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This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.
This Last Call Working Draft provides several significantdocument has undergone some small changes since the
previous version as an Ordinary Working Draftof 02 December 2008.21st April, 2009.
The specified semantics called "OWL 2 RDFBased Semantics", andOWL Working Group seeks
to gather experience from implementations in order to
increase confidence in the wholelanguage called "OWL 2 Full" as RDF graphs being interpreted using this semantics. The "Introduction" sectionand the texts describing the semantic conditions have been extended. The "Ontologies" section has been rewrittenmeet specific exit criteria. This document will remain a Candidate Recommendation until at least 30 July 2009. After that date, when
and now talks primarily about the RDF graph syntax forif the OWL 2 RDFBased Semantics.exit criteria are met, the definitiongroup intends to request Proposed
Recommendation status.
Please send reports of datatypes with facetsimplementation experience, and the semantic conditions for datatype restrictions have been better alignedother
feedback, to publicowlcomments@w3.org
(public
archive). Reports of any success or difficulty with the Direct Semantics. The correspondence theorem has been revised andtest cases are encouraged. Open discussion among developers is welcome at publicowldev@w3.org (public archive).
Publication as a proof sketch for the theorem has been constructed, and an elaborate example exists that motivatesCandidate Recommendation does not imply endorsement by the theorem and its proof. Several WG resolutions have been implemented (see also the list of OWL 2 changes ). A few errors have been corrected. Last Call The Working Group believes it has completed its design work for the technologies specified this document, so this is a "Last Call" draft. The design is not expected to change significantly, going forward, and now is the key time for external review, before the implementation phase. Please Comment By 12 May 2009 The OWL Working Group seeks public feedback on this Working Draft. Please send your comments to publicowlcomments@w3.org ( public archive ). If possible, please offer specific changes to the text that would address your concern. You may also wish to check the Wiki Version of this document and see if the relevant text has already been updated. No Endorsement Publication as a Working Draft does not imply endorsement by the W3C Membership. This is a draft documentW3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.
This document was produced by a group operating under the 5 February 2004 W3C Patent Policy. W3C maintains a public list of any patent disclosures made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent. An individual who has actual knowledge of a patent which the individual believes contains Essential Claim(s) must disclose the information in accordance with section 6 of the W3C Patent Policy.
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
RDFS"DEntailment"
(RDFS with datatype support ( "DEntailment" ),support),
as specified in
[RDF Semantics].
In other words,
the meaning given to an RDF graph by the OWL 2 RDFBased Semantics
includes the meaning given to the graph by the semantics of RDFS with datatypes,
and additional meaning is given to 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 same design principles
that have been 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 onlythose 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 OWL 2 RDFBased Semantics document.document at hand.
In particular,
the terminology used in the RDF Semantics
is reused in the document at hand,here,
except for cases
where a conflict exists with the rest of the OWL 2 specification.
The following paragraphs
outline thisthe document's structure and content,
and provide an overview of
some of the distinguishing features of the OWL 2 RDFBased Semantics.
According to Section 2,
the syntax
forover which the OWL 2 RDFBased Semantics is defined
is the set of all RDF graphs
[RDF Concepts].
Every such RDF graph
is given a precise formal meaning
by the OWL 2 RDFBased Semantics.
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 any risk of confusion.
The OWL 2 RDFBased Semantics
interprets the
RDF and RDFS vocabularies
[RDF Semantics]
and the OWL 2 RDFBased vocabulary (see Section 3 ). Also treated are,
together with an extended set of datatypes
and their constraining facets
for datatype restrictions.(see Section 3).
OWL 2 RDFBased interpretations (Section 4) are defined on a universe that is divided into parts, namely individuals, classes, and properties, which are identified with their RDF counterparts (see Figure 1). In particular, 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 further divided into subparts 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 entirely consisting of data values.
Finally,
the part of properties subsumes the parts of
object properties,
data properties,
annotationontology properties
and ontologyannotation properties.
In particular,
the part of object properties equals the whole part of properties,
and all other kinds of properties are therefore also object properties.
For the particular case ofannotations properties
it is important tonote that annotations cannot reallybe regarded as "semanticfree comments"considered "semanticfree"
under the OWL 2 RDFBased Semantics.
Just like every other triple or set of triples occurring in an RDF graph,
an annotation is actually an interpreted assertion that denotesassigned a truth value under theby any given OWL 2 RDFBased Semantics. This means thatinterpretation.
Hence,
although annotations are meant to be "semantically weak",
i.e. their formal meaning does not significantly exceed
that coming from the RDF Semantics specification,
adding an annotation
may still change the meaning of an ontology.
An analogA similar discussion holds for statements
that are built from ontology properties.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:class,
written as "ICEXT(C)".
An individual a is an instance of a given class C
exactly if a is a member of the class extension of C , written as "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,
for a class to 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,
though still being different individuals (i.e.,individuals,
i.e., they do not need to share the same properties).properties.
Similarly,
every property has a property extension,
written as "IEXT(p)",
associated with it, consistingit
that consists of pairs of individuals:individuals.
An individual a_{1} has a relationship to another individual a_{2}
based on a given property p,
exactly if the pair ⟨ a_{1} , a_{2} ⟩
is a member of the property extension of p , written as "IEXT( p )"..
Again, properties are distinguished from their property extensions.
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, since a class extension and a property extension may independently be associated with the same individual.
The main part of the OWL 2 RDFBased Semantics is Section 5,
which specifies
a formal meaning for all the actual semantic features in termsOWL 2 language constructs
by means of the
OWL 2 RDFBased semantic conditions.
These semantic conditions extend all the semantic conditions
given in [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,
there exist semantic conditions
that allow to interpret the RDF 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 actually 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 "rdfs:subClassOf rdf:type rdf:Property" or "rdfs:subClassOf rdfs:domain rdfs:Class". However, there is no explicit collection of additional axiomatic triples for the OWL 2 RDFBased Semantics but, 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 sub semantics 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].
The OWL 2 RDFBased Semantics actually deviates from the OWL 1 RDFCompatible Semanticsits predecessor in several aspects,
in most cases due to serious technical problems
that would have arisen
from a conservative semantic extension.
One important change is that,
while there still exist
so called "comprehension conditions"
for the OWL 2 RDFBased Semantics
(see Section 8),
these are not part of the
normative set of semantic conditions anymore.
The OWL 2 RDFBased Semantics also corrects several errors of theOWL 1 RDFCompatible Semantics.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].
This section determines the syntax
for the OWL 2 RDFBased Semantics.Semantics,
and gives an overview on typical content of ontologies
for ontology management tasks.
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]), i.e. on every set of RDF triples (Section 6.1 of [RDF Concepts]).
In accordance with the rest of the OWL 2 specification
(see Section 2.3 of
[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].
In contrast,
the RDF Semantics specification
[RDF Semantics]
is defined on RDF graphs containing URIs
[RFC 2396].
This change
is backwards compatible
with the RDF specification.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 [OWL 2 Specification]),
whereas the RDF Semantics specification
[RDF Semantics]
uses the term "URI reference" instead.reference".
According to [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 [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
inaccording to Section 6.1 of [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]),
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]),
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 orand future technologies and tools,
the document at hand
does not explicitly forbid the use of
generalized RDF graphs to containconsisting 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.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]
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".
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 componentsconstructs
that are aimed to support ontology management tasks.
Examples are
ontology headers
and
ontology IRIs,
as well as componentsconstructs 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 [OWL 2 Specification]
deals with these topics in greaterdetail,
and can therefore be used as a guide
on how to apply these componentsconstructs in OWL 2 Full ontologies accordingly.
The mappings of all these componentsconstructs to their respective RDF encodings
are defined in
[OWL 2 RDF Mapping].
This section specifies the OWL 2 RDFBased vocabulary, and lists the names of the datatypes and facets used under the OWL 2 RDFBased Semantics.
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# 
Table 3.2
lists the IRIs of the OWL 2 RDFBased vocabulary,
which is the set of vocabulary IRIs (or "vocabulary terms")terms
that are specific for the OWL 2 RDFBased Semantics.
