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  <title>OWL 2 Web Ontology Language RDF-Based Semantics</title>
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<a href="http://www.w3.org/"><img alt="W3C" height="48" src="http://www.w3.org/Icons/w3c_home" width="72" /></a><h1 id="title" style="clear:both">OWL 2 Web Ontology Language <br /><span id="short-title">RDF-Based Semantics</span></h1>

<h2 id="W3C-doctype">W3C Recommendation 27 October 2009</h2>

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<dl>
<dt>This version:</dt>
<dd><a href="http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/" id="this-version-url">http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/</a></dd>

<dt>Latest version (series 2):</dt>
<dd><a href="http://www.w3.org/TR/owl2-rdf-based-semantics/">http://www.w3.org/TR/owl2-rdf-based-semantics/</a></dd>

<dt>Latest Recommendation:</dt>
<dd><a href="http://www.w3.org/TR/owl-rdf-based-semantics">http://www.w3.org/TR/owl-rdf-based-semantics</a></dd>

<dt>Previous version:</dt>
<dd><a href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/</a> (<a href="http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/diff-from-20090922">color-coded diff</a>)</dd>
</dl>

<dl><dt>Editors:</dt><dd><a href="http://www.fzi.de/michael.schneider">Michael Schneider</a>, FZI Research Center for Information Technology</dd>
<dt>Contributors: (in alphabetical order)</dt><dd><a href="http://semanticweb.org/wiki/Jeremy_J._Carroll">Jeremy Carroll</a>, HP (now at TopQuadrant)</dd>
<dd><a href="http://www.w3.org/People/Ivan/">Ivan Herman</a>, W3C/ERCIM</dd>
<dd><a href="http://ect.bell-labs.com/who/pfps/">Peter F. Patel-Schneider</a>, Bell Labs Research, Alcatel-Lucent</dd>
</dl>

<p>Please refer to the <a href="http://www.w3.org/2007/OWL/errata"><strong>errata</strong></a> for this document, which may include some normative corrections.</p>

<p>This document is also available in these non-normative formats: <a href="http://www.w3.org/2009/pdf/REC-owl2-rdf-based-semantics-20091027.pdf">PDF version</a>.</p>

<p>See also <a href="http://www.w3.org/2007/OWL/translation/owl2-rdf-based-semantics">translations</a>.</p>

<p class="copyright"><a href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a> &copy; 2009 <a href="http://www.w3.org/"><acronym title="World Wide Web Consortium">W3C</acronym></a><sup>&reg;</sup> (<a href="http://www.csail.mit.edu/"><acronym title="Massachusetts Institute of Technology">MIT</acronym></a>, <a href="http://www.ercim.org/"><acronym title="European Research Consortium for Informatics and Mathematics">ERCIM</acronym></a>, <a href="http://www.keio.ac.jp/">Keio</a>), All Rights Reserved. W3C <a href="http://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>, <a href="http://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a> and <a href="http://www.w3.org/Consortium/Legal/copyright-documents">document use</a> rules apply.</p>

</div>
<hr />
<h2><a id="abstract" name="abstract">Abstract</a></h2>

<div>
<div><p>The OWL 2 Web Ontology Language, informally OWL 2, is an ontology language for the Semantic Web with formally defined meaning.  OWL 2 ontologies provide classes, properties, individuals, and data values and are stored as Semantic Web documents.  OWL 2 ontologies can be used along with information written in RDF, and OWL 2 ontologies themselves are primarily exchanged as RDF documents.  The OWL 2 <a href="http://www.w3.org/TR/2009/REC-owl2-overview-20091027/" title="Document Overview">Document Overview</a> describes the overall state of OWL 2, and should be read before other OWL 2 documents.</p><p>This document defines the RDF-compatible model-theoretic semantics of OWL 2.</p></div>
</div>

<h2 class="no-toc no-num">
<a id="status" name="status">Status of this Document</a>
</h2>
    
<h4 class="no-toc no-num" id="may-be">May Be Superseded</h4>
    
<p><em>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 <a href="http://www.w3.org/TR/">W3C technical reports index</a> at http://www.w3.org/TR/.</em></p>

    

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<h4 class="no-toc no-num" id="sotd-xml-dep">XML Schema Datatypes Dependency</h4>

