RDFBased Semantics
__NUMBEREDHEADINGS__
 Document title:
 OWL 2 Web Ontology Language
RDFBased Semantics (Second Edition)
 Editor
 Michael Schneider, FZI Research Center for Information Technology
 Contributors (alphabetical order)
 Jeremy Carroll, HP (now at TopQuadrant)
 Ivan Herman, W3C/ERCIM
 Peter F. PatelSchneider, Nuance Communications
 Abstract

The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning. OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents. OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents. The OWL 2 Document Overview describes the overall state of OWL 2, and should be read before other OWL 2 documents.
This document defines the RDFcompatible modeltheoretic semantics of OWL 2.
 Status of this Document
Copyright © 20082009 W3C^{®} (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark and document use rules apply.
Contents
 1 Introduction (Informative)
 2 Ontologies
 3 Vocabulary
 4 Interpretations
 5 Semantic Conditions
 5.1 Semantic Conditions for the Parts of the Universe
 5.2 Semantic Conditions for the Vocabulary Classes
 5.3 Semantic Conditions for the Vocabulary Properties
 5.4 Semantic Conditions for Boolean Connectives
 5.5 Semantic Conditions for Enumerations
 5.6 Semantic Conditions for Property Restrictions
 5.7 Semantic Conditions for Datatype Restrictions
 5.8 Semantic Conditions for the RDFS Vocabulary
 5.9 Semantic Conditions for Equivalence and Disjointness
 5.10 Semantic Conditions for Nary Disjointness
 5.11 Semantic Conditions for Sub Property Chains
 5.12 Semantic Conditions for Inverse Properties
 5.13 Semantic Conditions for Property Characteristics
 5.14 Semantic Conditions for Keys
 5.15 Semantic Conditions for Negative Property Assertions
 6 Appendix: Axiomatic Triples (Informative)
 7 Appendix: Relationship to the Direct Semantics (Informative)
 8 Appendix: Comprehension Conditions (Informative)
 8.1 Comprehension Conditions for Sequences
 8.2 Comprehension Conditions for Boolean Connectives
 8.3 Comprehension Conditions for Enumerations
 8.4 Comprehension Conditions for Property Restrictions
 8.5 Comprehension Conditions for Datatype Restrictions
 8.6 Comprehension Conditions for Inverse Properties
 9 Appendix: Changes from OWL 1 (Informative)
 10 Appendix: Change Log (Informative)
 11 Acknowledgments
 12 References
1 Introduction (Informative)
This document defines the RDFcompatible modeltheoretic semantics of OWL 2, referred to as the "OWL 2 RDFBased Semantics". The OWL 2 RDFBased Semantics gives a formal meaning to every RDF graph [RDF Concepts] and is fully compatible with the RDF Semantics specification [RDF Semantics]. The specification provided here is the successor to the original OWL 1 RDFCompatible Semantics specification [OWL 1 RDFCompatible Semantics].
Technically, the OWL 2 RDFBased Semantics is defined as a semantic extension of "DEntailment" (RDFS with datatype support), as specified in the RDF Semantics [RDF Semantics]. In other words, the meaning given to an RDF graph by the OWL 2 RDFBased Semantics includes the meaning provided by the semantics of RDFS with datatypes, and additional meaning is specified for all the language constructs of OWL 2, such as Boolean connectives, sub property chains and qualified cardinality restrictions (see the OWL 2 Structural Specification [OWL 2 Specification] for further information on all the language constructs of OWL 2). The definition of the semantics for the extra constructs follows the design principles as applied to the RDF Semantics.
The content of this document is not meant to be selfcontained but builds on top of the RDF Semantics document [RDF Semantics] by adding those aspects that are specific to OWL 2. Hence, the complete definition of the OWL 2 RDFBased Semantics is given by the combination of both the RDF Semantics document and the document at hand. In particular, the terminology used in the RDF Semantics is reused here except for cases where a conflict exists with the rest of the OWL 2 specification.
The remainder of this section provides an overview of some of the distinguishing features of the OWL 2 RDFBased Semantics and outlines the document's structure and content.
In Section 2, the syntax over which the OWL 2 RDFBased Semantics is defined is the set of all RDF graphs [RDF Concepts]. The OWL 2 RDFBased Semantics provides a precise formal meaning for every RDF graph. The language that is determined by RDF graphs being interpreted using the OWL 2 RDFBased Semantics is called "OWL 2 Full". In this document, RDF graphs are also called "OWL 2 Full ontologies", or simply "ontologies", unless there is risk of confusion.
The OWL 2 RDFBased Semantics interprets the RDF and RDFS vocabularies [RDF Semantics] and the OWL 2 RDFBased vocabulary together with an extended set of datatypes and their constraining facets (see Section 3).
OWL 2 RDFBased interpretations (Section 4) are defined on a universe (see Section 1.3 of the RDF Semantics specification [RDF Semantics] for an overview of the basic intuition of modeltheoretic semantics). The universe is divided into parts, namely individuals, classes, and properties, which are identified with their RDF counterparts (see Figure 1). The part of individuals equals the whole universe. This means that all classes and properties are also individuals in their own right. Further, every name interpreted by an OWL 2 RDFBased interpretation denotes an individual.
The three basic parts are divided into further parts as follows. The part of individuals subsumes the part of data values, which comprises the denotations of all literals. Also subsumed by the individuals is the part of ontologies. The part of classes subsumes the part of datatypes, which are classes consisting entirely of data values. Finally, the part of properties subsumes the parts of object properties, data properties, ontology properties and annotation properties. The part of object properties equals the whole part of properties, and therefore all other kinds of properties are also object properties.
For annotations properties note that annotations are not "semanticfree" under the OWL 2 RDFBased Semantics. Just like every other triple or set of triples occurring in an RDF graph, an annotation is assigned a truth value by any given OWL 2 RDFBased interpretation. Hence, although annotations are meant to be "semantically weak", i.e., their formal meaning does not significantly exceed that originating from the RDF Semantics specification, adding an annotation may still change the meaning of an ontology. A similar discussion holds for statements that are built from ontology properties, such as owl:imports, which are used to define relationships between two ontologies.
Every class represents a specific set of individuals, called the class extension of the class: an individual a is an instance of a class C, if a is a member of the class extension ICEXT(C). Since a class is itself an individual under the OWL 2 RDFBased Semantics, classes are distinguished from their respective class extensions. This distinction allows, for example, that a class may be an instance of itself by being a member of its own class extension. Also, two classes may be equivalent by sharing the same class extension, although being different individuals, e.g., they do not need to share the same properties. Similarly, every property has an associated property extension that consists of pairs of individuals: an individual a_{1} has a relationship to an individual a_{2} with respect to a property p if the pair ( a_{1} , a_{2} ) is a member of the property extension IEXT(p). Again, properties are distinguished from their property extensions. In general, if there are no further constraints, an arbitrary extension may be associated with a given class or property, and two interpretations may associate distinct extensions with the same class or property.
Individuals may play different "roles". For example, an individual can be both a data property and an annotation property, since the different parts of the universe of an OWL 2 RDFBased interpretation are not required to be mutually disjoint, or an individual can be both a class and a property by associating both a class extension and a property extension with it. In the latter case there will be no specific relationship between the class extension and the property extension of such an individual without further constraints. For example, the same individual can have an empty class extension while having a nonempty property extension.
The main part of the OWL 2 RDFBased Semantics is Section 5, which specifies a formal meaning for all the OWL 2 language constructs by means of the OWL 2 RDFBased semantic conditions. These semantic conditions extend all the semantic conditions given in the RDF Semantics [RDF Semantics]. The OWL 2 RDFBased semantic conditions effectively determine which sets of RDF triples are assigned a specific meaning and what this meaning is. For example, semantic conditions exist that allow one to interpret the triple "C owl:disjointWith D" to mean that the denotations of the IRIs C and D have disjoint class extensions.
There is usually no need to provide localizing information (e.g., by means of "typing triples") for the IRIs occurring in an ontology. As for the RDF Semantics, the OWL 2 RDFBased semantic conditions have been designed to ensure that the denotation of any IRI will be in the appropriate part of the universe. For example, the RDF triple "C owl:disjointWith D" is sufficient to deduce that the denotations of the IRIs C and D are actually classes. It is not necessary to explicitly add additional typing triples "C rdf:type rdfs:Class" and "D rdf:type rdfs:Class" to the ontology.
In the RDF Semantics, this kind of "automatic localization" was to some extent achieved by so called "axiomatic triples" [RDF Semantics], such as "rdf:type rdf:type rdf:Property" or "rdf:type rdfs:domain rdfs:Resource". However, there is no explicit normative collection of additional axiomatic triples for the OWL 2 RDFBased Semantics; instead, the specific axiomatic aspects of the OWL 2 RDFBased Semantics are determined by a subset of the OWL 2 RDFBased semantic conditions. Section 6 discusses axiomatic triples in general and provides an example set of axiomatic triples that is compatible with the OWL 2 RDFBased Semantics.
Section 7 compares the OWL 2 RDFBased Semantics with the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. While the OWL 2 RDFBased Semantics is based on the RDF Semantics specification [RDF Semantics], the OWL 2 Direct Semantics is a description logic style semantics. Several fundamental differences exist between the two semantics, but there is also a strong relationship basically stating that the OWL 2 RDFBased Semantics is able to reflect all logical conclusions of the OWL 2 Direct Semantics. This means that the OWL 2 Direct Semantics can in a sense be regarded as a semantics subset of the OWL 2 RDFBased Semantics. The precise relationship is given by the OWL 2 correspondence theorem.
Significant effort has been spent in keeping the design of the OWL 2 RDFBased Semantics as close as possible to that of the original specification of the OWL 1 RDFCompatible Semantics [OWL 1 RDFCompatible Semantics]. While this aim was achieved to a large degree, the OWL 2 RDFBased Semantics actually deviates from its predecessor in several aspects. In most cases, this is because of serious technical problems that would have arisen from a conservative semantic extension. One important change is that while so called "comprehension conditions" for the OWL 2 RDFBased Semantics (see Section 8) still exist, these are not part of the normative set of semantic conditions anymore. The OWL 2 RDFBased Semantics also corrects several errors of OWL 1. A list of differences between the two languages is given in Section 9.
The italicized keywords MUST, MUST NOT, SHOULD, SHOULD NOT, and MAY are used to specify normative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119 [RFC 2119].
Each node is labeled with a class IRI
that represents a part of the universe
of an OWL 2 RDFbased interpretation.
Arrows point from parts to their super parts.
2 Ontologies
This section determines the syntax for the OWL 2 RDFBased Semantics, and gives an overview on typical content of ontologies for ontology management tasks.
2.1 Syntax
Following Sections 0.2 and 0.3 of the RDF Semantics specification [RDF Semantics], the OWL 2 RDFBased Semantics is defined on every RDF graph (Section 6.2 of RDF Concepts [RDF Concepts]), i.e. on every set of RDF triples (Section 6.1 of RDF Concepts [RDF Concepts]).
In accordance with the rest of the OWL 2 specification (see Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification]), this document uses an extended notion of an RDF graph by allowing the RDF triples in an RDF graph to contain arbitrary IRIs ("Internationalized Resource Identifiers") according to RFC 3987 [RFC 3987]. In contrast, the RDF Semantics specification [RDF Semantics] is defined on RDF graphs containing URIs [RFC 2396]. This change is backward compatible with the RDF specification, since URIs are also IRIs.
Terminological note: The document at hand uses the term "IRI" in accordance with the rest of the OWL 2 specification (see Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification]), whereas the RDF Semantics specification [RDF Semantics] uses the term "URI reference". According to RFC 3987 [RFC 3987], the term "IRI" stands for an absolute resource identifier with optional fragment, which is what is being used throughout this document. In contrast, the term "IRI reference" additionally covers relative references, which are never used in this document.
Convention: In this document, IRIs are abbreviated in the way defined by Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification], i.e., the abbreviations consist of a prefix name and a local part, such as "prefix:localpart".
The definition of an RDF triple according to Section 6.1 of RDF Concepts [RDF Concepts] is restricted to cases where the subject of an RDF triple is an IRI or a blank node (Section 6.6 of RDF Concepts [RDF Concepts]), and where the predicate of an RDF triple is an IRI. As a consequence, the definition does not treat cases, where, for example, the subject of a triple is a literal (Section 6.5 of RDF Concepts [RDF Concepts]), as in "s" ex:p ex:o, or where the predicate of a triple is a blank node, as in ex:s _:p ex:o. In order to allow for interoperability with other existing and future technologies and tools, the document at hand does not explicitly forbid the use of generalized RDF graphs consisting of generalized RDF triples, which are defined to allow for IRIs, literals and blank nodes to occur in the subject, predicate and object position. Thus, an RDF graph MAY contain generalized RDF triples, but an implementation is not required to support generalized RDF graphs. Note that every RDF graph consisting entirely of RDF triples according to Section 6.1 of RDF Concepts [RDF Concepts] is also a generalized RDF graph.
Terminological notes: The term "OWL 2 Full" refers to the language that is determined by the set of all RDF graphs being interpreted using the OWL 2 RDFBased Semantics. Further, in this document the term "OWL 2 Full ontology" (or simply "ontology", unless there is any risk of confusion) will be used interchangeably with the term "RDF graph".
2.2 Content of Ontologies (Informative)
While there do not exist any syntactic restrictions on the set of RDF graphs that can be interpreted by the OWL 2 RDFBased Semantics, in practice an ontology will often contain certain kinds of constructs that are aimed to support ontology management tasks. Examples are ontology headers and ontology IRIs, as well as constructs that are about versioning, importing and annotating of ontologies, including the concept of incompatibility between ontologies.
These topics are outside the scope of this semantics specification. Section 3 of the OWL 2 Structural Specification [OWL 2 Specification] deals with these topics in detail, and can therefore be used as a guide on how to apply these constructs in OWL 2 Full ontologies accordingly. The mappings of all these constructs to their respective RDF encoding are defined in the OWL 2 RDF Mapping [OWL 2 RDF Mapping].
3 Vocabulary
This section specifies the OWL 2 RDFBased vocabulary, and lists the names of the datatypes and facets used under the OWL 2 RDFBased Semantics.
3.1 Standard Prefixes
Table 3.1 lists the standard prefix names and their prefix IRIs used in this document.
Prefix Name  Prefix IRI  

OWL  owl  http://www.w3.org/2002/07/owl# 
RDF  rdf  http://www.w3.org/1999/02/22rdfsyntaxns# 
RDFS  rdfs  http://www.w3.org/2000/01/rdfschema# 
XML Schema  xsd  http://www.w3.org/2001/XMLSchema# 
3.2 Vocabulary Terms
Table 3.2 lists the IRIs of the OWL 2 RDFBased vocabulary, which is the set of vocabulary terms that are specific for the OWL 2 RDFBased Semantics. This vocabulary extends the RDF and RDFS vocabularies as specified in Sections 3.1 and 4.1 of the RDF Semantics [RDF Semantics], respectively. Table 3.2 does not mention those IRIs that will be listed in Section 3.3 on datatype names or Section 3.4 on facet names.
Implementations are not required to support the IRI owl:onProperties, but MAY support it in order to realize nary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).
Note: The use of the IRI owl:DataRange has been deprecated as of OWL 2. The IRI rdfs:Datatype SHOULD be used instead.
owl:AllDifferent owl:AllDisjointClasses owl:AllDisjointProperties owl:allValuesFrom owl:annotatedProperty owl:annotatedSource owl:annotatedTarget 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:ObjectProperty owl:onClass owl:onDataRange owl:onDatatype owl:oneOf owl:onProperty owl:onProperties owl:Ontology owl:OntologyProperty owl:priorVersion owl:propertyChainAxiom owl:propertyDisjointWith owl:qualifiedCardinality owl:ReflexiveProperty owl:Restriction owl:sameAs owl:someValuesFrom owl:sourceIndividual owl:SymmetricProperty owl:targetIndividual owl:targetValue owl:Thing owl:topDataProperty owl:topObjectProperty owl:TransitiveProperty owl:unionOf owl:versionInfo owl:versionIRI owl:withRestrictions 
3.3 Datatype Names
Table 3.3 lists the IRIs of the datatypes used in the OWL 2 RDFBased Semantics. The datatype rdf:XMLLiteral is described in Section 3.1 of the RDF Semantics [RDF Semantics]. All other datatypes are described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification]. The normative set of datatypes of the OWL 2 RDFBased Semantics equals the set of datatypes described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification].
xsd:anyURI xsd:base64Binary xsd:boolean xsd:byte xsd:dateTime xsd:dateTimeStamp 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 rdf:PlainLiteral xsd:positiveInteger owl:rational owl:real xsd:short xsd:string xsd:token xsd:unsignedByte xsd:unsignedInt xsd:unsignedLong xsd:unsignedShort rdf:XMLLiteral 
3.4 Facet Names
Table 3.4 lists the IRIs of the facets used in the OWL 2 RDFBased Semantics. Each datatype listed in Section 3.3 has a (possibly empty) set of constraining facets. All facets are described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification] in the context of their respective datatypes. The normative set of facets of the OWL 2 RDFBased Semantics equals the set of facets described in Section 4 of the OWL 2 Structural Specification [OWL 2 Specification].
In this specification, facets are used for defining datatype restrictions (see Section 5.7). For example, to refer to the set of all strings of length 5 one can restrict the datatype xsd:string (Section 3.3) by the facet xsd:length and the value 5.
rdf:langRange xsd:length xsd:maxExclusive xsd:maxInclusive xsd:maxLength xsd:minExclusive xsd:minInclusive xsd:minLength xsd:pattern 
4 Interpretations
The OWL 2 RDFBased Semantics provides vocabulary interpretations and vocabulary entailment (see Section 2.1 of the RDF Semantics [RDF Semantics]) for the RDF and RDFS vocabularies and for the OWL 2 RDFBased vocabulary. This section defines OWL 2 RDFBased datatype maps and OWL 2 RDFBased interpretations, and specifies what satisfaction of ontologies, consistency and entailment means under the OWL 2 RDFBased Semantics. In addition, the so called "parts" of the universe of an OWL 2 RDFBased interpretation are defined.
4.1 Datatype Maps
According to Section 5.1 of the RDF Semantics specification [RDF Semantics], a datatype d has the following components:
 LS(d), the lexical space of d, which is a set of lexical forms;
 VS(d), the value space of d, which is a set of data values;
 L2V(d), the lexicaltovalue mapping of d, which maps lexical forms in LS(d) to data values in VS(d).
Terminological notes: The document at hand uses the term "data value" in accordance with the rest of the OWL 2 specification (see Section 4 of the OWL 2 Structural Specification [OWL 2 Specification]), whereas the RDF Semantics specification [RDF Semantics] uses the term "datatype value" instead. Further, the names "LS" and "VS", which stand for the lexical space and the value space of a datatype, respectively, are not used in the RDF Semantics specification, but have been introduced here for easier reference.
In this document, the basic definition of a datatype is extended to take facets into account. See Section 3.4 for information and an example on facets. Note that Section 5.1 of the RDF Semantics specification [RDF Semantics] explicitly permits that semantic extensions may impose more elaborate datatyping conditions than those listed above.
A datatype with facets d is a datatype that has the following additional components:
 FS(d), the facet space of d, which is a set of pairs of the form ( F , v ), where F is an IRI called the constraining facet and v is an arbitrary data value called the constraining value;
 F2V(d), the facettovalue mapping of d, which maps each facetvalue pair ( F , v ) in FS(d) to a subset of VS(d).
Note that it is not further specified what the nature of the denotation of a facet IRI is, i.e. it is only known that a facet IRI denotes some individual. Semantic extensions MAY impose further restrictions on the denotations of facets. In fact, Section 5.3 will define additional restrictions on facets.
Also note that for a datatype d and a facetvalue pair ( F , v ) in FS(d) the value v is not required to be included in the value space VS(d) of d itself. For example, the datatype xsd:string (Section 3.3) has the facet xsd:length (Section 3.4), which takes nonnegative integers as its constraining values rather than strings.
In this document, it will always be assumed from now on that any datatype d is a datatype with facets. If the facet space FS(d) of a datatype d has not been explicitly defined, or if it is not derived from another datatype's facet space according to some well defined condition, then FS(d) is the empty set. Unless there is any risk of confusion, the term "datatype" will always refer to a datatype with facets.
Section 5.1 of the RDF Semantics specification [RDF Semantics] further defines a datatype map D to be a set of namedatatype pairs ( u , d ) consisting of an IRI u and a datatype d, such that no IRI appears twice in the set. As a consequence of what has been said before, in this document every datatype map D will entirely consist of datatypes with facets.
The following definition specifies what an OWL 2 RDFBased datatype map is.
Definition 4.1 (OWL 2 RDFBased Datatype Map): A datatype map D is an OWL 2 RDFBased datatype map, if and only if for every datatype name u listed in Section 3.3 and its respective set of constraining facets (Section 3.4) there is a namedatatype pair ( u, d ) in D with the specified lexical space LS(d), value space VS(d), lexicaltovalue mapping L2V(d), facet space FS(d) and facettovalue mapping F2V(d).
Note that Definition 4.1 does not prevent additional datatypes to be in an OWL 2 RDFBased datatype map. For the special case of an OWL 2 RDFBased datatype map D that exclusively contains the datatypes listed in Section 3.3, it is ensured that there are datatypes available for all the facet values, i.e., for every namedatatype pair ( u , d ) in D and for every facetvalue pair ( F , v ) in FS(d) there exists a namedatatype pair ( u^{*} , d^{*} ) in D such that v is in VS(d^{*}).
4.2 Vocabulary Interpretations
From the RDF Semantics specification [RDF Semantics], let V be a set of literals and IRIs containing the RDF and RDFS vocabularies, and let D be a datatype map according to Section 5.1 of the RDF Semantics [RDF Semantics] (and accordingly Section 4.1). A Dinterpretation I of V with respect to D is a tuple
I = ( IR , IP , IEXT , IS , IL , LV ) .
IR is the universe of I, i.e., a nonempty set that contains at least the denotations of literals and IRIs in V. IP is a subset of IR, the properties of I. LV, the data values of I, is a subset of IR that contains at least the set of plain literals (see Section 6.5 of RDF Concepts [RDF Concepts]) in V, and the value spaces of each datatype of D. IEXT is used to associate properties with their property extension, and is a mapping from IP to the powerset of IR × IR. IS is a mapping from IRIs in V to their denotations in IR. In particular, IS(u) = d for any namedatatype pair ( u , d ) in D. IL is a mapping from typed literals "s"^^u in V to their denotations in IR, where IL("s"^^u) = L2V(d)(s), provided that d is a datatype of D, IS(u) = d, and s is in the lexical space LS(d); otherwise IL("s"^^u) is not in LV.
Convention: Following the practice introduced in Section 1.4 of the RDF Semantics [RDF Semantics], for a given interpretation I of a vocabulary V the notation "I(x)" will be used instead of "IL(x)" and "IS(x)" for the typed literals and IRIs x in V, respectively.
As detailed in the RDF Semantics [RDF Semantics], a Dinterpretation has to meet all the semantic conditions for ground graphs and blank nodes, those for RDF interpretations and RDFS interpretations, and the "general semantic conditions for datatypes".
In this document, the basic definition of a Dinterpretation is extended to take facets into account.
A Dinterpretation with facets I is a Dinterpretation for a datatype map D consisting entirely of datatypes with facets (Section 4.1), where I meets the following additional semantic conditions: for each namedatatype pair ( u , d ) in D and each facetvalue pair ( F , v ) in the facet space FS(d)
 F is in the vocabulary V of I;
 a namedatatype pair ( u^{*} , d^{*} ) exists in D, such that v is in the value space VS(d^{*}).
In this document, it will always be assumed from now on that any Dinterpretation I is a Dinterpretation with facets. Unless there is any risk of confusion, the term "Dinterpretation" will always refer to a Dinterpretation with facets.
The following definition specifies what an OWL 2 RDFBased interpretation is.
Definition 4.2 (OWL 2 RDFBased Interpretation): Let D be an OWL 2 RDFBased datatype map, and let V be a vocabulary that includes the RDF and RDFS vocabularies and the OWL 2 RDFBased vocabulary together with all the datatype and facet names listed in Section 3. An OWL 2 RDFBased interpretation, I = ( IR , IP , IEXT , IS , IL , LV ), of V with respect to D is a Dinterpretation of V with respect to D that meets all the extra semantic conditions given in Section 5.
4.3 Satisfaction, Consistency and Entailment
The following definitions specify what it means for an RDF graph to be satisfied by a given OWL 2 RDFBased interpretation, to be consistent under the OWL 2 RDFBased Semantics, and to entail another RDF graph.
The notion of satisfaction under the OWL 2 RDFBased Semantics is based on the notion of satisfaction for Dinterpretations and Simple interpretations, as defined in the RDF Semantics [RDF Semantics]. In essence, in order to satisfy an RDF graph, an interpretation I has to satisfy all the triples in the graph, i.e., for a triple "s p o" it is necessary that the relationship ( I(s) , I(o) ) ∈ IEXT(I(p)) holds (special treatment exists for blank nodes, as detailed in Section 1.5 of the RDF Semantics [RDF Semantics]). In other words, the given graph has to be compatible with the specific form of the IEXT mapping of I. The distinguishing aspect of OWL 2 RDFBased satisfaction is that an interpretation I needs to meet all the OWL 2 RDFBased semantic conditions (see Section 5), which have a constraining effect on the possible forms an IEXT mapping can have.
Definition 4.3 (OWL 2 RDFBased Satisfaction): Let G be an RDF graph, let D be an OWL 2 RDFBased datatype map, let V be a vocabulary that includes the RDF and RDFS vocabularies and the OWL 2 RDFBased vocabulary together with all the datatype and facet names listed in Section 3, and let I be a Dinterpretation of V with respect to D. I OWL 2 RDFBased satisfies G with respect to V and D if and only if I is an OWL 2 RDFBased interpretation of V with respect to D that satisfies G as a Dinterpretation of V with respect to D according to the RDF Semantics [RDF Semantics].
Definition 4.4 (OWL 2 RDFBased Consistency): Let S be a collection of RDF graphs, and let D be an OWL 2 RDFBased datatype map. S is OWL 2 RDFBased consistent with respect to D if and only if there is some OWL 2 RDFBased interpretation I with respect to D of some vocabulary V that includes the RDF and RDFS vocabularies and the OWL 2 RDFBased vocabulary together with all the datatype and facet names listed in Section 3, such that I OWL 2 RDFBased satisfies all the RDF graphs in S with respect to V and D.
Definition 4.5 (OWL 2 RDFBased Entailment): Let S_{1} and S_{2} be collections of RDF graphs, and let D be an OWL 2 RDFBased datatype map. S_{1} OWL 2 RDFBased entails S_{2} with respect to D if and only if for every OWL 2 RDFBased interpretation I with respect to D of any vocabulary V that includes the RDF and RDFS vocabularies and the OWL 2 RDFBased vocabulary together with all the datatype and facet names listed in Section 3 the following holds: If I OWL 2 RDFBased satisfies all the RDF graphs in S_{1} with respect to V and D, then I OWL 2 RDFBased satisfies all the RDF graphs in S_{2} with respect to V and D.
4.4 Parts of the Universe
Table 4.1 defines the "parts" of the universe of a given OWL 2 RDFBased interpretation I.
The second column tells the name of the part. The third column gives a definition of the part in terms of the mapping IEXT of I, and by referring to a particular term of the RDF, RDFS or OWL 2 RDFBased vocabulary.
As an example, the part of all datatypes is named "IDC", and it is defined as the set of all individuals x for which the relationship "( x , I(rdfs:Datatype) ) ∈ IEXT(I(rdf:type))" holds. According to the semantics of rdf:type, as defined in Section 4.1 of the RDF Semantics [RDF Semantics], this means that the name "IDC" denotes the class extension (see Section 4.5) of I(rdfs:Datatype).
Name of Part S 
Definition of S as { x ∈ IR  ( x , I(E) ) ∈ IEXT(I(rdf:type)) } where IRI E is  