This vocabulary
extends the RDF and RDFS vocabularies
as specified by Sections 3.1 and 4.1 of
[RDF Semantics],
respectively.
Table 3.2
excludes those IRIs
that arewill be mentioned in
eitherSection 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 Section 7 of [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 
Table 3.3
lists the IRIs of the datatypes used in the OWL 2 RDFBased Semantics.
The normative set of datatypes for the OWL 2 RDFBased Semantics equals the set of datatypesdatatype rdf:XMLLiteral is described in
Section 3.1 of
[RDF Semantics].
All other datatypes are described in
Section 4 of [OWL 2 Specification].
The meaningnormative set of rdf:XMLLiteral is described in Section 3.1datatypes of [ RDFthe OWL 2 RDFBased Semantics ]. All otherequals the set of datatypes
are asdescribed in Section 4 of
[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 
Feature At Risk #1: owl:rational support
Note: This feature is "at risk" and may be removed from this specification based on feedback. Please send feedback to publicowlcomments@w3.org. For the current status see features "at risk" in OWL 2
The owl:rational datatype might be removed from OWL 2 if implementation experience reveals problems with supporting this datatype.
Table 3.4
lists the IRIs of the facets used underin 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 [OWL 2 Specification]
in the context of their respective datatypes.
The normative set of facets forof the OWL 2 RDFBased Semantics equals the set of facets
described in Section 4 of
[OWL 2 Specification].
In this specification,
facets can beare used to restrict datatypes.for defining datatype restrictions
(see Section 5.7).
For example,
to refer to the set of all strings of length 5,5
one can restrict
the datatype xsd:string
(Section 3.3)
by the facet xsd:length
and the value 5.

The OWL 2 RDFBased Semantics provides
vocabulary interpretations and vocabulary entailment
(see Section 2.1 of [RDF Semantics])
for the RDF and RDFS vocabularies
and for the OWL 2 RDFBased vocabulary.
This section defines
the concepts of an
OWL 2 RDFBased datatype map
and an
OWL 2 RDFBased interpretation,
and specifies what
satisfaction of ontologies,
consistency and entailment
meanmeans under the OWL 2 RDFBased Semantics.
In addition,
the so called "parts" of the universe
of an OWL 2 RDFBased interpretation
are defined.
According to Section 5.1 of the RDF semantics specification [RDF Semantics], a datatype d has the following components:
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 [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.
For information and an example on facets, seefacets
(see Section 3.4 .).
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:
Note that
it is not further specified
ofwhat kindthe denotationnature of a facet IRIIRI's denotation is,
i.e. it is only known that a facet IRI denotes some individual.
The definition of datatypes with facets does not suggest a certain kind of object.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 values, rather than strings.
In this document, it will always be assumed from now on that every 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.
According toAs a previous comment,consequence of what has 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 respectivespecified
lexical space LS(d),
value space VS(d),
lexicaltovalue mapping L2V(d),
facet space FS(d) and
facettovalue mapping F2V(d ) specified according).
Note that Definition 4.1
does not prevent additional datatypes
to the definitionsbe in Section 4 of [an OWL 2 Specification ] (for the facets listed in Section 3.4 ). NoteRDFBased datatype map.
For the special case of
an OWL 2 RDFBased datatype map D
that exclusively contains no otherthe datatypes than thoselisted 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 the facet spaceFS(d)
there exists a namedatatype pair ⟨ u^{*} , d^{*} ⟩ in D
such that v is in the value spaceVS(d^{*}).
From the RDF Semantics specification
[RDF Semantics],
let V be a set of IRIs andliterals and IRIs
containing the RDF and RDFS vocabularies,
and let D be a datatype map according to Section 5.1 of
[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 IRIs andliterals 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]),
and
the value spaces forof 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,
as also introduced in
Section 1.4 of [RDF Semantics],
for a given interpretation I of a vocabulary V
the notation
"I(x)"
will be used to
denote "IS("IL(x)" and "IL("IS(x)"
for the IRIs x in V and thetyped literals and IRIs x in V,
respectively.
As detailed in [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)
In this document, it will always be assumed from now on that every 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.
The following definitions specify
what it means thatfor an RDF graph isto be satisfied
by a given OWL 2 RDFBased interpretation,
what it means that an RDF graph isto be consistent
under the OWL 2 RDFBased Semantics,
and what it means that an RDF graph entailsto entail another RDF graph under the OWL 2 RDFBased Semantics.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 [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 of the form "s p o"
it is necessary that the relationship
⟨ I(s) , I(o) ⟩ ∈ IEXT(I(p))
holds
(special treatment is necessaryexists for blank nodes,
as detailed in
Section 1.5 of [RDF Semantics]).
In other words,
the given graph has to be compatible with
the specific form of the IEXT mapping of the interpretationI.
The maindistinguishing 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 actually constrainhave the effect of constraining the possible forms
the mappingan IEXT of Imapping 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 [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
(ofof 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
(ofof 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
) thatthe following holds:
If I
OWL 2 RDFBased satisfies all the RDF graphs in S_{1}
with respect to V and D also,
then I
OWL 2 RDFBased satisfies all the RDF graphs in S_{2}
with respect to V and D.
Table 4.1 defines the "parts" of the universe of a given OWL 2 RDFBased interpretation I.
The firstsecond column tells the name of the part.
The secondthird column gives a definition of the part
in terms of the mapping IEXT of I,
and by referring to particular terms of
the RDF, RDFS and OWL 2 RDFBased vocabularies.
The third column notes what the defined part is meant to be.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 [RDF Semantics],
this actuallymeans that the name "IDC"
denotes the set of all members of theclass 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  

 IR  rdfs:Resource 
 LV  rdfs:Literal 
ontologies  IX  owl:Ontology 
 IC  rdfs:Class 
 IDC  rdfs:Datatype 
 IP  rdf:Property 
data properties  IODP  owl:DatatypeProperty 
 IOXP  owl:OntologyProperty 
 IOAP  owl:AnnotationProperty 
The mapping ICEXT from IC to P(IR) (where P isthe power set),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)) } for every c ∈ IC..
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].
Section 5.1
specifies semantic conditions for the different parts of the universe
(defined(as defined in Section 4.4)
of the OWL 2 RDFBased interpretation being considered.
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 shorthandshortform for "if and only if".
Conjunctive commas:
A comma
(",")
that separatesseparating 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
an expression such as"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 explicitlygiven for a variable "x" ,"x",
as in "∀ x : …" or "{ x  … }",
then "x" is unconstrained,
which means thatx ∈ IR,
i.e. "x""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 a list of n ≥ 0
elements,individuals a_{1} , … , a_{n},
all of them being members of the set S.
Precisely,
s = I(rdf:nil) for n = 0,0;
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 [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 for certain setsare used as convenient abbreviations:abbreviations for certain sets:
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]
withinunder the different constraints the two semantics have to adhere to.meet.
The way this semantic alignment is actuallydescribed
is via the OWL 2 Correspondence Theorem
in Section 7.2.
For this theorem to hold,
the semantic conditions
that treat the RDF encodings
of OWL 2 axioms
(Section(compare Section 3.2.5 of [OWL 2 RDF Mapping]
and
Section 9 of [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 these construct encodings.
On the other hand,
the RDF encodings
of OWL 2 expressions
(Section(compare Section 3.2.4 of [OWL 2 RDF Mapping]
and
Sections 6 – 8 of [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
enumerations
of individuals.
These language constructs actuallybuild OWL 2 expressions.
However,But for backwards compatibility reasons
there are also RDF encodings of axioms
based on the vocabulary for these language constructs
(see Table 18 in Section 3.2.5 of
[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
intoto 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}.
So,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 amultitriple representationrepresentations in RDF,
where the different triples share a common "root node",
such as the blank node
"_:x"
in the examplefollowing example:
_:x rdf:type owl:AllDisjointClasses .
_:x owl:members ( ex:c_{1} ex:c_{2} ) .
In essence,
the semantic conditions for the encodings 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 of the form
"∃ z ∈ IR".