<p>OWL 2 is defined to use datatypes defined in the <a href="http://www.w3.org/TR/xmlschema-2/">XML Schema Definition Language (XSD)</a>.  As of this writing, the latest W3C Recommendation for XSD is version 1.0, with <a href="http://www.w3.org/TR/xmlschema11-1/">version 1.1</a> progressing toward Recommendation.  OWL 2 has been designed to take advantage of the new datatypes and clearer explanations available in XSD 1.1, but for now those advantages are being partially put on hold.  Specifically, until XSD 1.1 becomes a W3C Recommendation, the elements of OWL 2 which are based on it should be considered <em>optional</em>, as detailed in <a href="http://www.w3.org/TR/2009/REC-owl2-conformance-20091027/#XML_Schema_Datatypes">Conformance, section 2.3</a>.  Upon the publication of XSD 1.1 as a W3C Recommendation, those elements cease to be optional and are to be considered required as otherwise specified.</p>

<p>We suggest that for now developers and users follow the <a href="http://www.w3.org/TR/2009/CR-xmlschema11-1-20090430/">XSD 1.1 Candidate Recommendation</a>.  Based on discussions between the Schema and OWL Working Groups, we do not expect any implementation changes will be necessary as XSD 1.1 advances to Recommendation.</p>
</div>



           <h4 class="no-toc no-num" id="status-changes">Summary of Changes</h4>

            <div>There have been no <a href="http://www.w3.org/2005/10/Process-20051014/tr#substantive-change">substantive</a> changes since the <a href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">previous version</a>.   For details on the minor changes see the <a href="#changelog">change log</a> and <a href="http://www.w3.org/TR/2009/REC-owl2-rdf-based-semantics-20091027/diff-from-20090922">color-coded diff</a>.</div>



<h4 class="no-toc no-num" id="please">Please Send Comments</h4><p>Please send any comments to <a class="mailto" href="mailto:public-owl-comments@w3.org">public-owl-comments@w3.org</a>
    (<a class="http" href="http://lists.w3.org/Archives/Public/public-owl-comments/">public
    archive</a>).  Although work on this document by the <a href="http://www.w3.org/2007/OWL/">OWL Working Group</a> is complete, comments may be addressed in the <a href="http://www.w3.org/2007/OWL/errata">errata</a> or in future revisions.  Open discussion among developers is welcome at <a class="mailto" href="mailto:public-owl-dev@w3.org">public-owl-dev@w3.org</a> (<a class="http" href="http://lists.w3.org/Archives/Public/public-owl-dev/">public archive</a>).</p>
    
<h4 class="no-toc no-num" id="endorsement">Endorsed By W3C</h4>
    
<p><em>This document has been reviewed by W3C Members, by software developers, and by other W3C groups and interested parties, and is endorsed by the Director as a W3C Recommendation. It is a stable document and may be used as reference material or cited from another document. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.</em></p>


<h4 class="no-toc no-num" id="patents">Patents</h4>
    
<p><em>This document was produced by a group operating under the <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/">5 February 2004 W3C Patent Policy</a>. W3C maintains a <a href="http://www.w3.org/2004/01/pp-impl/41712/status" rel="disclosure">public list of any patent disclosures</a> made in connection with the deliverables of the group; that page also includes instructions for disclosing a patent.</em></p>