individuals  IR  rdfs:Resource 
data values  LV  rdfs:Literal 
ontologies  IX  owl:Ontology 
classes  IC  rdfs:Class 
datatypes  IDC  rdfs:Datatype 
properties  IP  rdf:Property 
data properties  IODP  owl:DatatypeProperty 
ontology properties  IOXP  owl:OntologyProperty 
annotation properties  IOAP  owl:AnnotationProperty 
4.5 Class Extensions
The mapping ICEXT from IC to the powerset of IR, which associates classes with their class extension, is defined for every c ∈ IC as
ICEXT(c) = { x ∈ IR  ( x , c ) ∈ IEXT(I(rdf:type)) } .
5 Semantic Conditions
This section defines the semantic conditions of the OWL 2 RDFBased Semantics. The semantic conditions presented here are basically only those for the specific constructs of OWL 2. The complete set of semantic conditions for the OWL 2 RDFBased Semantics is the combination of the semantic conditions presented here and the semantic conditions for Simple Entailment, RDF, RDFS and DEntailment, as specified in the RDF Semantics specification [RDF Semantics].
All semantic conditions in this section are defined with respect to an interpretation I. Section 5.1 specifies semantic conditions for the different parts of the universe of the interpretation being considered (compare Section 4.4). Section 5.2 and Section 5.3 list semantic conditions for the classes and the properties of the OWL 2 RDFBased vocabulary. In the rest of this section, the OWL 2 RDFBased semantic conditions for the different language constructs of OWL 2 are specified.
Conventions used in this Section
iff: Throughout this section the term "iff" is used as a shortform for "if and only if".
Conjunctive commas: A comma (",") separating two assertions in a semantic condition, as in "c ∈ IC , p ∈ IP", is read as a logical "and". Further, a comma separating two variables, as in "c, d ∈ IC", is used for abbreviating two comma separated assertions, "c ∈ IC , d ∈ IC" in this example.
Unscoped variables: If no explicit scope is given for a variable "x", as in "∀ x : …" or "{ x  … }", then "x" is unconstrained, which means x ∈ IR, i.e. "x" denotes an arbitrary individual in the universe.
Set cardinality: For a set S, an expression of the form "#S" means the number of elements in S.
Sequence expressions: An expression of the form "s sequence of a_{1} , … , a_{n} ∈ S" means that "s" represents an RDF list of n ≥ 0 individuals a_{1} , … , a_{n}, all of them being members of the set S. Precisely, s = I(rdf:nil) for n = 0; and for n > 0 there exist z_{1} ∈ IR , … , z_{n} ∈ IR, such that
s = z_{1} ,
a_{1} ∈ S ,
( z_{1} , a_{1} ) ∈ IEXT(I(rdf:first)) ,
( z_{1} , z_{2} ) ∈ IEXT(I(rdf:rest)) ,
… ,
a_{n} ∈ S,
( z_{n} , a_{n} ) ∈ IEXT(I(rdf:first)) ,
( z_{n} , I(rdf:nil) ) ∈ IEXT(I(rdf:rest)) .
Note, as mentioned in Section 3.3.3 of the RDF Semantics [RDF Semantics], there are no semantic constraints that enforce "wellformed" sequence structures. So, for example, it is possible for a sequence head s to refer to more than one sequence.
Set names: The following names are used as convenient abbreviations for certain sets:
 ISEQ: The set of all sequences. This set equals the class extension of rdf:List, i.e., ISEQ := ICEXT(I(rdf:List)).
 INNI: The set of all nonnegative integers. This set equals the value space of the datatype xsd:nonNegativeInteger, i.e., INNI := ICEXT(I(xsd:nonNegativeInteger)), but is also subsumed by the value spaces of other numerical datatypes, such as xsd:integer.
Notes on the Form of Semantic Conditions (Informative)
One design goal of OWL 2 was to ensure an appropriate degree of alignment between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics [OWL 2 Direct Semantics] under the different constraints the two semantics have to meet. The way this semantic alignment is described is via the OWL 2 correspondence theorem in Section 7.2. For this theorem to hold, the semantic conditions that treat the RDF encoding of OWL 2 axioms (compare Section 3.2.5 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] and Section 9 of the OWL 2 Structural Specification [OWL 2 Specification]), such as inverse property axioms, must have the form of "iff" ("ifandonlyif") conditions. This means that these semantic conditions completely determine the semantics of the encoding of these constructs. On the other hand, the RDF encoding of OWL 2 expressions (compare Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] and Sections 6 – 8 of the OWL 2 Structural Specification [OWL 2 Specification]), such as property restrictions, are treated by "ifthen" conditions. These weaker semantic conditions for expressions are sufficient for the correspondence theorem to hold, so there is no necessity to define stronger "iff" conditions under the OWL 2 RDFBased Semantics for these language constructs.
Special cases are the semantic conditions for Boolean connectives of classes and for enumerations. These language constructs build OWL 2 expressions. But for backward compatibility reasons there is also RDF encoding of axioms based on the vocabulary for these language constructs (see Table 18 in Section 3.2.5 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping]). For example, an RDF expression of the form
ex:c_{1} owl:unionOf ( ex:c_{2} ex:c_{3} ) .
is mapped by the reverse RDF mapping to an OWL 2 axiom that states the equivalence of the class denoted by ex:c_{1} with the union of the classes denoted by ex:c_{2} and ex:c_{3}. In order to ensure that the correspondence theorem holds, and in accordance with the original OWL 1 RDFCompatible Semantics specification [OWL 1 RDFCompatible Semantics], the semantic conditions for the mentioned language constructs are therefore "iff" conditions.
Further, special treatment exists for OWL 2 axioms that have a multitriple representation in RDF, where the different triples share a common "root node", such as the blank node "_:x" in the following example:
_:x rdf:type owl:AllDisjointClasses .
_:x owl:members ( ex:c_{1} ex:c_{2} ) .
In essence, the semantic conditions for the encoding of these language constructs are "iff" conditions, as usual for axioms. However, in order to cope with the specific syntactic aspect of a "root node", the "iff" conditions of these language constructs have been split into two "ifthen" conditions, where the "ifthen" condition representing the righttoleft direction contains an additional premise having the form "∃ z ∈ IR". The purpose of this premise is to ensure the existence of an individual that is needed to satisfy the root node under the OWL 2 RDFBased semantics. The language constructs in question are nary disjointness axioms in Section 5.10, and negative property assertions in Section 5.15.
The "ifthen" semantic conditions in this section sometimes do not explicitly list all typing statements in their consequent that one might expect. For example, the semantic condition for owl:someValuesFrom restrictions in Section 5.6 does not list the statement "x ∈ ICEXT(I(owl:Restriction))" on its right hand side. Consequences are generally not mentioned, if they can already be deduced by other means. Often, these redundant consequences follow from the semantic conditions for vocabulary classes and vocabulary properties in Section 5.2 and Section 5.3, respectively, occasionally in connection with the semantic conditions for the parts of the universe in Section 5.1. In the example above, the omitted consequence can be obtained from the third column of the entry for owl:someValuesFrom in the table in Section 5.3, which determines that IEXT(I(owl:someValuesFrom)) ⊆ ICEXT(I(owl:Restriction)) × IC.
5.1 Semantic Conditions for the Parts of the Universe
Table 5.1 lists the semantic conditions for the parts of the universe of the OWL 2 RDFBased interpretation being considered. Additional semantic conditions affecting these parts are given in Section 5.2.
The first column tells the name of the part, as defined in Section 4.4. The second column defines certain conditions on the part. In most cases, the column specifies for the part by which other part it is subsumed, and thus the position of the part in the "parts hierarchy" of the universe is narrowed down. The third column provides further information about the instances of those parts that consist of classes or properties. In general, if the part consists of classes, then for the class extensions of the member classes is specified by which part of the universe they are subsumed. If the part consists of properties, then the domains and ranges of the member properties are determined.
Name of Part S 
Conditions on S  Conditions on Instances x of S 

IR  S ≠ ∅  
LV  S ⊆ IR  
IX  S ⊆ IR  
IC  S ⊆ IR  ICEXT(x) ⊆ IR 
IDC  S ⊆ IC  ICEXT(x) ⊆ LV 
IP  S ⊆ IR  IEXT(x) ⊆ IR × IR 
IODP  S ⊆ IP  IEXT(x) ⊆ IR × LV 
IOXP  S ⊆ IP  IEXT(x) ⊆ IX × IX 
IOAP  S ⊆ IP  IEXT(x) ⊆ IR × IR 
5.2 Semantic Conditions for the Vocabulary Classes
Table 5.2 lists the semantic conditions for the classes that have IRIs in the OWL 2 RDFBased vocabulary. In addition, the table contains all those classes with IRIs in the RDF and RDFS vocabularies that represent parts of the universe of the OWL 2 RDFBased interpretation being considered (Section 4.4). The semantic conditions for the remaining classes with names in the RDF and RDFS vocabularies can be found in the RDF Semantics specification [RDF Semantics].
The first column tells the IRI of the class. The second column defines of what particular kind a class is, i.e. whether it is a general class (a member of the part IC) or a datatype (a member of IDC). The third column specifies for the class extension of the class by which part of the universe (Section 4.4) it is subsumed: from an entry of the form "ICEXT(I(C)) ⊆ S", for a class IRI C and a set S, and given an RDF triple of the form "u rdf:type C", one can deduce that the relationship "I(u) ∈ S" holds. Note that some entries are of the form "ICEXT(I(C)) = S", which means that the class extension is exactly specified to be that set. See Section 5.1 for further semantic conditions on those classes that represent parts.
Not included in this table are the datatypes of the OWL 2 RDFBased Semantics with IRIs listed in Section 3.3. For each such datatype IRI E, the following semantic conditions hold (as a consequence of the fact that E is a member of the datatype map of every OWL 2 RDFBased interpretation according to Definition 4.2, and by the "general semantic conditions for datatypes" listed in Section 5.1 of the RDF Semantics [RDF Semantics]):
 I(E) ∈ IDC
 ICEXT(I(E)) ⊆ LV
IRI E  I(E)  ICEXT(I(E)) 

owl:AllDifferent  ∈ IC  ⊆ IR 
owl:AllDisjointClasses  ∈ IC  ⊆ IR 
owl:AllDisjointProperties  ∈ IC  ⊆ IR 
owl:Annotation  ∈ IC  ⊆ IR 
owl:AnnotationProperty  ∈ IC  = IOAP 
owl:AsymmetricProperty  ∈ IC  ⊆ IP 
owl:Axiom  ∈ IC  ⊆ IR 
rdfs:Class  ∈ IC  = 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 
5.3 Semantic Conditions for the Vocabulary Properties
Table 5.3 lists the semantic conditions for the properties that have IRIs in the OWL 2 RDFBased vocabulary. In addition, the table contains all those properties with IRIs in the RDFS vocabulary that are specified to be annotation properties under the OWL 2 RDFBased Semantics. The semantic conditions for the remaining properties with names in the RDF and RDFS vocabularies can be found in the RDF Semantics specification [RDF Semantics].
The first column tells the IRI of the property. The second column defines of what particular kind a property is, i.e. whether it is a general property (a member of the part IP), a datatype property (a member of IODP), an ontology property (a member of IOXP) or an annotation property (a member of IOAP). The third column specifies the domain and range of the property: from an entry of the form "IEXT(I(p)) ⊆ S_{1} × S_{2}", for a property IRI p and sets S_{1} and S_{2}, and given an RDF triple "s p o", one can deduce the relationships "I(s) ∈ S_{1}" and "I(o) ∈ S_{2}". Note that some entries are of the form "IEXT(I(p)) = S_{1} × S_{2}", which means that the property extension is exactly specified to be the Cartesian product of the two sets.
Not included in this table are the facets of the OWL 2 RDFBased Semantics with IRIs listed in Section 3.4, which are used to specify datatype restrictions (see Section 5.7). For each such facet IRI E, the following semantic conditions extend the basic semantics specification that has been given for datatypes with facets in Section 4.1:
 I(E) ∈ IODP
 IEXT(I(E)) ⊆ IR × LV
Implementations are not required to support the semantic condition for owl:onProperties, but MAY support it in order to realize nary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).
Informative notes:
owl:topObjectProperty relates every two individuals in the universe with each other. Likewise, owl:topDataProperty relates every individual with every data value. Further, owl:bottomObjectProperty and owl:bottomDataProperty stand both for the empty relationship.
The ranges of the properties owl:deprecated and owl:hasSelf are not restricted in any form, and, in particular, they are not restricted to Boolean values. The actual object values of these properties do not have any intended meaning, but could as well have been defined to be of any other value. Therefore, the semantics given here are of a form that the values can be arbitrarily chosen without leading to any nontrivial semantic conclusions. It is, however, recommended to still use an object literal of the form "true"^^xsd:boolean in ontologies, in order to not get in conflict with the required usage of these properties in scenarios that ask for applying the reverse RDF mapping (compare Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping] for owl:hasSelf, and Section 5.5 of the OWL 2 Structural Specification [OWL 2 Specification] for owl:deprecated).
The range of the property owl:annotatedProperty is unrestricted, i.e. it is not specified as the set of properties. Annotations are meant to be "semantically weak", i.e. their formal meaning should not significantly exceed that originating from the RDF Semantics specification.
Several properties, such as owl:priorVersion, have been specified as both ontology properties and annotation properties, in order to be in line with both the original OWL 1 RDFCompatible Semantics specification [OWL 1 RDFCompatible Semantics] and the rest of the OWL 2 specification (see Section 5.5 of the OWL 2 Structural Specification [OWL 2 Specification]).
IRI E  I(E)  IEXT(I(E)) 

owl:allValuesFrom  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IC 
owl:annotatedProperty  ∈ IP  ⊆ IR × IR 
owl:annotatedSource  ∈ IP  ⊆ IR × IR 
owl:annotatedTarget  ∈ IP  ⊆ IR × IR 
owl:assertionProperty  ∈ IP  ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IP 
owl:backwardCompatibleWith  ∈ IOXP , ∈ IOAP 
⊆ IX × IX 
owl:bottomDataProperty  ∈ IODP  = ∅ 
owl:bottomObjectProperty  ∈ IP  = ∅ 
owl:cardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
rdfs:comment  ∈ IOAP  ⊆ IR × LV 
owl:complementOf  ∈ IP  ⊆ IC × IC 
owl:datatypeComplementOf  ∈ IP  ⊆ IDC × IDC 
owl:deprecated  ∈ IOAP  ⊆ IR × IR 
owl:differentFrom  ∈ IP  ⊆ IR × IR 
owl:disjointUnionOf  ∈ IP  ⊆ IC × ISEQ 
owl:disjointWith  ∈ IP  ⊆ IC × IC 
owl:distinctMembers  ∈ IP  ⊆ ICEXT(I(owl:AllDifferent)) × ISEQ 
owl:equivalentClass  ∈ IP  ⊆ IC × IC 
owl:equivalentProperty  ∈ IP  ⊆ IP × IP 
owl:hasKey  ∈ IP  ⊆ IC × ISEQ 
owl:hasSelf  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IR 
owl:hasValue  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IR 
owl:imports  ∈ IOXP  ⊆ IX × IX 
owl:incompatibleWith  ∈ IOXP , ∈ IOAP 
⊆ IX × IX 
owl:intersectionOf  ∈ IP  ⊆ IC × ISEQ 
owl:inverseOf  ∈ IP  ⊆ IP × IP 
rdfs:isDefinedBy  ∈ IOAP  ⊆ IR × IR 
rdfs:label  ∈ IOAP  ⊆ IR × LV 
owl:maxCardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
owl:maxQualifiedCardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
owl:members  ∈ IP  ⊆ IR × ISEQ 
owl:minCardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
owl:minQualifiedCardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
owl:onClass  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IC 
owl:onDataRange  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IDC 
owl:onDatatype  ∈ IP  ⊆ IDC × IDC 
owl:oneOf  ∈ IP  ⊆ IC × ISEQ 
owl:onProperty  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IP 
owl:onProperties  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × ISEQ 
owl:priorVersion  ∈ IOXP , ∈ IOAP 
⊆ IX × IX 
owl:propertyChainAxiom  ∈ IP  ⊆ IP × ISEQ 
owl:propertyDisjointWith  ∈ IP  ⊆ IP × IP 
owl:qualifiedCardinality  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × INNI 
owl:sameAs  ∈ IP  ⊆ IR × IR 
rdfs:seeAlso  ∈ IOAP  ⊆ IR × IR 
owl:someValuesFrom  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IC 
owl:sourceIndividual  ∈ IP  ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR 
owl:targetIndividual  ∈ IP  ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IR 
owl:targetValue  ∈ IP  ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × LV 
owl:topDataProperty  ∈ IODP  = IR × LV 
owl:topObjectProperty  ∈ IP  = IR × IR 
owl:unionOf  ∈ IP  ⊆ IC × ISEQ 
owl:versionInfo  ∈ IOAP  ⊆ IR × IR 
owl:versionIRI  ∈ IOXP  ⊆ IX × IX 
owl:withRestrictions  ∈ IP  ⊆ IDC × ISEQ 
5.4 Semantic Conditions for Boolean Connectives
Table 5.4 lists the semantic conditions for Boolean connectives, including intersections, unions and complements of classes and datatypes. An intersection or a union of a collection of datatypes or a complement of a datatype is itself a datatype. While a complement of a class is created w.r.t. the whole universe, a datatype complement is created for a datatype w.r.t. the set of data values only.
Informative notes: Of the three pairs of semantic conditions in the table every first is an "iff" condition, since the corresponding OWL 2 language constructs are both class expressions and axioms. In contrast, the semantic condition on datatype complements is an "ifthen" condition, since it only corresponds to a datarange expression. See the notes on the form of semantic conditions for further information. For the remaining semantic conditions that treat the cases of intersections and unions of datatypes it is sufficient to have "ifthen" conditions, since stronger "iff" conditions would be redundant due to the more general "iff" conditions that already exist for classes. Note that the datatype related semantic conditions do not apply to empty sets, but one can still receive a datatype from an empty set by explicitly asserting the resulting class to be an instance of class rdfs:Datatype.
if s sequence of c_{1} , … , c_{n} ∈ IR then  