The singlepurpose 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 5.115.10,
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:allValuesFromowl: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 classes and 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:allValuesFromowl:someValuesFrom
in the table in
Section 5.3,
which determines that
IEXT(I( owl:allValuesFromowl:someValuesFrom))
⊆
ICEXT(I(owl:Restriction)) × IC.
Table 5.1
lists the semantic conditions
for the parts of the universe
of the OWL 2 RDFBased interpretation being considered, as definedconsidered.
Additional semantic conditions affecting the parts
are given in Section 4.45.2.
The first column tells the name of the part.part,
as defined in
Section 4.4.
The second column defines
certain conditions on the part .part.
In most cases,
it is specifiedthe column specifies for the part
by which other part it is subsumed,
and by thisthus 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 it is specifiedfor the class extensions of the member classes
it 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 
 S ⊆ IP  IEXT(x) ⊆ 
IOAP  S ⊆ IP  IEXT(x) ⊆ 
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 thethose 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 name 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 . Thus,:
from an entry of the form
"ICEXT(I(C )) ⊆S",)) ⊆ S",
for a class name C
and a set S,S,
and given an RDF triple of the form
"u rdf:type C",
one can deduce
that the relationship
given by"I(u ) ∈ S") ∈ 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"
givenlisted in Section 5.1 of [RDF Semantics]):
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 
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 thethose properties
with IRIs in the RDF andRDFS vocabulariesvocabulary
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 andRDFS vocabulariesvocabulary
can be found in the RDF Semantics specification
[RDF Semantics].
The first column tells the name 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. Thus,property:
from an entry of the form
"IEXT(I(p )) ⊆ S)) ⊆ S_{1} × S × S_{2}",
for a property name p
and sets S_{1} and S_{2},
and given an RDF triple of the form
"s p o",
one can deduce
that the relationships
given by"I(s ) ∈ S) ∈ S_{1}"
and
"I(o ) ∈ S) ∈ S_{2}"
hold.
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 datatype 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 datatype facet IRI E, the following semantic conditions extend the basic semantics specification that has been given for datatypes with facets in Section 4.1:
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 Section 7 of [OWL 2 Specification] for further information).
Informative notes:
owl:topObjectProperty
relates every two individuals in the universe towith each other.
Likewise, owl:topDataProperty
relates every individual towith 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 be 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 [OWL 2 RDF Mapping] for owl:hasSelf, and Section 5.5 of [OWL 2 Specification] for owl:deprecated).
The range of the property owl:annotatedProperty is unrestricted in order to avoid undesired semantic side effects from an annotation, when the annotated axiom or annotation is not contained in the ontology.
IRI E  I(E)  IEXT(I(E)) 

owl:allValuesFrom  ∈ IP  ⊆ ICEXT(I(owl:Restriction)) × IC 
 ∈ IP  ⊆ 
owl:annotatedSource  ∈  ⊆ 
owl:annotatedTarget  ∈ IP  ⊆ IR × IR 
owl:assertionProperty  ∈ IP  ⊆ ICEXT(I(owl:NegativePropertyAssertion)) × IP 
owl:backwardCompatibleWith  ∈ IOXP  ⊆ 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  ⊆ 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 
 ∈ 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 
 ∈ IOXP  ⊆ 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 
 ∈ 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 
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:
Every first semantic condition of the three condition pairs in the table is an "iff" condition,
since the treatedcorresponding OWL 2 language constructs
have the status ofare both
class expressions and axioms of OWL 2.axioms.
In contrast,
the firstsemantic condition on datatype complements
is an "ifthen" condition,
since it only has the status ofcorresponds to a datarange expression.
See the
notes on the form of semantic conditions
for further information on this topic.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 obsoleted byredundant
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) 
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 treatedcorresponding OWL 2 language construct
has the status ofis both a class expression and an axiom of OWL 2.axiom.
See the
notes on the form of semantic conditions
for further information on this topic.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 obsoleted byredundant
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 
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 that the property has a specifically defined value.
By placing a self restriction on some given property
one only regardsconsiders 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
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 Section 7 of [OWL 2 Specification] for further information).
Informative notes:
All the semantic conditions are "ifthen" conditions,
since the treatedcorresponding OWL 2 language constructs
have the status ofare class expressions of OWL 2.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 on these topics.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 } 
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
viaby means of a set of facetvalue pairs.
For information and an example on facets, seefacets
(see Section 3.4 . Note that).
Certain special cases exist:
If no facetvalue pair is applied to a given datatype at all,
then the resulting datatype will be equivalent to the original datatype.
Note further that,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 havingwith an empty facet space (usually,space.
The set IFS(d)
for a nonempty facet space only exists for those datatypes belonging to thedatatype map). The set IFSd
is defined by
IFS(d) := { ⟨ I(F) , v ⟩  ⟨ F , v ⟩ ∈ FS(d) } ,
where
d is a datatype,F is the IRI of a facet,
and v is a value of the facet.
This set corresponds to the facet space, FS, ofspace FS(d ,),
as defined in Section 4.1,
but rather consists of
pairs of the denotation of a facet and its value.
The mapping IF2VIF2V(d)
for a datatype d
is defined by
IF2V(d)(⟨ I(F) , v ⟩) := F2V(d)(⟨ F , v ⟩) ,
where
d is a datatype,F is the IRI of a facet,
and v is a value of the facet.
This mapping corresponds to the facettovalue mapping, F2V, ofmapping 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 its value.
Informative notes:
The semantic condition is an "ifthen" condition,
since the treatedcorresponding OWL 2 language construct
has the status ofis a datarange expression of OWL 2.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 on these topics.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})  z , d ∈ IDC , 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} ⟩) 
Table 5.8
extends the RDFS semantic conditions
for parts of the RDFS vocabulary, includingsubclass axioms, subproperty axioms, domain axioms and range axioms,axioms.
The semantic conditions provided here are "iff" conditions,
while the original semantic conditions,
as originally definedspecified in
Section 4.1 of [RDF Semantics ].],
were weaker "ifthen" conditions.
Only the additional semantic conditions are given here
and the other conditions on theof RDF and RDFS
vocabulariesare retained.
Informative notes:
All the semantic conditions are "iff" conditions,
since the treatedcorresponding OWL 2 language constructs
have the status of axioms of OWL 2.are axioms.
See the
notes on the form of semantic conditions
for further information on this topic. Note that the original semantics for these RDFS axioms only provide for weaker "ifthen" semantic conditions.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) 
Table 5.9
lists the semantic conditions for sub property chains, which allow forspecifying
complex property subsumption axioms. As an example, one can define a sub property chain axiom, for which the chain consisting of the extensions of the properties ex:hasFatherthat two individuals are equal or different from each other,
and ex:hasBrother is covered by the extension of the property ex:hasUncle .that either two classes or two properties
are equivalent or disjoint with each other,
respectively.
Also treated here are disjoint union axioms.
Informative notes:
All the semantic condition is anconditions are "iff" condition,conditions,
since the treated language construct has the status of an axiom ofcorresponding OWL 2.2 language constructs
are axioms.
See the
notes on the form of semantic conditions
for further information on this topic.information.
Also note that the
semantics has been specified in a way to allow for a sub property chain axiomIRI owl:equivalentClass
is used to be satisfiable without requiring the existence of a property that represents the property chain. Table 5.9: Semantic Conditions for Sub Property Chains 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 ) 5.10 Semantic Conditions for Equivalence and Disjointness Axioms Table 5.10 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. Also treated here are disjoint union axioms that allow for stating that a given class is equivalent to the union of a given collection of mutually disjoint classes. Informative notes: All the semantic conditions are "iff" conditions, since the treated language constructs have the status of axiomsformulate datatype definitions
(see Section 9.4 of
[OWL 2. See the notes on the form of semantic conditions2 Specification]
for furtherinformation on this topic.about datatype definitions).
 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 
Table 5.115.10
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 eitherseveral given classes or severalproperties
are mutually disjoint with each other.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.
There is no difference in theBoth variants have an equivalent formal meaning of these two variants.meaning.