<hr title="Separator After Status Section" />

 
<table class="toc" id="toc" summary="Contents"><tr><td><div id="toctitle"><h2>Table of Contents</h2></div>
<ul>
<li class="toclevel-1"><a href="#Introduction_.28Informative.29"><span class="tocnumber">1</span> <span class="toctext">Introduction (Informative)</span></a></li>
<li class="toclevel-1"><a href="#Ontologies"><span class="tocnumber">2</span> <span class="toctext">Ontologies</span></a>
<ul>
<li class="toclevel-2"><a href="#Syntax"><span class="tocnumber">2.1</span> <span class="toctext">Syntax</span></a></li>
<li class="toclevel-2"><a href="#Content_of_Ontologies_.28Informative.29"><span class="tocnumber">2.2</span> <span class="toctext">Content of Ontologies (Informative)</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Vocabulary"><span class="tocnumber">3</span> <span class="toctext">Vocabulary</span></a>
<ul>
<li class="toclevel-2"><a href="#Standard_Prefixes"><span class="tocnumber">3.1</span> <span class="toctext">Standard Prefixes</span></a></li>
<li class="toclevel-2"><a href="#Vocabulary_Terms"><span class="tocnumber">3.2</span> <span class="toctext">Vocabulary Terms</span></a></li>
<li class="toclevel-2"><a href="#Datatype_Names"><span class="tocnumber">3.3</span> <span class="toctext">Datatype Names</span></a></li>
<li class="toclevel-2"><a href="#Facet_Names"><span class="tocnumber">3.4</span> <span class="toctext">Facet Names</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Interpretations"><span class="tocnumber">4</span> <span class="toctext">Interpretations</span></a>
<ul>
<li class="toclevel-2"><a href="#Datatype_Maps"><span class="tocnumber">4.1</span> <span class="toctext">Datatype Maps</span></a></li>
<li class="toclevel-2"><a href="#Vocabulary_Interpretations"><span class="tocnumber">4.2</span> <span class="toctext">Vocabulary Interpretations</span></a></li>
<li class="toclevel-2"><a href="#Satisfaction.2C_Consistency_and_Entailment"><span class="tocnumber">4.3</span> <span class="toctext">Satisfaction, Consistency and Entailment</span></a></li>
<li class="toclevel-2"><a href="#Parts_of_the_Universe"><span class="tocnumber">4.4</span> <span class="toctext">Parts of the Universe</span></a></li>
<li class="toclevel-2"><a href="#Class_Extensions"><span class="tocnumber">4.5</span> <span class="toctext">Class Extensions</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Semantic_Conditions"><span class="tocnumber">5</span> <span class="toctext">Semantic Conditions</span></a>
<ul>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_the_Parts_of_the_Universe"><span class="tocnumber">5.1</span> <span class="toctext">Semantic Conditions for the Parts of the Universe</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_the_Vocabulary_Classes"><span class="tocnumber">5.2</span> <span class="toctext">Semantic Conditions for the Vocabulary Classes</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_the_Vocabulary_Properties"><span class="tocnumber">5.3</span> <span class="toctext">Semantic Conditions for the Vocabulary Properties</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Boolean_Connectives"><span class="tocnumber">5.4</span> <span class="toctext">Semantic Conditions for Boolean Connectives</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Enumerations"><span class="tocnumber">5.5</span> <span class="toctext">Semantic Conditions for Enumerations</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Property_Restrictions"><span class="tocnumber">5.6</span> <span class="toctext">Semantic Conditions for Property Restrictions</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Datatype_Restrictions"><span class="tocnumber">5.7</span> <span class="toctext">Semantic Conditions for Datatype Restrictions</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_the_RDFS_Vocabulary"><span class="tocnumber">5.8</span> <span class="toctext">Semantic Conditions for the RDFS Vocabulary</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Equivalence_and_Disjointness"><span class="tocnumber">5.9</span> <span class="toctext">Semantic Conditions for Equivalence and Disjointness</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_N-ary_Disjointness"><span class="tocnumber">5.10</span> <span class="toctext">Semantic Conditions for N-ary Disjointness</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Sub_Property_Chains"><span class="tocnumber">5.11</span> <span class="toctext">Semantic Conditions for Sub Property Chains</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Inverse_Properties"><span class="tocnumber">5.12</span> <span class="toctext">Semantic Conditions for Inverse Properties</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Property_Characteristics"><span class="tocnumber">5.13</span> <span class="toctext">Semantic Conditions for Property Characteristics</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Keys"><span class="tocnumber">5.14</span> <span class="toctext">Semantic Conditions for Keys</span></a></li>
<li class="toclevel-2"><a href="#Semantic_Conditions_for_Negative_Property_Assertions"><span class="tocnumber">5.15</span> <span class="toctext">Semantic Conditions for Negative Property Assertions</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Appendix:_Axiomatic_Triples_.28Informative.29"><span class="tocnumber">6</span> <span class="toctext">Appendix: Axiomatic Triples (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="#Axiomatic_Triples_in_RDF"><span class="tocnumber">6.1</span> <span class="toctext">Axiomatic Triples in RDF</span></a></li>
