interpretation. An RDF graph G is said to be satisfied by a D-interpretation I, if I(G) = true.

Definition 4.1 (OWL 2 Full Datatype Map): Let D be a datatype map as defined in Section 5.1 ofThe RDF Semantics . D isfollowing definition specifies what a OWL 2 Full datatype map , if it contains at least all datatypes listed inis. First, Table 2.2 . Definition4.1 defines sets that relate datatypes with their facets, and with the values a facet is allowed to take in combination with a certain datatype.

Table 4.1: Sets that relate datatypes, facets and facet values
Name of Set S Definition
IFP(d) The set of all facets allowed for datatype d.
IFV(d,f) The set of all facet values allowed for the combination of datatype d and facet f.
IFEXT(d,f,u) The subset of the class extension of datatype d that results from applying facet f with facet value u to d.
Editor's Note: The sets given in this table need to be exchanged for sets that have counter parts in OWL 2 DL, in order to proof the Correspondence Theorem. The semantic conditions and comprehension principles for datatype restrictions have then to be adjusted as well.
Editor's Note: The definitions in this table should be less informal. They should be similar to the definitions given in the OWL 2 DL Specification.

Definition 4.1 (OWL 2 Full Datatype Map): Let D be a datatype map as defined in Section 5.1 of [RDF Semantics]. D is a OWL 2 Full datatype map, if it contains at least all datatypes listed in Table 2.2, and if it defines the sets listed in Table 4.1 for each contained datatype.

The next definition specifies what a OWL 2 Full interpretation is.

Definition 4.2 (OWL 2 Full Interpretation): Let D be a OWL 2 Full datatype map, and let V be a vocabulary that includes the RDF and RDFS vocabularies, and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2. An OWL 2 Full interpretation, I = ⟨ IR, IP, IEXT, IS, IL, LV ⟩, of V with respect to D, is a D-interpretation of V that satisfies all the extra semantic conditions given in Section 5.

Definition 4.3 (OWL 2 Full Consistency): Let K be a collectionTable 4.2 defines the "parts" of RDF graphs, and let D be athe OWL 2 Full datatype map. K isuniverse in terms of the mapping IEXT of an OWL 2 Full consistent with respect to Dinterpretation and by referring to the RDF, RDFS and OWL 2 Full vocabularies.

Table 4.2: Parts of the OWL 2 Full Universe
Name of
Part S		 Definition of S as
{x ∈ IR | ⟨x,I(U)⟩ ∈ IEXT(I(rdf:type))}
where URI U is		 Explanation
IR rdfs:Resource individuals
LV rdfs:Literal datatype values
IX owl:Ontology ontologies
IC rdfs:Class classes
IDC rdfs:Datatype datatypes
IP rdf:Property properties
IODP owl:DatatypeProperty data properties
IOAP owl:AnnotationProperty annotation properties
IOXP owl:OntologyProperty ontology properties

Further, the mapping ICEXT from IC to P(IR) that associates classes with their class extension, is defined as

ICEXT(c) = { x ∈ IR | ⟨x,c⟩ ∈ IEXT(I(rdf:type)) }

for c ∈ IC.

The following definitions specify what a consistent OWL 2 Full ontology is, and what it means that an OWL 2 Full ontology entails another OWL 2 Full Ontology.

Definition 4.3 (OWL 2 Full Consistency): Let K be a collection of RDF graphs, and let D be a OWL 2 Full datatype map. K is OWL 2 Full consistent with respect to D iff there is some OWL 2 Full interpretation Iwith respect to D of(of some vocabulary Vthat includes the RDF and RDFS vocabularies, and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2 , where I) that satisfies all the RDF graphs in K.

Definition 4.4 (OWL 2 Full Entailment): Let K and Q be collections of RDF graphs, and let D be a OWL 2 Full datatype map. K OWL 2 Full entails Q with respect to D iff every OWL 2 Full interpretation Iwith respect to D of(of any vocabulary V that includes the RDF and RDFS vocabulariesvocabularies, and the OWL 2 Full vocabulary together with all the datatype and facet names listed in Section 2 , and where I) that satisfies all the RDF graphs in K, then IK also satisfies all the RDF graphs in Q.

5 Semantic Conditions

This section defines the semantic conditions of OWL 2 Full. The semantic conditions presented here are only those for the specific tofeatures of OWL 2. The complete set of semantic conditions for OWL 2 Full vocabulary . This setis the combination of the semantic conditions is not self-contained, but has to be regarded in conjunction withpresented here and the semantic conditions given for Simple Entailment, RDF, RDFS and D-Entailment in the[RDF Semantics .].

Table 5.1 on "Basic Sets" enumeratesspecifies semantic conditions for the different parts of the OWL 2 Full universe, suchas the sets of classes, properties, etc., which are referred to by many semantic conditionsdefined in this section.Section 4. Table 5.2 lists several "Convenient Abbreviations" for sets that are often used within semantic conditions. Table 5.3and Table 5.45.3 list basicsemantic conditions for the classes and the properties of the OWL 2 Full vocabulary, which can be regarded as the OWL 2 Full "axiomatic triples" (this is further explained by the introduction of these tables).vocabulary. The remaining tables in this section specify the OWL 2 Full semantic conditions for the different language features of OWL 2.

Most semantic conditions are "iff" conditions, which exactlycompletely specify the semantics of the respective language feature. For some language featuresfeatures, however, there are only have"if-then" conditions in order to avoid certain semantic paradoxes and other problems with the semantics. Several language features with "iff" conditions, namely Sub Property Chains in Table 5.105.9, N-ary Axioms in Table 5.125.11, and Negative Property Assertions in Table 5.165.15, have a multi-triple representation in RDF, where the different triples share a common "root node" x. In order to treat this specific syntactic aspect technically, the "iff" conditions of these language features arehave been split into two "if-then" conditions, and the right-to-left "if" condition contains an additional premise of the form "∃x ∈ IR", which has the single purpose to provide the needed "root node" x.

Conventions used in this section:

Several conventions are used when presenting logic expressions in the below tables.

Having a comma between two statementsassertions in a semantic condition, as in

c ∈ IC , p ∈ IP

means a logical "and".

If no scope is explicitly given for a variable x, as in "∀x:…" or in "{x|…}", then x is unconstrained, which means that x ∈ IR.

An expression of the form "l sequence of u1,…, un ∈ S" means that l represents a list of n elements, all of them being instances of the class S. Precisely, u1 ∈ S,… , un ∈ S, and thatthere exist x1 ∈ IR,…, xn ∈ IR, such that

I(l) ∈ ICEXT(I(rdf:List)),
I(l) = I(x1),
⟨x1,u1⟩ ∈ IEXT(I(rdf:first)), ⟨x1,x2⟩ ∈ IEXT(I(rdf:rest)),
…,
⟨xn,un⟩ ∈ IEXT(I(rdf:first)), ⟨xn,I(rdf:nil)⟩ ∈ IEXT(I(rdf:rest)).

The following names for certain sets are used in addition to those given in Section 4 :as convenient abbreviations throughout this and the following sections:

  • ISEQ: The set of all sequences. This set equals the class extension of rdf:List.
  • INNI: The set of all non-negative integers. This set equals the value space of xsd:NonNegativeInteger, but is also contained in the value spaces of other numerical datatypes, such as xsd:integer.

The semantic conditions in the following tables belowsometimes do not explicitly list typing statements in their consequent that one would normally expect. This may beFor example, the casesemantic condition for owl:allValuesFrom restrictions in Table 5.6 does not list the statement x ∈ ICEXT(I(owl:Restriction)) on its right hand side. Consequents are generally not mentioned, if these statementsthey can already be deduced by means of the semantic conditions given in Table 5.35.2 and Table 5.45.3, occasionally in connection with Table 5.1. For example, the semantic condition for owl:allValuesFrom restrictionsIn Table 5.7 does not havethe statement x ∈ ICEXT(I( owl:Restriction )) on its right hand side.example above, the reason is that this resultommitted consequent can alreadybe obtained from the third column of the entry for owl:allValuesFrom in Table 5.45.3, which determines that IEXT(I(owl:allValuesFrom)) ⊆ ICEXT(I(owl:Restriction)) × IC.

Table 5.1 presents basic sets used inlists the semantic conditions for the parts of the OWL 2 Full, their relationshipFull universe, as defined by Table 4.2 in Section 4. The semantic conditions say how the parts are related to other sets,parts, and propertiesthey further specify the semantics for the instances of their instances.some of the parts.

Table 5.1: Basic SetsSemantic Conditions for the Parts of the OWL 2 Full Universe
Name of
SetPart S		 Conditions on SetS Conditions on
Instances x of S
ExplanationIR S ≠ ∅
all individuals (or resources)LV S ⊆ IR
all datatype valuesIX S ⊆ IR
all ontologiesIC S ⊆ IR ICEXT(x) ⊆ IR
all classesIDC S ⊆ IC ICEXT(x) ⊆ LV
all datatypesIP S ⊆ IR IEXT(x) ⊆ IR × IR
all (object) propertiesIODP S ⊆ IP IEXT(x) ⊆ IR × LV
all data propertiesIOAP S ⊆ IP IEXT(x) ⊆ IR × IR
all annotation propertiesIOXP S ⊆ IP IEXT(x) ⊆ IX × IX

all ontology propertiesTable 5.2 provides abbreviationslists the semantic conditions for additional sets, which are used throughout this document. Table 5.2: Convenient Abbreviations Name of Set S Explanation IFP(d)the setclasses of all facets allowed for datatype d. IFV(d,f)the set of all facet values allowed forOWL 2 Full vocabulary, and certain classes from RDF and RDFS. It tells the combinationsort of datatype dclass, and facet f. IFEXT(d,f,u)specifies the subsetpart of the classOWL 2 Full universe the extension of datatype d that results from applying facet f with facet value u to d. Editor'seach class belongs to. As a specific note: The definitionsFor owl:NamedIndividual that there is no way in this table should be less informal. Table 5.3 lists the classes of theOWL 2 Full vocabulary (and certain classes from RDF and RDFS), together with their relationshipto other classes.restrict the set of individuals to only those being named by a URI, hence the extension of this class has been specified to equal the whole domain.