Informative notes:
The semantic conditions essentially represent "iff" conditions,
since the treated language constructs have the status of axioms ofcorresponding OWL 2.2 language constructs
are axioms.
However,
there are actually two suchsemantic conditions for each language construct,construct
due to the multitriple RDF encoding of these language constructs.
The actual"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 on these topics.information.
if  then 

s sequence of a_{1} , … , a_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDifferent)) , ⟨ z , s ⟩ ∈ IEXT(I(  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( 
if  then 
s sequence of a_{1} , … , a_{n} ∈ IR , z ∈ ICEXT(I(owl:AllDifferent)) , ⟨ z , s ⟩ ∈ IEXT(I(  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( 
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)) 
Table 5.125.11
lists the semantic conditions for inversesub property chains,
which allow for specifying complex property subsumption axioms.
The inverse ofAs an example,
one can define a givensub property ischain axiom
that specifies
the corresponding property with subjectchain consisting of the extensions of the properties
ex:hasFather
and
object swapped for eachex:hasBrother
to be a sub relation of
the extension of the property
assertion built from it.ex:hasUncle.
Informative notes:
The semantic condition is an "iff" condition,
since the treatedcorresponding OWL 2 language construct
has the status ofis an axiom of OWL 2.axiom.
See the
notes on the form of semantic conditions
for further information on this topic.information.
The semantics has been specified in a way
that allows a sub property chain axiom to be satisfiable
without requiring the existence of a property
that represents the property chain.
if s sequence of p_{1} , … , p_{ 2n} ∈ IR then  

⟨ p , s ⟩ ∈ IEXT(I(  iff  p ∈ IP , p_{1} , … , p_{ 2n} ∈ IP , ∀ y_{0} , … , y_{n} : ⟨ y_{0} , y_{1} ⟩ ∈ IEXT(p_{1}) 
Table 5.135.12
lists the semantic conditions for inverse property characteristics. Ifaxioms.
The inverse of a given property
is functional , then at most one distinct value can be assigned to any given individual via this property. An inverse functionalthe 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}) } 

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 at all.
If two individuals are related by a symmetric property,
then this property also relates them by the reverse relationship,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 treatedcorresponding OWL 2 language constructs
have the status of axioms of OWL 2.are axioms.
See the
notes on the form of semantic conditions
for further information on this topic.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) 
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 withinbeing instances of the class,
and no assumption is made
about individuals
external to the class, orfor which it is unknown whether they are instancesmembership of the class.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.
PleaseNote that
keys are not functional by default
under the OWL 2 RDFBased Semantics.
Informative notes:
The semantic condition is an "iff" condition,
since the treatedcorresponding OWL 2 language construct
has the status ofis an axiom of OWL 2.axiom.
See the
notes on the form of semantic conditions
for further information on this topic.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 
Table 5.15
lists the semantic conditions for negative property assertions.
They allow to state that
an individual a 1 doestwo given individuals are not stand in a relationship p with another individualrelated by a 2 .given property.
The second form based on owl:targetValue
is more specific than the first form based on owl:targetIndividual
,in that it is restricted to the case of negative data property assertions.
Note that the second form
will coerce the target individual of a negative property assertion
into a data value,
due to the range defined for the property
owl:targetValue
in
Section 5.3.
Informative notes:
The semantic conditions essentially represent "iff" conditions,
since the treatedcorresponding OWL 2 language constructs
have the status of axioms of OWL 2.are axioms.
However,
there are actually two suchsemantic conditions for each language construct,
due to the multitriple RDF encoding of these language constructs.
The actual"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 on these topics.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)) 
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 providenormatively specify any axiomatic triples.
It might not be possible to give a set of RDF triples
that captures all "axiomatic aspects"
of the OWL 2 RDFBased Semantics,
just as one cannot expect
to define the whole OWL 2 RDFBased Semantics specification
in terms of RDF entailment rules only.
Furthermore,
axiomatic triples for the OWL 2 RDFBased Semantics could,
in principle,
contain arbitrarily complex class or propertyexpressions,
e.g. the union of several classes,
and by this it becomes nonobvious
which oneof several possible inequivalentnonequivalent sets of axiomatic triples
should be selected.
However,
the OWL 2 RDFBased Semantics includes many semantic conditions
that can in a sense be regarded to beas 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,
the Sections 6.2
and
6.3
will discuss how the "axiomatic" semantic conditions
of the OWL 2 RDFBased Semantics
relate to axiomatic triples,
resulting in
an example set of axiomatic triples
that is compatible with the OWL 2 RDFBased Semantics.
In RDF and RDFS
[RDF Semantics],
axiomatic triples are used
to provide basic meaning
tofor all the vocabulary terms
of the respective language.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
the denotation of a vocabulary term belongs to (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 by axiomatic triples.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 specifically applying the
RDFS semantic conditions
for the RDFS vocabularygiven 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
canappear there.
of the RDFBased SemanticsThe semantic conditions for vocabulary classes
in Table 5.2
of 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,
which specifies the semantic conditions for classes,if the second column contains an entry
of the form
"I(E) ∈ S"
for some set S,
then this entry corresponds to some RDF triple of the form "E rdf:type C",
where C is the IRI of some 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 typically be
one of
"rdfs:Class",
"owl:Class"
(S=IC in both cases)
or "rdfs:Datatype" (S=IDC).
For example, the semantic condition for the IRI "owl:FunctionalProperty", given by
I(owl:FunctionalProperty) ∈ IC ,
would have 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 some RDF triple of the form "E rdfs:subClassOf C" (or "E owl:equivalentClass C"), where C is the IRI of some class with ICEXT(I(C)) = S. In every case, S will be either one of the parts of the universe of I or the empty set.
For example, the semantic condition
ICEXT(I(owl:FunctionalProperty)) ⊆ IP
would have 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
, which specifies the semantic conditions for the parts of the universe,is of the form
"S_{1} ⊆ S_{2}"
for some sets S_{1} and S_{2},
then this corresponds to some RDF triple
of the form
"C_{1} owl:subClassOf C_{2}",
where C_{1} and C_{2} are the IRIs
of some classes with
ICEXT(I(C_{1})) = S_{1}
and
ICEXT(I(C_{2})) = S_{2},
respectively,
according to
Section 5.2.
JustAs stated in
Section 6.1
for the RDF axiomatic triples,
all the axiomatic triples for classes
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 [RDF Semantics].
of the RDFBased SemanticsThe semantic conditions for vocabulary properties
in Table 5.3
of 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,
which specifies the semantic conditions for properties,if the second column contains an entry
of the form
"I(E) ∈ S" for some set S,
then this entry corresponds to some RDF triple of the form
"E rdf:type C",
where C is the IRI of some 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 typically be
one of
"rdf:Property",
"owl:ObjectProperty",
(S=IP in both cases),
"owl:DatatypeProperty" (S=IODP),
" owl:AnnotationPropertyowl:OntologyProperty" (S =IOAP)=IOXP)
or " owl:OntologyPropertyowl:AnnotationProperty" (S =IOXP).=IOAP).
For example, the semantic condition for the IRI "owl:disjointWith", given by
I(owl:disjointWith) ∈ IP ,
would have 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 some RDF triples of the forms "E rdfs:domain C_{1}" and "E rdfs:range C_{2}", where C_{1} and C_{2} are the IRIs of some 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
would have the corresponding axiomatic triples
owl:disjointWith rdfs:domain rdfs:Classowl:Class .
owl:disjointWith rdfs:range rdfs:Classowl: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 that of owl:topObjectProperty.
JustAs stated in
Section 6.1
for the RDF axiomatic triples,
all the axiomatic triples for properties
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).
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 sub semantics 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 [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 can be transformed
by applying
the reverse OWL 2 RDF mapping
[OWL 2 RDF Mapping]
into corresponding
OWL 2 DL ontologies in Functional Syntax form
according to the Functional style syntax defined in
[OWL 2 FunctionalSpecification],
and which must further meet
all the restrictions on OWL 2 DL ontologies
that are specified in
Section 3 of [OWL 2 FunctionalSpecification].
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
theOWL 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.