( z , s ) ∈ IEXT(I(owl:intersectionOf))  iff  z , c_{1} , … , c_{n} ∈ IC , ICEXT(z) = ICEXT(c_{1}) ∩ … ∩ ICEXT(c_{n})  
if  then  
s sequence of d_{1} , … , d_{n} ∈ IDC , n ≥ 1 , ( z , s ) ∈ IEXT(I(owl:intersectionOf)) 
z ∈ IDC  
if s sequence of c_{1} , … , c_{n} ∈ IR then  
( z , s ) ∈ IEXT(I(owl:unionOf))  iff  z , c_{1} , … , c_{n} ∈ IC , ICEXT(z) = ICEXT(c_{1}) ∪ … ∪ ICEXT(c_{n})  
if  then  
s sequence of d_{1} , … , d_{n} ∈ IDC , n ≥ 1 , ( z , s ) ∈ IEXT(I(owl:unionOf)) 
z ∈ IDC  
( z , c ) ∈ IEXT(I(owl:complementOf))  iff  z , c ∈ IC , ICEXT(z) = IR \ ICEXT(c)  
if  then  
( z , d ) ∈ IEXT(I(owl:datatypeComplementOf))  ICEXT(z) = LV \ ICEXT(d) 
5.5 Semantic Conditions for Enumerations
Table 5.5 lists the semantic conditions for enumerations, i.e. classes that consist of an explicitly given finite set of instances. In particular, an enumeration entirely consisting of data values is a datatype.
Informative notes: The first semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is both a class expression and an axiom. See the notes on the form of semantic conditions for further information. For the remaining semantic condition that treats the case of enumerations of data values it is sufficient to have an "ifthen" condition, since a stronger "iff" condition would be redundant due to the more general "iff" condition that already exists for individuals. Note that the data value related semantic condition does not apply to empty sets, but one can still receive a datatype from an empty set by explicitly asserting the resulting class to be an instance of class rdfs:Datatype.
if s sequence of a_{1} , … , a_{n} ∈ IR then  

( z , s ) ∈ IEXT(I(owl:oneOf))  iff  z ∈ IC , ICEXT(z) = { a_{1} , … , a_{n} }  
if  then  
s sequence of v_{1} , … , v_{n} ∈ LV , n ≥ 1 , ( z , s ) ∈ IEXT(I(owl:oneOf)) 
z ∈ IDC 
5.6 Semantic Conditions for Property Restrictions
Table 5.6 lists the semantic conditions for property restrictions.
Value restrictions require that some or all of the values of a certain property must be instances of a given class or data range, or that the property has a specifically defined value. By placing a self restriction on some given property one only considers those individuals that are reflexively related to themselves via this property. Cardinality restrictions determine how often a certain property is allowed to be applied to a given individual. Qualified cardinality restrictions are more specific than cardinality restrictions in that they determine the quantity of a property application with respect to a particular class or data range from which the property values are taken.
Implementations are not required to support the semantic conditions for owl:onProperties, but MAY support them in order to realize nary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).
Informative notes: All the semantic conditions are "ifthen" conditions, since the corresponding OWL 2 language constructs are class expressions. The "ifthen" conditions generally only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information. Note that the semantic condition for self restrictions does not constrain the right hand side of a owl:hasSelf assertion to be the Boolean value "true"^^xsd:boolean. See Section 5.3 for an explanation.
if  then 

( z , c ) ∈ IEXT(I(owl:someValuesFrom)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  ∃ y : ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(c) } 
s sequence of p_{1} , … , p_{n} ∈ IR , n ≥ 1 , ( z , c ) ∈ IEXT(I(owl:someValuesFrom)) , ( z , s ) ∈ IEXT(I(owl:onProperties)) 
p_{1} , … , p_{n} ∈ IP , ICEXT(z) = { x  ∃ y_{1} , … , y_{n} : ( x , y_{k} ) ∈ IEXT(p_{k}) for each 1 ≤ k ≤ n and ( y_{1} , … , y_{n} ) ∈ ICEXT(c) } 
( z , c ) ∈ IEXT(I(owl:allValuesFrom)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  ∀ y : ( x , y ) ∈ IEXT(p) implies y ∈ ICEXT(c) } 
s sequence of p_{1} , … , p_{n} ∈ IR , n ≥ 1 , ( z , c ) ∈ IEXT(I(owl:allValuesFrom)) , ( z , s ) ∈ IEXT(I(owl:onProperties)) 
p_{1} , … , p_{n} ∈ IP , ICEXT(z) = { x  ∀ y_{1} , … , y_{n} : ( x , y_{k} ) ∈ IEXT(p_{k}) for each 1 ≤ k ≤ n implies ( y_{1} , … , y_{n} ) ∈ ICEXT(c) } 
( z , a ) ∈ IEXT(I(owl:hasValue)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  ( x , a ) ∈ IEXT(p) } 
( z , v ) ∈ IEXT(I(owl:hasSelf)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  ( x , x ) ∈ IEXT(p) } 
( z , n ) ∈ IEXT(I(owl:minCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) } ≥ n } 
( z , n ) ∈ IEXT(I(owl:maxCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) } ≤ n } 
( z , n ) ∈ IEXT(I(owl:cardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) } = n } 
( z , n ) ∈ IEXT(I(owl:minQualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , c ) ∈ IEXT(I(owl:onClass)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(c) } ≥ n } 
( z , n ) ∈ IEXT(I(owl:minQualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) 
p ∈ IODP , ICEXT(z) = { x  #{ y ∈ LV  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(d) } ≥ n } 
( z , n ) ∈ IEXT(I(owl:maxQualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , c ) ∈ IEXT(I(owl:onClass)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(c) } ≤ n } 
( z , n ) ∈ IEXT(I(owl:maxQualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) 
p ∈ IODP , ICEXT(z) = { x  #{ y ∈ LV  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(d) } ≤ n } 
( z , n ) ∈ IEXT(I(owl:qualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , c ) ∈ IEXT(I(owl:onClass)) 
ICEXT(z) = { x  #{ y  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(c) } = n } 
( z , n ) ∈ IEXT(I(owl:qualifiedCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) 
p ∈ IODP , ICEXT(z) = { x  #{ y ∈ LV  ( x , y ) ∈ IEXT(p) and y ∈ ICEXT(d) } = n } 
5.7 Semantic Conditions for Datatype Restrictions
Table 5.7 lists the semantic conditions for datatype restrictions, which are used to define sub datatypes of existing datatypes by restricting the original datatype by means of a set of facetvalue pairs. See Section 3.4 for information and an example on constraining facets.
Certain special cases exist: If no facetvalue pair is applied to a given datatype, then the resulting datatype will be equivalent to the original datatype. Further, if a facetvalue pair is applied to a datatype without being a member of the datatype's facet space, then the ontology cannot be satisfied and will therefore be inconsistent. In particular, a datatype restriction with one or more specified facetvalue pairs will result in an inconsistent ontology, if applied to a datatype with an empty facet space.
The set IFS is defined by IFS(d) := { ( I(F) , v )  ( F , v ) ∈ FS(d) } , where d is a datatype, F is the IRI of a constraining facet, and v is a constraining value of the facet. This set corresponds to the facet space FS(d), as defined in Section 4.1, but rather consists of pairs of the denotation of a facet and a value.
The mapping IF2V is defined by IF2V(d)(( I(F) , v )) := F2V(d)(( F , v )) , where d is a datatype, F is the IRI of a constraining facet, and v is a constraining value of the facet. This mapping corresponds to the facettovalue mapping F2V(d), as defined in Section 4.1, resulting in the same subsets of the value space VS(d), but rather applies to pairs of the denotation of a facet and a value.
Informative notes: The semantic condition is an "ifthen" condition, since the corresponding OWL 2 language construct is a datarange expression. The "ifthen" condition only lists those consequences on its right hand side that are specific for the condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.
if  then 

s sequence of z_{1} , … , z_{n} ∈ IR , f_{1} , … , f_{n} ∈ IP , ( z , d ) ∈ IEXT(I(owl:onDatatype)) , ( z , s ) ∈ IEXT(I(owl:withRestrictions)) , ( z_{1} , v_{1} ) ∈ IEXT(f_{1}) , … , ( z_{n} , v_{n} ) ∈ IEXT(f_{n}) 
f_{1} , … , f_{n} ∈ IODP , v_{1} , … , v_{n} ∈ LV , ( f_{1} , v_{1} ) , … , ( f_{n} , v_{n} ) ∈ IFS(d) , ICEXT(z) = ICEXT(d) ∩ IF2V(d)(( f_{1} , v_{1} )) ∩ … ∩ IF2V(d)(( f_{n} , v_{n} )) 
5.8 Semantic Conditions for the RDFS Vocabulary
Table 5.8 extends the RDFS semantic conditions for subclass axioms, subproperty axioms, domain axioms and range axioms. The semantic conditions provided here are "iff" conditions, while the original semantic conditions, as specified in Section 4.1 of the RDF Semantics [RDF Semantics], are weaker "ifthen" conditions. Only the additional semantic conditions are given here and the other conditions of RDF and RDFS are retained.
Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.
( c_{1} , c_{2} ) ∈ IEXT(I(rdfs:subClassOf))  iff  c_{1} , c_{2} ∈ IC , ICEXT(c_{1}) ⊆ ICEXT(c_{2}) 

( p_{1} , p_{2} ) ∈ IEXT(I(rdfs:subPropertyOf))  p_{1} , p_{2} ∈ IP , IEXT(p_{1}) ⊆ IEXT(p_{2})  
( p , c ) ∈ IEXT(I(rdfs:domain))  p ∈ IP , c ∈ IC , ∀ x , y : ( x , y ) ∈ IEXT(p) implies x ∈ ICEXT(c)  
( p , c ) ∈ IEXT(I(rdfs:range))  p ∈ IP , c ∈ IC , ∀ x , y : ( x , y ) ∈ IEXT(p) implies y ∈ ICEXT(c) 
5.9 Semantic Conditions for Equivalence and Disjointness
Table 5.9 lists the semantic conditions for specifying that two individuals are equal or different from each other, and that either two classes or two properties are equivalent or disjoint with each other, respectively. The property owl:equivalentClass is also used to formulate datatype definitions (see Section 9.4 of the OWL 2 Structural Specification [OWL 2 Specification] for information about datatype definitions). In addition, the table treats disjoint union axioms.
Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.
( a_{1} , a_{2} ) ∈ IEXT(I(owl:sameAs))  iff  a_{1} = a_{2} 

( a_{1} , a_{2} ) ∈ IEXT(I(owl:differentFrom))  a_{1} ≠ a_{2}  
( c_{1} , c_{2} ) ∈ IEXT(I(owl:equivalentClass))  c_{1} , c_{2} ∈ IC , ICEXT(c_{1}) = ICEXT(c_{2})  
( c_{1} , c_{2} ) ∈ IEXT(I(owl:disjointWith))  c_{1} , c_{2} ∈ IC , ICEXT(c_{1}) ∩ ICEXT(c_{2}) = ∅  
( p_{1} , p_{2} ) ∈ IEXT(I(owl:equivalentProperty))  p_{1} , p_{2} ∈ IP , IEXT(p_{1}) = IEXT(p_{2})  
( p_{1} , p_{2} ) ∈ IEXT(I(owl:propertyDisjointWith))  p_{1} , p_{2} ∈ IP , IEXT(p_{1}) ∩ IEXT(p_{2}) = ∅  
if s sequence of c_{1} , … , c_{n} ∈ IR then  
( c , s ) ∈ IEXT(I(owl:disjointUnionOf))  iff  c , c_{1} , … , c_{n} ∈ IC , ICEXT(c) = ICEXT(c_{1}) ∪ … ∪ ICEXT(c_{n}) , ICEXT(c_{j}) ∩ ICEXT(c_{k}) = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
5.10 Semantic Conditions for Nary Disjointness
Table 5.10 lists the semantic conditions for specifying nary diversity and disjointness axioms, i.e. that several given individuals are mutually different from each other, and that several given classes or properties are mutually disjoint with each other, respectively.
Note that there are two alternative ways to specify owl:AllDifferent axioms, by using either the property owl:members that is used for all other constructs, too, or by applying the legacy property owl:distinctMembers. Both variants have an equivalent formal meaning.
Informative notes: The semantic conditions essentially represent "iff" conditions, since the corresponding OWL 2 language constructs are axioms. However, there are actually two semantic conditions for each language construct due to the multitriple RDF encoding of these language constructs. The "ifthen" conditions only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.
if  then 

s sequence of a_{1} , … , a_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDifferent)) , ( z , s ) ∈ IEXT(I(owl:members)) 
a_{j} ≠ a_{k} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
if  then exists z ∈ IR 
s sequence of a_{1} , … , a_{n} ∈ IR , a_{j} ≠ a_{k} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
z ∈ ICEXT(I(owl:AllDifferent)) , ( z , s ) ∈ IEXT(I(owl:members)) 
if  then 
s sequence of a_{1} , … , a_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDifferent)) , ( z , s ) ∈ IEXT(I(owl:distinctMembers)) 
a_{j} ≠ a_{k} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
if  then exists z ∈ IR 
s sequence of a_{1} , … , a_{n} ∈ IR , a_{j} ≠ a_{k} for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
z ∈ ICEXT(I(owl:AllDifferent)) , ( z , s ) ∈ IEXT(I(owl:distinctMembers)) 
if  then 
s sequence of c_{1} , … , c_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDisjointClasses)) , ( z , s ) ∈ IEXT(I(owl:members)) 
c_{1} , … , c_{n} ∈ IC , ICEXT(c_{j}) ∩ ICEXT(c_{k}) = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
if  then exists z ∈ IR 
s sequence of c_{1} , … , c_{n} ∈ IC , ICEXT(c_{j}) ∩ ICEXT(c_{k}) = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
z ∈ ICEXT(I(owl:AllDisjointClasses)) , ( z , s ) ∈ IEXT(I(owl:members)) 
if  then 
s sequence of p_{1} , … , p_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDisjointProperties)) , ( z , s ) ∈ IEXT(I(owl:members)) 
p_{1} , … , p_{n} ∈ IP , IEXT(p_{j}) ∩ IEXT(p_{k}) = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
if  then exists z ∈ IR 
s sequence of p_{1} , … , p_{n} ∈ IP , IEXT(p_{j}) ∩ IEXT(p_{k}) = ∅ for each 1 ≤ j ≤ n and each 1 ≤ k ≤ n such that j ≠ k 
z ∈ ICEXT(I(owl:AllDisjointProperties)) , ( z , s ) ∈ IEXT(I(owl:members)) 
5.11 Semantic Conditions for Sub Property Chains
Table 5.11 lists the semantic conditions for sub property chains, which allow for specifying complex property subsumption axioms.
As an example, one can define a sub property chain axiom that specifies the chain consisting of the property extensions of properties ex:hasFather and ex:hasBrother to be a sub relation of the extension of the property ex:hasUncle.
Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information. The semantics has been specified in a way such that a sub property chain axiom can be satisfied without requiring the existence of a property that has the property chain as its property extension.
if s sequence of p_{1} , … , p_{n} ∈ IR then  

( p , s ) ∈ IEXT(I(owl:propertyChainAxiom))  iff  p ∈ IP , p_{1} , … , p_{n} ∈ IP , ∀ y_{0} , … , y_{n} : ( y_{0} , y_{1} ) ∈ IEXT(p_{1}) and … and ( y_{n1} , y_{n} ) ∈ IEXT(p_{n}) implies ( y_{0} , y_{n} ) ∈ IEXT(p) 
5.12 Semantic Conditions for Inverse Properties
Table 5.12 lists the semantic conditions for inverse property axioms. The inverse of a given property is the corresponding property with subject and object swapped for each property assertion built from it.
Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information.
( p_{1} , p_{2} ) ∈ IEXT(I(owl:inverseOf))  iff  p_{1} , p_{2} ∈ IP , IEXT(p_{1}) = { ( x , y )  ( y , x ) ∈ IEXT(p_{2}) } 

5.13 Semantic Conditions for Property Characteristics
Table 5.13 lists the semantic conditions for property characteristics.
If a property is functional, then at most one distinct value can be assigned to any given individual via this property. An inverse functional property can be regarded as a "key" property, i.e. no two different individuals can be assigned the same value via this property. A reflexive property relates every individual in the universe to itself, whereas an irreflexive property does not relate any individual with itself. If two individuals are related by a symmetric property, then this property also relates them reversely, while this is never the case for an asymmetric property. A transitive property that relates an individual a with an individual b, and the latter with an individual c, also relates a with c.
Informative notes: All the semantic conditions are "iff" conditions, since the corresponding OWL 2 language constructs are axioms. See the notes on the form of semantic conditions for further information.
p ∈ ICEXT(I(owl:FunctionalProperty))  iff  p ∈ IP , ∀ x , y_{1} , y_{2} : ( x , y_{1} ) ∈ IEXT(p) and ( x , y_{2} ) ∈ IEXT(p) implies y_{1} = y_{2} 

p ∈ ICEXT(I(owl:InverseFunctionalProperty))  p ∈ IP , ∀ x_{1} , x_{2} , y : ( x_{1} , y ) ∈ IEXT(p) and ( x_{2} , y ) ∈ IEXT(p) implies x_{1} = x_{2}  
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) implies ( y , x ) ∈ IEXT(p)  
p ∈ ICEXT(I(owl:AsymmetricProperty))  p ∈ IP , ∀ x , y : ( x , y ) ∈ IEXT(p) implies ( y , x ) ∉ IEXT(p)  
p ∈ ICEXT(I(owl:TransitiveProperty))  p ∈ IP , ∀ x , y , z : ( x , y ) ∈ IEXT(p) and ( y , z ) ∈ IEXT(p) implies ( x , z ) ∈ IEXT(p) 
5.14 Semantic Conditions for Keys
Table 5.14 lists the semantic conditions for Keys.
Keys provide an alternative to inverse functional properties (see Section 5.13). They allow for defining a property as a key local to a given class: the specified property will have the features of a key only for individuals being instances of the class, and no assumption is made about individuals for which membership of the class cannot be entailed. Further, it is possible to define "compound keys", i.e. several properties can be combined into a single key applicable to composite values. Note that keys are not functional by default under the OWL 2 RDFBased Semantics.
Informative notes: The semantic condition is an "iff" condition, since the corresponding OWL 2 language construct is an axiom. See the notes on the form of semantic conditions for further information.
if s sequence of p_{1} , … , p_{n} ∈ IR then  

( c , s ) ∈ IEXT(I(owl:hasKey))  iff  c ∈ IC , p_{1} , … , p_{n} ∈ IP , ∀ x , y , z_{1} , … , z_{n} : if x ∈ ICEXT(c) and y ∈ ICEXT(c) and ( x , z_{k} ) ∈ IEXT(p_{k}) and ( y , z_{k} ) ∈ IEXT(p_{k}) for each 1 ≤ k ≤ n then x = y 
5.15 Semantic Conditions for Negative Property Assertions
Table 5.15 lists the semantic conditions for negative property assertions. They allow to state that two given individuals are not related by a given property.
The second form based on owl:targetValue is more specific than the first form based on owl:targetIndividual in that the second form is restricted to the case of negative data property assertions. Note that the second form will coerce the target value of a negative property assertion into a data value, due to the range defined for the property owl:targetValue in Section 5.3.
Informative notes: The semantic conditions essentially represent "iff" conditions, since the corresponding OWL 2 language constructs are axioms. However, there are actually two semantic conditions for each language construct, due to the multitriple RDF encoding of these language constructs. The "ifthen" conditions only list those consequences on their right hand side that are specific for the respective condition, i.e. consequences that do not already follow by other means. See the notes on the form of semantic conditions for further information.
if  then 