Not included in this table are the differentdatatypes ,of OWL 2 Full, as given in Table 2.2. For a datatype URI DU, I(D)the following semantic conditions hold: I(U) ∈ IDC, and ICEXT(I(D))ICEXT(I(U)) ⊆ LV.

Informative Note: TheTable 5.2: Semantic Conditions given here can be regarded as the OWL 2 Full "axiomatic triples" for classes:for eachClasses
Vocabulary URI U occurring in the first column of the table, if the second column contains an entry "I(U)I(U) ICEXT(I(U))
owl:AllDifferent ∈ S" for some set S, then this entry corresponds to some RDF triple of the form "U rdf:type C", where C is the URI of some class with ICEXT(I(C))IC ⊆ IR
owl:AllDisjointClasses ∈ IC ⊆ IR
owl:AllDisjointProperties ∈ IC ⊆ IR
owl:Annotation ∈ IC ⊆ IR
owl:AnnotationProperty ∈ IC = S. In this table, S will always be either the setIOAP
owl:AsymmetricProperty ∈ IC of all classes, or a subset of IC. Hence, in a corresponding RDF triple the URI C will typically be one of "rdfs:Class" or "owl:Class" (S=IC in both cases), or "rdfs:Datatype" (S=IDC). Further, for each URI U in the first column, if the third column contains an entry "ICEXT(I(U))⊆ S" ("ICEXT(I(U))IP
owl:Axiom ∈ IC ⊆ IR
rdfs:Class ∈ IC = S")IC
owl:Class ∈ IC = IC
owl:DataRange ∈ IC = IDC
rdfs:Datatype ∈ IC = IDC
owl:DatatypeProperty ∈ IC = IODP
owl:DeprecatedClass ∈ IC ⊆ IC
owl:DeprecatedProperty ∈ IC ⊆ IP
owl:FunctionalProperty ∈ IC ⊆ IP
owl:InverseFunctionalProperty ∈ IC ⊆ IP
owl:IrreflexiveProperty ∈ IC ⊆ IP
rdfs:Literal ∈ IDC = LV
owl:NamedIndividual ∈ IC = IR
owl:NegativePropertyAssertion ∈ IC ⊆ IR
owl:Nothing ∈ IC = ∅
owl:ObjectProperty ∈ IC = IP
owl:Ontology ∈ IC = IX
owl:OntologyProperty ∈ IC = IOXP
rdf:Property ∈ IC = IP
owl:ReflexiveProperty ∈ IC ⊆ IP
rdfs:Resource ∈ IC = IR
owl:Restriction ∈ IC ⊆ IC
owl:SymmetricProperty ∈ IC ⊆ IP
owl:Thing ∈ IC = IR
owl:TransitiveProperty ∈ IC ⊆ IP

Table 5.3 lists the semantic conditions for some set S, then this entry corresponds to some RDF triplethe properties of the form "U rdfs:subClassOf C" ("U owl:equivalentClass C"), where C isOWL 2 Full vocabulary and certain properties from RDFS. It tells the URIsort of some class with ICEXT(I(C)) = S. Additionally, the conditions onproperty, and specifies the sets givendomain and range for each property. As specific notes: owl:topObjectProperty relates every two individuals in Table 5.1 havethe universe to be taken into account. In particular, if an entry of Table 5.1 states S 1 ⊆ S 2 for some sets S 1 and S 2 , then this correspondseach other. Likewise, owl:topDataProperty relates every individual to some RDF triple C 1 owl:subClassOf C 2 , where C 1every datavalue. owl:bottomObjectProperty and C 2 areowl:bottomDataProperty do not relate any individuals to each other at all. The URIsranges of some classes with ICEXT(I(C 1 )) = S 1the properties owl:deprecated and ICEXT(I(C 2 )) = S 2 , respectively, accordingowl:hasSelf are not restricted to Table 5.3 . Note that somebe boolean values, so it is possible for these properties to have objects of the RDF triples receivedarbitrary type.

Not included in this way already follow from the RDFS Semantics . These axiomatic triplestable are "simple" in the following sense: For every set S mentioned in the second andthe third columndatatype facets of theOWL 2 Full, as given in Table there exists2.3. For a facet URI C of some class inU, the vocabularies for RDF, RDFS or those given in Section 2following semantic conditions hold: I(U) ∈ IP, for which S = ICEXT(I(C))and IEXT(I(U)) ⊆ IR × LV.

Table 5.3: ClassesSemantic Conditions for Properties
Vocabulary URI U I(U) ICEXT(I(U)) owl:AllDifferentIEXT(I(U))
owl:allValuesFrom ∈ ICIP ⊆ IR owl:AllDisjointClasses ∈ICEXT(I(owl:Restriction)) × IC
⊆ IR owl:AllDisjointPropertiesowl:assertionProperty ∈ ICIP ⊆ IR owl:AnnotationICEXT(I(owl:NegativePropertyAssertion)) × IP
owl:backwardCompatibleWith ∈ ICIOXP ⊆ IR owl:AnnotationPropertyIX × IX
owl:bottomDataProperty ∈ ICIODP = IOAP owl:AsymmetricProperty∅
owl:bottomObjectProperty ∈ IC ⊆IP owl:Axiom= ∅
owl:cardinality ∈ ICIP ⊆ ICEXT(I(owl:Restriction)) × INNI
rdfs:comment ∈ IOAP ⊆ IR rdfs:Class× LV
owl:complementOf ∈ IP ⊆ IC =× IC
owl:Classowl:datatypeComplementOf ∈ IC = IC owl:DataRange ∈ IC =IP ⊆ IDC rdfs:Datatype ∈ IC =× IDC
owl:DatatypePropertyowl:deprecated ∈ IC = IODP owl:DeprecatedClassIOAP ⊆ IR × IR
owl:differentFrom ∈ ICIP ⊆ IC owl:DeprecatedPropertyIR × IR
owl:disjointUnionOf ∈ IP ⊆ IC × ISEQ
owl:disjointWith ∈ IP ⊆ IC × IC
owl:distinctMembers ∈ IP owl:FunctionalProperty⊆ ICEXT(I(owl:AllDifferent)) × ISEQ
owl:equivalentClass ∈ IP ⊆ IC × IC
owl:equivalentProperty ∈ IP ⊆ IP owl:InverseFunctionalProperty× IP
owl:hasKey ∈ IP ⊆ IC × ISEQ
owl:hasSelf ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IR
owl:hasValue ∈ IP owl:IrreflexiveProperty⊆ ICEXT(I(owl:Restriction)) × IR
owl:imports ∈ IOXP ⊆ IX × IX
owl:incompatibleWith ∈ IOXP ⊆ IX × IX
owl:intersectionOf ∈ IP ⊆ IC × ISEQ
owl:inverseOf ∈ IP ⊆ IP rdfs:Literal× IP
rdfs:isDefinedBy ∈ IDC = LV owl:NamedIndividualIOAP ⊆ IR × IR
rdfs:label ∈ IC =IOAP ⊆ IR owl:NegativePropertyAssertion× LV
owl:maxCardinality ∈ ICIP ⊆ ICEXT(I(owl:Restriction)) × INNI
owl:maxQualifiedCardinality ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI
owl:members ∈ IP ⊆ IR owl:Nothing× ISEQ
owl:minCardinality ∈ IC = ∅ owl:ObjectPropertyIP ⊆ ICEXT(I(owl:Restriction)) × INNI
owl:minQualifiedCardinality ∈ IC =IP owl:Ontology⊆ ICEXT(I(owl:Restriction)) × INNI
owl:object ∈ IC = IX owl:OntologyPropertyIP ⊆ IR × IR
owl:onClass ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IC
= IOXP rdf:Propertyowl:onDataRange ∈ IC =IP owl:ReflexiveProperty⊆ ICEXT(I(owl:Restriction)) × IDC
owl:onDatatype ∈ IP ⊆ IDC × IDC
owl:oneOf ∈ IP ⊆ IC × ISEQ
owl:onProperty ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IP
rdfs:Resourceowl:onProperties ∈ IC =IP ⊆ ICEXT(I(owl:Restriction)) × ISEQ
owl:predicate ∈ IP ⊆ IR owl:Restriction× IP
owl:priorVersion ∈ ICIOXP ⊆ IC owl:SymmetricPropertyIX × IX
owl:propertyChain ∈ ICIP ⊆ IP owl:Thing× ISEQ
owl:propertyDisjointWith ∈ IC = IR owl:TransitivePropertyIP ⊆ IP × IP
owl:qualifiedCardinality ∈ ICIP ⊆ ICEXT(I(owl:Restriction)) × INNI
owl:sameAs ∈ IP Table 5.4 lists the properties of the OWL 2 Full vocabulary (and certain properties from RDF and RDFS), together with their domain and range. Note that the ranges of the properties owl:deprecated and owl:hasSelf are not restricted to be boolean values, in order to allow assertions with one of these properties to have objects of arbitrary type. Not included in this table are the different datatype facets , as given in Table 2.3 . For a facet URI F , I(F)⊆ IR × IR
rdfs:seeAlso ∈ IOAP ⊆ IR × IR
owl:someValuesFrom ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IC
owl:sourceIndividual ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR
owl:subject ∈ IP ⊆ IR × IR
owl:targetIndividual ∈ IP , and IEXT(I(F))⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR
owl:targetValue ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × LV
. Informative Note:owl:topDataProperty ∈ IODP = IR × LV
owl:topObjectProperty ∈ IP = IR × IR
owl:unionOf ∈ IP ⊆ IC × ISEQ
owl:versionInfo ∈ IOAP ⊆ IR × IR
owl:withRestrictions ∈ IP ⊆ IDC × ISEQ

Table 5.4 lists the semantic conditions given here can be regarded as the OWL 2 Full "axiomatic triples" for properties:for each URI U occurring in the first columnboolean class expressions, including complements, intersections, and unions of the table, if the second column containsclasses. An entry "I(U) ∈ S" for some set S, then this entry corresponds to some RDF tripleintersection or union of the form "U rdf:type C", where Ca collection of datatypes is the URIitself a datatype. While a complement of somea class with ICEXT(I(C)) = S. In this table, S will always be eitheris created w.r.t. to the set IP of all properties, orwhole domain, a subset of IP. Hence, indatatype complement is created for a corresponding RDF tripledatatype w.r.t. the URI C will typically be oneset of "rdf:Property" or "owl:ObjectProperty" (S=IPdata values only, and results itself in both cases), "owl:DatatypeProperty" (S=IODP), "owl:AnnotationProperty" (S=IOAP), or "owl:OntologyProperty" (S=IOXP). Further,a datatype.