<li class="toclevel-2"><a href="#Axiomatic_Triples_for_the_Vocabulary_Classes"><span class="tocnumber">6.2</span> <span class="toctext">Axiomatic Triples for the Vocabulary Classes</span></a></li>
<li class="toclevel-2"><a href="#Axiomatic_Triples_for_the_Vocabulary_Properties"><span class="tocnumber">6.3</span> <span class="toctext">Axiomatic Triples for the Vocabulary Properties</span></a></li>
<li class="toclevel-2"><a href="#A_Set_of_Axiomatic_Triples"><span class="tocnumber">6.4</span> <span class="toctext">A Set of Axiomatic Triples</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29"><span class="tocnumber">7</span> <span class="toctext">Appendix: Relationship to the Direct Semantics (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="#Example_on_Semantic_Differences"><span class="tocnumber">7.1</span> <span class="toctext">Example on Semantic Differences</span></a></li>
<li class="toclevel-2"><a href="#Correspondence_Theorem"><span class="tocnumber">7.2</span> <span class="toctext">Correspondence Theorem</span></a></li>
<li class="toclevel-2"><a href="#Proof_for_the_Correspondence_Theorem"><span class="tocnumber">7.3</span> <span class="toctext">Proof for the Correspondence Theorem</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Appendix:_Comprehension_Conditions_.28Informative.29"><span class="tocnumber">8</span> <span class="toctext">Appendix: Comprehension Conditions (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Sequences"><span class="tocnumber">8.1</span> <span class="toctext">Comprehension Conditions for Sequences</span></a></li>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Boolean_Connectives"><span class="tocnumber">8.2</span> <span class="toctext">Comprehension Conditions for Boolean Connectives</span></a></li>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Enumerations"><span class="tocnumber">8.3</span> <span class="toctext">Comprehension Conditions for Enumerations</span></a></li>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Property_Restrictions"><span class="tocnumber">8.4</span> <span class="toctext">Comprehension Conditions for Property Restrictions</span></a></li>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Datatype_Restrictions"><span class="tocnumber">8.5</span> <span class="toctext">Comprehension Conditions for Datatype Restrictions</span></a></li>
<li class="toclevel-2"><a href="#Comprehension_Conditions_for_Inverse_Properties"><span class="tocnumber">8.6</span> <span class="toctext">Comprehension Conditions for Inverse Properties</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Appendix:_Changes_from_OWL_1_.28Informative.29"><span class="tocnumber">9</span> <span class="toctext">Appendix: Changes from OWL 1 (Informative)</span></a></li>
<li class="toclevel-1"><a href="#Appendix:_Change_Log_.28Informative.29"><span class="tocnumber">10</span> <span class="toctext">Appendix: Change Log (Informative)</span></a>
<ul>
<li class="toclevel-2"><a href="#Changes_Since_Proposed_Recommendation"><span class="tocnumber">10.1</span> <span class="toctext">Changes Since Proposed Recommendation</span></a></li>
<li class="toclevel-2"><a href="#Changes_Since_Candidate_Recommendation"><span class="tocnumber">10.2</span> <span class="toctext">Changes Since Candidate Recommendation</span></a></li>
<li class="toclevel-2"><a href="#Changes_Since_Last_Call"><span class="tocnumber">10.3</span> <span class="toctext">Changes Since Last Call</span></a></li>
</ul>
</li>
<li class="toclevel-1"><a href="#Acknowledgments"><span class="tocnumber">11</span> <span class="toctext">Acknowledgments</span></a></li>
<li class="toclevel-1"><a href="#References"><span class="tocnumber">12</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2"><a href="#Normative_References"><span class="tocnumber">12.1</span> <span class="toctext">Normative References</span></a></li>
<li class="toclevel-2"><a href="#Nonnormative_References"><span class="tocnumber">12.2</span> <span class="toctext">Nonnormative References</span></a></li>
</ul>
</li>
</ul>
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<p><br />
</p>
<a name="Introduction_.28Informative.29"></a><h2> <span class="mw-headline">1  Introduction (Informative) </span></h2>
<div id="topic-intro-purpose"></div>
<p>This document defines the RDF-compatible model-theoretic semantics of OWL 2, 
referred to as the <i>"OWL 2 RDF-Based Semantics"</i>.
The OWL 2 RDF-Based Semantics gives a formal meaning
to every <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax"><i>RDF graph</i></a>
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
and is fully compatible with the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/"><i>RDF Semantics specification</i></a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The specification provided here
is the successor to 
the original <a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i> specification</a>
[<cite><a href="#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
</p>
<div id="topic-intro-rdfcompatible"></div>
<p>Technically,
the OWL 2 RDF-Based Semantics 
is defined as a 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension"><i>semantic extension</i></a> 
of
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#dtype_interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#dtype_interp">"D-Entailment"</a>