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
of RDF graphs⟨ 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:comment "an annotation"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]
intoto 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:commentrdfs:label ex:c3 "an annotation""c3" )
(7) )
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. 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 triple (4) in G_{2}^{*}. The consequences are analog to those described for reason 1.
Resolution of Reason 2. The declaration triple (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 triple. The changed RDF graphs will still be OWL 2 DL ontologies in RDF graph form, since adding the entity declaration does not hurt any of the restrictions on OWL 2 DL ontologies. 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 this new 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.
In general, the union of two arbitrary sets is always ensured to exist as a set as well, since it is an element of the powerset of the universe. Hence the class denoted by the class expression in axiom (5) of F( G 2 * ) is warranted to exist with respect to the OWL 2 Direct Semantics.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 generalguarantee
that there will also alwaysexist
an individual in the universe
that has this set union as its class extension, i.e. it cannot be guaranteed that there will always be a union class for every union set.extension.
Under the OWL 2 RDFBased Semantics,
triple (7) of G_{2}^{*}
claims that
there isa class exists being
the union of two other classes.
But since
the existence of such a union class
is not alreadyensured by G_{1}^{*},
there will existbe 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".
If an OWL 2 RDFBased interpretation satisfies the modified graph G_{1}^{*},
then the triples (6) and (7) of G_{2}^{*}
will also be satisfied.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the whole set of added triples
encodes a proper OWL 2 DL axiom.
Further,
for the IRI
ex:c3,
which occurs in the union class expression
but not in G_{1}^{*},
an entity declaration has already beenis added
to G_{1}^{*}
by the resolution of reason 2.
Also, this operation will not change
the formal meaning of the ontologies
under the OWL 2 Direct Semantics,
since the added equivalence axiom
is actuallya 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}^{*}:
G_{1} and G_{2} are again OWL 2 DL ontologies in RDF graph form and can be mapped by the reverse RDF mapping to the following 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) 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.
In particular,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.
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 claimsstates 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.entailment.
However,
as discussed in
Section 7.1,
there exist semantic differences
between the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics thatSemantics,
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,
without changingwhile 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 of the theorem
in Section 7.3
will be more concrete
in that it will substitute each given OWL 2 DL entailment query
bywith 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 [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:
Let D be an OWL 2 RDFBased datatype map
according to
Definition 4.1.
The corresponding OWL 2 Direct datatype map
[ OWL 2 Direct Semantics ]F(D) := (
N_{DT} ,
N_{LS} ,
N_{FS} ,
⋅ ^{DT} ,
⋅ ^{LS} ,
⋅ ^{FS}
)
[OWL 2 Direct Semantics]
is given by
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 [OWL 2 Direct Semantics]
that corresponds to D w.r.t.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 OWL 2 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 [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 conditions 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 OWL 2 RDF mapping to G_{1} and G_{2}, respectively:
This is a sketch of a proof for Theorem 7.1 (OWL 2 Correspondence Theorem), stated 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 taking the full amount of language constructs of OWL 2 into account. A complete proof can make use of the observation that the definitions of the OWL 2 Direct Semantics and the OWL 2 RDFBased Semantics are actually closely aligned for all the different language constructs of OWL 2.
The proof sketch will make use of an approach that will be called "balancing" throughout this appendix, and which will now be introduced. A concrete example for how this approach can be applied is 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 additional 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 OWL 2 RDF mapping [OWL 2 RDF Mapping] to G_{1} and G_{2}, respectively:
Balancing Lemma: An algorithm exists that terminates on every input and that has the following input/output behavior:
Let the input of the algorithm be a pair 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 OWL 2 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 [OWL 2 Specification].
Then the output of the algorithm
will be 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 conditions 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 OWL 2 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 [OWL 2 Direct Semantics]
that corresponds to D w.r.t.according to the
technical note on corresponding datatype maps
in Section 7.2:
Proof of the Balancing Lemma:
Let G_{1}^{*} and G_{2}^{*} be OWL 2 DL ontologies in RDF graph form, with F(G_{1}^{*}) and F(G_{2}^{*}) being the corresponding OWL 2 DL ontologies in Functional Syntax form that result from applying the reverse OWL 2 RDF mapping to G_{1}^{*} and G_{2}^{*}, respectively, such that F(G_{1}^{*}) and F(G_{2}^{*}) mutually meet the restrictions on OWL 2 DL ontologies. The resulting RDF graphs G_{1} and G_{2} are constructed as follows.
The initial versions of G_{1} and G_{2} are copies of G_{1}^{*} and G_{2}^{*}, respectively.
A preprocessing step will substitute all blank nodes in G_{1} for fresh blank nodes that do not occur in G_{2}. One can therefore assume from now on that G_{1} and G_{2} have no common blank nodes.
Since G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, the canonical parsing process for computing the reverse OWL 2 RDF mapping, as described in Section 3 of [OWL 2 RDF Mapping], can be applied to map the graphs G_{1} and G_{2} to corresponding OWL 2 DL ontologies in Functional Syntax form. For the resulting ontologies it is then algorithmically possible to determine for every occurring IRI and anonymous individual all the entity types. By this, all missing declaration triples are added to G_{1} and G_{2}.
Further, since G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, the canonical parsing process can also be applied to safely identify all subgraphs of G_{1} and G_{2} that correspond to language constructs described in [OWL 2 Specification], including all the language constructs considered in the theorem.
Based on these observations, the following steps are performed on every subgraph g ⊆ G_{2} that has been identified by the canonical parsing process:
In the following it is shown that all the claims of the theorem hold.
A: Existence of a terminating algorithm. An algorithm exists for mapping the input pair ⟨ G_{1}^{*} , G_{2}^{*} ⟩ to the output pair ⟨ G_{1} , G_{2} ⟩, since the canonical parsing process for the determination of the missing entity declarations and for the identification of the language construct subgraphs is described in the form of an algorithm in [OWL 2 RDF Mapping]. All other operations described above can obviously be performed algorithmically. The algorithm terminates, since the canonical parsing process terminates (including termination on invalid input), and since all other operations described above are executed by a finite number of steps, respectively.
B: The resulting RDF graphs are OWL 2 DL ontologies. Since the original RDF graphs G_{1}^{*} and G_{2}^{*} are OWL 2 DL ontologies in RDF graph form, this is also the case for G_{1} and G_{2}, since each of the steps above transforms a pair of OWL 2 DL ontologies in RDF graph form again into a pair of OWL 2 DL ontologies in RDF graph form, for the following reasons:
C: The resulting pair of RDF graphs is balanced. Property (1) of the theorem requires that the pair ⟨ G_{1} , G_{2} ⟩ is balanced. The following list checks that all the properties of the definition are satisfied.
D: The resulting ontologies are semantically equivalent with the original ontologies. Property (2) of the theorem requires that F(G_{1}) is semantically equivalent with F(G_{1}^{*}). This is the case, since F(G_{1}) differs from F(G_{1}^{*}) only by
Further, property (3) of the theorem requires that F(G_{2}) is semantically equivalent with F(G_{2}^{*}). This is the case, since F(G_{2}) differs from F(G_{2}^{*}) only by additional entity declarations, and missing annotations including deprecation annotations (due to (a) and (b)), which all have no formal meaning under the OWL 2 Direct Semantics.
End of the Proof of the Balancing Lemma.
In the following, the correspondence theorem will be proven.
Assume that the premises of the correspondence theorem hold for given RDF graphs G_{1}^{*} and G_{2}^{*}. This allows for applying the balancing lemma, which provides the existence of certain RDF graphs G_{1} and G_{2} that are OWL 2 DL ontologies in RDF graph form. Hence, it is possible to build OWL 2 DL ontologies in Functional Syntax form F(G_{1}) and F(G_{2}) by applying the reverse OWL 2 RDF mapping to G_{1} and G_{2}, respectively.
The balancing lemma further provides that the pair ⟨ G_{1} , G_{2} ⟩ is balanced.
The claimed property (1) of the correspondence theorem follows directly from property (1) of the balancing lemma and from property (1) of the "Balanced"definition. The claimed properties (2) and (3) of the correspondence theorem follow directly from properties (2) and (3) of the balancing lemma, respectively.