Table 5.4: Semantic Conditions for each URI U in the first column,Boolean Class Expressions
⟨c,d⟩ ∈ IEXT(I(owl:complementOf)) iff c, d ∈ IC
ICEXT(c) = IR \ ICEXT(d)
⟨c,d⟩ ∈ IEXT(I(owl:datatypeComplementOf)) c, d ∈ IDC,
ICEXT(c) = LV \ ICEXT(d)
if the third column contains an entry "IEXT(I(U)) ⊆ Sl sequence of d1 × S 2 " for some sets S,…, dn ∈ IR then
⟨c,l⟩ ∈ IEXT(I(owl:intersectionOf)) iff c, d1 and S 2 ,,…, dn ∈ IC,
ICEXT(c) = ICEXT(d1) ∩…∩ ICEXT(dn)
⟨c,l⟩ ∈ IEXT(I(owl:unionOf)) c, d1,…, dn ∈ IC,
ICEXT(c) = ICEXT(d1) ∪…∪ ICEXT(dn)
if then
this entry corresponds to some RDF triplesl sequence of the forms "U rdfs:domain C 1 " and "U rdfs:range C 2 ", where Cd1 and,…, dn ∈ IDC, n ≥ 1,
⟨c,l⟩ ∈ IEXT(I(owl:intersectionOf))		 c 2 are the URIs∈ IDC
l sequence of some classes with ICEXT(I(C 1 )) = Sd1 and ICEXT(I(C 2 )) = S 2 , respectively. Exceptions are,…, dn ∈ IDC, n ≥ 1,
⟨c,l⟩ ∈ IEXT(I(owl:unionOf))		 c ∈ IDC

Table 5.5 lists the semantic conditions "IEXT(I(owl:topObjectProperty)) = IR × IR" and "IEXT(I(owl:topDataProperty)) = IR × LV",for which there are no corresponding domain and range triples. These axiomatic triples are "simple" in the following sense: For everyenumerations, i.e. classes that consist of an explicitly given finite set S mentionedof instances. In the second columnparticular, an enumeration entirely consisting of datatype values is a datatype.

Table 5.5: Semantic Conditions for Enumerations
if l sequence of u1,…, un ∈ IR then
⟨c,l⟩ ∈ IEXT(I(owl:oneOf)) iff c ∈ IC,
ICEXT(c) = { u1,…, un }
if then
l sequence of u1,…, un ∈ LV, n ≥ 1,
⟨c,l⟩ ∈ IEXT(I(owl:oneOf))		 c ∈ IDC

Table 5.6 lists the table,semantic conditions for property restrictions, including value restrictions, cardinality restrictions, and asself restrictions. There are also semantic conditions for value restrictions dealing with n-ary datatypes. Note that the semantic condition for self restrictions does not entail the left orright hand side of a Cartesian product in the third column of the table there existsowl:hasSelf assertion to be a URI Cboolean value, so it is possible to have right hand sides of some class in the vocabularies for RDF, RDFS or those given in Section 2 , for which S = ICEXT(I(C)) .arbitrary type.

Table 5.4: Properties Vocabulary URI U I(U) IEXT(I(U)) owl:allValuesFrom5.6: Semantic Conditions for Property Restrictions
if then
⟨x,u⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IC owl:assertionPropertyIEXT(I(owl:hasSelf)),
⟨x,p⟩ ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IP owl:backwardCompatibleWithIEXT(I(owl:onProperty))		 ICEXT(x) = {y | ⟨y,y⟩ ∈ IOXP ⊆ IX × IX owl:bottomDataPropertyIEXT(p)}
⟨x,c⟩ ∈ IODP = ∅ owl:bottomObjectPropertyIEXT(I(owl:allValuesFrom)),
⟨x,p⟩ ∈ IPIEXT(I(owl:onProperty))		 ICEXT(x) = ∅ owl:cardinality{y | ∀z : ⟨y,z⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI rdfs:commentIEXT(p) → z ∈ IOAP ⊆ IR × LV owl:complementOfICEXT(c)}
l sequence of p1,…, pn ∈ IP ⊆ IC × IC owl:datatypeComplementOfIR,
⟨x,c⟩ ∈ IP ⊆ IDC × IDC owl:deprecatedIEXT(I(owl:allValuesFrom)),
⟨x,l⟩ ∈ IOAP ⊆ IR × IR owl:differentFromIEXT(I(owl:onProperties))		 p1,…, pn ∈ IP ⊆ IR × IR owl:disjointUnionOfIODP,
c ∈ IP ⊆ IC × ISEQ owl:disjointWithIDC,
ICEXT(x) = {y | ∀z1,…,zn ∈ IP ⊆ IC × IC owl:distinctMembersLV : ⟨y,z1⟩ ∈ IP ⊆ ICEXT(I(owl:AllDifferent)) × ISEQ owl:equivalentClassIEXT(p1) ∧…∧ ⟨y,zn⟩ ∈ IP ⊆ IC × IC owl:equivalentPropertyIEXT(pn) → ⟨z1,…,zn⟩ ∈ IP ⊆ IP × IP owl:hasKeyICEXT(c)}
⟨x,c⟩ ∈ IP ⊆ IC × ISEQ owl:hasSelfIEXT(I(owl:someValuesFrom)),
⟨x,p⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IR owl:hasValueIEXT(I(owl:onProperty))		 ICEXT(x) = {y | ∃z : ⟨y,z⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IR owl:importsIEXT(p) ∧ z ∈ IOXP ⊆ IX × IX owl:incompatibleWithICEXT(c)}
l sequence of p1,…, pn ∈ IOXP ⊆ IX × IX owl:intersectionOfIR,
⟨x,c⟩ ∈ IP ⊆ IC × ISEQ owl:inverseOfIEXT(I(owl:someValuesFrom)),
⟨x,l⟩ ∈ IP ⊆ IP × IP rdfs:isDefinedByIEXT(I(owl:onProperties))		 p1,…, pn ∈ IOAP ⊆ IR × IR rdfs:labelIODP,
c ∈ IOAP ⊆ IR × LV owl:maxCardinalityIDC,
ICEXT(x) = {y | ∃z1,…,zn ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI owl:maxQualifiedCardinalityLV : ⟨y,z1⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI owl:membersIEXT(p1) ∧…∧ ⟨y,zn⟩ ∈ IP ⊆ IR × ISEQ owl:minCardinalityIEXT(pn) ∧ ⟨z1,…,zn⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI owl:minQualifiedCardinalityICEXT(c)}
⟨x,u⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI owl:objectIEXT(I(owl:hasValue)),
⟨x,p⟩ ∈ IP ⊆ IR × IR owl:onClassIEXT(I(owl:onProperty))		 ICEXT(x) = {y | ⟨y,u⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IC owl:onDataRangeIEXT(p)}
⟨x,n⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IDC owl:onDatatypeIEXT(I(owl:cardinality)),
⟨x,p⟩ ∈ IP ⊆ IDC × IDC owl:oneOfIEXT(I(owl:onProperty))		 ICEXT(x) = {y | #{z|⟨y,z⟩ ∈ IP ⊆ IC × ISEQ owl:onPropertyIEXT(p)} = n}
⟨x,n⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IP owl:onPropertiesIEXT(I(owl:minCardinality)),
⟨x,p⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × ISEQ owl:predicateIEXT(I(owl:onProperty))		 ICEXT(x) = {y | #{z|⟨y,z⟩ ∈ IP ⊆ IR × IP owl:priorVersionIEXT(p)} ≥ n}
⟨x,n⟩ ∈ IOXP ⊆ IX × IX owl:propertyChainIEXT(I(owl:maxCardinality)),
⟨x,p⟩ ∈ IP ⊆ IP × ISEQ owl:propertyDisjointWithIEXT(I(owl:onProperty))		 ICEXT(x) = {y | #{z|⟨y,z⟩ ∈ IP ⊆ IP × IP owl:qualifiedCardinalityIEXT(p)} ≤ n}
⟨x,n⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × INNI owl:sameAsIEXT(I(owl:qualifiedCardinality)),
⟨x,c⟩ ∈ IP ⊆ IR × IR rdfs:seeAlsoIEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IOAP ⊆ IR × IR owl:someValuesFromIEXT(I(owl:onProperty))		 ICEXT(x) = {y | #{z|⟨y,z⟩ ∈ IP ⊆ ICEXT(I(owl:Restriction)) × IC owl:sourceIndividualIEXT(p) ∧ z ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR owl:subjectICEXT(c)} = n}
⟨x,n⟩ ∈ IP ⊆ IR × IR owl:targetIndividualIEXT(I(owl:qualifiedCardinality)),
⟨x,c⟩ ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR owl:targetValueIEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IP ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × LV owl:topDataPropertyIEXT(I(owl:onProperty))		 p ∈ IODPIODP,
ICEXT(x) = IR × LV owl:topObjectProperty{y | #{z ∈ IPLV|⟨y,z⟩ ∈ IEXT(p) ∧ z ∈ ICEXT(c)} = IR × IR owl:unionOfn}
⟨x,n⟩ ∈ IP ⊆ IC × ISEQ owl:versionInfoIEXT(I(owl:minQualifiedCardinality)),
⟨x,c⟩ ∈ IOAP ⊆ IR × IR owl:withRestrictionsIEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IP ⊆ IDC × ISEQ Table 5.5 lists the semantic conditions for boolean class expressions, which exist for building complements, intersections, and unions of classes. An intersection or union of a set of datatypes is itself a datatype. While a "normal" complement of a class is created w.r.t. to the whole domain, a datatype complement is created for a datatype w.r.t. the set of data values only, and results itself in a datatype. Table 5.5: Boolean Class Expressions ⟨c,d⟩ ∈ IEXT(I(owl:complementOf)) iff c, d ∈ IC ICEXT(c)IEXT(I(owl:onProperty))		 ICEXT(x) = IR \ ICEXT(d) ⟨c,d⟩{y | #{z|⟨y,z⟩ ∈ IEXT(I(owl:datatypeComplementOf)) c, dIEXT(p) ∧ z ∈ IDC, ICEXT(c) = LV \ ICEXT(d) if l sequence of d 1 ,…, d nICEXT(c)} ≥ n}
⟨x,n⟩ ∈ IR then ⟨c,l⟩IEXT(I(owl:minQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:intersectionOf)) iff c, d 1 ,…, d nIEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IC, ICEXT(c)IEXT(I(owl:onProperty))		 p ∈ IODP,
ICEXT(x) = ICEXT(d 1 ) ∩…∩ ICEXT(d n ) ⟨c,l⟩{y | #{z ∈ IEXT(I(owl:unionOf)) c, d 1 ,…, d nLV|⟨y,z⟩ ∈ IC, ICEXT(c) = ICEXT(d 1 ) ∪…∪ ICEXT(d n ) if then l sequence of d 1 ,…, d nIEXT(p) ∧ z ∈ IDC, nICEXT(c)} ≥ 1, ⟨c,l⟩n}
⟨x,n⟩ ∈ IEXT(I(owl:intersectionOf)) cIEXT(I(owl:maxQualifiedCardinality)),
⟨x,c⟩ ∈ IDC l sequence of d 1 ,…, d nIEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IDC, n ≥ 1, ⟨c,l⟩IEXT(I(owl:onProperty))		 ICEXT(x) = {y | #{z |⟨y,z⟩ ∈ IEXT(I(owl:unionOf)) cIEXT(p) ∧ z ∈ IDC Table 5.6 lists the semantic conditions for enumerations, i.e. classes, which have an explicitly defined, finite set of instances. An enumeration entirely consisting of datatype values is a datatype. Table 5.6: Enumeration Classes if l sequence of u 1 ,…, u nICEXT(c)} ≤ n}
⟨x,n⟩ ∈ IR then ⟨c,l⟩IEXT(I(owl:maxQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:oneOf)) iff cIEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IC, ICEXT(c)IEXT(I(owl:onProperty))		 p ∈ IODP,
ICEXT(x) = { u 1 ,…, u n } if then l sequence of u 1 ,…, u n{y | #{z ∈ LV, n ≥ 1, ⟨c,l⟩LV|⟨y,z⟩ ∈ IEXT(I(owl:oneOf)) cIEXT(p) ∧ z ∈ IDCICEXT(c)} ≤ n}
Editor's Note: There is an open issue with OWL 2 Full's support of n-ary datatypes. I suppose that we need to introduce a new sort of extension, a "datarange extension of arity n", which associates an individual with a set of n-tuples of individuals. Affected parts of the document would be: Interpretations, the Semantic Conditions and the Comprehension Principles for the Datatype Complement, and the Restrictions on N-ary Datatypes ("owl:onProperties"). Should this be the case, then the latter semantic conditions have to be considered broken at the moment, since n-tuples of individuals then cannot be instances of class extensions.