( z , a_{1} ) ∈ IEXT(I(owl:sourceIndividual)) , ( z , p ) ∈ IEXT(I(owl:assertionProperty)) , ( z , a_{2} ) ∈ IEXT(I(owl:targetIndividual)) 
( a_{1} , a_{2} ) ∉ IEXT(p) 
if  then exists z ∈ IR 
a_{1} ∈ IR , p ∈ IP , a_{2} ∈ IR , ( a_{1} , a_{2} ) ∉ IEXT(p) 
( z , a_{1} ) ∈ IEXT(I(owl:sourceIndividual)) , ( z , p ) ∈ IEXT(I(owl:assertionProperty)) , ( z , a_{2} ) ∈ IEXT(I(owl:targetIndividual)) 
if  then 
( z , a ) ∈ IEXT(I(owl:sourceIndividual)) , ( z , p ) ∈ IEXT(I(owl:assertionProperty)) , ( z , v ) ∈ IEXT(I(owl:targetValue)) 
p ∈ IODP , ( a , v ) ∉ IEXT(p) 
if  then exists z ∈ IR 
a ∈ IR , p ∈ IODP , v ∈ LV , ( a , v ) ∉ IEXT(p) 
( z , a ) ∈ IEXT(I(owl:sourceIndividual)) , ( z , p ) ∈ IEXT(I(owl:assertionProperty)) , ( z , v ) ∈ IEXT(I(owl:targetValue)) 
6 Appendix: Axiomatic Triples (Informative)
The RDF Semantics specification [RDF Semantics] defines so called "axiomatic triples" as part of the semantics of RDF and RDFS. Unlike the RDF Semantics, the OWL 2 RDFBased Semantics does not normatively specify any axiomatic triples, since one cannot expect to find a set of RDF triples that fully captures all "axiomatic aspects" of the OWL 2 RDFBased Semantics. Furthermore, axiomatic triples for the OWL 2 RDFBased Semantics could, in principle, contain arbitrarily complex class expressions, e.g. the union of several classes, and by this it becomes nonobvious which of several possible nonequivalent sets of axiomatic triples should be selected. However, the OWL 2 RDFBased Semantics includes many semantic conditions that can in a sense be regarded as being "axiomatic", and thus can be considered a replacement for the missing axiomatic triples. After an overview on axiomatic triples for RDF and RDFS in Section 6.1, Sections 6.2 and 6.3 will discuss how the "axiomatic" semantic conditions of the OWL 2 RDFBased Semantics relate to axiomatic triples. Based on this discussion, an explicit example set of axiomatic triples that is compatible with the OWL 2 RDFBased Semantics will be provided in Section 6.4.
6.1 Axiomatic Triples in RDF
In RDF and RDFS [RDF Semantics], axiomatic triples are used to provide basic meaning for all the vocabulary terms of the two languages. This formal meaning is independent of any given RDF graph, and it even holds for vocabulary terms, which do not occur in a graph that is interpreted by an RDF or RDFS interpretation. As a consequence, all the axiomatic triples of RDF and RDFS are entailed by the empty graph, when being interpreted under the semantics of RDF or RDFS, respectively.
Examples of RDF and RDFS axiomatic triples are:
(1) rdf:type rdf:type rdf:Property .
(2) rdf:type rdfs:domain rdfs:Resource .
(3) rdf:type rdfs:range rdfs:Class .
(4) rdfs:Datatype rdfs:subClassOf rdfs:Class .
(5) rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .
As shown by these examples, axiomatic triples are typically used by the RDF Semantics specification to determine the part of the universe to which the denotation of a vocabulary term belongs (1). In the case of a property, the domain (2) and range (3) is specified as well. Also, in some cases, hierarchical relationships between classes (4) or properties (5) of the vocabulary are determined.
Under the OWL 2 RDFBased Semantics, all the axiomatic triples of RDF and RDFS could, in principle, be replaced by "axiomatic" semantic conditions that have neither premises nor bound variables. By applying the RDFS semantic conditions given in Section 5.8, the example axiomatic triples (1) – (5) can be equivalently restated as:
I(rdf:type) ∈ ICEXT(I(rdf:Property)) ,
IEXT(I(rdf:type)) ⊆ ICEXT(I(rdfs:Resource)) × ICEXT(I(rdfs:Class)) ,
ICEXT(I(rdfs:Datatype)) ⊆ ICEXT(I(rdfs:Class)) ,
IEXT(I(rdfs:isDefinedBy)) ⊆ IEXT(I(rdfs:seeAlso)) .
All the axiomatic triples of RDF and RDFS can be considered "simple" in the sense that they have in their object position only single terms from the RDF and RDFS vocabularies, and no complex class or property expressions appear there.
6.2 Axiomatic Triples for the Vocabulary Classes
The semantic conditions for vocabulary classes in Section 5.2 can be considered as corresponding to a set of axiomatic triples for the classes in the vocabulary of the OWL 2 RDFBased Semantics.
First, for each IRI E occurring in the first column of Table 5.2, if the second column contains an entry of the form "I(E) ∈ S" for some set S, then this entry corresponds to an RDF triple of the form "E rdf:type C", where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In the table, S will always be either the part IC of all classes, or some sub part of IC. Hence, in a corresponding RDF triple the IRI C will be one of "rdfs:Class", "owl:Class" (S=IC in both cases) or "rdfs:Datatype" (S=IDC).
For example, for the IRI "owl:FunctionalProperty", the semantic condition
I(owl:FunctionalProperty) ∈ IC
has the corresponding axiomatic triple
owl:FunctionalProperty rdf:type rdfs:Class .
Further, for each IRI E in the first column of the table, if the third column contains an entry of the form "ICEXT(I(E)) ⊆ S" (or "ICEXT(I(E)) = S") for some set S, then this entry corresponds to an RDF triple of the form "E rdfs:subClassOf C" (or additionally "C rdfs:subClassOf E"), where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In each case, S will be one of the parts of the universe of I.
For example, the semantic condition
ICEXT(I(owl:FunctionalProperty)) ⊆ IP
has the corresponding axiomatic triple
owl:FunctionalProperty rdfs:subClassOf rdf:Property .
In addition, the semantic conditions for the parts of the universe in Table 5.1 of Section 5.1 have to be taken into account. In particular, if an entry in the second column of Table 5.1 is of the form "S_{1} ⊆ S_{2}" for some sets S_{1} and S_{2}, then this corresponds to an RDF triple of the form "C_{1} owl:subClassOf C_{2}", where C_{1} and C_{2} are the IRIs of vocabulary classes with ICEXT(I(C_{1})) = S_{1} and ICEXT(I(C_{2})) = S_{2}, respectively, according to Section 5.2.
Section 5.2 also specifies semantic conditions for all the datatypes of the OWL 2 RDFBased Semantics, as listed in Section 3.3. For each datatype IRI E, such as E := "xsd:string", for the semantic conditions "I(E) ∈ IDC" and "ICEXT(I(E)) ⊆ LV" the corresponding axiomatic triples are of the form
E rdf:type rdfs:Datatype .
E rdfs:subClassOf rdfs:Literal .
In analogy to Section 6.1 for the RDF axiomatic triples, all the axiomatic triples for the vocabulary classes (including datatypes) can be considered "simple" in the sense that they will have in their object position only single terms from the RDF, RDFS and OWL 2 RDFBased vocabularies (Section 3.2).
Note that some of the axiomatic triples obtained in this way already follow from the semantics of RDF and RDFS, as defined in the RDF Semantics [RDF Semantics].
6.3 Axiomatic Triples for the Vocabulary Properties
The semantic conditions for vocabulary properties in Section 5.3 can be considered as corresponding to a set of axiomatic triples for the properties in the vocabulary of the OWL 2 RDFBased Semantics.
First, for each IRI E occurring in the first column of Table 5.3, if the second column contains an entry of the form "I(E) ∈ S" for some set S, then this entry corresponds to an RDF triple of the form "E rdf:type C", where C is the IRI of a vocabulary class with ICEXT(I(C)) = S. In the table, S will always be either the part IP of all properties, or some sub part of IP. Hence, in a corresponding RDF triple the IRI C will be one of "rdf:Property", "owl:ObjectProperty", (S=IP in both cases), "owl:DatatypeProperty" (S=IODP), "owl:OntologyProperty" (S=IOXP) or "owl:AnnotationProperty" (S=IOAP).
For example, for the IRI "owl:disjointWith", the semantic condition
I(owl:disjointWith) ∈ IP
has the corresponding axiomatic triple
owl:disjointWith rdf:type rdf:Property .
Further, for each IRI E in the first column of the table, if the third column contains an entry of the form "IEXT(I(E)) ⊆ S_{1} × S_{2}" for some sets S_{1} and S_{2}, then this entry corresponds to RDF triples of the form "E rdfs:domain C_{1}" and "E rdfs:range C_{2}", where C_{1} and C_{2} are the IRIs of vocabulary classes with ICEXT(I(C_{1})) = S_{1} and ICEXT(I(C_{2})) = S_{2}, respectively. Note that the sets S_{1} and S_{2} do not always correspond to any of the parts of the universe of I.
For example, the semantic condition
IEXT(I(owl:disjointWith)) ⊆ IC × IC
has the corresponding axiomatic triples
owl:disjointWith rdfs:domain owl:Class .
owl:disjointWith rdfs:range owl:Class .
Exceptions are the semantic conditions "IEXT(I(owl:topObjectProperty)) = IR × IR" and "IEXT(I(owl:topDataProperty)) = IR × LV", since the exactly specified property extensions of these properties cannot be expressed solely by domain and range axiomatic triples. For example, the domain and range axiomatic triples for owl:sameAs are equal to those for owl:topObjectProperty, but the property extension of owl:sameAs is different from the property extension of owl:topObjectProperty.
Section 5.3 also specifies semantic conditions for all the facets of the OWL 2 RDFBased Semantics, as listed in Section 3.4. For each facet IRI E, such as E := "xsd:length", for the semantic conditions "I(E) ∈ IODP" and "IEXT(I(E)) ⊆ IR × LV" the corresponding axiomatic triples are of the form
E rdf:type owl:DatatypeProperty .
E rdfs:domain rdfs:Resource .
E rdfs:range rdfs:Literal .
In analogy to Section 6.1 for the RDF axiomatic triples, all the axiomatic triples for the vocabulary properties (including facets) can be considered "simple" in the sense that they will have in their object position only single terms from the RDF, RDFS and OWL 2 RDFBased vocabularies (Section 3.2).
6.4 A Set of Axiomatic Triples
This section provides a concrete example set of axiomatic triples based on the discussion in the Sections 6.2 and 6.3. The axiomatic triples are grouped by different tables for the classes and the properties of the OWL 2 RDFBased vocabulary, for the datatypes and the facets of the OWL 2 RDFBased Semantics, and for some of the classes and properties of the RDFS vocabulary. Note that this set of axiomatic triples is not meant to be free of redundancy.
owl:AllDifferent rdf:type rdfs:Class . owl:AllDifferent rdfs:subClassOf rdfs:Resource . 
owl:AllDisjointClasses rdf:type rdfs:Class . owl:AllDisjointClasses rdfs:subClassOf rdfs:Resource . 
owl:AllDisjointProperties rdf:type rdfs:Class . owl:AllDisjointProperties rdfs:subClassOf rdfs:Resource . 
owl:Annotation rdf:type rdfs:Class . owl:Annotation rdfs:subClassOf rdfs:Resource . 
owl:AnnotationProperty rdf:type rdfs:Class . owl:AnnotationProperty rdfs:subClassOf rdf:Property . 
owl:AsymmetricProperty rdf:type rdfs:Class . owl:AsymmetricProperty rdfs:subClassOf owl:ObjectProperty . 
owl:Axiom rdf:type rdfs:Class . owl:Axiom rdfs:subClassOf rdfs:Resource . 
owl:Class rdf:type rdfs:Class . owl:Class rdfs:subClassOf rdfs:Class . 
owl:DataRange rdf:type rdfs:Class . owl:DataRange rdfs:subClassOf rdfs:Datatype . 
owl:DatatypeProperty rdf:type rdfs:Class . owl:DatatypeProperty rdfs:subClassOf rdf:Property . 
owl:DeprecatedClass rdf:type rdfs:Class . owl:DeprecatedClass rdfs:subClassOf rdfs:Class . 
owl:DeprecatedProperty rdf:type rdfs:Class . owl:DeprecatedProperty rdfs:subClassOf rdf:Property . 
owl:FunctionalProperty rdf:type rdfs:Class . owl:FunctionalProperty rdfs:subClassOf rdf:Property . 
owl:InverseFunctionalProperty rdf:type rdfs:Class . owl:InverseFunctionalProperty rdfs:subClassOf owl:ObjectProperty . 
owl:IrreflexiveProperty rdf:type rdfs:Class . owl:IrreflexiveProperty rdfs:subClassOf owl:ObjectProperty . 
owl:NamedIndividual rdf:type rdfs:Class . owl:NamedIndividual rdfs:subClassOf owl:Thing . 
owl:NegativePropertyAssertion rdf:type rdfs:Class . owl:NegativePropertyAssertion rdfs:subClassOf rdfs:Resource . 
owl:Nothing rdf:type owl:Class . owl:Nothing rdfs:subClassOf owl:Thing . 
owl:ObjectProperty rdf:type rdfs:Class . owl:ObjectProperty rdfs:subClassOf rdf:Property . 
owl:Ontology rdf:type rdfs:Class . owl:Ontology rdfs:subClassOf rdfs:Resource . 
owl:OntologyProperty rdf:type rdfs:Class . owl:OntologyProperty rdfs:subClassOf rdf:Property . 
owl:ReflexiveProperty rdf:type rdfs:Class . owl:ReflexiveProperty rdfs:subClassOf owl:ObjectProperty . 
owl:Restriction rdf:type rdfs:Class . owl:Restriction rdfs:subClassOf owl:Class . 
owl:SymmetricProperty rdf:type rdfs:Class . owl:SymmetricProperty rdfs:subClassOf owl:ObjectProperty . 
owl:Thing rdf:type owl:Class . 
owl:TransitiveProperty rdf:type rdfs:Class . owl:TransitiveProperty rdfs:subClassOf owl:ObjectProperty . 
owl:allValuesFrom rdf:type rdf:Property . owl:allValuesFrom rdfs:domain owl:Restriction . owl:allValuesFrom rdfs:range rdfs:Class . 
owl:annotatedProperty rdf:type rdf:Property . owl:annotatedProperty rdfs:domain rdfs:Resource . owl:annotatedProperty rdfs:range rdfs:Resource . 
owl:annotatedSource rdf:type rdf:Property . owl:annotatedSource rdfs:domain rdfs:Resource . owl:annotatedSource rdfs:range rdfs:Resource . 
owl:annotatedTarget rdf:type rdf:Property . owl:annotatedTarget rdfs:domain rdfs:Resource . owl:annotatedTarget rdfs:range rdfs:Resource . 
owl:assertionProperty rdf:type rdf:Property . owl:assertionProperty rdfs:domain owl:NegativePropertyAssertion . owl:assertionProperty rdfs:range rdf:Property . 
owl:backwardCompatibleWith rdf:type owl:AnnotationProperty . owl:backwardCompatibleWith rdf:type owl:OntologyProperty . owl:backwardCompatibleWith rdfs:domain owl:Ontology . owl:backwardCompatibleWith rdfs:range owl:Ontology . 
owl:bottomDataProperty rdf:type owl:DatatypeProperty . owl:bottomDataProperty rdfs:domain owl:Thing . owl:bottomDataProperty rdfs:range rdfs:Literal . 
owl:bottomObjectProperty rdf:type owl:ObjectProperty . owl:bottomObjectProperty rdfs:domain owl:Thing . owl:bottomObjectProperty rdfs:range owl:Thing . 
owl:cardinality rdf:type rdf:Property . owl:cardinality rdfs:domain owl:Restriction . owl:cardinality rdfs:range xsd:nonNegativeInteger . 
owl:complementOf rdf:type rdf:Property . owl:complementOf rdfs:domain owl:Class . owl:complementOf rdfs:range owl:Class . 
owl:datatypeComplementOf rdf:type rdf:Property . owl:datatypeComplementOf rdfs:domain rdfs:Datatype . owl:datatypeComplementOf rdfs:range rdfs:Datatype . 
owl:deprecated rdf:type owl:AnnotationProperty . owl:deprecated rdfs:domain rdfs:Resource . owl:deprecated rdfs:range rdfs:Resource . 
owl:differentFrom rdf:type rdf:Property . owl:differentFrom rdfs:domain owl:Thing . owl:differentFrom rdfs:range owl:Thing . 
owl:disjointUnionOf rdf:type rdf:Property . owl:disjointUnionOf rdfs:domain owl:Class . owl:disjointUnionOf rdfs:range rdf:List . 
owl:disjointWith rdf:type rdf:Property . owl:disjointWith rdfs:domain owl:Class . owl:disjointWith rdfs:range owl:Class . 
owl:distinctMembers rdf:type rdf:Property . owl:distinctMembers rdfs:domain owl:AllDifferent . owl:distinctMembers rdfs:range rdf:List . 
owl:equivalentClass rdf:type rdf:Property . owl:equivalentClass rdfs:domain rdfs:Class . owl:equivalentClass rdfs:range rdfs:Class . 
owl:equivalentProperty rdf:type rdf:Property . owl:equivalentProperty rdfs:domain rdf:Property . owl:equivalentProperty rdfs:range rdf:Property . 
owl:hasKey rdf:type rdf:Property . owl:hasKey rdfs:domain owl:Class . owl:hasKey rdfs:range rdf:List . 
owl:hasSelf rdf:type rdf:Property . owl:hasSelf rdfs:domain owl:Restriction . owl:hasSelf rdfs:range rdfs:Resource . 
owl:hasValue rdf:type rdf:Property . owl:hasValue rdfs:domain owl:Restriction . owl:hasValue rdfs:range rdfs:Resource . 
owl:imports rdf:type owl:OntologyProperty . owl:imports rdfs:domain owl:Ontology . owl:imports rdfs:range owl:Ontology . 
owl:incompatibleWith rdf:type owl:AnnotationProperty . owl:incompatibleWith rdf:type owl:OntologyProperty . owl:incompatibleWith rdfs:domain owl:Ontology . owl:incompatibleWith rdfs:range owl:Ontology . 
owl:intersectionOf rdf:type rdf:Property . owl:intersectionOf rdfs:domain rdfs:Class . owl:intersectionOf rdfs:range rdf:List . 
owl:inverseOf rdf:type rdf:Property . owl:inverseOf rdfs:domain owl:ObjectProperty . owl:inverseOf rdfs:range owl:ObjectProperty . 
owl:maxCardinality rdf:type rdf:Property . owl:maxCardinality rdfs:domain owl:Restriction . owl:maxCardinality rdfs:range xsd:nonNegativeInteger . 
owl:maxQualifiedCardinality rdf:type rdf:Property . owl:maxQualifiedCardinality rdfs:domain owl:Restriction . owl:maxQualifiedCardinality rdfs:range xsd:nonNegativeInteger . 
owl:members rdf:type rdf:Property . owl:members rdfs:domain rdfs:Resource . owl:members rdfs:range rdf:List . 
owl:minCardinality rdf:type rdf:Property . owl:minCardinality rdfs:domain owl:Restriction . owl:minCardinality rdfs:range xsd:nonNegativeInteger . 
owl:minQualifiedCardinality rdf:type rdf:Property . owl:minQualifiedCardinality rdfs:domain owl:Restriction . owl:minQualifiedCardinality rdfs:range xsd:nonNegativeInteger . 
owl:onClass rdf:type rdf:Property . owl:onClass rdfs:domain owl:Restriction . owl:onClass rdfs:range owl:Class . 
owl:onDataRange rdf:type rdf:Property . owl:onDataRange rdfs:domain owl:Restriction . owl:onDataRange rdfs:range rdfs:Datatype . 
owl:onDatatype rdf:type rdf:Property . owl:onDatatype rdfs:domain rdfs:Datatype . owl:onDatatype rdfs:range rdfs:Datatype . 
owl:oneOf rdf:type rdf:Property . owl:oneOf rdfs:domain rdfs:Class . owl:oneOf rdfs:range rdf:List . 
owl:onProperty rdf:type rdf:Property . owl:onProperty rdfs:domain owl:Restriction . owl:onProperty rdfs:range rdf:Property . 
owl:onProperties rdf:type rdf:Property . owl:onProperties rdfs:domain owl:Restriction . owl:onProperties rdfs:range rdf:List . 
owl:priorVersion rdf:type owl:AnnotationProperty . owl:priorVersion rdf:type owl:OntologyProperty . owl:priorVersion rdfs:domain owl:Ontology . owl:priorVersion rdfs:range owl:Ontology . 
owl:propertyChainAxiom rdf:type rdf:Property . owl:propertyChainAxiom rdfs:domain owl:ObjectProperty . owl:propertyChainAxiom rdfs:range rdf:List . 
owl:propertyDisjointWith rdf:type rdf:Property . owl:propertyDisjointWith rdfs:domain rdf:Property . owl:propertyDisjointWith rdfs:range rdf:Property . 
owl:qualifiedCardinality rdf:type rdf:Property . owl:qualifiedCardinality rdfs:domain owl:Restriction . owl:qualifiedCardinality rdfs:range xsd:nonNegativeInteger . 
owl:sameAs rdf:type rdf:Property . owl:sameAs rdfs:domain owl:Thing . owl:sameAs rdfs:range owl:Thing . 
owl:someValuesFrom rdf:type rdf:Property . owl:someValuesFrom rdfs:domain owl:Restriction . owl:someValuesFrom rdfs:range rdfs:Class . 
owl:sourceIndividual rdf:type rdf:Property . owl:sourceIndividual rdfs:domain owl:NegativePropertyAssertion . owl:sourceIndividual rdfs:range owl:Thing . 
owl:targetIndividual rdf:type rdf:Property . owl:targetIndividual rdfs:domain owl:NegativePropertyAssertion . owl:targetIndividual rdfs:range owl:Thing . 
owl:targetValue rdf:type rdf:Property . owl:targetValue rdfs:domain owl:NegativePropertyAssertion . owl:targetValue rdfs:range rdfs:Literal . 
owl:topDataProperty rdf:type owl:DatatypeProperty . owl:topDataProperty rdfs:domain owl:Thing . owl:topDataProperty rdfs:range rdfs:Literal . 
owl:topObjectProperty rdf:type rdf:ObjectProperty . owl:topObjectProperty rdfs:domain owl:Thing . owl:topObjectProperty rdfs:range owl:Thing . 
owl:unionOf rdf:type rdf:Property . owl:unionOf rdfs:domain rdfs:Class . owl:unionOf rdfs:range rdf:List . 
owl:versionInfo rdf:type owl:AnnotationProperty . owl:versionInfo rdfs:domain rdfs:Resource . owl:versionInfo rdfs:range rdfs:Resource . 
owl:versionIRI rdf:type owl:OntologyProperty . owl:versionIRI rdfs:domain owl:Ontology . owl:versionIRI rdfs:range owl:Ontology . 
owl:withRestrictions rdf:type rdf:Property . owl:withRestrictions rdfs:domain rdfs:Datatype . owl:withRestrictions rdfs:range rdf:List . 
xsd:anyURI rdf:type rdfs:Datatype . xsd:anyURI rdfs:subClassOf rdfs:Literal . 
xsd:base64Binary rdf:type rdfs:Datatype . xsd:base64Binary rdfs:subClassOf rdfs:Literal . 
xsd:boolean rdf:type rdfs:Datatype . xsd:boolean rdfs:subClassOf rdfs:Literal . 
xsd:byte rdf:type rdfs:Datatype . xsd:byte rdfs:subClassOf rdfs:Literal . 
xsd:dateTime rdf:type rdfs:Datatype . xsd:dateTime rdfs:subClassOf rdfs:Literal . 
xsd:dateTimeStamp rdf:type rdfs:Datatype . xsd:dateTimeStamp rdfs:subClassOf rdfs:Literal . 
xsd:decimal rdf:type rdfs:Datatype . xsd:decimal rdfs:subClassOf rdfs:Literal . 
xsd:double rdf:type rdfs:Datatype . xsd:double rdfs:subClassOf rdfs:Literal . 
xsd:float rdf:type rdfs:Datatype . xsd:float rdfs:subClassOf rdfs:Literal . 
xsd:hexBinary rdf:type rdfs:Datatype . xsd:hexBinary rdfs:subClassOf rdfs:Literal . 
xsd:int rdf:type rdfs:Datatype . xsd:int rdfs:subClassOf rdfs:Literal . 
xsd:integer rdf:type rdfs:Datatype . xsd:integer rdfs:subClassOf rdfs:Literal . 
xsd:language rdf:type rdfs:Datatype . xsd:language rdfs:subClassOf rdfs:Literal . 
xsd:long rdf:type rdfs:Datatype . xsd:long rdfs:subClassOf rdfs:Literal . 
xsd:Name rdf:type rdfs:Datatype . xsd:Name rdfs:subClassOf rdfs:Literal . 
xsd:NCName rdf:type rdfs:Datatype . xsd:NCName rdfs:subClassOf rdfs:Literal . 
xsd:negativeInteger rdf:type rdfs:Datatype . xsd:negativeInteger rdfs:subClassOf rdfs:Literal . 
xsd:NMTOKEN rdf:type rdfs:Datatype . xsd:NMTOKEN rdfs:subClassOf rdfs:Literal . 
xsd:nonNegativeInteger rdf:type rdfs:Datatype . xsd:nonNegativeInteger rdfs:subClassOf rdfs:Literal . 
xsd:nonPositiveInteger rdf:type rdfs:Datatype . xsd:nonPositiveInteger rdfs:subClassOf rdfs:Literal . 
xsd:normalizedString rdf:type rdfs:Datatype . xsd:normalizedString rdfs:subClassOf rdfs:Literal . 
rdf:PlainLiteral rdf:type rdfs:Datatype . rdf:PlainLiteral rdfs:subClassOf rdfs:Literal . 
xsd:positiveInteger rdf:type rdfs:Datatype . xsd:positiveInteger rdfs:subClassOf rdfs:Literal . 
owl:rational rdf:type rdfs:Datatype . owl:rational rdfs:subClassOf rdfs:Literal . 
owl:real rdf:type rdfs:Datatype . owl:real rdfs:subClassOf rdfs:Literal . 
xsd:short rdf:type rdfs:Datatype . xsd:short rdfs:subClassOf rdfs:Literal . 
xsd:string rdf:type rdfs:Datatype . xsd:string rdfs:subClassOf rdfs:Literal . 
xsd:token rdf:type rdfs:Datatype . xsd:token rdfs:subClassOf rdfs:Literal . 
xsd:unsignedByte rdf:type rdfs:Datatype . xsd:unsignedByte rdfs:subClassOf rdfs:Literal . 
xsd:unsignedInt rdf:type rdfs:Datatype . xsd:unsignedInt rdfs:subClassOf rdfs:Literal . 
xsd:unsignedLong rdf:type rdfs:Datatype . xsd:unsignedLong rdfs:subClassOf rdfs:Literal . 
xsd:unsignedShort rdf:type rdfs:Datatype . xsd:unsignedShort rdfs:subClassOf rdfs:Literal . 
rdf:XMLLiteral rdf:type rdfs:Datatype . rdf:XMLLiteral rdfs:subClassOf rdfs:Literal . 
rdf:langRange rdf:type owl:DatatypeProperty . rdf:langRange rdfs:domain rdfs:Resource . rdf:langRange rdfs:range rdfs:Literal . 
xsd:length rdf:type owl:DatatypeProperty . xsd:length rdfs:domain rdfs:Resource . xsd:length rdfs:range rdfs:Literal . 
xsd:maxExclusive rdf:type owl:DatatypeProperty . xsd:maxExclusive rdfs:domain rdfs:Resource . xsd:maxExclusive rdfs:range rdfs:Literal . 
xsd:maxInclusive rdf:type owl:DatatypeProperty . xsd:maxInclusive rdfs:domain rdfs:Resource . xsd:maxInclusive rdfs:range rdfs:Literal . 
xsd:maxLength rdf:type owl:DatatypeProperty . xsd:maxLength rdfs:domain rdfs:Resource . xsd:maxLength rdfs:range rdfs:Literal . 
xsd:minExclusive rdf:type owl:DatatypeProperty . xsd:minExclusive rdfs:domain rdfs:Resource . xsd:minExclusive rdfs:range rdfs:Literal . 
xsd:minInclusive rdf:type owl:DatatypeProperty . xsd:minInclusive rdfs:domain rdfs:Resource . xsd:minInclusive rdfs:range rdfs:Literal . 
xsd:minLength rdf:type owl:DatatypeProperty . xsd:minLength rdfs:domain rdfs:Resource . xsd:minLength rdfs:range rdfs:Literal . 
xsd:pattern rdf:type owl:DatatypeProperty . xsd:pattern rdfs:domain rdfs:Resource . xsd:pattern rdfs:range rdfs:Literal . 
rdfs:Class rdfs:subClassOf owl:Class . 
rdfs:comment rdf:type owl:AnnotationProperty . rdfs:comment rdfs:domain rdfs:Resource . rdfs:comment rdfs:range rdfs:Literal . 
rdfs:Datatype rdfs:subClassOf owl:DataRange . 
rdfs:isDefinedBy rdf:type owl:AnnotationProperty . rdfs:isDefinedBy rdfs:domain rdfs:Resource . rdfs:isDefinedBy rdfs:range rdfs:Resource . 
rdfs:label rdf:type owl:AnnotationProperty . rdfs:label rdfs:domain rdfs:Resource . rdfs:label rdfs:range rdfs:Literal . 
rdfs:Literal rdf:type rdfs:Datatype . 
rdf:Property rdfs:subClassOf owl:ObjectProperty . 
rdfs:Resource rdfs:subClassOf owl:Thing . 
rdfs:seeAlso rdf:type owl:AnnotationProperty . rdfs:seeAlso rdfs:domain rdfs:Resource . rdfs:seeAlso rdfs:range rdfs:Resource . 
7 Appendix: Relationship to the Direct Semantics (Informative)
This section compares the OWL 2 RDFBased Semantics with the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. While the OWL 2 RDFBased Semantics is based on the RDF Semantics specification [RDF Semantics], the OWL 2 Direct Semantics is a description logic style semantics. Several fundamental differences exist between the two semantics, but there is also a strong relationship basically stating that the OWL 2 RDFBased Semantics is able to reflect all logical conclusions of the OWL 2 Direct Semantics. This means that the OWL 2 Direct Semantics can in a sense be regarded as a semantics subset of the OWL 2 RDFBased Semantics.
Technically, the comparison will be performed by comparing the sets of entailments that hold for each of the two semantics, respectively. The definition of an OWL 2 RDFBased entailment was given in Section 4.3 of this document, while the definition of an OWL 2 Direct entailment is provided in Section 2.5 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. In both cases, entailments are defined for pairs of ontologies, and such an ordered pair of two ontologies will be called an entailment query in this section.
Comparing the two semantics by means of entailments will only be meaningful if the entailment queries allow for applying both the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics to them. In order to ensure this, the comparison will be restricted to entailment queries, for which the lefthand side and righthand side ontologies are both OWL 2 DL ontologies in RDF graph form. These are RDF graphs that, by applying the reverse RDF mapping [OWL 2 RDF Mapping], can be transformed into corresponding OWL 2 DL ontologies in Functional Syntax form according to the functional style syntax defined in the OWL 2 Structural Specification [OWL 2 Specification], and which must further meet all the restrictions on OWL 2 DL ontologies that are specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification]. In fact, these restrictions must be mutually met by both ontologies that occur in an entailment query, i.e. all these restrictions need to be satisfied as if the two ontologies would be part of a single ontology. Any entailment query that adheres to the conditions defined here will be called an OWL 2 DL entailment query.
Ideally, the relationship between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics would be of the form that every OWL 2 DL entailment query that is an OWL 2 Direct entailment is also an OWL 2 RDFBased entailment. However, this desirable relationship cannot hold in general due to a variety of differences that exist between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, as demonstrated in Section 7.1.
Fortunately, the problems resulting from these semantic differences can be overcome in a way that for every OWL 2 DL entailment query there is another one for which the desired entailment relationship indeed holds, and the new entailment query is semantically equivalent to the original entailment query under the OWL 2 Direct Semantics. This is the gist of the OWL 2 correspondence theorem, which will be presented in Section 7.2. The proof of this theorem, given in Section 7.3, will further demonstrate that such a substitute OWL 2 DL entailment query can always be algorithmically constructed by means of simple syntactic transformations.
7.1 Example on Semantic Differences
This section will show that differences exist between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, and it will be demonstrated how these semantic differences complicate a comparison of the two semantics in terms of entailments. An example OWL 2 DL entailment query will be given, which will happen to be an OWL 2 Direct entailment without being an OWL 2 RDFBased entailment. The section will explain the different reasons and will provide a resolution of each of them. It will turn out that the example entailment query can be syntactically transformed into another OWL 2 DL entailment query that is both an OWL 2 Direct entailment and an OWL 2 RDFBased entailment, while being semantically unchanged compared to the original entailment query under the OWL 2 Direct Semantics. This example will motivate the OWL 2 correspondence theorem in Section 7.2 and its proof in Section 7.3.