The rest of this proof will treat the claimed property (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.
Let I be an OWL 2 RDFBased interpretation w.r.t. an OWL 2 RDFBased datatype map D of a vocabulary V_{I} that covers all the names (IRIs and literals) occurring in the RDF graphs G_{1} and G_{2}, and let I OWL 2 RDFBased satisfy G_{1}. It will be shown that I OWL 2 RDFBased satisfies G_{2}.
As a first step, an OWL 2 Direct interpretation J w.r.t. the corresponding OWL 2 Direct datatype map F(D) will be constructed for a vocabulary V_{J} that covers all the names (IRIs and literals) occurring in the OWL 2 DL ontologies in Functional Syntax form F(G_{1}) and F(G_{2}). J will be defined in a way such that it closely corresponds to I on those parts of the vocabularies V_{I} and V_{J} that cover G_{1} and G_{2}, and F(G_{1}) and F(G_{2}), respectively.
G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form that are mapped by the reverse RDF mapping to F(G_{1}) and F(G_{2}), respectively. This means that the same literals are used in both G_{1} and F(G_{1}), and in both G_{2} and F(G_{2}), respectively. Further, since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, according to property (2) of the "Balanced"definition there are entity declarations in F(G_{1}) and F(G_{2}) for all the entity types of every nonbuiltin IRI occurring in G_{1} and G_{2}, respectively. For each entity declaration of the form Declaration(T(u)) in F(G_{1}) and F(G_{2}), where T is the entity type for some IRI u, a typing triple of the form u rdf:type t exists in G_{1} or G_{2}, respectively, where t denotes the class representing the part of the universe that corresponds to T; and vice versa.
Since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, all the entity declarations of F(G_{2}) are also contained in F(G_{1}), and therefore all the typing triples of G_{2} that correspond to some entity declaration in F(G_{2}) are also contained in G_{1}. Since I OWL 2 RDFBased satisfies G_{1}, all these "declaring" typing triples are OWL 2 RDFBased satisfied by I, and thus all nonbuiltin IRIs in G_{1} and G_{2} are actually instances of their declared parts of the universe.
Based on these observations, the OWL 2 Direct interpretation J and its vocabulary V_{J} for the datatype map F(D) can now be defined.
The vocabulary V_{J} := ( V_{C} , V_{OP} , V_{DP} , V_{I} , V_{DT} , V_{LT} , V_{FA} ) is defined as follows.
The sets V_{DT} of datatype names, V_{LT} of literals, and V_{FA} of facetliteral pairs are defined according to Section 2.1 of [OWL 2 Direct Semantics] w.r.t. the datatype map F(D). Specifically, V_{DT} includes all IRIs that are declared as datatypes in F(G_{1}).
The interpretation J := ( Δ_{I} , Δ_{D} , ⋅ ^{C} , ⋅ ^{OP} , ⋅ ^{DP} , ⋅ ^{I} , ⋅ ^{DT} , ⋅ ^{LT} , ⋅ ^{FA} ) is 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 datatype interpretation function ⋅ ^{DT}, the literal interpretation function ⋅ ^{LT}, and the facet interpretation function ⋅ ^{FA} are defined according to Section 2.2 of [OWL 2 Direct Semantics]. Specifically, ⋅ ^{DT} interprets all IRIs that are declared as datatypes in F(G_{1}) according the following definition. For every nonbuiltin IRI u occurring in F(G_{1}):
Note that G_{1} may also contain declarations of annotation properties, but they will not be interpreted by the OWL 2 Direct Semantics and are therefore ignored here. This will not lead to problems, since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, and therefore G_{2} does not contain any annotations.
Further, note that the definition of J is compatible with the concept of a nonseparated vocabulary in OWL 2 DL (also called "punning", see Section 5.9 of [OWL 2 Specification]). Since G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, it is allowed that the same IRI u is declared to be all of an individual name, and a class name, and either an object property name or a data property name. According to I, the IRI u will always denote the same individual in the universe IR, where I(u) will be both a class and a property. Under J, however, the individual name u will denote an individual, the class name u will denote a subset of Δ_{I}, and the property name u will denote a subset of Δ_{I} × Δ_{I}.
Literals occurring in G_{1} and G_{2} are mapped by the OWL 2 RDF mapping to the same literals in the corresponding interpreted language constructs of F(G_{1}) and F(G_{2}), which comprise data enumerations, hasvalue restrictions with a data value, cardinality restrictions, datatype restrictions, data property assertions, and negative data property assertions. Also, the semantics of literals is strictly analog for both the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics. Therefore, literals need no further treatment in this proof.
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 to show that J OWL 2 Direct satisfies every axiom occurring in F(G_{1}). Let A be an axiom occurring in F(G_{1}), and let g_{A} be the subgraph of G_{1} that is mapped to A by the reverse OWL 2 RDF mapping. It is possible to prove that J OWL 2 Direct satisfies A by showing that the meaning, which is given to A by the OWL 2 Direct Semantics, is compatible with the semantic relationship that, according to J, holds between the denotations of the names occurring in A. The basic idea is as follows:
Since I OWL 2 RDFBased satisfies G_{1}, all the triples occurring in g_{A} are OWL 2 RDFBased satisfied by I. Also, since I is an OWL 2 RDFBased interpretation, all the OWL 2 RDFBased semantic conditions are met by I. Hence, the lefttoright directions of all the semantic conditions that are "matched" by the triples in g_{A} will apply. This will reveal certain semantic relationships that, according to I, hold between the denotations of the names occurring in g_{A}. These semantic relationships are, roughly speaking, the semantic consequences of the axiom that is encoded by the triples in g_{A}.
Since the denotations w.r.t. J of all the names occurring in A have been defined in terms of the denotations and class and property extensions w.r.t. I of the same names occurring in g_{A}, and since the meaning of the axiom A w.r.t. the OWL 2 Direct Semantics turns out to be fully covered by the semantic consequences of the subgraph g_{A} w.r.t. the OWL 2 RDFBased Semantics, one can eventually show that J OWL 2 Direct satisfies A.
A special note is necessary for anonymous individuals occurring in an assertion A. These have the form of the same blank node b both in A and in g_{A}. Both the OWL 2 RDFBased Semantics and the OWL 2 Direct Semantics treat blank nodes as existential variables in an ontology. Since I satisfies g_{A}, b can be mapped to an individual x in IR such that g_{A} becomes true under I (see Section 1.5 in [RDF Semantics] for the precise definition on how blank nodes are treated in RDF based languages). The same mapping from b to x can also be used for J in order to OWL 2 Direct satisfy 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 could be constructed in principle. A complete proof would need to take every language construct of OWL 2 into account, as well as additional aspects such as datatype maps and facets. As in the example below, such a proof can make use of the observation that the definitions of the OWL 2 Direct Semantics and the OWL 2 RDFBased Semantics are actually closely aligned for all the different language constructs of OWL 2.
Let A = SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3)) for IRIs ex:c1, ex:c2 and ex:c3 that are declared to be classes elsewhere in F(G_{1}).
Due to the reverse OWL 2 RDF mapping, g_{A} has the form
g_{A} :
ex:c1 rdfs:subClassOf _:x .
_:x rdf:type owl:Class .
_:x owl:unionOf ( ex:c2 ex:c3 ) .
Since I is an OWL 2 RDFBased interpretation, it meets all the OWL 2 RDFBased semantic conditions. Since I OWL 2 RDFBased satisfies G_{1}, all the triples in g_{A} are OWL 2 RDFBased satisfied, and this triggers the lefttoright directions of the semantic conditions for subclass axioms (rdfs:subClassOf) and union class expressions (owl:unionOf). This reveals that the denotations of the names in g_{A} are actually classes
I(ex:c1) ∈ IC ,
I(ex:c2) ∈ IC ,
I(ex:c3) ∈ IC ,
and that the following semantic relationship holds between the extensions of these classes:
ICEXT(I(ex:c1)) ⊆ ICEXT(I(ex:c2)) ∪ ICEXT(I(ex:c3)) .