However, it has been proposed that there is no need to introduce such an additional sort of extension.

Table 5.7 lists the semantic conditions for property restrictions, including self restrictions, valuedatatype restrictions, which are specified for a datatype, and (qualified) cardinality restrictions.for a set of facets with their facet values applied to that datatype. Note that if no facet is applied to a given datatype, then the resulting datatype will be equivalent to the original datatype. Note further that the semantic conditionconditions are specified in a way that applying a facet to a datatype, for self restrictions doeswhich it is not entail the right hand sidedefined, will lead to an unsatisfiable ontology. Likewise, adding an inapplicable facet value to a certain combination of a owl:hasSelf assertion to bedatatype a boolean value, in orderfacet will lead to allow such assertions having right hand sidesan unsatisfiable ontology. As a consequence, a datatype restriction with one or more specified facets will lead to an unsatisfiable ontology if applied to a datatype for which no facets are defined (usually a set of arbitrary type.facets only exists for datatypes contained in the datatype map).

Table 5.7: PropertySemantic Conditions for Datatype Restrictions
if then
⟨x,u⟩ ∈ IEXT(I(owl:hasSelf)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | ⟨y,y⟩ ∈ IEXT(p)} ⟨x,c⟩ ∈ IEXT(I(owl:allValuesFrom)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | ∀z : ⟨y,z⟩ ∈ IEXT(p) → z ∈ ICEXT(c)}l sequence of py1,…, pyn ∈ IR,
⟨x,c⟩ ∈ IEXT(I(owl:allValuesFrom)), ⟨x,l⟩ ∈ IEXT(I(owl:onProperties)) pf1,…, pfn ∈ IODP, cIP,
⟨c,d⟩ ∈ IDC, ICEXT(x) = {y | ∀zIEXT(I(owl:onDatatype)),
⟨c,l⟩ ∈ IEXT(I(owl:withRestrictions)),
⟨y1 ,…,z n  : ⟨y,z,u1⟩ ∈ IEXT(pIEXT(f1 ) ∧…∧ ⟨y,z n ⟩ ∈ IEXT(p),
…,
⟨yn ) → ⟨z 1 ,…,z,un⟩ ∈ ICEXT(c)} ⟨x,c⟩ ∈ IEXT(I(owl:someValuesFrom)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | ∃z : ⟨y,z⟩ ∈ IEXT(p) ∧ z ∈ ICEXT(c)} l sequence of p 1 ,…, pIEXT(fn)		 c, d ∈ IR, ⟨x,c⟩ ∈ IEXT(I(owl:someValuesFrom)), ⟨x,l⟩ ∈ IEXT(I(owl:onProperties)) p 1 ,…, p nIDC,
fi ∈ IODP, cIFP(d) for 1≤i≤n,
ui ∈ IDC, ICEXT(x)IFV(d,fi) for 1≤i≤n,
ICEXT(c) = {y | ∃z 1 ,…,z n  : ⟨y,zICEXT(d) ∩ IFEXT(d,f1 ⟩ ∈ IEXT(p,u1) ∧…∧ ⟨y,z∩…∩ IFEXT(d,fn ⟩ ∈ IEXT(p,un)

∧ ⟨z 1 ,…,z n ⟩ ∈ ICEXT(c)} ⟨x,u⟩ ∈ IEXT(I(owl:hasValue)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | ⟨y,u⟩ ∈ IEXT(p)} ⟨x,n⟩ ∈ IEXT(I(owl:cardinality)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z|⟨y,z⟩ ∈ IEXT(p)} = n} ⟨x,n⟩ ∈ IEXT(I(owl:minCardinality)), ⟨x,p⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z|⟨y,z⟩Table 5.8 extends the semantic conditions for the RDFS vocabulary. The original semantics for the language features regarded here are specified in [RDF Semantics], and they only provide for "if-then" semantic conditions, while OWL 2 Full specifies stronger "iff" semantic conditions. Note that only the additional semantic conditions are given here and that the other conditions on the RDF and RDFS vocabularies are retained.

Table 5.8: Extended Semantic Conditions for the RDFS Vocabulary
⟨c,d⟩ ∈ IEXT(p)} ≥ n} ⟨x,n⟩IEXT(I(rdfs:subClassOf)) iff c, d ∈ IEXT(I(owl:maxCardinality)), ⟨x,p⟩IC,
ICEXT(c) ⊆ ICEXT(d)
⟨p,q⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z|⟨y,z⟩IEXT(I(rdfs:subPropertyOf)) p, q ∈ IEXT(p)} ≤ n} ⟨x,n⟩IP,
IEXT(p) ⊆ IEXT(q)
⟨p,c⟩ ∈ IEXT(I(owl:qualifiedCardinality)), ⟨x,c⟩IEXT(I(rdfs:domain)) p ∈ IEXT(I(owl:onClass)), ⟨x,p⟩IP, c ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z|⟨y,z⟩IC,
∀x,y : ⟨x,y⟩ ∈ IEXT(p) ∧ z ∈ ICEXT(c)} = n} ⟨x,n⟩ ∈ IEXT(I(owl:qualifiedCardinality)), ⟨x,c⟩→ x ∈ IEXT(I(owl:onDataRange)), ⟨x,p⟩ICEXT(c)
⟨p,c⟩ ∈ IEXT(I(owl:onProperty))IEXT(I(rdfs:range)) p ∈ IODP, ICEXT(x) = {y | #{zIP, c ∈ LV|⟨y,z⟩IC,
∀x,y : ⟨x,y⟩ ∈ IEXT(p) ∧ z→ y ∈ ICEXT(c)} = n} ⟨x,n⟩ICEXT(c)

Table 5.9 lists the semantic conditions for sub property chains. The semantics have been specified in a way to allow a sub property chain axiom to be satisfiable without requiring the existence of a property that represents the property chain. In particular, the property on the left hand side of the sub property assertion does not necessarily represent the property chain.