The example entailment query consists of the following pair ( G_{1}^{*} , G_{2}^{*} ) of RDF graphs:
G_{1}^{*} :
(1) ex:o1 rdf:type owl:Ontology .
(2) ex:c1 rdf:type owl:Class .
(3) ex:c2 rdf:type owl:Class .
(4) ex:c1 rdfs:subClassOf ex:c2 .
G_{2}^{*} :
(1) ex:o2 rdf:type owl:Ontology .
(2) ex:c1 rdf:type owl:Class .
(3) ex:c2 rdf:type owl:Class .
(4) ex:c3 rdf:type owl:Class .
(5) ex:c1 rdfs:subClassOf _:x .
(6) _:x rdf:type owl:Class .
(7) _:x owl:unionOf ( ex:c2 ex:c3 ) .
(8) ex:c3 rdfs:label "c3" .
Both G_{1}^{*} and G_{2}^{*} are OWL 2 DL ontologies in RDF graph form and can therefore be mapped by the reverse RDF mapping [OWL 2 RDF Mapping] to the following two OWL 2 DL ontologies in Functional Syntax form F(G_{1}^{*}) and F(G_{2}^{*}):
F(G_{1}^{*}) :
(1) Ontology( ex:o1
(2) Declaration( Class( ex:c1 ) )
(3) Declaration( Class( ex:c2 ) )
(4) SubClassOf( ex:c1 ex:c2 )
(5) )
F(G_{2}^{*}) :
(1) Ontology( ex:o2
(2) Declaration( Class( ex:c1 ) )
(3) Declaration( Class( ex:c2 ) )
(4) Declaration( Class( ex:c3 ) )
(5) SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )
(6) AnnotationAssertion( rdfs:label ex:c3 "c3" )
(7) )
Note that F(G_{1}^{*}) and F(G_{2}^{*}) mutually meet the restrictions on OWL 2 DL ontologies as specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification]. For example, none of the IRIs being declared as a class in F(G_{1}^{*}) is declared as a datatype in F(G_{2}^{*}), since this would not be allowed for an OWL 2 DL entailment query.
It follows that F(G_{1}^{*}) OWL 2 Direct entails F(G_{2}^{*}). To show this, only the axioms (4) of F(G_{1}^{*}) and (5) of F(G_{2}^{*}) have to be considered. None of the other statements in the two ontologies are relevant for this OWL 2 Direct entailment to hold, since they do not have a formal meaning under the OWL 2 Direct Semantics. However, it turns out that the RDF graph G_{1}^{*} does not OWL 2 RDFBased entail G_{2}^{*}, for reasons discussed in detail now.
Reason 1: An Annotation in F(G_{2}^{*}). The ontology F(G_{2}^{*}) contains an annotation (6). The OWL 2 Direct Semantics does not give a formal meaning to annotations. In contrast, under the OWL 2 RDFBased Semantics every RDF triple occurring in an RDF graph has a formal meaning, including the corresponding annotation triple (8) in G_{2}^{*}. Since this annotation triple only occurs in G_{2}^{*} but not in G_{1}^{*}, there will exist OWL 2 RDFBased interpretations that satisfy G_{1}^{*} without satisfying triple (8) of G_{2}^{*}. Hence, G_{1}^{*} does not OWL 2 RDFBased entail G_{2}^{*}.
Resolution of Reason 1. The annotation triple (8) in G_{2}^{*} will be removed, which will avoid requiring OWL 2 RDFBased interpretations to interpret this triple. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since annotations are strictly optional in OWL 2 DL ontologies and may therefore be omitted. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since annotations do not have a formal meaning under this semantics.
Reason 2: An Entity Declaration exclusively in F(G_{2}^{*}). The ontology F(G_{2}^{*}) contains an entity declaration for the class IRI ex:c3 (4), for which there is no corresponding entity declaration in F(G_{1}^{*}). The OWL 2 Direct Semantics does not give a formal meaning to entity declarations, while the OWL 2 RDFBased Semantics gives a formal meaning to the corresponding declaration statement (4) in G_{2}^{*}. The consequences are analog to those described for reason 1.
Resolution of Reason 2. The declaration statement (4) in G_{2}^{*} will be copied to G_{1}^{*}. An OWL 2 RDFBased interpretation that satisfies the modified graph G_{1}^{*} will then also satisfy the declaration statement. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since the copied declaration statement is not in conflict with any of the other entity declarations in G_{1}^{*}. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since entity declarations do not have a formal meaning under this semantics.
Reason 3: Different Ontology IRIs in F(G_{1}^{*}) and F(G_{2}^{*}). The ontology IRIs for the two ontologies, given by (1) in F(G_{1}^{*}) and by (1) in F(G_{2}^{*}), differ from each other. The OWL 2 Direct Semantics does not give a formal meaning to ontology headers, while the OWL 2 RDFBased Semantics gives a formal meaning to the corresponding header triples (1) in G_{1}^{*} and (1) in G_{2}^{*}. Since these header triples differ from each other, the consequences are analog to those described for reason 1.
Resolution of Reason 3. The IRI in the subject position of the header triple (1) in G_{2}^{*} is changed into a blank node. Due to the existential semantics of blank nodes under the OWL 2 RDFBased Semantics the resulting triple will then be entailed by triple (1) in G_{1}^{*}. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since an ontology IRI is optional for an OWL 2 DL ontology. (Note, however, that it would have been an error to simply remove triple (1) from G_{2}^{*}, since an OWL 2 DL ontology is required to contain an ontology header.) Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since ontology headers do not have a formal meaning under this semantics.
Reason 4: A Class Expression in F(G_{2}^{*}). Axiom (5) of F(G_{2}^{*}) contains a class expression that represents the union of the two classes denoted by ex:c2 and ex:c3. Within G_{2}^{*}, this class expression is represented by the triples (6) and (7), both having the blank node "_:x" in their respective subject position. The way the OWL 2 RDFBased Semantics interprets these two triples differs from the way the OWL 2 Direct Semantics treats the class expression in axiom (5) of F(G_{2}^{*}).
The OWL 2 Direct Semantics treats classes as sets, i.e. subsets of the universe. Thus, the IRIs ex:c2 and ex:c3 in F(G_{2}^{*}) denote two sets, and the class expression in axiom (5) of F(G_{2}^{*}) therefore represents the set that consists of the union of these two sets.
The OWL 2 RDFBased Semantics, on the other hand, treats classes as individuals, i.e. members of the universe. While every class under the OWL 2 RDFBased Semantics represents a certain subset of the universe, namely its class extension, this set is actually distinguished from the class itself. For two given classes it is ensured under the OWL 2 RDFBased Semantics, just as for the OWL 2 Direct Semantics, that the union of their class extensions will always exist as a subset of the universe. However, there is no guarantee that there will also exist an individual in the universe that has this set union as its class extension.
Under the OWL 2 RDFBased Semantics, triple (7) of G_{2}^{*} essentially claims that a class exists being the union of two other classes. But since the existence of such a union class is not ensured by G_{1}^{*}, there will be OWL 2 RDFBased interpretations that satisfy G_{1}^{*} without satisfying triple (7) of G_{2}^{*}. Hence, G_{1}^{*} does not OWL 2 RDFBased entail G_{2}^{*}.
Resolution of Reason 4. The triples (6) and (7) of G_{2}^{*} are copied to G_{1}^{*} together with the new triple "_:x owl:equivalentClass _:x". In addition, for the IRI ex:c3, which only occurs in the union class expression but not in G_{1}^{*}, an entity declaration is added to G_{1}^{*} by the resolution of reason 2. If an OWL 2 RDFBased interpretation satisfies the modified graph G_{1}^{*}, then the triples (6) and (7) of G_{2}^{*} will now be satisfied. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since the whole set of added triples validly encodes an OWL 2 axiom, and since none of the restrictions on OWL 2 DL ontologies is hurt. Also, this operation will not change the formal meaning of the ontologies under the OWL 2 Direct Semantics, since the added equivalence axiom is a tautology under this semantics.
Note that it would have been an error to simply copy the triples (6) and (7) of G_{2}^{*} to G_{1}^{*}, without also adding the new triple "_:x owl:equivalentClass _:x". This would have produced a class expression that has no connection to any axiom in the ontology. An OWL 2 DL ontology is basically a set of axioms and does not allow for the occurrence of "dangling" class expressions. This is the reason for actually "embedding" the class expression in an axiom. It would have also been wrong to use an arbitrary axiom for such an embedding, since it has to be ensured that the formal meaning of the original ontology does not change under the OWL 2 Direct Semantics. However, any tautological axiom that contains the original class expression would have been sufficient for this purpose as well.
Complete Resolution: The Transformed Entailment Query.
Combining the resolutions of all the above reasons leads to the following new pair of RDF graphs ( G_{1} , G_{2} ):
G_{1} :
(1) ex:o1 rdf:type owl:Ontology .
(2) ex:c1 rdf:type owl:Class .
(3) ex:c2 rdf:type owl:Class .
(4) ex:c3 rdf:type owl:Class .
(5) ex:c1 rdfs:subClassOf ex:c2 .
(6) _:x owl:equivalentClass _:x .
(7) _:x rdf:type owl:Class .
(8) _:x owl:unionOf ( ex:c2 ex:c3 ) .
G_{2} :
(1) _:o rdf:type owl:Ontology .
(2) ex:c1 rdf:type owl:Class .
(3) ex:c2 rdf:type owl:Class .
(4) ex:c3 rdf:type owl:Class .
(5) ex:c1 rdfs:subClassOf _:x .
(6) _:x rdf:type owl:Class .
(7) _:x owl:unionOf ( ex:c2 ex:c3 ) .
The following list reiterates the changes compared to the original RDF graphs G_{1}^{*} and G_{2}^{*}:
 Resolution of Reason 1 (Annotation): Triple (8) in G_{2}^{*} has been removed, i.e. there is no corresponding annotation triple in G_{2}.
 Resolution of Reason 2 (Entity Declaration): Triple (4) in G_{2}^{*} has been copied to G_{1}^{*}, becoming triple (4) in G_{1}.
 Resolution of Reason 3 (Ontology IRIs): The IRI in the subject position of triple (1) in G_{2}^{*} has been changed into a blank node, becoming triple (1) in G_{2}.
 Resolution of Reason 4 (Class Expression): Triples (6) and (7) in G_{2}^{*} have been copied to G_{1}^{*} together with the new triple "_:x owl:equivalentClass _:x", becoming triples (6), (7) and (8) in G_{1}.
G_{1} and G_{2} are again OWL 2 DL ontologies in RDF graph form and can be mapped to the following OWL 2 DL ontologies in Functional Syntax form F(G_{1}) and F(G_{2}), which again mutually meet the restrictions on OWL 2 DL ontologies:
F(G_{1}) :
(1) Ontology( ex:o1
(2) Declaration( Class( ex:c1 ) )
(3) Declaration( Class( ex:c2 ) )
(4) Declaration( Class( ex:c3 ) )
(5) SubClassOf( ex:c1 ex:c2 )
(6) EquivalentClasses( ObjectUnionOf( ex:c2 ex:c3 ) ObjectUnionOf( ex:c2 ex:c3 ) )
(7) )
F(G_{2}) :
(1) Ontology(
(2) Declaration( Class( ex:c1 ) )
(3) Declaration( Class( ex:c2 ) )
(4) Declaration( Class( ex:c3 ) )
(5) SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )
(6) )
As said earlier, all the applied changes preserve the formal meaning of the original OWL 2 DL ontologies under the OWL 2 Direct Semantics. Hence, it is still the case that F(G_{1}) OWL 2 Direct entails F(G_{2}). However, due to the syntactic transformation the situation has changed for the OWL 2 RDFBased Semantics: it is now possible to show, by following the lines of argumentation for the resolutions of the different reasons given above, that G_{1} OWL 2 RDFBased entails G_{2} as well.
7.2 Correspondence Theorem
This section presents the OWL 2 correspondence theorem, which compares the semantic expressivity of the OWL 2 RDFBased Semantics with that of the OWL 2 Direct Semantics. The theorem basically states that the OWL 2 RDFBased Semantics is able to reflect all the semantic conclusions of the OWL 2 Direct Semantics, where the notion of a "semantic conclusion" is technically expressed in terms of an entailment.
However, as discussed in Section 7.1, there exist semantic differences between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, which do not allow for stating that any OWL 2 DL entailment query that is an OWL 2 Direct entailment will always also be an OWL 2 RDFBased entailment. Nevertheless, it can still be ensured that any given OWL 2 DL entailment query can be substituted by another OWL 2 DL entailment query in a way that for the substitute entailment query the desired relationship will really hold, while preserving the formal meaning compared to the original entailment query under the OWL 2 Direct Semantics.
In fact, the theorem only makes the seemingly weak assertion that such a substitute entailment query will always exist. But the actual proof for the theorem in Section 7.3 will be more concrete in that it will substitute each given OWL 2 DL entailment query with a variant that can be algorithmically constructed by applying a set of simple syntactic transformations to the original entailment query. One can get an idea of how this works from Section 7.1.
Technical Note on Corresponding Datatype Maps. A distinction exists between the format of an OWL 2 RDFBased datatype map, as defined by Definition 4.1, and the format of an OWL 2 Direct datatype map, as defined in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. It is, however, possible to translate between an OWL 2 RDFBased datatype map D and the corresponding OWL 2 Direct datatype map F(D) in the following way:
For an OWL 2 RDFBased datatype map D, the corresponding OWL 2 Direct datatype map F(D) := ( N_{DT} , N_{LS} , N_{FS} , ⋅ ^{DT} , ⋅ ^{LS} , ⋅ ^{FS} ) [OWL 2 Direct Semantics] is given by
 Datatype Names: N_{DT} is defined as the set of all IRIs u, for which there is a datatype d, such that ( u , d ) ∈ D.
 Lexical Space: For each datatype name u ∈ N_{DT}, set N_{LS}(u) := LS(d), where ( u , d ) ∈ D.
 Facet Space: For each datatype name u ∈ N_{DT}, set N_{FS}(u) := FS(d), where ( u , d ) ∈ D.
 Value Space: For each datatype name u ∈ N_{DT}, set (u) ^{DT} := VS(d), where ( u , d ) ∈ D.
 LexicaltoValue Mapping: For each datatype name u ∈ N_{DT} and each lexical form a ∈ N_{LS}(u), set ( a , u ) ^{LS} := L2V(d)(a), where ( u , d ) ∈ D.
 FacettoValue Mapping: For each datatype name u ∈ N_{DT} and each facetvalue pair ( F , v ) ∈ N_{FS}(u), set ( F , v ) ^{FS} := F2V(d)(( F , v )), where ( u , d ) ∈ D.
Theorem 7.1 (OWL 2 Correspondence Theorem):
Let D be an OWL 2 RDFBased datatype map according to Definition 4.1, with F(D) being the OWL 2 Direct datatype map according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] that corresponds to D according to the technical note on corresponding datatype maps. Let G_{1}^{*} and G_{2}^{*} be RDF graphs that are OWL 2 DL ontologies in RDF graph form, with F(G_{1}^{*}) and F(G_{2}^{*}) being the OWL 2 DL ontologies in Functional Syntax form [OWL 2 Specification] that result from applying the reverse RDF mapping [OWL 2 RDF Mapping] to G_{1}^{*} and G_{2}^{*}, respectively. Let F(G_{1}^{*}) and F(G_{2}^{*}) mutually meet the restrictions on OWL 2 DL ontologies as specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification].
Then, there exist RDF graphs G_{1} and G_{2} that are OWL 2 DL ontologies in RDF graph form, such that all the following relationships hold, with F(G_{1}) and F(G_{2}) being the OWL 2 DL ontologies in Functional Syntax form that result from applying the reverse RDF mapping to G_{1} and G_{2}, respectively:
 F(G_{1}) and F(G_{2}) mutually meet the restrictions on OWL 2 DL ontologies.
 F(G_{1}) OWL 2 Direct entails F(G_{1}^{*}) with respect to F(D), and F(G_{1}^{*}) OWL 2 Direct entails F(G_{1}) with respect to F(D).
 F(G_{2}) OWL 2 Direct entails F(G_{2}^{*}) with respect to F(D), and F(G_{2}^{*}) OWL 2 Direct entails F(G_{2}) with respect to F(D).
 If F(G_{1}) OWL 2 Direct entails F(G_{2}) with respect to F(D), then G_{1} OWL 2 RDFBased entails G_{2} with respect to D.
7.3 Proof for the Correspondence Theorem
This is the sketch of a proof for Theorem 7.1 (OWL 2 Correspondence Theorem) in Section 7.2. The proof sketch provides the basic line of argumentation for showing the theorem. However, for complexity reasons, some technical aspects of the theorem are only coarsely treated, and the proof sketch also refrains from considering the full amount of OWL 2 language constructs. For certain steps of the proof there are example calculations that focus only on a small fraction of language constructs, but which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDFBased Semantics could be constructed in principle. A complete proof could make use of the observation that the definitions of the OWL 2 Direct Semantics and the OWL 2 RDFBased Semantics, despite their technical differences as outlined in Section 7.1, are closely aligned with respect to the different language constructs of OWL 2.
The proof sketch will make use of an approach that will be called "balancing" throughout this section, and which will now be introduced. The basic idea is to substitute the original pair of RDF graphs in an OWL 2 DL entailment query by another entailment query having the same semantic characteristics under the OWL 2 Direct Semantics, but for which the technical differences between the two semantics specifications have no relevant consequences under the OWL 2 RDFBased Semantics anymore. A concrete example for the application of this approach was given in Section 7.1.
Definition (Balanced): A pair of RDF graphs ( G_{1} , G_{2} ) is called balanced, if and only if G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, such that all the following conditions hold, with F(G_{1}) and F(G_{2}) being the OWL 2 DL ontologies in Functional Syntax form [OWL 2 Specification] that result from applying the reverse RDF mapping [OWL 2 RDF Mapping] to G_{1} and G_{2}, respectively:
 F(G_{1}) and F(G_{2}) mutually meet the restrictions on OWL 2 DL ontologies as specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification].
 Nodes in G_{1} and G_{2}:
 for every IRI u occurring in G_{1} or G_{2} that corresponds to a nonbuiltin entity in F(G_{1}) or F(G_{2}), respectively, the graph contains, for every entity type T of u, a declaration statement of the form "u rdf:type t", where t is the vocabulary class IRI corresponding to T (see Table 7 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping] and Section 5.8 of the OWL 2 Structural Specification [OWL 2 Specification]);
 every plain or typed literal occurring in G_{2} also occurs in G_{1} (see Section 4 of the OWL 2 Structural Specification [OWL 2 Specification]).
 G_{2} contains exactly one ontology header consisting of a single RDF triple of the form "x rdf:type owl:Ontology", where x is either a blank node or, if an ontology IRI is used in G_{1}, may alternatively equal that ontology IRI (see Table 4 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]).
 G_{2} does not contain:
 the RDF encoding of an annotation (see Sections 3.2.2 and 3.2.3, and Table 17 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 a statement with an ontology property such as "owl:imports";
 a deprecation statement based on "owl:DeprecatedClass", "owl:DeprecatedProperty" and "owl:deprecated" (see Table 16 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 an annotation property axiom based on "rdfs:subClassOf", "rdfs:domain" and "rdfs:range" (see Table 16 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]).
 Any of the following sub graphs of G_{2} is also a sub graph of G_{1}:
 the RDF encoding of an entity declaration (see Table 7 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 the RDF encoding of a property expression (see Table 11 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 the RDF encoding of a class expression (see Tables 13 and 15 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 the RDF encoding of a data range expression (see Tables 12 and 14 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]);
 an RDF sequence (see Table 3 in the OWL 2 RDF Mapping [OWL 2 RDF Mapping]).
Balancing Lemma: An algorithm exists that terminates on every valid input and that has the following input/output behavior:
The valid input of the algorithm is given by all the pairs of RDF graphs ( G_{1}^{*} , G_{2}^{*} ), where G_{1}^{*} and G_{2}^{*} are OWL 2 DL ontologies in RDF graph form, with F(G_{1}^{*}) and F(G_{2}^{*}) being the OWL 2 DL ontologies in Functional Syntax form [OWL 2 Specification] that result from applying the reverse RDF mapping [OWL 2 RDF Mapping] to G_{1}^{*} and G_{2}^{*}, respectively. Further, F(G_{1}^{*}) and F(G_{2}^{*}) have to mutually meet the restrictions on OWL 2 DL ontologies as specified in Section 3 of the OWL 2 Structural Specification [OWL 2 Specification].
For a valid input, the output of the algorithm is a pair of RDF graphs ( G_{1} , G_{2} ), where G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, such that for any OWL 2 RDFBased datatype map D according to Definition 4.1 all the following relationships hold, with F(G_{1}) and F(G_{2}) being the OWL 2 DL ontologies in Functional Syntax form that result from applying the reverse RDF mapping to G_{1} and G_{2}, respectively, and with F(D) being the OWL 2 Direct datatype map according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] that corresponds to D according to the technical note on corresponding datatype maps in Section 7.2:
 The pair ( G_{1} , G_{2} ) is balanced.
 F(G_{1}) OWL 2 Direct entails F(G_{1}^{*}) with respect to F(D), and F(G_{1}^{*}) OWL 2 Direct entails F(G_{1}) with respect to F(D).
 F(G_{2}) OWL 2 Direct entails F(G_{2}^{*}) with respect to F(D), and F(G_{2}^{*}) OWL 2 Direct entails F(G_{2}) with respect to F(D).
Proof for the Balancing Lemma:
Let the graph pair ( G_{1}^{*} , G_{2}^{*} ) be a valid input. The resulting RDF graphs G_{1} and G_{2} are constructed as follows, starting from copies of G_{1}^{*} and G_{2}^{*}, respectively.
Since the initial versions of G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, the canonical parsing process (CP) for computing the reverse RDF mapping, as described in Section 3 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping], can be applied. Based on CP, it is possible to identify within these graphs
 all entity types for every nonbuiltin IRI,
 all blank nodes that correspond to anonymous individuals, and
 all sub graphs that correspond to OWL 2 language constructs (ontology headers, declarations, expressions, axioms and annotations) as described in the OWL 2 Structural Specification [OWL 2 Specification].
Based on this observation, the following steps are performed:
 Consistently substitute all blank nodes in G_{2} such that G_{1} and G_{2} have no common blank nodes.
 Apply CP to G_{1} and G_{2} (without changing these graphs) to identify the entity types of the IRIs, the anonymous individuals, and the sub graphs encoding OWL 2 language constructs.
 For each sub graph g of G_{2}: remove g from G_{2}, if g is the RDF encoding of
 an annotation, or
 a deprecation statement, or
 an annotation property axiom.
 For the sub graph g of G_{2} corresponding to the ontology header in F(G_{2}): substitute g in G_{2} by a triple of the form "x rdf:type owl:Ontology", where x is a new blank node not yet used in G_{2}.
 For each nonbuiltin IRI u in G_{1} and G_{2} and for each entity type T of u identified by CP: add to G_{1} or G_{2}, respectively, the RDF triple "u rdf:type t", where t is the vocabulary class IRI corresponding to T.
 For each plain or typed literal L in G_{2}: add to G_{1} the RDF triple "o rdfs:comment L", where o is the IRI or blank node of the ontology header triple "o rdf:type owl:Ontology" in G_{1}.
 For each sub graph g of G_{2} that is the RDF encoding of an entity declaration: add g to G_{1}.
 For each sub graph g of G_{2} that is the RDF encoding of a property expression with root blank node x: add g to G_{1} together with the RDF triple "x owl:equivalentProperty x".
 For each sub graph g of G_{2} that is the RDF encoding of a class expression with root blank node x: add g to G_{1} together with the RDF triple "x owl:equivalentClass x".
 For each sub graph g of G_{2} that is the RDF encoding of a data range expression with root blank node x:
 If g is part of a data property restriction expression, then nothing needs to be done, since the comprising restriction expression is covered by the treatment of class expressions, and therefore g occurs in G_{1} as well.
 Otherwise, add a declaration triple to G_{1} for a new data property p that does not yet occur in G_{1} and G_{2}. Then, the RDF encoding r of a universal data property restriction expression on property p is created for g. Let r have the new root blank node y. Add r to G_{1} together with the RDF triple "y owl:equivalentClass y".
 For each sub graph g of G_{2} that is an RDF sequence with root blank node x, which does not occur in the RDF encoding of language constructs already treated by one of the earlier steps, i.e. g is part of the encoding of an axiom: create the RDF encoding r of an enumeration class expression with a new root blank node y having the main RDF triple "y owl:oneOf x". Then, add r to G_{1} together with the RDF triple "y owl:equivalentClass y". Additionally, for every IRI u being a member of the RDF sequence, add to G_{1} a typing triple "u rdf:type owl:NamedIndividual". If one of the sequence members is a blank node z that is the root node of some property expression or class expression e, then select a new IRI w not yet occurring in G_{1}, consistently replace z by w everywhere in r, add to G_{1} the triple "w owl:equivalentProperty z" or "w owl:equivalentClass z", respectively, and add to G_{1} the two triples "w rdf:type owl:NamedIndividual" and "w rdf:type t", where t is the vocabulary class IRI that represents the appropriate entity type of the expression e. No further treatment of e is needed, since e is treated by the earlier steps covering expressions.
In the following it is shown that all the claims of the balancing lemma hold.
A: Existence of a Terminating Algorithm. An algorithm exists for mapping the input graph pair ( G_{1}^{*} , G_{2}^{*} ) to the output graph pair ( G_{1} , G_{2} ), since CP (applied in step 2) is described in the form of an algorithm in the OWL 2 RDF Mapping [OWL 2 RDF Mapping], and since all other steps can obviously be performed algorithmically. The algorithm terminates, since CP terminates on arbitrary input graphs, and since all other steps can obviously be executed in finite time.
B: The Resulting RDF Graphs are OWL 2 DL Ontologies. The RDF graphs G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form that mutually meet the restrictions on OWL 2 DL ontologies, since the original RDF graphs G_{1}^{*} and G_{2}^{*} have this feature, and since each of the steps described above transforms a pair of RDF graphs with this feature again into a pair of RDF graphs with this feature, for the following reasons:
 The consistent substitution of blank nodes in step 1 does not change the structure of an OWL 2 DL ontology.
 The application of CP in step 2 does not change the graphs.
 Annotations, deprecation statements and annotation property axioms are optional information in an OWL 2 DL ontology and can therefore be omitted in step 3.
 The ontology header of an OWL 2 DL ontology does neither require the existence of an ontology IRI nor of any ontology properties, and so the substitution of the ontology header in step 4 is a valid operation.
 If an entity has some particular entity type for which there is no explicitly given entity declaration, then the entity declaration may be added, as done in step 5.
 It is allowed to add arbitrary annotations to the ontology header of an OWL 2 DL ontology, as done in step 6.
 Entity declarations may be copied from G_{2} to G_{1} in step 7 without conflict, since the original ontologies have been assumed to mutually meet the restrictions on OWL 2 DL ontologies regarding different entity declarations for the same IRI (e.g. that one IRI must not be the name of both an object property and a data property).
 Adding to G_{1} an axiom that claims equivalence of some property expression (step 8) or class expression (step 9) with itself, where the expression already occurs in G_{2}, is an allowed operation, since the original ontologies are assumed to mutually meet the restrictions on OWL 2 DL ontologies concerning property and class expressions, and since no syntactic restrictions exist on this specific use of equivalence axioms.