From applying the definition of J follows that the following semantic relationship, w.r.t. J, holds between the denotations of the class names occurring in A:
(ex:c1) ^{C} ⊆ (ex:c2) ^{C} ∪ (ex:c3) ^{C} .
This semantic relationship equals the meaning of the axiom A = SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3)) w.r.t. the OWL 2 Direct Semantics. Hence, J OWL 2 Direct satisfies A.
Since J OWL 2 Direct satisfies F(G_{1}), and since F(G_{1}) OWL 2 Direct entails F(G_{2}), it follows that J OWL 2 Direct satisfies F(G_{2}).
The next step will be to show that I OWL 2 RDFBased satisfies G_{2}. For this to hold, I needs to OWL 2 RDFBased satisfy all the triples occurring in G_{2}, taking into account the premise that an OWL 2 RDFBased interpretation is required to meet all the OWL 2 RDFBased semantic conditions.
Since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, G_{2} contains a single ontology header consisting of a single triple "b rdf:type owl:Ontology" with a blank node b, and it does neither contain annotations nor deprecation statements. Hence, F(G_{2}) only consists of entity declarations and axioms, and does not have any ontology IRI and no ontology version, annotations or import directives. Further, since G_{2} is an OWL 2 DL ontology in RDF graph form, every triple occurring in G_{2}, which is not the ontology header triple, belongs to some subgraph of G_{2} that is mapped by the reverse OWL 2 RDF mapping to one of the entity declarations or axioms contained in F(G_{2}).
For the ontology header triple "b rdf:type owl:Ontology" in G_{2}: Since G_{1} is an OWL 2 DL ontology in RDF graph form, G_{1} contains an ontology header containing a triple "x rdf:type owl:Ontology", where x is either an IRI or a blank node. Since I OWL 2 RDFBased satisfies G_{1}, this particular triple is satisfied by I. From the semantic conditions of "Simple Entailment", as defined in [RDF Semantics], follows that the triple "b rdf:type owl:Ontology" with the existentially interpreted blank node b is satisfied by I, too.
For entity declarations, let A be an entity declaration in F(G_{2}), and let g_{A} be the corresponding subgraph of G_{2}. Since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, A occurs in F(G_{1}), and hence g_{A} is a subgraph of G_{1}. Since I OWL 2 RDFBased satisfies G_{1}, I in particular OWL 2 RDFBased satisfies g_{A}.
For axioms, let A be an axiom in F(G_{2}), and let g_{A} be the corresponding subgraph of G_{2}. It is possible to prove that I OWL 2 RDFBased satisfies g_{A}, by showing that all the premises for the righttoleft hand side of the particular semantic conditions, which are associated with the sort of axiom represented by g_{A}, are met. This will allow to apply the semantic condition, from which will follow that all the triples in g_{A} are OWL 2 RDFBased satisfied by I. The premises of the semantic condition generally require that the denotations of all the nonbuiltin names in g_{A} are contained in the appropriate part of the universe, and that the semantic relationship that is expressed on the right hand side of the semantic condition actually holds between the denotations of all these names w.r.t. I. Special care has to be taken regarding the blank nodes occurring in g_{A}. The basic idea is 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, property (2) of the "Balanced"definition provides that there are entity declarations in F(G_{2}) for all the entity types of u, each being of the form E := "Declaration(T(u))" for some entity type T. From the reverse RDF mapping follows that for each such declaration E a typing triple e exists in G_{2}, being of the form e := "u rdf:type t", where t is the name of a class representing the part of the universe corresponding to the entity type T. It has already been shown that for E being an entity declaration in F(G_{2}), and e being the corresponding subgraph in G_{2}, I OWL 2 RDFBased satisfies e. Hence, I(u) is an individual contained in the appropriate part of the universe.
Further, since J OWL 2 Direct satisfies F(G_{2}), J OWL 2 Direct satisfies A. Therefore, the semantic relationship that is represented by A according to the OWL 2 Direct Semantics actually holds between the denotations of the names occurring in A w.r.t. J. Since the denotations of these names w.r.t. J have been defined in terms of the denotations and class and property extensions w.r.t. I of the same names in G_{2}, by applying the definition of J it will turn out that the analog relationship also holds between the denotations of the same names occurring in g_{A}.
Finally, for the blank nodes occurring in g_{A}, it becomes clear from the fact that G_{2} is an OWL 2 DL ontology in RDF graph form that only certain kinds of subgraphs of g_{A} can occur having blank nodes.
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 could be constructed in principle. A complete proof would need to take every language construct of OWL 2 into account, as well as additional aspects such as datatype maps and facets. As in the example below, such a proof can make use of the observation that the definitions of the OWL 2 Direct Semantics and the OWL 2 RDFBased Semantics are actually closely aligned for all the different language constructs of OWL 2.
Let A = SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3)) for IRIs ex:c1, ex:c2 and ex:c3 that are declared to be classes elsewhere in F(G_{2}).
Due to the reverse OWL 2 RDF mapping, g_{A} has the form
g_{A} :
ex:c1 rdfs:subClassOf _:x .
_:x rdf:type owl:Class .
_:x owl:unionOf ( ex:c2 ex:c3 ) .
The entity declarations for the class names ex:c1, ex:c2 and ex:c3 occurring in both A and g_{A} correspond to the typing triples
ex:c1 rdf:type owl:Class .
ex:c2 rdf:type owl:Class .
ex:c3 rdf:type owl:Class .
in G_{2}, respectively. Based on the premise that the pair ⟨ G_{1} , G_{2} ⟩ is balanced, all these typing triples are OWL 2 RDFBased satisfied by I. Hence, all of the IRIs denote classes:
I(ex:c1) ∈ IC ,
I(ex:c2) ∈ IC and
I(ex:c3) ∈ IC .
Since J OWL 2 Direct satisfies A, the following semantic relationship holds between the denotations of the class names in A w.r.t. J:
(ex:c1) ^{C} ⊆ (ex:c2) ^{C} ∪ (ex:c3) ^{C} .
Applying the definition of J results in the following semantic relationship w.r.t. I that holds between the denotations of the names in g_{A}:
ICEXT(I(ex:c1)) ⊆ ICEXT(I(ex:c2)) ∪ ICEXT(I(ex:c3)) .
The subgraph g_{E} of g_{A}, given by
g_{E} :
_:x rdf:type owl:Class .
_:x owl:unionOf ( c2 c3 ) .
corresponds to a union class expression in A. Since the pair ⟨ G_{1} , G_{2} ⟩ is balanced, g_{E} is also a subgraph of G_{1} (it will be assumed that the same blank nodes are used in both instances of g_{E} in order to simplify the argument). Since both G_{1} and G_{2} are OWL 2 DL ontologies in RDF graph form, the blank nodes occurring in g_{E} do not occur outside of g_{E}, neither in G_{1} nor in G_{2}.
Since I OWL 2 RDFBased satisfies G_{1}, according to the semantic conditions for RDF graphs with blank nodes (see Section 1.5 of [RDF Semantics]), a mapping B from blank(g_{E}) to IR exists, where blank(g_{E}) is the set of all blank nodes in g_{E}, such that the extended interpretation I+B OWL 2 RDFBased satisfies all the triples in g_{E}. An analog argument holds for all the blank nodes occurring in the sequence expression ( c2 c3 ).
This allows to apply the lefttoright direction of the semantic condition for union class expressions (owl:unionOf), providing:
[I+B](_:x) ∈ IC ,
ICEXT([I+B](_:x))
=
ICEXT(I(ex:c2))
∪
ICEXT(I(ex:c3)) .
Together with the intermediate results from above, it follows:
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 of subclass axioms (rdfs:subClassOf), which results in
⟨ I(ex:cl) , [I+B](_:x) ⟩ ∈ IEXT(I(rdfs:subClassOf)) .
So, the triple
ex:c1 rdfs:subClassOf _:x .
is OWL 2 RDFBased satisfied by I+B, where "_:x" is the same blank node as the root blank node of the union class expression in g_{A}.