Table 5.9: Semantic Conditions for Sub Property Chains
if then
l sequence of p1,…, pn ∈ IEXT(I(owl:minQualifiedCardinality)), ⟨x,c⟩IR,
⟨x,q⟩ ∈ IEXT(I(owl:onClass)), ⟨x,p⟩IEXT(I(rdfs:subPropertyOf)),
⟨x,l⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z|⟨y,z⟩IEXT(I(owl:propertyChain))		 p1,…, pn ∈ IEXT(p) ∧ zIP,
q ∈ ICEXT(c)} ≥ n} ⟨x,n⟩IP,
∀y0,…,yn : ⟨y0,y1⟩ ∈ IEXT(I(owl:minQualifiedCardinality)), ⟨x,c⟩IEXT(p1) ∧…∧ ⟨yn-1,yn⟩ ∈ IEXT(I(owl:onDataRange)), ⟨x,p⟩IEXT(pn) → ⟨y0,yn⟩ ∈ IEXT(I(owl:onProperty))IEXT(q)
if then exists x ∈ IR
l sequence of p1,…, pn ∈ IODP, ICEXT(x) = {y | #{zIP,
q ∈ LV|⟨y,z⟩IP,
∀y0,…,yn : ⟨y0,y1⟩ ∈ IEXT(p) ∧ zIEXT(p1) ∧…∧ ⟨yn-1,yn⟩ ∈ ICEXT(c)} ≥ n} ⟨x,n⟩IEXT(pn) → ⟨y0,yn⟩ ∈ IEXT(I(owl:maxQualifiedCardinality)), ⟨x,c⟩IEXT(q)		 ⟨x,q⟩ ∈ IEXT(I(owl:onClass)), ⟨x,p⟩IEXT(I(rdfs:subPropertyOf)),
⟨x,l⟩ ∈ IEXT(I(owl:onProperty)) ICEXT(x) = {y | #{z |⟨y,z⟩IEXT(I(owl:propertyChain))

Table 5.10 lists the semantic conditions for equal and different individuals, equivalent and disjoint classes, and equivalent and disjoint properties. Also treated here are disjoint union axioms.

Table 5.10: Semantic Conditions for Equivalence and Disjointness Axioms
⟨u,w⟩ ∈ IEXT(p) ∧ zIEXT(I(owl:sameAs)) iff u = w
⟨u,w⟩ ∈ ICEXT(c)} ≤ n} ⟨x,n⟩IEXT(I(owl:differentFrom)) u ≠ w
⟨c,d⟩ ∈ IEXT(I(owl:maxQualifiedCardinality)), ⟨x,c⟩IEXT(I(owl:equivalentClass)) c, d ∈ IEXT(I(owl:onDataRange)), ⟨x,p⟩IC,
ICEXT(c) = ICEXT(d)
⟨c,d⟩ ∈ IEXT(I(owl:onProperty)) pIEXT(I(owl:disjointWith)) c, d ∈ IODP, ICEXT(x)IC,
ICEXT(c) ∩ ICEXT(d) = {y | #{z∅
⟨p,q⟩ ∈ LV|⟨y,z⟩IEXT(I(owl:equivalentProperty)) p, q ∈ IP,
IEXT(p) ∧ z= IEXT(q)
⟨p,q⟩ ∈ ICEXT(c)} ≤ n} Table 5.8 lists the semantic conditions for datatype restrictions, which are specified by a setIEXT(I(owl:propertyDisjointWith)) p, q ∈ IP,
IEXT(p) ∩ IEXT(q) = ∅
if l sequence of facets with their facet values applied to a datatype. Note that if no facet is applied to a given datatype,d1,…, dn ∈ IR then
the resulting datatype will be equivalent to the original datatype. Note further that⟨c,l⟩ ∈ IEXT(I(owl:disjointUnionOf)) iff c, d1,…, dn ∈ IC,
ICEXT(c) = ICEXT(d1) ∪…∪ ICEXT(dn),
ICEXT(di) ∩ ICEXT(dk) = ∅ for 1 ≤ i ≠ k ≤ n

Table 5.11 lists the semantic condition is specified in a way that applying a facet to a datatype, for which it is not defined, will lead to an unsatisfiable ontology. Likewise, adding an inapplicable facet value to a certain combination of a datatype a facet will lead to an unsatisfiable ontology. As a consequence, a datatype restriction with one or more specified facets will lead to an unsatisfiable ontology when applied to a datatypeconditions for which no facetsn-ary axioms on different individuals, disjoint classes, and disjoint properties. Note that there are defined (usually a settwo alternative ways to specify owl:AllDifferent axioms, both of facets only exists for datatypes contained inthem having the datatype map).same model-theoretic meaning.

Table 5.8: Datatype Restrictions5.11: Semantic Conditions for N-ary Axioms
if then
l sequence of yu1,…, yun ∈ IR,
fx ∈ ICEXT(I(owl:AllDifferent)),
⟨x,l⟩ ∈ IEXT(I(owl:distinctMembers))		 ui ≠ uk for 1 ≤ i ≠ k ≤ n
l sequence of u1,…, fun ∈ IP, ⟨c,d⟩IR,
x ∈ IEXT(I(owl:onDatatype)), ⟨c,l⟩ICEXT(I(owl:AllDifferent)),
⟨x,l⟩ ∈ IEXT(I(owl:withRestrictions)), ⟨yIEXT(I(owl:members))		 ui ≠ uk for 1 ,u≤ i ≠ k ≤ n
l sequence of c1 ⟩,…, cn ∈ IEXT(f 1 ), …, ⟨y n ,u n ⟩ ∈ IEXT(f n ) c, dIR,
x ∈ IDC, f iICEXT(I(owl:AllDisjointClasses)),
⟨x,l⟩ ∈ IFP(d) for 1≤i≤n, u iIEXT(I(owl:members))		 c1,…, cn ∈ IFV(d,fIC,
ICEXT(ci) for 1≤i≤n, ICEXT(c) = ICEXT(d)∩ IFEXT(d,f 1 ,u 1 ) ∩…∩ IFEXT(d,f n ,u nICEXT(ck) Table 5.9 extends the semantic conditions for the RDFS vocabulary. The original semantics for RDFS, as specified in the RDF Semantics , only provide "if-then" semantic conditions, while OWL 2 Full specifies stronger "iff" semantic conditions. Note that only the additional semantic conditions are given here and that the other conditions on the RDF and RDFS vocabulary are retained. Table 5.9: Extended Conditions for the RDFS Vocabulary ⟨c,d⟩ ∈ IEXT(I(rdfs:subClassOf)) iff c, d ∈ IC, ICEXT(c) ⊆ ICEXT(d) ⟨p,q⟩ ∈ IEXT(I(rdfs:subPropertyOf)) p, q ∈ IP, IEXT(p) ⊆ IEXT(q) ⟨p,c⟩ ∈ IEXT(I(rdfs:domain)) p ∈ IP, c ∈ IC, ∀x,y : ⟨x,y⟩ ∈ IEXT(p) → x ∈ ICEXT(c) ⟨p,c⟩ ∈ IEXT(I(rdfs:range)) p ∈ IP, c ∈ IC, ∀x,y : ⟨x,y⟩ ∈ IEXT(p) → y ∈ ICEXT(c) Table 5.10 lists the semantic conditions= ∅ for sub property chains. Table 5.10: Sub Property Chains if then1 ≤ i ≠ k ≤ n
l sequence of p1,…, pn ∈ IR,
⟨x,q⟩x ∈ IEXT(I(rdfs:subPropertyOf)),ICEXT(I(owl:AllDisjointProperties)),
⟨x,l⟩ ∈ IEXT(I(owl:propertyChain))IEXT(I(owl:members))		 p1,…, pn ∈ IP,
∀y 0 ,…,y n  : ⟨y 0 ,y 1 ⟩ ∈IEXT(p 1i) ∧…∧ ⟨y n-1 ,y n ⟩ ∈∩ IEXT(p nk) → ⟨y 0 ,y= ∅ for 1 ≤ i ≠ k ≤ n
⟩ ∈ IEXT(q)if then exists x ∈ IR
l sequence of pu1,…, pun ∈ IP, ∀y 0 ,…,y n  : ⟨y 0 ,y 1 ⟩ ∈ IEXT(p 1 ) ∧…∧ ⟨y n-1 ,y n ⟩ ∈ IEXT(p n ) → ⟨y 0 ,y n ⟩ ∈ IEXT(q) ⟨x,q⟩ ∈ IEXT(I(rdfs:subPropertyOf)), ⟨x,l⟩ ∈ IEXT(I(owl:propertyChain)) Table 5.11 lists the semantic conditions for equal and different individuals, equivalent and disjoint classes, and equivalent and disjoint properties. Also treated here are axioms to specify disjoint unions. Table 5.11: Equivalence and Disjointness ⟨u,w⟩ ∈ IEXT(I(owl:sameAs)) iff u = w ⟨u,w⟩ ∈ IEXT(I(owl:differentFrom))IR,
u ≠ w ⟨c,d⟩ ∈ IEXT(I(owl:equivalentClass)) c, d ∈ IC, ICEXT(c) = ICEXT(d) ⟨c,d⟩ ∈ IEXT(I(owl:disjointWith)) c, d ∈ IC, ICEXT(c) ∩ ICEXT(d) = ∅ ⟨p,q⟩ ∈ IEXT(I(owl:equivalentProperty)) p, q ∈ IP, IEXT(p) = IEXT(q) ⟨p,q⟩ ∈ IEXT(I(owl:propertyDisjointWith)) p, q ∈ IP, IEXT(p) ∩ IEXT(q) = ∅ if l sequence of d 1 ,…, d n ∈ IR then ⟨c,l⟩ ∈ IEXT(I(owl:disjointUnionOf)) iff c, d 1 ,…, d n ∈ IC, ICEXT(c) = ICEXT(d 1 ) ∪…∪ ICEXT(d n ), ICEXT(di ) ∩ ICEXT(d≠ uk ) = ∅for 1 ≤ i ≠ k ≤ n		 Table 5.12 lists the semantic conditions for the n-ary versions of the axioms for different individuals, disjoint classes, and disjoint properties. Table 5.12: N-ary Axioms if then l sequence of u 1 ,…, u n ∈ IR,x ∈ ICEXT(I(owl:AllDifferent)),
⟨x,l⟩ ∈ IEXT(I(owl:distinctMembers))
u i ≠ u k for 1 ≤ i ≠ k ≤ nl sequence of u1,…, un ∈ IR,
x ∈ ICEXT(I(owl:AllDifferent)), ⟨x,l⟩ ∈ IEXT(I(owl:members))ui ≠ uk for 1 ≤ i ≠ k ≤ n		 l sequence of c 1 ,…, c n ∈ IR,x ∈ ICEXT(I(owl:AllDisjointClasses)),ICEXT(I(owl:AllDifferent)),
⟨x,l⟩ ∈ IEXT(I(owl:members))
l sequence of c1,…, cn ∈ IC,
ICEXT(ci) ∩ ICEXT(ck) = ∅ for 1 ≤ i ≠ k ≤ n		 l sequence of p 1 ,…, p n ∈ IR,x ∈ ICEXT(I(owl:AllDisjointProperties)),ICEXT(I(owl:AllDisjointClasses)),
⟨x,l⟩ ∈ IEXT(I(owl:members))
p 1 ,…, p n ∈ IP, IEXT(p i ) ∩ IEXT(p k ) = ∅ for 1 ≤ i ≠ k ≤ n if then exists x ∈ IR l sequence of u 1 ,…, u n ∈ IR, u i ≠ u k for 1 ≤ i ≠ k ≤ n x ∈ ICEXT(I(owl:AllDifferent)), ⟨x,l⟩ ∈ IEXT(I(owl:distinctMembers)) l sequence of u 1 ,…, u n ∈ IR, u i ≠ u k for 1 ≤ i ≠ k ≤ n x ∈ ICEXT(I(owl:AllDifferent)), ⟨x,l⟩ ∈ IEXT(I(owl:members)) l sequence of c 1 ,…, c n ∈ IC, ICEXT(c i ) ∩ ICEXT(c k ) = ∅ for 1 ≤ i ≠ k ≤ n x ∈ ICEXT(I(owl:AllDisjointClasses)), ⟨x,l⟩ ∈ IEXT(I(owl:members)) l sequence ofl sequence of p1,…, pn ∈ IP,
IEXT(pi) ∩ IEXT(pk) = ∅ for 1 ≤ i ≠ k ≤ n		 x ∈ ICEXT(I(owl:AllDisjointProperties)),
⟨x,l⟩ ∈ IEXT(I(owl:members))