(RDFS with datatype support),
as specified in the <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
In other words,
the meaning given to an RDF graph by the OWL 2 RDF-Based Semantics
includes the meaning provided by the semantics of RDFS with datatypes,
and additional meaning is specified for all the language constructs of OWL 2,
such as Boolean connectives, 
sub property chains 
and qualified cardinality restrictions
(see the <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
for further information
on all the language constructs of OWL 2). 
The definition of the semantics for the extra constructs
follows the design principles 
as applied to the RDF Semantics.
</p>
<div id="topic-intro-documentcontent"></div>
<p>The content of this document is not meant to be self-contained
but builds on top of the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics document</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
by adding those aspects 
that are specific to OWL 2.
Hence,
the complete definition of the OWL 2 RDF-Based Semantics 
is given by 
the <i>combination</i> of both
the RDF Semantics document 
and the document at hand.
In particular,
the terminology used in the RDF Semantics
is reused here
except for cases 
where a conflict exists with the rest of the OWL 2 specification.
</p>
<div id="topic-intro-outline"></div>
<p>The remainder of this section
provides an overview
of some of the distinguishing features 
of the OWL 2 RDF-Based Semantics
and outlines the document's structure and content.
</p>
<div id="topic-intro-ontologies"></div>
<p>In <a href="#Ontologies" title="">Section 2</a>,
the <i>syntax</i>
over which the OWL 2 RDF-Based Semantics is defined
is the set of all 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax"><i>RDF graphs</i></a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>].
The OWL 2 RDF-Based Semantics
provides a precise formal meaning
for every RDF graph.
The language
that is determined
by RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics
is called 
<i>"OWL 2 Full"</i>.
In this document,
RDF graphs are also called
<i>"OWL 2 Full ontologies"</i>,
or simply <i>"ontologies"</i>,
unless there is risk of confusion. 
</p>
<div id="topic-intro-vocabulary"></div>
<p>The OWL 2 RDF-Based Semantics
interprets the
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFV" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFV">RDF</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFSV" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defRDFSV">RDFS <i>vocabularies</i></a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
and the <i>OWL 2 RDF-Based vocabulary</i>
together with an extended set of <i>datatypes</i>
and their constraining <i>facets</i> 
(see <a href="#Vocabulary" title="">Section 3</a>).
</p>
<div id="topic-intro-interpretation"></div>
<p><i>OWL 2 RDF-Based interpretations</i> 
(<a href="#Interpretations" title="">Section 4</a>) 
are defined on a <i>universe</i>
(see <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">Section 1.3 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
for an overview of
the basic intuition of model-theoretic semantics).
The universe is divided into <i>parts</i>,
namely <i>individuals</i>, <i>classes</i>, and <i>properties</i>,
which are identified with their RDF counterparts
(see <a href="#fig-partshierarchy" title="">Figure 1</a>).
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 RDF-Based interpretation
denotes an individual.
</p>
<div id="topic-intro-subparts"></div>
<p>The three basic parts are divided into further parts as follows.
The part of individuals subsumes the part of <i>data values</i>,
which comprises the denotations of all literals.
Also subsumed by the individuals is the part of <i>ontologies</i>.
The part of classes subsumes the part of <i>datatypes</i>,
which are classes 
consisting entirely of data values.
Finally, 
the part of properties subsumes the parts of
<i>object properties</i>, 
<i>data properties</i>,
<i>ontology properties</i> 
and <i>annotation properties</i>.
The part of object properties equals the whole part of properties,
and therefore all other kinds of properties are also object properties.
</p>
<div id="topic-intro-annotations"></div>
<p>For <i>annotations properties</i> 
note that annotations are not "semantic-free"
under the OWL 2 RDF-Based Semantics.
Just like every other triple or set of triples occurring in an RDF graph,
an annotation is assigned a truth value by any given OWL 2 RDF-Based interpretation.
Hence,
although annotations are meant to be "semantically weak",
i.e., their formal meaning does not significantly exceed 
that originating from the RDF Semantics specification,
adding an annotation 
may still change the meaning of an ontology.
A similar discussion holds for statements 
that are built from <i>ontology properties</i>,
such as <span class="name">owl:imports</span>,
which are used to define relationships between two ontologies.
</p>
<div id="topic-intro-extensions"></div>
<p>Every class represents a specific set of individuals,
called the <i>class extension</i> of the class:
an individual <i>a</i> is an instance of a class <i>C</i>,
if <i>a</i> is a member of the class extension ICEXT(<i>C</i>).
Since a class is itself an individual under the OWL 2 RDF-Based Semantics,
classes are distinguished from their respective class extensions.
This distinction allows,
for example, 