Hence, w.r.t. existential blank node semantics, I OWL 2 RDFBased satisfies all the triples in g_{A}.
To conclude, for every OWL 2 RDFBased interpretation I that OWL 2 RDFBased satisfies G_{1} it turns out that I also OWL 2 RDFBased satisfies G_{2}. Hence, G_{1} OWL 2 RDFBased entails G_{2}.
Q.E.D.
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 formallyconsist 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" orand "useful" entailments,
or for what it means thatwhen a givengraph transformation
iscan be considered "harmless" or not.
As discussed in
Section 7.1,
a core obstacle for the correspondence theorem to hold
waswere RDF encodings of OWL 2 expressions,
such as union class expressions,
whichwhen 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 thatexists,
which represents the denotation of such an expression.
The "comprehension principles"conditions" defined in this section
are additional semantic conditions
that provide forthe necessary individuals
for every possible list,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 OWL 2 DLentailment queries, while at the same time allowing for the interpretationqueries consisting of arbitrary entailment queries.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.
Table 8.1
lists the comprehension conditions for sequences,
i.e. RDF lists.
These comprehension conditions provide the existence
of sequences
built from any possiblefinite 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)) 
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 complement classes for any class,
and of unions andintersections and unions
built from any possiblefinite set of classes
contained in the universe.
if  then exists z ∈ IR 

s sequence of c_{ ∈ IC ⟨ z1} , … , c_{ ⟩ ∈ IEXT( I ( owl:complementOf )) dn} ∈  ⟨ z , 
s sequence of c_{1} , … , c_{n} ∈ IC  ⟨ z , s ⟩ ∈ IEXT(I( 
 ⟨ z , c 
d ∈ IDC  ⟨ z , 
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 possiblefinite 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)) 
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 restriction classesrestrictions
on any property and for any nonnegative integer,
as well as value restriction classesrestrictions
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 [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 Section 7 of [OWL 2 Specification] for further information).
if  then exists z ∈ IR 

p ∈ IP  ⟨ z , c ⟩ ∈ IEXT(I( ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
c ∈ IC , s sequence of p_{1} , … , p_{n} ∈ IP , n ≥ 1  ⟨ z , c ⟩ ∈ IEXT(I( ⟨ z , s ⟩ ∈ IEXT(I(owl:onProperties)) 
c ∈ IC , p ∈ IP  ⟨ z , c ⟩ ∈ IEXT(I( ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
c ∈ IC , s sequence of p_{1} , … , p_{n} ∈ IP , n ≥ 1  ⟨ z , c ⟩ ∈ IEXT(I( ⟨ 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( ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , p ∈ IP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , p ∈ IP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , c ⟩ ∈ IEXT(I(owl:onClass)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , d ⟩ ∈ IEXT(I(owl:onDataRange)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , c ⟩ ∈ IEXT(I(owl:onClass)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , d ⟩ ∈ IEXT(I(owl:onDataRange)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , c ∈ IC , p ∈ IP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , c ⟩ ∈ IEXT(I(owl:onClass)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
n ∈ INNI , d ∈ IDC , p ∈ IODP  ⟨ z , n ⟩ ∈ IEXT(I( ⟨ z , d ⟩ ∈ IEXT(I(owl:onDataRange)) , ⟨ z , p ⟩ ∈ IEXT(I(owl:onProperty)) 
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 possiblefinite 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}) 
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 variablesvariable
on the left hand side of an inverse property axioms. However,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.
ButHowever,
the OWL 2 RDFBased Semantics doesnormative semantic conditions
do not permit this conclusion,
since it is not ensured that
for every property p
there is an individual in the universe
that happens to be the inverse property of p.
if  then exists z ∈ IR 

p ∈ IP  ⟨ z , p ⟩ ∈ IEXT(I(owl:inverseOf)) 
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 the OWL 1 RDFCompatible Semanticsits predecessor in several aspects,
in most cases due to serious technical problems
that would arisehave 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:
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, seespecification
(see Section 2.4 of [OWL 2 Specification ].]).
In addition,
the OWL 2 RDFBased Semantics
MAYis 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 theOWL 1 RDFCompatible Semantics,1,
since all RDF graphs that were legal forunder the OWL 1 RDFCompatible Semantics
are still legal under the OWL 2 RDFBased Semantics.
Datatype Facets [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
in Section 4 of the OWL 2 RDFBased Semanticsto take datatypeconstraining facets into account. Facets are appliedaccount
(see Section 4),
in connection with a new language construct of OWL 2, called "datatype restrictions"order to allow for datatype restrictions
as specified in Section 5.7.
This change is compatible with theOWL 1 RDFCompatible Semantics,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 comprehensionsemantic conditions of the OWL 1 RDFCompatible Semantics included
a set of so called
"comprehension conditions" .,
which allowed to show the original
"correspondence theorem"
stating that every entailment of OWL 2 RDFBased Semantics extends this set by semantic1 DL was also an entailment of OWL 1 Full.
The document at hand adds comprehension conditions
for the new language constructs of OWL 2, see2
(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 languagesemantics
(Issue 119).
Further,In the OWL 1 RDFCompatible Semantics specification, the comprehension conditions were primarily used to ensure the correctness of the so called "correspondence theorem" . However,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 semantic 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,conditions
and whichcan be seen as a technical refinement
of an idea
originally discussed by the WebOnt Working Group
(email).
This change is an incompatible deviation from theOWL 1 RDFCompatible Semantics,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
hadwere defined in a flawed definitions thatform,
which lead to formal inconsistency of the OWL 1 RDFCompatiblesemantics
(Issue 120;
see also an unofficial
problem description).
TheseThe affected semantic conditions have been corrected for the OWL 2 RDFBased Semantics by changing the form of the semantic conditions,revised;
see
Section 5.4
and
Section 5.5.
This change is an incompatible deviation from theOWL 1 RDFCompatible Semantics,1,
since the semantics has formally been weakened
by eliminatingin 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 an unofficial
problem description).
The missing semantic condition
has been added to the OWL 2 RDFBased Semantics
in(see Section 5.11 .5.10).
This change is compatible with theOWL 1 RDFCompatible Semantics,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
in the OWL 2 RDFBased Semantics(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
ofin the OWL 1 RDFCompatible Semantics
has been analyzed in(email).
This change is compatible with the OWL 1 RDFCompatible Semantics according to this analysis. Weakened Semantics ofNonEmpty Data Value Enumerations [DEV]:
The application of thesemantic condition for enumerations of data values
in Section 5.5
is now restricted to nonempty sets of data values only.values.
This prevents the class owl:Nothing
from unintentionally becoming an instance
of the class rdfs:Datatype,
as analyzed in
(email).
This weakeningrestriction of the semantics
is an incompatible deviation from theOWL 1 RDFCompatible Semantics. Note1.
Note, however,
that this change doesit is still permit the definition ofpossible
to define an empty enumeration with an explicitly given type rdfs:Datatypeof data values,
as explained in Section 5.5.
Terminological Clarifications [NOM]:
TheThis 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 whole language
that is determined
by the set of 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 OWL 2 RDFBased Semantics.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}"
have been changed toreplaced 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 editorialextension
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.
Additional names for class extensions,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.
Deprecated Vocabulary Terms [NOM]: The following vocabulary terms have been deprecated as of OWL 2 by the Working Group, and SHOULD NOT be used in new ontologies anymore:
This section lists significantChanges sincefrom the
Second PublicLast Call Working Draft
of 02 December 2008. Section 1 "Introduction": Better clarifies the document's purpose at the beginning of the section. Reflects the chronology of the document21 April 2009:
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),
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),
Elisa Kendall (Sandpiper Software),
Markus Krötzsch (FZI),
Carsten Lutz (Universität Bremen),
Deborah L. McGuinness (RPI),
Boris Motik (Oxford University),
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),
andZhe Wu (Oracle Corporation).Corporation), and
Antoine Zimmermann (DERI Galway).
We would also like to thank past members of the working group:
Jeremy Carroll,
Jim Hendler,
Vipul Kashyap.