Table 5.135.12 lists the semantic conditionconditions for inverse property axioms.

Table 5.13:5.12: Semantic Conditions for Inverse PropertiesProperty Axioms
⟨p,q⟩ ∈ IEXT(I(owl:inverseOf)) iff p, q ∈ IP,
IEXT(p) = { ⟨x,y⟩{⟨x,y⟩ | ⟨y,x⟩ ∈ IEXT(q) }IEXT(q)}

Table 5.145.13 lists the semantic conditions for property characteristics, i.e. functionality and inverse functionality, reflexivity and irreflexivity, symmetry and asymmetry, and transitivity of properties.

Table 5.14:5.13: Semantic Conditions for Property Characteristics
p ∈ ICEXT(I(owl:FunctionalProperty)) iff p ∈ IP,
∀x,y,z : ⟨x,y⟩, ⟨x,z⟩ ∈ IEXT(p) → y = z
p ∈ ICEXT(I(owl:InverseFunctionalProperty)) p ∈ IP,
∀x,y,z : ⟨y,x⟩, ⟨z,x⟩ ∈ IEXT(p) → y = z
p ∈ ICEXT(I(owl:ReflexiveProperty)) p ∈ IP,
∀x : ⟨x,x⟩ ∈ IEXT(p)
p ∈ ICEXT(I(owl:IrreflexiveProperty)) p ∈ IP,
∀x : ⟨x,x⟩ ∉ IEXT(p)
p ∈ ICEXT(I(owl:SymmetricProperty)) p ∈ IP,
∀x,y : ⟨x,y⟩ ∈ IEXT(p) → ⟨y,x⟩ ∈ IEXT(p)
p ∈ ICEXT(I(owl:AsymmetricProperty)) p ∈ IP,
∀x,y : ⟨x,y⟩ ∈ IEXT(p) → ⟨y,x⟩ ∉ IEXT(p)
p ∈ ICEXT(I(owl:TransitiveProperty)) p ∈ IP,
∀x,y,z : ⟨x,y⟩, ⟨y,z⟩ ∈ IEXT(p) → ⟨x,z⟩ ∈ IEXT(p)

Table 5.155.14 lists the semantic conditionconditions for Keys. Keys are an alternative to inverse functional properties (see Table 5.15:5.13). They provide for compound keys, and they allow to specify the class of individuals for which a property plays the role of a key feature.

Table 5.14: Semantic Conditions for Keys
if l sequence of p1,…, pn ∈ IR then
⟨c,l⟩ ∈ IEXT(I(owl:hasKey)) iff c ∈ IC,
p1,…, pn ∈ IP,
∀x,y,z1,…,zn :
 x, y ∈ ICEXT(c),
 ⟨x,zi⟩, ⟨y,zi⟩ ∈ IEXT(pi), 1 ≤ i ≤ n
 → x = y

Table 5.165.15 lists the semantic conditions for negative property assertions. Table 5.16: Negative Property Assertions if then ⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)), ⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)), ⟨x,w⟩ ∈ IEXT(I(owl:targetIndividual)) x ∈ ICEXT(I(owl:NegativePropertyAssertion)), ⟨u,w⟩ ∉ IEXT(p) ⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)), ⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)), ⟨x,w⟩ ∈ IEXT(I(owl:targetValue)) x ∈ ICEXT(I(owl:NegativePropertyAssertion)), p ∈ IODP, ⟨u,w⟩ ∉ IEXT(p) if then exists x ∈ IRThey allow to state that an individual u ∈ IR,does not stand in a relationship p ∈ IP,with another individual w ∈ IR, ⟨u,w⟩ ∉ IEXT(p) ⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)), ⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)), ⟨x,w⟩ ∈ IEXT(I(owl:targetIndividual)) u. The second form based on owl:targetValue is more specific than the first form based on owl:targetIndividual in that it is restricted to the case of negative data property assertions. Note that the second form will coerce the target individual of a negative property assertion into a data value, due to the range defined for the property owl:targetValue in Table 5.3.

Table 5.15: Semantic Conditions for Negative Property Assertions
if then
⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)),
⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)),
⟨x,w⟩ ∈ IEXT(I(owl:targetIndividual))		 ⟨u,w⟩ ∉ IEXT(p)
⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)),
⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)),
⟨x,w⟩ ∈ IEXT(I(owl:targetValue))		 p ∈ IODP,
⟨u,w⟩ ∉ IEXT(p)
if then exists x ∈ IR
u ∈ IR,
p ∈ IP,
w ∈ IR,
⟨u,w⟩ ∉ IEXT(p)		 ⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)),
⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)),
⟨x,w⟩ ∈ IEXT(I(owl:targetIndividual))
u ∈ IR,
p ∈ IODP,
w ∈ LV,
⟨u,w⟩ ∉ IEXT(p)		 ⟨x,u⟩ ∈ IEXT(I(owl:sourceIndividual)),
⟨x,p⟩ ∈ IEXT(I(owl:assertionProperty)),
⟨x,w⟩ ∈ IEXT(I(owl:targetValue))

6 Relationship to OWL 2 DL

This section is concerned with a strong relationship that holds between the RDF-Based Semantics ofOWL 2 Full and the Direct Semantics of OWL 2 DL.[OWL 2 Direct Semantics].

6.1 A Difference to the Direct Semantics (Informative)

One design goal of OWL 2 has been that OWL 2 Full should reflect every logical consequence of the Direct Semantics of OWL 2 DL,[OWL 2 Direct Semantics], as long as this consequence and all its premises can be represented as valid OWL 2 DL ontologies in RDF graph form. However, a fundamental semantic difference exists between OWL 2 DLthe Direct Semantics and OWL 2 Full, which complicates a comparison of thetheir semantic expressiveness ofexpressiveness. The two languages. OWL 2 DLDirect Semantics treats classes as sets, i.e. subsets of the universe, whileuniverse. Classes in OWL 2 FullFull, however, are individuals ,in the universe, which have such a set associated to them as their class extension. Hence, under OWL 2 Full semantics,Full, all classes are instances of the universe, but this cannot generally be assumed under OWL 2the Direct Semantics. An analog difference existsdistinction holds for properties.

An effect of this difference is that certain logical conclusions of OWL 2 DL do not become "visible" under theOWL 2 Full semantics,Full, although they are reflected by OWL 2 Full at a set theoretical level. For example, under OWLconsider the following two RDF graphs G1 and G2 Direct Semantics,(RDF graphs are presented here in the style used in [OWL 2 RDF graphMapping]):

G1 := {

ex:C rdf:type owl:Class .
ex:D rdf:type owl:Class .
ex:C rdfs:subClassOf ex:D .

}

entails the RDF graphG2 := {

ex:C rdf:type owl:Class .
ex:D rdf:type owl:Class .
_:x owl:intersectionOf (SEQ ex:C ex:D) .
_:x rdfs:subClassOf ex:D .

}.}

Both graphs are OWL 2 Full, onDL ontologies in RDF graph form, and G1 entails G2 under the other hand,Direct Semantics. However, under OWL 2 Full this entailment does not hold. Actually, OWL 2 Full interprets G1 in a way such that the set theoretical relationship

ICEXT(I(ex:C)) ∩ ICEXT(I(ex:D)) ⊆ ICEXT(I(ex:D))

can be concluded. But since OWL 2 Full distinguishes between classes as individuals and their class extensions, Gextensions being the actual sets, G2 is not entailed, unless there exists some additional "helper" individual, which hasindividual w, having the set S, defined by

S := ICEXT(w) = ICEXT(I(ex:C)) ∩ ICEXT(I(ex:D))

as its class extension. Whether such a helper individual exists or not has no effect on the answer to the question, whether the basic logical conclusion that existsat the set theoretical level holds or not. The individual is, however needed,however, required to represent this conclusion inas the particular form given byRDF graph G2.