that a class may be an instance of itself 
by being a member of its own class extension.
Also,
two classes may be equivalent 
by sharing the same class extension, 
although being different individuals, 
e.g., they do not need to share the same properties.
Similarly,
every property has an associated <i>property extension</i>
that consists of pairs of individuals:
an individual <i>a<sub>1</sub></i> 
has a relationship to an individual <i>a<sub>2</sub></i> 
with respect to a property <i>p</i>
if the pair 
( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> )
is a member of the property extension IEXT(<i>p</i>). 
Again, properties are distinguished from their property extensions.
In general, 
if there are no further constraints,
an arbitrary extension may be associated with
a given class or property,
and two interpretations may associate 
distinct extensions
with the same class or property.
</p>
<div id="topic-intro-roleplay"></div>
<p>Individuals may <i>play different "roles"</i>.
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 RDF-Based interpretation 
are not required to be mutually disjoint,
or an individual can be
both a class and a property
by associating
both a class extension and a property extension
with it.
In the latter case
there will be no specific relationship
between the class extension and the property extension
of such an individual
without further constraints.
For example,
the same individual 
can have an empty class extension
while having a nonempty property extension.
</p>
<div id="topic-intro-conditions"></div>
<p>The main part of the OWL 2 RDF-Based Semantics is <a href="#Semantic_Conditions" title="">Section 5</a>,
which specifies 
a formal meaning for all the OWL 2 language constructs 
by means of the
<i>OWL 2 RDF-Based semantic conditions</i>.
These semantic conditions extend all the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">semantic conditions given in the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The OWL 2 RDF-Based semantic conditions effectively determine
which sets of RDF triples are assigned a specific meaning
and what this meaning is.
For example,
semantic conditions exist
that allow one to interpret the triple 
"<i>C</i> <span class="name">owl:disjointWith</span> <i>D</i>"
to mean that the denotations of the IRIs 
<i>C</i> and <i>D</i> 
have disjoint class extensions.
</p>
<div id="topic-intro-localization"></div>
<p>There is usually no need to provide <i>localizing information</i> 
(e.g., by means of "typing triples")
for the IRIs occurring in an ontology.
As for the RDF Semantics,
the OWL 2 RDF-Based semantic conditions have been designed 
to ensure that the denotation of any IRI
will be in the appropriate part of the universe.
For example,
the RDF triple
"<i>C</i> <span class="name">owl:disjointWith</span> <i>D</i>"
is sufficient to deduce that 
the denotations of the IRIs 
<i>C</i> and <i>D</i>
are actually <i>classes</i>.
It is not necessary to explicitly add additional typing triples
"<i>C</i> <span class="name">rdf:type rdfs:Class</span>"
and
"<i>D</i> <span class="name">rdf:type rdfs:Class</span>"
to the ontology.
</p>
<div id="topic-intro-axiomatic"></div>
<p>In the RDF Semantics,
this kind of "automatic localization" 
was to some extent achieved by so called 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples"><i>"axiomatic triples"</i></a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
such as
"<span class="name">rdf:type rdf:type rdf:Property</span>"
or
"<span class="name">rdf:type rdfs:domain rdfs:Resource</span>".
However, 
there is no explicit normative collection 
of additional axiomatic triples 
for the OWL 2 RDF-Based Semantics;
instead, 
the specific axiomatic aspects of the OWL 2 RDF-Based Semantics 
are determined by a subset of the OWL 2 RDF-Based semantic conditions.
<a href="#Appendix:_Axiomatic_Triples_.28Informative.29" title="">Section 6</a> 
discusses axiomatic triples in general
and provides an example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-intro-correspondence"></div>
<p><a href="#Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29" title="">Section 7</a> compares
the OWL 2 RDF-Based Semantics
with the <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/" title="Direct Semantics"><i>OWL 2 Direct Semantics</i></a>
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
While 
the OWL 2 RDF-Based Semantics is based on the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 Direct Semantics
is a <i>description logic</i> style semantics.
Several fundamental differences
exist between the two semantics,
but 
there is also a strong relationship 
basically stating that the OWL 2 RDF-Based Semantics 
is able to reflect all logical conclusions 
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.
The precise relationship is given by the 
<a href="#Correspondence_Theorem" title=""><i>OWL 2 correspondence theorem</i></a>. 
</p>
<div id="topic-intro-changes"></div>
<p>Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics 
as close as possible 
to that of the original specification of the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i></a>
[<cite><a href="#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
While this aim was achieved to a large degree, 
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases, 
this is because of serious technical problems 
that would have arisen 
from a conservative 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension">semantic extension</a>.