 For the case of data ranges (step 10) it is sufficient to note that placing universal property restrictions on arbitrary (simple or complex) property expressions is allowed in OWL 2 DL. The rest of the argumentation follows the lines of the treatment of class expressions in step 9.
 For the treatment of RDF sequences in step 11: First, the enumeration class expressions being constructed from the RDF sequences are syntactically valid in OWL 2 DL, since all enumerated entries are IRIs by construction. Second, there is no restriction in OWL 2 DL disallowing axioms that claim equivalence of enumeration class expressions with themselves. Third, punning in OWL 2 DL allows a given nonbuiltin IRI of any entity type to be additionally declared as a named individual. Forth, there is no OWL 2 DL restriction forbidding to add an entity declaration for a new (i.e. not elsewhere used) IRI and to assert the denotation of this new IRI to be equivalent to some existing property or class expression. Hence, the resulting ontologies still mutually meet all syntactic restrictions on OWL 2 DL ontologies.
C: The Resulting Pair of RDF Graphs is Balanced. All the conditions of balanced pairs of RDF graphs are met by the pair ( G_{1} , G_{2} ) for the following reasons:
 Condition 1: It has already been shown in paragraph B that G_{1} and G_{2} mutually meet the restrictions on OWL 2 DL ontologies.
 Conditions 2.1 and 2.2 on nodes in G_{1} and G_{2} are met by steps 5 and 6, respectively.
 Condition 3 on ontology headers in G_{2} is satisfied by step 4, always applying an anonymous ontology header.
 Conditions 4.1, 4.3 and 4.4 on annotations, deprecation statements and annotation property axioms in G_{2}, respectively, are all satisfied by step 3.
 Condition 4.2 on statements with ontology properties is implicitly satisfied by step 4, since the substitution of the ontology header in G_{2} removes all existing statements with ontology properties.
 Condition 5.1 on entity declarations in G_{2} being reflected in G_{1} is satisfied by step 7.
 Conditions 5.2, 5.3 and 5.4 on property, class and data range expressions in G_{2}, respectively, being reflected in G_{1} are met by steps 8, 9 and 10, respectively.
 Condition 5.5 on RDF sequences in G_{2} being reflected in G_{1} is satisfied by step 11.
D: The Resulting Ontologies are semantically equivalent with the Original Ontologies under the OWL 2 Direct Semantics. F(G_{1}) is semantically equivalent with F(G_{1}^{*}), since F(G_{1}) differs from F(G_{1}^{*}) only by (potentially):
 additional entity declarations (steps 5, 7 and 11), which have no formal meaning under the OWL 2 Direct Semantics;
 additional annotations (step 6), which have no formal meaning;
 additional tautological axioms (steps 8, 9, 10 and 11), which do not change the formal meaning;
F(G_{2}) is semantically equivalent with F(G_{2}^{*}), since F(G_{2}) differs from F(G_{2}^{*}) only by (potentially):
 differently labeled anonymous individuals (step 1), by which the formal meaning under the OWL 2 Direct Semantics keeps unchanged, since anonymous individuals are existentially interpreted;
 missing annotations, deprecation statements and annotation property axioms (step 3), which have no formal meaning;
 a modified ontology header (step 4), which has no formal meaning;
 additional entity declarations (step 5), which have no formal meaning.
End of Proof for the Balancing Lemma.
In the following, the correspondence theorem will be proven.
Assume that the premises of the correspondence theorem are true for a given pair ( G_{1}^{*} , G_{2}^{*} ) of RDF graphs. This allows for applying the balancing lemma, which provides the existence of corresponding RDF graphs G_{1} and G_{2} that are OWL 2 DL ontologies in RDF graph form, and which meet the definition of balanced graph pairs. Let F(G_{1}) and F(G_{2}) be the corresponding OWL 2 DL ontologies in Functional Syntax form. Then, the claimed relationship 1 of the correspondence theorem follows directly from relationship 1 of the balancing lemma and from condition 1 of the definition of balanced graph pairs. Further, the claimed relationships 2 and 3 of the correspondence theorem follow directly from the relationships 2 and 3 of the balancing lemma, respectively.
The rest of this proof will treat the claimed relationship 4 of the correspondence theorem, which states that if F(G_{1}) OWL 2 Direct entails F(G_{2}) with respect to F(D), then G_{1} OWL 2 RDFBased entails G_{2} with respect to D. For this to see, an arbitrary OWL 2 RDFBased interpretation I will be selected that OWL 2 RDFBased satisfies G_{1}. For I, a closely corresponding OWL 2 Direct interpretation J will be constructed, and it will then be shown that J OWL 2 Direct satisfies F(G_{1}). Since it was assumed that F(G_{1}) OWL 2 Direct entails F(G_{2}), it will follow that J OWL 2 Direct satisfies F(G_{2}). Based on this result, it will then be possible to show that I also OWL 2 RDFBased satisfies G_{2}. Since I was arbitrarily selected, this will mean that G_{1} OWL 2 RDFBased entails G_{2}.
Step 1: Selection of a Pair of Corresponding Interpretations.
Let F(G_{1}) OWL 2 Direct entail F(G_{2}) w.r.t. F(D), and let I be an OWL 2 RDFBased interpretation of a vocabulary V^{I} w.r.t. D, such that I OWL 2 RDFBased satisfies G_{1}.
Since the pair ( G_{1} , G_{2} ) is balanced, there exist entity declarations in F(G_{1}) for each entity type of every nonbuiltin IRI occurring in G_{1}: For each entity declaration of the form "Declaration(T(u))" in F(G_{1}), such that T is the entity type for some IRI u, a typing triple of the form "u rdf:type t" exists in G_{1}, where t is the vocabulary class IRI representing the part of the universe of I that corresponds to T. Since I OWL 2 RDFBased satisfies G_{1}, all these declaration typing triples are OWL 2 RDFBased satisfied by I, and thus all nonbuiltin IRIs in G_{1} are instances of all their declared parts of the universe of I.
The vocabulary V^{J} := ( V^{J}_{C} , V^{J}_{OP} , V^{J}_{DP} , V^{J}_{I} , V^{J}_{DT} , V^{J}_{LT} , V^{J}_{FA} ) of the OWL 2 Direct interpretation J w.r.t. the datatype map F(D) is now constructed as follows.
 The set V^{J}_{C} of classes contains all IRIs in V^{I} that are declared as classes in F(G_{1}), together with all the required class names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
 The set V^{J}_{OP} of object properties contains all IRIs in V^{I} that are declared as object properties in F(G_{1}), together with all the required object property names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
 The set V^{J}_{DP} of data properties contains all IRIs in V^{I} that are declared as data properties in F(G_{1}), together with all the required data property names listed in Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics].
 The set V^{J}_{I} of individuals contains all IRIs in V^{I} that are declared as named individuals in F(G_{1}), and additionally all anonymous individuals occurring in F(G_{1}) and F(G_{2}).
 The set V^{J}_{DT} of datatypes is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D), together with all other IRIs in V^{I} that are declared as datatypes in F(G_{1}).
 The set V^{J}_{LT} of literals is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D).
 The set V^{J}_{FA} of facetliteral pairs is defined according to Section 2.1 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics] w.r.t. the datatype map F(D).
The OWL 2 Direct interpretation J := ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} ) is now defined as follows. The object and data domains of J are identified with the universe IR and the set of data values LV of I, respectively, i.e., Δ_{I} := IR and Δ_{D} := LV. The class interpretation function ⋅ ^{C}, the object property interpretation function ⋅ ^{OP}, the data property interpretation function ⋅ ^{DP}, the datatype interpretation function ⋅ ^{DT}, the literal interpretation function ⋅ ^{LT}, and the facet interpretation function ⋅ ^{FA} are defined according to Section 2.2 of the OWL 2 Direct Semantics [OWL 2 Direct Semantics]. Specifically, for every nonbuiltin IRI u occurring in F(G_{1}):
 If u is declared as a class, then set u^{C} := ICEXT(I(u)), since G_{1} contains the triple "u rdf:type owl:Class", i.e., I(u) ∈ IC.
 If u is declared as an object property, then set u^{OP} := IEXT(I(u)), since G_{1} contains the triple "u rdf:type owl:ObjectProperty", i.e., I(u) ∈ IP.
 If u is declared as a data property, then set u^{DP} := IEXT(I(u)), since G_{1} contains the triple "u rdf:type owl:DatatypeProperty", i.e., I(u) ∈ IODP.
 If u is declared as a named individual, then set u^{I} := I(u), since G_{1} contains the triple "u rdf:type owl:NamedIndividual", i.e., I(u) ∈ IR.
 If u is declared as a datatype, then set u^{DT} := ICEXT(I(u)), since G_{1} contains the triple "u rdf:type rdfs:Datatype", i.e., I(u) ∈ IDC.
Notes:
 A literal occurring in G_{1} is mapped by the reverse RDF mapping to the same literal in F(G_{1}), and the formal meaning of a wellformed literal is analog for both the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics.
 A blank node b occurring in G_{1} that represents an anonymous individual is written as the same blank node b in F(G_{1}). Both the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics treat anonymous individuals in an analog way as existential variables defined locally to a given ontology, i.e. some individual x exists in the universe to which all occurrences of b in the ontology can be mapped (see Section 1.5 in the RDF Semantics [RDF Semantics] for the precise definition on how blank nodes are treated under the OWL 2 RDFBased Semantics). Hence, the same mapping from b to x can be used with both I and J.
 G_{1} may also contain declarations for annotation properties. Since annotation properties have no formal meaning under the OWL 2 Direct Semantics, the OWL 2 Direct interpretation J does not treat them.
 With the above definition it is possible for J to have a nonseparated vocabulary according to Section 5.9 of the OWL 2 Structural Specification [OWL 2 Specification]. Since G_{1} is an OWL 2 DL ontology in RDF graph form, it is allowed that the same IRI u may be declared as one or more of an individual name, either a class name or a datatype name, and either an object property name or a data property name. For the OWL 2 RDFBased interpretation I, the IRI u will always denote the same individual in the universe IR, where I(u) may additionally have a class extension or a property extension, or both. For the OWL 2 Direct interpretation J, however, u will denote as an individual name an element of Δ_{I}, as a class name a subset of Δ_{I}, as a datatype name a subset of Δ_{D}, as an object property name a subset of Δ_{I} × Δ_{I}, and as a data property name a subset of Δ_{I} × Δ_{D}.
Step 2: Satisfaction of F(G_{1}) by the OWL 2 Direct Interpretation.
Based on the premise that I OWL 2 RDFBased satisfies G_{1}, it has to be shown that J OWL 2 Direct satisfies F(G_{1}). For this to hold, it will be sufficient that J OWL 2 Direct satisfies every axiom A occurring in F(G_{1}). Let g_{A} be the sub graph of G_{1} that is mapped to A by the reverse RDF mapping. The basic idea can roughly be described as follows:
Since I is an OWL 2 RDFBased interpretation, all the OWL 2 RDFBased semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, OWL 2 RDFBased semantic conditions exist that semantically correspond to the definition of the interpretation of the axiom A. In particular, the antecedent of one of these semantic conditions will become true, if the RDFencoding of A, i.e. the graph g_{A}, is satisfied (in the case of an "ifandonlyif" semantic condition this will generally be the lefttoright direction of that condition). Now, all the RDF triples in g_{A} are OWL 2 RDFBased satisfied by I, since I OWL 2 RDFBased satisfies G_{1}. Hence, the antecedent of the semantic condition becomes true, and therefore its consequent becomes true as well. This will reveal a certain semantic relationship that according to I holds between the denotations of the IRIs, literals and anonymous individuals occurring in g_{A}, which, roughly speaking, expresses the meaning of the OWL 2 axiom A. Because of the close semantic correspondence of the OWL 2 Direct interpretation J to I, the analog semantic relationship holds according to J between the denotations of the IRIs, literals and anonymous individuals occurring in A. This semantic relationship turns out to be compatible with the formal meaning of the axiom A as specified by the OWL 2 Direct Semantics, i.e. J satisfies A.
This basic idea is now demonstrated in more detail for a single example axiom A in F(G_{1}), which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDFBased Semantics could be constructed in principle.
Let A be the following OWL 2 axiom in F(G_{1}):
A : SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))
and let g_{A} be the corresponding sub graph in G_{1} that is being mapped to A via the reverse RDF mapping, namely
g_{A} :
ex:c1 rdfs:subClassOf _:x .
_:x rdf:type owl:Class .
_:x owl:unionOf ( ex:c2 ex:c3 ) .
Since the pair ( G_{1} , G_{2} ) is balanced, G_{1} contains the typing triples
ex:c1 rdf:type owl:Class .
ex:c2 rdf:type owl:Class .
ex:c3 rdf:type owl:Class .
that correspond to class entity declarations in F(G_{1}) for the IRIs "ex:c1", "ex:c2", and "ex:c3", respectively. All these declaration typing triples are OWL 2 RDFBased satisfied by I, since it has been postulated that I OWL 2 RDFBased satisfies G_{1}. Hence, by applying the semantics of rdf:type (see Section 4.1 of the RDF Semantics [RDF Semantics]), all the IRIs denote classes, precisely:
I(ex:c1) ∈ IC ,
I(ex:c2) ∈ IC , and
I(ex:c3) ∈ IC .
Since I is an OWL 2 RDFBased interpretation, it meets all the OWL 2 RDFBased semantic conditions, and since I OWL 2 RDFBased satisfies G_{1}, all the triples in g_{A} are OWL 2 RDFBased satisfied. This meets the lefttoright directions of the semantic conditions for subclass axioms ("rdfs:subClassOf", see Section 5.8) and union class expressions ("owl:unionOf", see Section 5.4), which results in the following semantic relationship that holds between the extensions of the classes above according to I:
ICEXT(I(ex:c1)) ⊆ ICEXT(I(ex:c2)) ∪ ICEXT(I(ex:c3)) .
By applying the definition of J, one can conclude that the following semantic relationship holds between the denotations of the class names occurring in A according to J:
(ex:c1) ^{C} ⊆ (ex:c2) ^{C} ∪ (ex:c3) ^{C} .
This semantic relationship is compatible with the formal meaning of the axiom A under the OWL 2 Direct Semantics. Hence, J OWL 2 Direct satisfies A.
Since J OWL 2 Direct satisfies F(G_{1}), and since it has been postulated that F(G_{1}) OWL 2 Direct entails F(G_{2}), it follows that J OWL 2 Direct satisfies F(G_{2}).
Step 3: Satisfaction of G_{2} by the OWL 2 RDFBased Interpretation.
The last step will be to show that I OWL 2 RDFBased satisfies G_{2}. For this to hold, I needs to OWL 2 RDFBased satisfy every triple occurring in G_{2}. The basic idea can roughly be described as follows:
First: According to the "semantic conditions for ground graphs" in Section 1.4 of the RDF Semantics specification [RDF Semantics], all the IRIs and literals used in RDF triples in G_{2} need to be in the vocabulary V^{I} of I. This is true for the following reason: Since the pair ( G_{1} , G_{2} ) is balanced, all IRIs and literals occurring in G_{2} do also occur in G_{1}. Since I satisfies G_{1}, all IRIs and literals in G_{1}, including those in G_{2}, are contained in V^{I} due to the semantic conditions for ground graphs.
Second: If a set of RDF triples encodes an OWL 2 language construct that is not interpreted by the OWL 2 Direct Semantics, such as annotations, then G_{2} should contain such a set of RDF triples only if they are also included in G_{1}. The reason is that with such triples there will, in general, exist OWL 2 RDFBased interpretations only satisfying the graph G_{1} but not G_{2}, which will render the pair ( G_{1} , G_{2} ) into a nonentailment (an exception are RDF triples that are true under every OWL 2 RDFBased interpretation). Since the pair ( G_{1} , G_{2} ) is balanced, G_{2} will not contain the RDF encoding for any annotations, statements with ontology properties, deprecation statements or annotation property axioms. Hence, there are no corresponding RDF triples that need to be satisfied by I.
Third: Since G_{2} is an OWL 2 DL ontology in RDF graph form, the graph is partitioned by the reverse RDF mapping [OWL 2 RDF Mapping] into sub graphs corresponding to either ontology headers, entity declarations or axioms, where axioms may further consist of different kinds of expressions, such as Boolean class expressions. It has to be shown that all the triples in each such sub graph are OWL 2 RDFBased satisfied by I.
For ontology headers: Let A be the ontology header of F(G_{2}) and let g_{A} be the corresponding sub graph of G_{2}. Since the pair ( G_{1} , G_{2} ) is balanced, g_{A} is encoded as a single RDF triple of the form "x rdf:type owl:Ontology", where x is either an IRI or a blank node. Since G_{1} is an OWL 2 DL ontology in RDF graph form, G_{1} also contains the encoding of an ontology header including a triple g_{1} of the form "y rdf:type owl:Ontology", where y is either an IRI or a blank node. Since I OWL 2 RDFBased satisfies G_{1}, g_{1} is satisfied by I. If both y and x are IRIs, then, due to balancing, x equals y, and therefore g_{A} equals g_{1}, i.e. g_{A} is OWL 2 RDFBased satisfied by I. Otherwise, balancing forces x to be a blank node, i.e. x is treated as an existential variable under the OWL 2 RDFBased Semantics according to the "semantic conditions for blank nodes" [RDF Semantics]. From this observation, and from the premise that I satisfies g_{1}, it follows that g_{A} is OWL 2 RDFBased satisfied by I.
For entity declarations: Let A be an entity declaration in F(G_{2}), and let g_{A} be the corresponding sub graph of G_{2}. Since the pair ( G_{1} , G_{2} ) is balanced, A occurs in F(G_{1}), and hence g_{A} is a sub graph of G_{1}. Since I OWL 2 RDFBased satisfies G_{1}, I OWL 2 RDFBased satisfies g_{A}.
For axioms: Let A be an axiom in F(G_{2}), and let g_{A} be the corresponding sub graph of G_{2}. Since I is an OWL 2 RDFBased interpretation, all the OWL 2 RDFBased semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, OWL 2 RDFBased semantic conditions exist that semantically correspond to the definition of the interpretation of the axiom A. In particular, the consequent of one of these semantic conditions corresponds to the RDFencoding of A, i.e. the graph g_{A}, except for declaration typing triples, for which satisfaction has already been shown (in the case of an "ifandonlyif" semantic condition this will generally be the righttoleft direction of that condition). Hence, in order to show that g_{A} is OWL 2 RDFBased satisfied by I, it will be sufficient to show that the antecedent of this semantic condition is true. In general, several requirements have to be met to ensure this:
Requirement 1: The denotations of all the nonbuiltin IRIs in g_{A} have to be contained in the appropriate part of the universe of I. This can be shown as follows. For every nonbuiltin IRI u occurring in g_{A}, u also occurs in A. Since the pair ( G_{1} , G_{2} ) is balanced, there are entity declarations in F(G_{2}) for all the entity types of u, each being of the form D := "Declaration(T(u))" for some entity type T. From the reverse RDF mapping follows that for each such declaration D a typing triple d exists in G_{2}, being of the form d := "u rdf:type t", where t is the vocabulary class IRI representing the part of the universe of I that corresponds to the entity type T. It has already been shown that, for D being an entity declaration in F(G_{2}) and d being the corresponding sub graph in G_{2}, I OWL 2 RDFBased satisfies d. Hence, I(u) is an individual contained in the appropriate part of the universe.
Requirement 2: For every expression E occurring in A, with the RDF encoding g_{E} in g_{A}, an individual has to exist in the universe of I that appropriately represents the denotation of E. Since I is an OWL 2 RDFBased interpretation, all the OWL 2 RDFBased semantic conditions are met by I. Due to the close alignment between the definitions in the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics, OWL 2 RDFBased semantic conditions exist that semantically correspond to the definition of the interpretation of the expression E. In particular, the antecedent of one of these semantic conditions will become true, if the RDFencoding of E, i.e. the graph g_{E}, is satisfied (in the case of an "ifandonlyif" semantic condition this will generally be the lefttoright direction of that condition). Now, since the pair ( G_{1} , G_{2} ) is balanced, g_{E} also occurs in G_{1}. So, since I OWL 2 RDFBased satisfies G_{1}, g_{E} is OWL 2 RDFBased satisfied by I. Hence, the antecedent of the semantic condition becomes true, and therefore its consequent becomes true as well. This will result in the existence of an individual with the required properties, when taking into account existential blank node semantics.
Requirement 3: A semantic relationship has to hold between the denotations of the IRIs, literals and anonymous individuals occurring in g_{A} with respect to I, which, roughly speaking, expresses the meaning of the OWL 2 axiom A. This is the case for the following reasons: First, the literals and anonymous individuals occurring in A and g_{A}, respectively, are interpreted in an analog way under the OWL 2 Direct Semantics and the OWL 2 RDFBased Semantics. Second, it was assumed that the OWL 2 Direct interpretation J OWL 2 Direct satisfies A, and therefore a semantic relationship with the desired properties holds with respect to J. Third, J has been defined in close correspondence to I, so that for the semantic relationship expressed by J an analog semantic relationship holds with respect to I.
This basic idea is now demonstrated in more detail for a single example axiom A in F(G_{2}), which can be taken as a hint on how a complete proof taking into account every feature of the OWL 2 RDFBased Semantics could be constructed in principle.
Let A be the following OWL 2 axiom in F(G_{2}):
A : SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))
and let g_{A} be the corresponding sub graph in G_{2} that is being mapped to A via the reverse RDF mapping, namely
g_{A} :
ex:c1 rdfs:subClassOf _:x .
_:x rdf:type owl:Class .
_:x owl:unionOf ( ex:c2 ex:c3 ) .
First, since the pair ( G_{1} , G_{2} ) is balanced, G_{2} contains the typing triples
ex:c1 rdf:type owl:Class .
ex:c2 rdf:type owl:Class .
ex:c3 rdf:type owl:Class .
that correspond to class entity declarations in F(G_{2}) for the IRIs "ex:c1", "ex:c2", and "ex:c3", respectively. All these declaration typing triples are OWL 2 RDFBased satisfied by I, since due to balancing the typing triples exist in G_{1} as well, and since it has been postulated that I OWL 2 RDFBased satisfies all triples in G_{1}. Hence, by applying the semantics of rdf:type (see Section 4.1 of the RDF Semantics [RDF Semantics]), all the IRIs denote classes, and therefore the denotations of the IRIs are included in the appropriate part of the universe of I, precisely:
I(ex:c1) ∈ IC ,
I(ex:c2) ∈ IC , and
I(ex:c3) ∈ IC .
Second, g_{A} contains the sub graph g_{E}, given by
g_{E} :
_:x rdf:type owl:Class .
_:x owl:unionOf ( c2 c3 ) .
which corresponds to the union class expression E in A, given by
E : ObjectUnionOf(ex:c2 ex:c3)
Since the pair ( G_{1} , G_{2} ) is balanced, g_{E} occurs as a sub graph of G_{1} as well. g_{E} contains blank nodes and, since I satisfies G_{1}, the semantic conditions for RDF graphs with blank nodes apply (see Section 1.5 of the RDF Semantics [RDF Semantics]). This provides the existence of a mapping B from blank(g_{E}) to IR, where blank(g_{E}) is the set of all blank nodes occurring in g_{E}. It follows that the extended interpretation I+B OWL 2 RDFBased satisfies all the triples in g_{E}. Further, since I is an OWL 2 RDFBased interpretation, I meets all the OWL 2 RDFBased semantic conditions. Thus, the lefttoright direction of the semantic condition for union class expressions ("owl:unionOf", see Section 5.4) applies, providing:
[I+B](_:x) ∈ IC ,
ICEXT([I+B](_:x))
=
ICEXT(I(ex:c2))
∪
ICEXT(I(ex:c3)) .
Third, since the OWL 2 Direct interpretation J OWL 2 Direct satisfies A, the following semantic relationship holds between the denotations of the class names in A according to J:
(ex:c1) ^{C} ⊆ (ex:c2) ^{C} ∪ (ex:c3) ^{C} .
By applying the definition of the OWL 2 Direct interpretation J, one can conclude that the following semantic relationship holds between the extensions of the classes above according to I:
ICEXT(I(ex:c1)) ⊆ ICEXT(I(ex:c2)) ∪ ICEXT(I(ex:c3)) .
Finally, combining all intermediate results gives
I(ex:c1) ∈ IC ,
[I+B](_:x) ∈ IC ,
ICEXT(I(ex:c1))
⊆
ICEXT([I+B](_:x)) .
Therefore, all the premises are met to apply the righttoleft direction of the semantic condition for subclass axioms ("rdfs:subClassOf", see Section 5.8), which results in
( I(ex:cl) , [I+B](_:x) ) ∈ IEXT(I(rdfs:subClassOf)) .
So, the remaining triple
ex:c1 rdfs:subClassOf _:x .
in g_{A} is OWL 2 RDFBased satisfied by I+B, where "_:x" is the root blank node of the union class expression g_{E}. Hence, w.r.t. existential blank node semantics, I OWL 2 RDFBased satisfies all the triples in g_{A}.
To conclude, for any OWL 2 RDFBased interpretation I that OWL 2 RDFBased satisfies G_{1}, I also OWL 2 RDFBased satisfies G_{2}. Hence, G_{1} OWL 2 RDFBased entails G_{2}, and therefore relationship 4 of the correspondence theorem holds. Q.E.D.
8 Appendix: Comprehension Conditions (Informative)
The correspondence theorem in Section 7.2 shows that it is possible for the OWL 2 RDFBased Semantics to reflect all the entailments of the OWL 2 Direct Semantics [OWL 2 Direct Semantics], provided that one allows for certain "harmless" syntactic transformations on the RDF graphs being considered. This makes numerous potentially desirable and useful entailments available that would otherwise be outside the scope of the OWL 2 RDFBased Semantics, for the technical reasons discussed in Section 7.1. It seems natural to ask for similar entailments even when an entailment query does not consist of OWL 2 DL ontologies in RDF graph form. However, the correspondence theorem does not apply to such cases, and thus the OWL 2 Direct Semantics cannot be taken as a reference frame for "desirable" and "useful" entailments, or for when a graph transformation can be considered "harmless" or not.
As discussed in Section 7.1, a core obstacle for the correspondence theorem to hold was the RDF encoding of OWL 2 expressions, such as union class expressions, when they appear on the right hand side of an entailment query. Under the OWL 2 RDFBased Semantics it is not generally ensured that an individual exists, which represents the denotation of such an expression. The "comprehension conditions" defined in this section are additional semantic conditions that provide the necessary individuals for every sequence, class and property expression. By this, the combination of the normative semantic conditions of the OWL 2 RDFBased Semantics (Section 5) and the comprehension conditions can be regarded to "simulate" the semantic expressivity of the OWL 2 Direct Semantics on entailment queries consisting of arbitrary RDF graphs.
The combined semantics is, however, not primarily intended for use in actual implementations. The comprehension conditions add significantly to the complexity and expressivity of the basic semantics and, in fact, have proven to lead to formal inconsistency. But the combined semantics can still be seen as a generalized reference frame for "desirable" and "useful" entailments, and this can be used, for example, to evaluate methods that syntactically transform unrestricted entailment queries in order to receive additional entailments under the OWL 2 RDFBased Semantics. Such a concrete method is, however, outside the scope of this specification.
Note: The conventions in the introduction of Section 5 ("Semantic Conditions") apply to the current section as well.
8.1 Comprehension Conditions for Sequences
Table 8.1 lists the comprehension conditions for sequences, i.e. RDF lists. These comprehension conditions provide the existence of sequences built from any finite combination of individuals contained in the universe.
if  then exists z_{1} , … , z_{n} ∈ IR 