The following subsection introduces a set of "comprehension principles", which have the purpose to provide the missing "helper" individuals.

6.2 Comprehension Principles

This section lists the set of comprehension principles of OWL 2 Full. These comprehension principles are not part of the set of semantic conditions given in Section 5, and therefore do not need to be met by a OWL 2 Full interpretation as defined in Section 4. They are, however, needed for the correspondence theorem, stated in the secondnext subsection, to hold, since the correspondence theorem compares OWL 2 Full and OWL 2 DLand the Direct Semantics based on entailments.

6.1 Comprehension PrinciplesTable 6.1 lists the comprehension principleprinciples for lists, which provides the existence of RDF lists for each finite combination of individuals.

Table 6.1: Comprehension PrinciplePrinciples for Lists
if then exists x1,…, xn ∈ IR
u1,…, un ∈ IR ⟨x1,u1⟩ ∈ IEXT(I(rdf:first)), ⟨x1,x2⟩ ∈ IEXT(I(rdf:rest)),
…,
⟨xn,un⟩ ∈ IEXT(I(rdf:first)), ⟨xn,I(rdf:nil)⟩ ∈ IEXT(I(rdf:rest))

Table 6.2 lists the comprehension principles for boolean class expressions, which provide the existence of classes representing the complement of each class, and the datatype complement of each datatype, and the union and intersection of each finite set of classes.

Table 6.2: Comprehension Principles for Boolean Class Expressions
if then exists x ∈ IR
c ∈ IC ⟨x,c⟩ ∈ IEXT(I(owl:complementOf))
c ∈ IDC ⟨x,c⟩ ∈ IEXT(I(owl:datatypeComplementOf))
l sequence of c1,…, cn ∈ IC ⟨x,l⟩ ∈ IEXT(I(owl:unionOf))
l sequence of c1,…, cn ∈ IC ⟨x,l⟩ ∈ IEXT(I(owl:intersectionOf))

Table 6.3 lists the comprehension principles for enumeration classes, which provide the existence of classes representing each finite set of individuals.

Table 6.3: Comprehension Principles for Enumeration ClassesEnumerations
if then exists x ∈ IR
l sequence of u1,…, un ∈ IR ⟨x,l⟩ ∈ IEXT(I(owl:oneOf))

Table 6.4 lists the comprehension principles for property restrictions, which provide the existence of self restrictions,value restrictions, cardinality restrictions, and qualified cardinalityself restrictions for each property, class and individual for which such a restriction is meaningful. There are also comprehension principles for value restrictions dealing with n-ary datatypes.

Table 6.4: Comprehension Principles for Property Restrictions
if then exists x ∈ IR
p ∈ IP ⟨x,I("true"^^xsd:boolean)⟩ ∈ IEXT(I(owl:hasSelf)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
c ∈ IC,
p ∈ IP		 ⟨x,c⟩ ∈ IEXT(I(owl:allValuesFrom)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
c ∈ IDC,
l sequence of p1,…, pn ∈ IODP		 ⟨x,c⟩ ∈ IEXT(I(owl:allValuesFrom)),
⟨x,l⟩ ∈ IEXT(I(owl:onProperties))
c ∈ IC,
p ∈ IP		 ⟨x,c⟩ ∈ IEXT(I(owl:someValuesFrom)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
c ∈ IDC,
l sequence of p1,…, pn ∈ IODP		 ⟨x,c⟩ ∈ IEXT(I(owl:someValuesFrom)),
⟨x,l⟩ ∈ IEXT(I(owl:onProperties))
u ∈ IR,
p ∈ IP		 ⟨x,u⟩ ∈ IEXT(I(owl:hasValue)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:cardinality)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:minCardinality)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:maxCardinality)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IC,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:qualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IDC,
p ∈ IODP		 ⟨x,n⟩ ∈ IEXT(I(owl:qualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IC,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:minQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IDC,
p ∈ IODP		 ⟨x,n⟩ ∈ IEXT(I(owl:minQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IC,
p ∈ IP		 ⟨x,n⟩ ∈ IEXT(I(owl:maxQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onClass)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))
n ∈ INNI,
c ∈ IDC,
p ∈ IODP		 ⟨x,n⟩ ∈ IEXT(I(owl:maxQualifiedCardinality)),
⟨x,c⟩ ∈ IEXT(I(owl:onDataRange)),
⟨x,p⟩ ∈ IEXT(I(owl:onProperty))

Table 6.5 lists6.5 lists the comprehension principles for datatype restrictions, which provides the existence of restrictions for each datatype, and for each combination of facets and facet values for which such a restriction is meaningful.

Table 6.5: Comprehension Principles for Datatype Restrictions
if then exists x ∈ IR, l sequence of y1,…,yn ∈ IR
c ∈ IDC,
f1,…,fn facets,
u1,…,un ∈ LV		 ⟨x,c⟩ ∈ IEXT(I(owl:onDatatype)),
⟨x,l⟩ ∈ IEXT(I(owl:withRestrictions)),
⟨y1,u1⟩ ∈ IEXT(f1),
…,
⟨yn,un⟩ ∈ IEXT(fn)

6.3 Correspondence Theorem

Theorem 6.1 (Correspondence Theorem): Let D be a OWL 2 Full datatype map, and let K and Q be collections of valid OWL 2 DL ontologies in RDF graph form that are imports closed, and without annotations occurring in Q. Let F(K) and F(Q) be the collections of OWL 2 DL ontologies in Functional Syntax that result from applying the reverse RDF mapping [OWL 2 RDF Mapping] to K and Q, respectively. If F(K) entails F(Q) with respect to the OWL 2 Direct Semantics [OWL 2 Direct Semantics] and with respect to D, then K entails Q with respect to OWL 2 Full extended by the comprehension principles, and with respect to D.

Editor's Note: In the given form, the theorem is trivally true, since combining the set of all OWL 2 Full semantic conditions with all the comprehension principles leads to inconsistency. Further, in this form the theorem does not give very useful hints to implementers on how to best exploit the set of comprehension principles in order to build good entailment checkers that also cover all of OWL 2 DL. Currently, an alternative formulation of the theorem is under investigation.

A sketch of a proof for this theorem is given in Appendix B.

7 Appendix A: Axiomatic Triples (Informative)

Unlike the RDF Semantics, OWL 2 Full does not contain an explicit list of "axiomatic triples". One should note that it might not be possible to give a definition of OWL 2 Full that captures all "axiomatic aspects" of the language in the form of sets of RDF triples, just as it is not possible to define the whole semantics of OWL 2 Full in the form of a set of RDF triple rules. However, Section 5 contains a large set of semantic conditions that are in some sense "axiomatic", i.e. these semantic conditions are true in every OWL 2 Full ontology, including the empty ontology. This appendix shows how these semantic conditions relate to axiomatic triples.

The semantic conditions given in Table 5.2 can be regarded as a set of OWL 2 Full "axiomatic triples" for classes: For each URI U occurring in the first column of the table, if the second column contains an entry "I(U) ∈ S" for some set S, then this entry corresponds to some RDF triple of the form "U rdf:type C", where C is the URI of some class with ICEXT(I(C)) = S. In this table, S will always be either the set IC of all classes, or some subset of IC. Hence, in a corresponding RDF triple the URI C will typically be one of "rdfs:Class" or "owl:Class" (S=IC in both cases), or "rdfs:Datatype" (S=IDC). Further, for each URI U in the first column, if the third column contains an entry "ICEXT(I(U)) ⊆ S" ("ICEXT(I(U)) = S") for some set S, then this entry corresponds to some RDF triple of the form "U rdfs:subClassOf C" ("U owl:equivalentClass C"), where C is the URI of some class with ICEXT(I(C)) = S. Additionally, the conditions on the sets given in Table 5.1 have to be taken into account. In particular, if an entry of Table 5.1 states S1 ⊆ S2 for some sets S1 and S2, then this corresponds to some RDF triple C1 owl:subClassOf C2, where C1 and C2 are the URIs of some classes with ICEXT(I(C1)) = S1 and ICEXT(I(C2)) = S2, respectively, according to Table 5.2. Note that some of the comprehension principlesRDF triples received in this way already follow from the RDFS semantics [RDF Semantics]. These axiomatic triples are "simple" in the following sense: For every set S mentioned in the second and the third column of the table there exists a URI C of some class in the vocabularies for RDF, RDFS or those given in Section 2, for datatype restrictions,which providesS = ICEXT(I(C)).

The existencesemantic conditions given in Table 5.3 can be regarded as a set of restrictionsOWL 2 Full "axiomatic triples" for each datatype, andproperties: For each combinationURI U occurring in the first column of facets and facet valuesthe table, if the second column contains an entry "I(U) ∈ S" for which such a restrictionsome set S, then this entry corresponds to some RDF triple of the form "U rdf:type C", where C is meaningful. Table 6.5: Comprehension Principlesthe URI of some class with ICEXT(I(C)) = S. In this table, S will always be either the set IP of all properties, or some subset of IP. Hence, in a corresponding RDF triple the URI C will typically be one of "rdf:Property" or "owl:ObjectProperty" (S=IP in both cases), "owl:DatatypeProperty" (S=IODP), "owl:AnnotationProperty" (S=IOAP), or "owl:OntologyProperty" (S=IOXP). Further, for Datatype Restrictionseach URI U in the first column, if the third column contains an entry "IEXT(I(U)) ⊆ S1 × S2" for some sets S1 and S2, then exists x ∈ IR, l sequencethis entry corresponds to some RDF triples of y 1 ,…,y n ∈ IRthe forms "U rdfs:domain C ∈ IDC, f 1 ,…,f n facets, u1 ,…,u n ∈ LV ⟨x,c⟩ ∈ IEXT(I(owl:onDatatype)), ⟨x,l⟩ ∈ IEXT(I(owl:withRestrictions)), ⟨y" and "U rdfs:range C2", where C1 ,uand C2 are the URIs of some classes with ICEXT(I(C1 ⟩ ∈ IEXT(f)) = S1 ), …, ⟨y n ,u n ⟩ ∈ IEXT(f n ) 6.2 Correspondence Theorem Theorem 6.1 (Correspondence Theorem): Let D be a OWLand ICEXT(I(C2)) = S2 Full datatype map, respectively. Exceptions are the semantic conditions "IEXT(I(owl:topObjectProperty)) = IR × IR" and let K and Q be collections of valid OWL 2 DL ontologies in RDF graph form that"IEXT(I(owl:topDataProperty)) = IR × LV", for which there are imports closed,no corresponding domain and without annotations occurringrange triples. These axiomatic triples are "simple" in Q. Let F(K)the following sense: For every set S mentioned in the second column of the table, and F(Q) beas the collectionsleft or right hand side of OWL 2 DL ontologiesa Cartesian product in Functional Syntax that result from applyingthe Reverse RDF Mapping to K and Q, respectively. If F(K) entails F(Q) with respect tothird column of the OWL 2 Direct Semantics and with respect to D, then K entails Q with respect totable there exists a URI C of some class in the OWL 2 RDF-Based Semantics extended byvocabularies for RDF, RDFS or those given in Section 2, for which S = ICEXT(I(C)).