One important change is that
while so called <i>"comprehension conditions"</i>
for the OWL 2 RDF-Based Semantics
(see <a href="#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a>)
still exist,
these are <i>not</i> part of the 
normative set of semantic conditions anymore.
The OWL 2 RDF-Based Semantics also corrects several errors of OWL 1.
A list of differences between the two languages is given in 
<a href="#Appendix:_Changes_from_OWL_1_.28Informative.29" title="">Section 9</a>.
</p>
<div id="topic-intro-rfc2119"></div>
<p>The italicized keywords <em class="RFC2119" title="MUST in RFC 2119 context">MUST</em>, <em class="RFC2119" title="MUST NOT in RFC 2119 context">MUST NOT</em>, <em class="RFC2119" title="SHOULD in RFC 2119 context">SHOULD</em>, <em class="RFC2119" title="SHOULD NOT in RFC 2119 context">SHOULD NOT</em>, and <em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> are used to specify normative features of OWL 2 documents and tools, and are interpreted as specified in RFC 2119 [<cite><a href="#ref-rfc-2119" title="">RFC 2119</a></cite>].
</p>
<div class="image left" id="fig-partshierarchy">
<p><img alt="Parts Hierarchy of the OWL 2 RDF-Based Semantics. Each node is labeled with a class IRI that represents a part of the universe of an OWL 2 RDF-based interpretation. Arrows point from parts to their super parts." border="0" height="318" src="Owl2RdfBasedSemanticsPartsHierarchy.png" width="600" /><br />
<span class="caption">Figure 1: Parts Hierarchy of the OWL 2 RDF-Based Semantics</span><br />
Each <i>node</i> is labeled with a class IRI 
that represents a part of the universe
of an OWL 2 RDF-based interpretation.
<i>Arrows</i> point from parts to their super parts.
</p>
</div>
<a name="Ontologies"></a><h2> <span class="mw-headline">2  Ontologies </span></h2>
<p>This section determines the <i>syntax</i> 
for the OWL 2 RDF-Based Semantics,
and gives an overview on typical <i>content of ontologies</i>
for ontology management tasks.
</p>
<a name="Syntax"></a><h3> <span class="mw-headline">2.1  Syntax </span></h3>
<p><span id="topic-ont-rdfgraph"></span>
</p><p>Following <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax">Sections 0.2 and 0.3 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 RDF-Based Semantics 
is defined on every <i><b>RDF graph</b></i>
(<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-rdf-graph" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-rdf-graph">Section 6.2 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
i.e. on every set of <i><b>RDF triples</b></i>
(<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]).
</p><p><span id="topic-ont-iri"></span>
</p><p>In accordance with the rest of the OWL 2 specification
(see <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
this document
uses an extended notion of an RDF graph
by allowing the RDF triples in an RDF graph
to contain arbitrary <i><b>IRIs</b></i>
("Internationalized Resource Identifiers")
according to <a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987</a> 
[<cite><a href="#ref-rfc-3987" title="">RFC 3987</a></cite>].
In contrast,
the <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit">RDF Semantics specification</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
is defined on RDF graphs containing 
<a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt"><i>URIs</i></a>
[<cite><a href="#ref-rfc-2396" title="">RFC 2396</a></cite>].
This change 
is backward compatible
with the RDF specification,
since URIs are also IRIs. 
</p><p><span id="topic-ont-noteiriref"></span>
</p><p><i>Terminological note:</i>
The document at hand 
uses the term "IRI"
in accordance with the rest of the OWL 2 specification 
(see <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]), 
whereas the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#urisandlit">RDF Semantics specification</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>] 
uses the term "URI reference".
According to 
<a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987</a> 
[<cite><a href="#ref-rfc-3987" title="">RFC 3987</a></cite>],
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 <i>relative</i> references,
which are never used in this document.
</p><p><span id="topic-ont-iriabbrev"></span>
</p><p><i>Convention:</i>
In this document,
IRIs are abbreviated
in the way defined by 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>], 
i.e., the abbreviations consist of 
a <i>prefix name</i> and a <i>local part</i>,
such as
"<span class="name">prefix:localpart</span>".
</p><p><span id="topic-ont-generalrdf"></span>
</p><p>The definition of an RDF triple 
according to 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
is restricted to cases
where the <i>subject</i> of an RDF triple is 
an IRI
or a
<i>blank node</i>
(<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-blank-nodes" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-blank-nodes">Section 6.6 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
and where the <i>predicate</i> 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 <i>literal</i> 
(<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal">Section 6.5 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]),
as in
<span class="name">"s" ex:p ex:o</span>,
or where the predicate of a triple is a blank node,
as in 
<span class="name">ex:s _:p ex:o</span>.