a_{1} , … , a_{n} ∈ IR  ( z_{1} , a_{1} ) ∈ IEXT(I(rdf:first)) , ( z_{1} , z_{2} ) ∈ IEXT(I(rdf:rest)) , … , ( z_{n} , a_{n} ) ∈ IEXT(I(rdf:first)) , ( z_{n} , I(rdf:nil) ) ∈ IEXT(I(rdf:rest)) 
8.2 Comprehension Conditions for Boolean Connectives
Table 8.2 lists the comprehension conditions for Boolean connectives (see Section 5.4 for the corresponding semantic conditions). These comprehension conditions provide the existence of complements for any class and datatype, and of intersections and unions built from any finite set of classes contained in the universe.
if  then exists z ∈ IR 

s sequence of c_{1} , … , c_{n} ∈ IC  ( z , s ) ∈ IEXT(I(owl:intersectionOf)) 
s sequence of c_{1} , … , c_{n} ∈ IC  ( z , s ) ∈ IEXT(I(owl:unionOf)) 
c ∈ IC  ( z , c ) ∈ IEXT(I(owl:complementOf)) 
d ∈ IDC  ( z , d ) ∈ IEXT(I(owl:datatypeComplementOf)) 
8.3 Comprehension Conditions for Enumerations
Table 8.3 lists the comprehension conditions for enumerations (see Section 5.5 for the corresponding semantic conditions). These comprehension conditions provide the existence of enumeration classes built from any finite set of individuals contained in the universe.
if  then exists z ∈ IR 