8 Appendix B: Proof of the comprehension principles, and with respect to D. Proof.Correspondence Theorem (Informative)

Editor's Note: TODO: The proof still needs to be constructed.

79 Appendix C: Changes 7.1(Informative)

9.1 Changes since First Public Working Draft

This section lists significant changes since the First Public Working Draft.

  • Added datatype "owl:rational", marking it "at risk" (WG resolution of Issue 87).
  • The RDF syntax of self restrictions has been changed: The class owl:SelfRestriction has been replaced by the property owl:hasSelf (per WG resolution).
  • Removed the semantic conditions for axiom annotations (WG resolution of Issue 144).
  • Added semantic conditions inferring a union or intersection of datatypes into a datatype (following WG resolution of Issue 147).
  • The URIs owl:TopObjectProperty, owl:BottomObjectProperty, owl:TopDataProperty and owl:BottomDataProperty have been renamed to their lower-case variants, respectively.respectively (per WG decision).
  • The datatypes xsd:ID, xsd:IDREF and xsd:ENTITY have been removed (per WG resolution).
  • Changed the semantic conditions for axiom annotations .the purposen-ary value restrictions to infer the type of these semantic conditionsthe properties p1,...,pn (IODP) and the type of the class c (IDC).
  • Corrected definitions of consistency and entailment: The vocabulary V was a global parameter of the definitions. Now the form of the definitions is close to "reconstruct"the respective definitions in OWL 1.
  • Corrected the semantic condition for sub property chains: missing "base triple"premise "q in IP" in the second condition.
  • Removed redundant statements in the consequent of an axiom when it is annotated. Thesethe semantic conditions became redundant byfor negative property assertions.
  • For D-Interpretations, the range of the mapping IL has been changed to IR instead of LV, with a reference to the RDF Semantics. This was a bug, since in both the RDF Semantics and in OWL 1 the range of IL has been IR.
  • Splitted the table on "Parts of the Universe" in the "Semantic Conditions" section into a table defining the parts (now in the "Interpretations" section), and a table that specifies the semantic conditions for those parts.
  • The resolutiondefinition of Issue 144 , which demandsOWL 2 Full datatype maps now include the different facet-related sets that have formerly been part of the RDF mapping preserves"abbreviations" table in the base triples of annotated axioms and annotated annotations."Semantic Conditions" section.
  • The nomenclature for datatype maps has been aligned with the one used in the RDF Semantics. In particular, the concept being called an "interpretation of a literal" is now being called a "datatype value", and the concept being called an "interpretation of a datatype" is now being called a "datatype" or the "value space" of a datatype, depending on whether the datatype itself or its class extension is meant.
  • Moved "Ontologies" section from Section 5 to Section 3. In "Ontologies section": Said something about ontology headers and ontology versions, but removed every text referring to the semantic meaning of a OWL 2 Full ontology. Changed the semantic conditions for the n-ary value restrictions to infer the type of the properties p 1 ,..., p n (IODP) and the type of the class c (IDC).Replaced all applications of the URI-mapping 'IS(.)' by the more general interpretation function 'I(.)'. This usage is now in line with the usage in the RDF Semantics document, and is backed by the table in Section 1.4 of the RDF Semantics. Further,document. Also, there werehave formerly been applications of IS, where it was not guaranteed that the argument is a URI in every case, so using IS there was even not justified. For D-Interpretations, the range of the mapping IL has been changed to IR instead of LV, withURI.
  • Marked several sections as "Informative", as requestion by a referenceprevious review.
  • Added to the RDF Semantics. This was a bug, since in both the RDF Semantics and in OWL 1 the range of IL has been IR. Corrected definitions of consistency"Ontologies" section some text about ontology headers and entailment: The vocabulary V was a global parameter of the definitions. Now the form of the definitions is closeontology versions, but removed every text referring to the respective definitions in OWL 1. Addedsemantic conditions inferring a union or intersectionmeaning of datatypes intoa datatype ( Issue 147 "propositional data ranges").OWL 2 Full ontology.
  • Moved the datatypes xsd:ID, xsd:IDREF and xsd:ENTITY have been removed (per WG resolution)."Ontologies" section from Section 5 to Section 3.
  • Moved the discussion on axiomatic triples from the RDF syntax of self restrictions has been changed:section on "Semantic Conditions" to a dedicated appendix.
  • The class owl:SelfRestriction"Introduction" section has been replaced byrevised.
  • The property owl:hasSelf (per WG resolution). Added datatype "owl:rational", but marking it "at risk" (per WG resolutiondescriptions of Issue 87 ). 7.2the semantic condition tables have been revised.

9.2 Differences to OWL Full

This section lists significant differences between OWL 2 Full and the original version of OWL Full, as defined in Section 5 of the[OWL Semantics and Abstract Syntax .].

Editor's Note: This section needs to be written after the document is considered to be finished (beforecompleted before publication as a Last Call).Call working draft. The following topicsitems are suggested:currently under consideration:
  • Role of the Comprehension Principles
  • Changes to the correspondence theorem
  • Appendix on "Axiomatic triples": complete set for all classes and properties; design strategy "simple axiomatictriples"
  • Imports closure definition
  • more specific definition of owl:DataRange
  • deprecated URIs (owl:DataRange, …)
  • data-version of oneOf semantic condition now requires lists of length ≥ 1
  • Bugfixes
    • "sequence-based" constructs
    • missing semantic condition for AllDifferent
  • Editorial changes
    • different table layoutName and usage of the interpretation function and its components ('IS'('IEXT' instead of S_I,EXTI; using always I(.) instead of specific mappings, as in RDF Semantics)
    • Naming convention for "Basic Sets""Parts of the Universe" (RDFS takes precedence over OWL 1 Full, but keep OWL 1 Full names in every other case; reduced set of abbreviations)

8 References [RDF Semantics] RDF Semantics . Patrick Hayes, Editor. W3C Recommendation10 February 2004.Acknowledgements

Editor's Note: This section will include a list of certain working group members.

11 References

[OWL Semantics2 Conformance and Abstract Syntax]Test Cases]
OWL 2 Web Ontology Language: Semantics and Abstract SyntaxTest Cases. /LIST OF EDITORS/, eds., /DOCUMENT STATE/, /DATE/.
Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, Editors. W3C Recommendation, 10 February 2004.[OWL 2 RDF Mapping] Mapping to RDF GraphsDirect Semantics]
Direct Semantics Boris Motik, Peter F. Patel-Schneider, Boris Motik,Bernardo Cuenca Grau, eds. W3C Editor's Draft, 2126 November 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-mapping-to-rdf-20081121/http://www.w3.org/2007/OWL/draft/ED-owl2-semantics-20081126/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-mapping-to-rdf/http://www.w3.org/2007/OWL/draft/owl2-semantics/.
[OWL 2 Direct Semantics] Direct Semantics Boris Motik,RDF Mapping]
Mapping to RDF Graphs Peter F. Patel-Schneider, Bernardo Cuenca Grau,Boris Motik, eds. W3C Editor's Draft, 2126 November 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-semantics-20081121/http://www.w3.org/2007/OWL/draft/ED-owl2-mapping-to-rdf-20081126/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-semantics/http://www.w3.org/2007/OWL/draft/owl2-mapping-to-rdf/.
[OWL 2 Structural Specification]
Structural Specification and Functional-Style Syntax Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Editor's Draft, 2126 November 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-syntax-20081121/http://www.w3.org/2007/OWL/draft/ED-owl2-syntax-20081126/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-syntax/.
[OWL 2 ConformanceSemantics and Test Cases]Abstract Syntax]
OWL 2Web Ontology Language: Test CasesSemantics and Abstract Syntax. /LIST OF EDITORS/ , Editors. /DOCUMENT STATE/ , /DATE/Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds., W3C Recommendation, 10 February 2004.
[RDF]
Resource Description Framework (RDF): Concepts and Abstract Syntax. Graham Klyne and Jeremy J. Carroll, eds., W3C Recommendation, 10 February 2004.
[RDF Semantics]
RDF Semantics. Patrick Hayes, ed., W3C Recommendation, 10 February 2004.
[RDF:TEXT]
rdf:text: A Datatype for Internationalized Text Jie Bao, Axel Polleres, Boris Motik. W3C Editor's Draft, 26 November 2008, http://www.w3.org/2007/OWL/draft/ED-owl2-rdf-text-20081126/. Latest version available at http://www.w3.org/2007/OWL/draft/owl2-rdf-text/.
[RFC 2119]
RFC 2119: Key words for use in RFCs to Indicate Requirement Levels. Network Working Group, S. Bradner. Internet Best Current Practice, March 1997.