s sequence of a_{1} , … , a_{n} ∈ IR  ( z , s ) ∈ IEXT(I(owl:oneOf)) 
8.4 Comprehension Conditions for Property Restrictions
Table 8.4 lists the comprehension conditions for property restrictions (see Section 5.6 for the corresponding semantic conditions). These comprehension conditions provide the existence of cardinality restrictions on any property and for any nonnegative integer, as well as value restrictions on any property and on any class contained in the universe.
Note that the comprehension conditions for self restrictions constrains the right hand side of the produced owl:hasSelf assertions to be the Boolean value "true"^^xsd:boolean. This is in accordance with Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping [OWL 2 RDF Mapping].
Implementations are not required to support the comprehension conditions for owl:onProperties, but MAY support them in order to realize nary dataranges with arity ≥ 2 (see Sections 7 and 8.4 of the OWL 2 Structural Specification [OWL 2 Specification] for further information).
if  then exists z ∈ IR 

c ∈ IC , p ∈ IP 
( z , c ) ∈ IEXT(I(owl:someValuesFrom)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
c ∈ IC , s sequence of p_{1} , … , p_{n} ∈ IP , n ≥ 1 
( z , c ) ∈ IEXT(I(owl:someValuesFrom)) , ( z , s ) ∈ IEXT(I(owl:onProperties)) 
c ∈ IC , p ∈ IP 
( z , c ) ∈ IEXT(I(owl:allValuesFrom)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
c ∈ IC , s sequence of p_{1} , … , p_{n} ∈ IP , n ≥ 1 
( z , c ) ∈ IEXT(I(owl:allValuesFrom)) , ( z , s ) ∈ IEXT(I(owl:onProperties)) 
a ∈ IR , p ∈ IP 
( z , a ) ∈ IEXT(I(owl:hasValue)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
p ∈ IP  ( z , I("true"^^xsd:boolean) ) ∈ IEXT(I(owl:hasSelf)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:minCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:maxCardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:cardinality)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:minQualifiedCardinality)) , ( z , c ) ∈ IEXT(I(owl:onClass)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP 
( z , n ) ∈ IEXT(I(owl:minQualifiedCardinality)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:maxQualifiedCardinality)) , ( z , c ) ∈ IEXT(I(owl:onClass)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP 
( z , n ) ∈ IEXT(I(owl:maxQualifiedCardinality)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP 
( z , n ) ∈ IEXT(I(owl:qualifiedCardinality)) , ( z , c ) ∈ IEXT(I(owl:onClass)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP 
( z , n ) ∈ IEXT(I(owl:qualifiedCardinality)) , ( z , d ) ∈ IEXT(I(owl:onDataRange)) , ( z , p ) ∈ IEXT(I(owl:onProperty)) 
8.5 Comprehension Conditions for Datatype Restrictions
Table 8.5 lists the comprehension conditions for datatype restrictions (see Section 5.7 for the corresponding semantic conditions). These comprehension conditions provide the existence of datatypes built from restricting any datatype contained in the universe by any finite set of facetvalue pairs contained in the facet space (see Section 4.1) of the original datatype.
The set IFS is defined in Section 5.7.
if  then exists z ∈ IR , s sequence of z_{1} , … , z_{n} ∈ IR 

d ∈ IDC , f_{1} , … , f_{n} ∈ IODP , v_{1} , … , v_{n} ∈ LV , ( f_{1} , v_{1} ) , … , ( f_{n} , v_{n} ) ∈ IFS(d) 
( z , d ) ∈ IEXT(I(owl:onDatatype)) , ( z , s ) ∈ IEXT(I(owl:withRestrictions)) , ( z_{1} , v_{1} ) ∈ IEXT(f_{1}) , … , ( z_{n} , v_{n} ) ∈ IEXT(f_{n}) 
8.6 Comprehension Conditions for Inverse Properties
Table 8.6 lists the comprehension conditions for inverse property expressions. These comprehension conditions provide the existence of an inverse property for any property contained in the universe.
Inverse property expressions can be used to build axioms with anonymous inverse properties, such as in the graph
_:x owl:inverseOf ex:p .
_:x rdfs:subPropertyOf owl:topObjectProperty .
Note that, to some extent, the OWL 2 RDFBased Semantics already covers the use of inverse property expressions by means of the semantic conditions of inverse property axioms (see Section 5.12), since these semantic conditions also apply to an existential variable on the left hand side of an inverse property axiom. Nevertheless, not all relevant cases are covered by this semantic condition. For example, one might expect the above example graph to be generally true. However, the normative semantic conditions do not permit this conclusion, since it is not ensured that for every property p there is an individual in the universe with a property extension being inverse to that of p.
if  then exists z ∈ IR 

p ∈ IP  ( z , p ) ∈ IEXT(I(owl:inverseOf)) 
9 Appendix: Changes from OWL 1 (Informative)
This section lists relevant differences between the OWL 2 RDFBased Semantics and the original specification of the OWL 1 RDFCompatible Semantics [OWL 1 RDFCompatible Semantics]. Significant effort has been spent in keeping the design of the OWL 2 RDFBased Semantics as close as possible to that of the OWL 1 RDFCompatible Semantics. While this aim was achieved to a large degree, the OWL 2 RDFBased Semantics actually deviates from its predecessor in several aspects. In most cases this is because of serious technical problems that would have arisen from a conservative semantic extension. Not listed are the new language constructs and the new datatypes of OWL 2.
The following markers are used:
 [DEV]: a deviation from OWL 1 that breaks backward compatibility
 [EXT]: a backward compatible extension to OWL 1
 [NOM]: a change of the nomenclature originally used in OWL 1
 [DPR]: a feature of OWL 1 that has been deprecated as of OWL 2
Generalized Graph Syntax [EXT]. The OWL 2 RDFBased Semantics allows RDF graphs to contain IRIs [RFC 3987] (see Section 2.1), whereas the OWL 1 RDFCompatible Semantics was restricted to RDF graphs with URIs [RFC 2396]. This change is in accordance with the rest of the OWL 2 specification (see Section 2.4 of the OWL 2 Structural Specification [OWL 2 Specification]). In addition, the OWL 2 RDFBased Semantics is now explicitly allowed to be applied to RDF graphs containing "generalized" RDF triples, i.e. triples that can consist of IRIs, literals or blank nodes in all three positions (Section 2.1), although implementations are not required to support this. In contrast, the OWL 1 RDFCompatible Semantics was restricted to RDF graphs conforming to the RDF Concepts specification [RDF Concepts]. These limitations of the OWL 1 RDFCompatible Semantics were actually inherited from the RDF Semantics specification [RDF Semantics]. The relaxations are intended to warrant interoperability with existing and future technologies and tools. Both changes are compatible with OWL 1, since all RDF graphs that were legal under the OWL 1 RDFCompatible Semantics are still legal under the OWL 2 RDFBased Semantics.
Facets for Datatypes [EXT]. The basic definitions of a datatype and a Dinterpretation, as defined by the RDF Semantics specification and as applied by the OWL 1 RDFCompatible Semantics, have been extended to take into account constraining facets (see Section 4), in order to allow for datatype restrictions as specified in Section 5.7. This change is compatible with OWL 1, since Section 5.1 of the RDF Semantics specification explicitly allows for extending the minimal datatype definition provided there.
Correspondence Theorem and Comprehension Conditions [DEV]. The semantic conditions of the OWL 1 RDFCompatible Semantics included a set of so called "comprehension conditions", which allowed to prove the original "correspondence theorem" stating that every entailment of OWL 1 DL was also an entailment of OWL 1 Full. The document at hand adds comprehension conditions for the new language constructs of OWL 2 (see Section 8). However, the comprehension conditions are not a normative aspect of the OWL 2 RDFBased Semantics anymore. It has turned out that combining the comprehension conditions with the normative set of semantic conditions in Section 5 would lead to formal inconsistency of the resulting semantics (Issue 119). In addition, it became clear that a correspondence theorem along the lines of the original theorem would not work for the relationship between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics [OWL 2 Direct Semantics], since it is not possible to "balance" the differences between the two semantics solely by means of additional comprehension conditions (see Section 7.1). Consequently, the correspondence theorem of the OWL 2 RDFBased Semantics (Section 7.2) follows an alternative approach that replaces the use of the comprehension conditions and can be seen as a technical refinement of an idea originally discussed by the WebOnt Working Group (email). This change is an incompatible deviation from OWL 1, since certain aspects of the originally normative definition of the semantics have been removed.
Flawed Semantics of Language Constructs with Argument Lists [DEV]. In the OWL 1 RDFCompatible Semantics, the semantic conditions for unions, intersections and enumerations of classes were defined in a flawed form, which lead to formal inconsistency of the semantics (Issue 120; see also this unofficial problem description). The affected semantic conditions have been revised; see Section 5.4 and Section 5.5. This change is an incompatible deviation from OWL 1, since the semantics has formally been weakened in order to eliminate a source of inconsistency.
Incomplete Semantics of owl:AllDifferent [EXT]. The OWL 1 RDFCompatible Semantics missed a certain semantic condition for axioms based on the vocabulary term "owl:AllDifferent" (see also this unofficial problem description). The missing semantic condition has been added to the OWL 2 RDFBased Semantics (see Section 5.10). This change is compatible with OWL 1, since the semantics has been conservatively extended.
Aligned Semantics of owl:DataRange and rdfs:Datatype [EXT]. The class owl:DataRange has been made an equivalent class to rdfs:Datatype (see Section 5.2). The main purpose for this change was to allow for the deprecation of the term owl:DataRange in favor of rdfs:Datatype. This change is compatible with OWL 1 according to an analysis of the relationship between the two classes in the OWL 1 RDFCompatible Semantics (email).
Ontology Properties as Annotation Properties [EXT]. Several properties that have been ontology properties in OWL 1, such as owl:priorVersion, have now been specified as both ontology properties and annotation properties, in order to be in line with the rest of the OWL 2 specification (see Section 5.5 of the OWL 2 Structural Specification [OWL 2 Specification]). This change is compatible with OWL 1, since the semantics has been conservatively extended: all the ontology properties of OWL 1 are still ontology properties in OWL 2.
Nonempty Data Value Enumerations [DEV]. The semantic condition for enumerations of data values in Section 5.5 is now restricted to nonempty sets of data values. This prevents the class owl:Nothing from unintentionally becoming an instance of the class rdfs:Datatype, as analyzed in (email). This restriction of the semantics is an incompatible deviation from OWL 1. Note, however, that it is still possible to define a datatype as an empty enumeration of data values, as explained in Section 5.5.
Terminological Clarifications [NOM]. This document uses the term "OWL 2 RDFBased Semantics" to refer to the specified semantics only. According to Section 2.1, the term "OWL 2 Full" refers to the language that is determined by the set of all RDF graphs (also called "OWL 2 Full ontologies") being interpreted using the OWL 2 RDFBased Semantics. OWL 1 has not been particularly clear on this distinction. Where the OWL 1 RDFCompatible Semantics specification talked about "OWL Full interpretations", "OWL Full satisfaction", "OWL Full consistency" and "OWL Full entailment", the OWL 2 RDFBased Semantics Specification talks in Section 4 about "OWL 2 RDFBased interpretations", "OWL 2 RDFBased satisfaction", "OWL 2 RDFBased consistency" and "OWL 2 RDFBased entailment", respectively, since these terms are primarily meant to be related to the semantics rather than the whole language.
Modified Abbreviations [NOM]. The names "R_{I}", "P_{I}", "C_{I}", "EXT_{I}", "CEXT_{I}", "S_{I}", "L_{I}" and "LV_{I}", which have been used in the OWL 1 RDFCompatible Semantics specification, have been replaced by the corresponding names defined in the RDF Semantics document [RDF Semantics], namely "IR", "IP", "IC", "IEXT", "ICEXT", "IS", "IL" and "LV", respectively. Furthermore, all uses of the IRI mapping "IS" have been replaced by the more general interpretation mapping "I", following the conventions in the RDF Semantics document. These changes are intended to support the use of the OWL 2 RDFBased Semantics document as an incremental extension of the RDF Semantics document. Names for the "parts of the universe" that were exclusively used in the OWL 1 RDFCompatible Semantics document, such as "IX" or "IODP", have not been changed. Other abbreviations, such as "IAD" for the class extension of owl:AllDifferent, have in general not been reused in the document at hand, but the explicit nonabbreviated form, such as "IEXT(I(owl:AllDifferent))", is used instead.
Modified Tuple Notation Style [NOM]. Tuples are written in the form "( … )" instead of "< … >", as in the other OWL 2 documents.
Deprecated Vocabulary Terms [DPR]. The following vocabulary terms have been deprecated as of OWL 2 by the Working Group, and SHOULD NOT be used in new ontologies anymore:
 owl:DataRange (per resolution of Issue 29)
10 Appendix: Change Log (Informative)
10.1 Changes Since Recommendation
This section summarizes the changes to this document since the Recommendation of 27 October, 2009.
 With the publication of the XML Schema Definition Language (XSD) 1.1 Part 2: Datatypes Recommendation of 5 April 2012, the elements of OWL 2 which are based on XSD 1.1 are now considered required, and the note detailing the optional dependency on the XSD 1.1 Candidate Recommendation of 30 April, 2009 has been removed from the "Status of this Document" section.
10.2 Changes Since Proposed Recommendation
This section summarizes the changes to this document since the Proposed Recommendation of 22 September, 2009.
 [editorial] Correction of grammar (punctuation, word order, etc.), mainly in the Introduction section.
 [editorial] Updated and corrected several hyperlinks.
10.3 Changes Since Candidate Recommendation
This section summarizes the changes to this document since the Candidate Recommendation of 11 June, 2009.
 [resolution] Redefinition of several ontology properties to be both ontology properties and annotation properties, in order to align the RDFBased Semantics with the rest of the OWL 2 specification, and in particular to avoid an equivocal definition of the OWL 2 RL/RDF rules (per WG resolution).
 [correction] Correction of the type of facets: Facets are intended to be data properties and have been used as such elsewhere in the document, but they were wrongly specified as unrestricted properties so far.
 [correction] Correction of a mismatch between the definition of Dinterpretations in the document at hand and the RDF Semantics specification: according to the definition of simple interpretations, LV contains all plain literals in the vocabulary V. The missing reference to "V" has been added.
 [nonnormative] Correction of an error in the formulation of the correspondence theorem.
 [nonnormative] The section on Axiomatic Triples has been extended by an explicit set of axiomatic triples, based on the discussion in the rest of the section.
 [nonnormative] The section on Axiomatic Triples now explicitly mentions axiomatic triples for datatypes and facets corresponding to the semantic conditions for datatypes and facets, respectively.
 [nonnormative] Refinement of the proof for the correspondence theorem and correction of several errors. Motivated by these changes, the example in Section 7.1 has been slightly revised as well.
 [editorial] Added a description and ALTattribute text to Figure 1 on the parts hierarchy.
 [editorial] Distinction between normative and nonnormative references, as in other OWL 2 documents.
 [editorial] Added some clarification to the introduction section.
 [editorial] Removed a redundant conclusion from the table presenting the semantic conditions for datatype restrictions, since this conclusion already follows from the semantic conditions for the vocabulary properties, and having the conclusion repeated would not match the general approach that is applied when presenting "ifthen" semantic conditions in this document.
 [editorial] Reworded the description of the markers in the section on changes from OWL 1, and added a marker "DPR" for the deprecated features.
 [editorial] Changed the presentation style of references and citations to a form used in all OWL 2 documents.
 [editorial] Changed the presentation style for tuples from "〈 … 〉" to "( … )", to follow the conventions used in the other OWL 2 documents.
 [editorial] Numerous minor corrections and stylistic improvements.
10.4 Changes Since Last Call
This section summarizes the changes to this document since the Last Call Working Draft of 21 April, 2009.
 [resolution] Renamed the annotation vocabulary terms "owl:subject", "owl:predicate" and "owl:object" to "owl:annotatedSource", "owl:annotatedProperty" and "owl:annotatedTarget", respectively (per WG resolution).
 [resolution] Replaced the datatype "rdf:text" by "rdf:PlainLiteral" (per WG resolution).
 [resolution] Replaced the facet "rdf:langPattern" by "rdf:langRange", following the same replacement in the original rdf:PlainLiteral specification.
 [correction] Changed the range of the property "owl:annotatedProperty" from IP to IR in order to avoid undesired semantic side effects from annotations. This was an oversight when the original semantic conditions for annotations of axioms and annotations were removed from the document.
 [nonnormative] The semantic conditions and comprehension conditions for the nary property restrictions have been changed to only cover property sequences of length greater than 0, since the meaning of an expression with an empty property set is not clear.
 [editorial] Explained the optional status of the semantic conditions concerned with the IRI "owl:onProperties", in accordance with the rest of the OWL 2 specification.
 [editorial] Shortened and clarified some section titles, moved the section on semantic conditions for sub property chains within Section 5, and aligned the entry order of all tables in Section 8 with those in Section 5.
 [editorial] Several clarifications, minor corrections and cosmetic changes.
11 Acknowledgments
The starting point for the development of OWL 2 was the OWL1.1 member submission, itself a result of user and developer feedback, and in particular of information gathered during the OWL Experiences and Directions (OWLED) Workshop series. The working group also considered postponed issues from the WebOnt Working Group.
This document has been produced by the OWL Working Group (see below), and its contents reflect extensive discussions within the Working Group as a whole. The editors extend special thanks to Jie Bao (RPI), Ivan Herman (W3C/ERCIM), Peter F. PatelSchneider (Bell Labs Research, AlcatelLucent) and Zhe Wu (Oracle Corporation) for their thorough reviews.
The regular attendees at meetings of the OWL Working Group at the time of publication of this document were: Jie Bao (RPI), Diego Calvanese (Free University of BozenBolzano), Bernardo Cuenca Grau (Oxford University Computing Laboratory), Martin Dzbor (Open University), Achille Fokoue (IBM Corporation), Christine Golbreich (Université de Versailles StQuentin and LIRMM), Sandro Hawke (W3C/MIT), Ivan Herman (W3C/ERCIM), Rinke Hoekstra (University of Amsterdam), Ian Horrocks (Oxford University Computing Laboratory), Elisa Kendall (Sandpiper Software), Markus Krötzsch (FZI), Carsten Lutz (Universität Bremen), Deborah L. McGuinness (RPI), Boris Motik (Oxford University Computing Laboratory), Jeff Pan (University of Aberdeen), Bijan Parsia (University of Manchester), Peter F. PatelSchneider (Bell Labs Research, AlcatelLucent), Sebastian Rudolph (FZI), Alan Ruttenberg (Science Commons), Uli Sattler (University of Manchester), Michael Schneider (FZI), Mike Smith (Clark & Parsia), Evan Wallace (NIST), Zhe Wu (Oracle Corporation), and Antoine Zimmermann (DERI Galway). We would also like to thank past members of the working group: Jeremy Carroll, Jim Hendler, and Vipul Kashyap.
12 References
12.1 Normative References
 [OWL 2 Specification]
 OWL 2 Web Ontology Language: Structural Specification and FunctionalStyle Syntax. Boris Motik, Peter F. PatelSchneider, and Bijan Parsia, eds., 2009.
 [RDF Concepts]
 Resource Description Framework (RDF): Concepts and Abstract Syntax. Graham Klyne and Jeremy J. Carroll, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/RECrdfconcepts20040210/. Latest version available as http://www.w3.org/TR/rdfconcepts/.
 [RDF Semantics]
 RDF Semantics. Patrick Hayes, ed., W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/RECrdfmt20040210/. Latest version available as http://www.w3.org/TR/rdfmt/.
 [RFC 2119]
 RFC 2119: Key words for use in RFCs to Indicate Requirement Levels. Network Working Group, S. Bradner. IETF, March 1997, http://www.ietf.org/rfc/rfc2119.txt
 [RFC 3987]
 RFC 3987: Internationalized Resource Identifiers (IRIs). M. Duerst and M. Suignard. IETF, January 2005, http://www.ietf.org/rfc/rfc3987.txt
12.2 Nonnormative References
 [OWL 2 Direct Semantics]
 OWL 2 Web Ontology Language: Direct Semantics. Boris Motik, Peter F. PatelSchneider, and Bernardo Cuenca Grau, eds., 2009.
 [OWL 2 RDF Mapping]
 OWL 2 Web Ontology Language: Mapping to RDF Graphs. Peter F. PatelSchneider and Boris Motik, eds., 2009.
 [OWL 1 RDFCompatible Semantics]
 OWL Web Ontology Language: Semantics and Abstract Syntax, Section 5. RDFCompatible ModelTheoretic Semantics. Peter F. PatelSchneider, Patrick Hayes, and Ian Horrocks, eds., W3C Recommendation, 10 February 2004.
 [RFC 2396]
 RFC 2396  Uniform Resource Identifiers (URI): Generic Syntax. T. BernersLee, R. Fielding, U.C. Irvine and L. Masinter. IETF, August 1998.