In order to allow for interoperability 
with other existing and future technologies and tools,
the document at hand
does not explicitly forbid the use of 
<i><b>generalized RDF graphs</b></i> consisting of <i><b>generalized RDF triples</b></i>,
which are defined to allow for
IRIs, literals and blank nodes
to occur in the subject, predicate and object position.
Thus,
an RDF graph
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
contain generalized RDF triples,
but an implementation is not required to support generalized RDF graphs.
Note that every RDF graph consisting entirely of RDF triples according to
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-triples">Section 6.1 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>]
is also a generalized RDF graph.
</p><p><span id="topic-ont-owl2full"></span>
</p><p><i>Terminological notes:</i>
The term 
<i><b>"OWL 2 Full"</b></i>
refers to the language
that is determined 
by the set of all RDF graphs
being interpreted using the OWL 2 RDF-Based Semantics.
Further,
in this document
the term 
<i><b>"OWL 2 Full ontology"</b></i>
(or simply <i><b>"ontology"</b></i>, 
unless there is any risk of confusion)
will be used interchangeably 
with the term "RDF graph".
</p>
<a name="Content_of_Ontologies_.28Informative.29"></a><h3> <span class="mw-headline">2.2  Content of Ontologies (Informative) </span></h3>
<p>While there do not exist any syntactic restrictions
on the set of RDF graphs
that can be interpreted by the OWL 2 RDF-Based Semantics,
in practice 
an ontology will often contain certain kinds of constructs 
that are aimed to support ontology management tasks.
Examples are 
<i><b>ontology headers</b></i> 
and 
<i><b>ontology IRIs</b></i>, 
as well as constructs that are about 
<i><b>versioning</b></i>, 
<i><b>importing</b></i>
and
<i><b>annotating</b></i> of ontologies, 
including the concept of <i><b>incompatibility</b></i> between ontologies.
</p><p>These topics are outside the scope of this semantics specification.
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
deals with these topics in detail,
and can therefore be used as a guide 
on how to apply these constructs in OWL 2 Full ontologies accordingly.
The mappings of all these constructs to their respective RDF encoding
are defined in
the <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/" title="Mapping to RDF Graphs">OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>].
</p>
<a name="Vocabulary"></a><h2> <span class="mw-headline">3  Vocabulary </span></h2>
<p>This section specifies the <i>OWL 2 RDF-Based vocabulary</i>,
and lists the names of the <i>datatypes</i> and <i>facets</i> 
used under the OWL 2 RDF-Based Semantics.
</p>
<a name="Standard_Prefixes"></a><h3> <span class="mw-headline">3.1  Standard Prefixes </span></h3>
<p><a href="#table-vocab-prefixes" title="">Table 3.1</a>
lists the standard prefix names 
and their prefix IRIs
used in this document.
</p>
<div class="left" id="table-vocab-prefixes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.1: Standard Prefixes</span>
</caption><tr>
<th style="text-align: center">
</th><th style="text-align: center"> Prefix Name
</th><th style="text-align: center"> Prefix IRI
</th></tr>
<tr>
<td> <span id="item-vocab-prefixes-owl"></span>OWL
</td><td class="name"> owl
</td><td class="name"> http://www.w3.org/2002/07/owl#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-rdf"></span>RDF
</td><td class="name"> rdf
</td><td class="name"> http://www.w3.org/1999/02/22-rdf-syntax-ns#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-rdfs"></span>RDFS
</td><td class="name"> rdfs
</td><td class="name"> http://www.w3.org/2000/01/rdf-schema#
</td></tr>
<tr>
<td> <span id="item-vocab-prefixes-xsd"></span>XML Schema
</td><td class="name"> xsd
</td><td class="name"> http://www.w3.org/2001/XMLSchema#
</td></tr>
</table>
</div>
<a name="Vocabulary_Terms"></a><h3> <span class="mw-headline">3.2  Vocabulary Terms </span></h3>
<p><a href="#table-vocab-owl" title="">Table 3.2</a> 
lists the IRIs of the <i>OWL 2 RDF-Based vocabulary</i>,
which is the set of vocabulary terms 
that are specific for the OWL 2 RDF-Based Semantics.
This vocabulary
extends the RDF and RDFS vocabularies
as specified in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">Sections 3.1</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">4.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
respectively.
<a href="#table-vocab-owl" title="">Table 3.2</a> 
does not mention those IRIs
that will be listed in 
<a href="#Datatype_Names" title="">Section 3.3</a> on datatype names 
or
<a href="#Facet_Names" title="">Section 3.4</a> on facet names. 
</p>
<div id="topic-vocab-narydatatype"></div>
<p>Implementations are <i>not</i> required 
to support the IRI <span class="name">owl:onProperties</span>, 
but 
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> 
support it 
in order to realize 
<i>n-ary dataranges</i> with arity &ge; 2
(see 
Sections
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Ranges" title="Syntax">7</a>
and
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for further information).
</p>
<div id="topic-vocab-deprecatedterms"></div>
<p><b>Note:</b> 
The use of the IRI <span class="name">owl:DataRange</span> has been deprecated as of OWL 2. 
The IRI <span class="name">rdfs:Datatype</span> 
<em class="RFC2119" title="SHOULD in RFC 2119 context">SHOULD</em> 
be used instead.
</p>
<div class="left" id="table-vocab-owl">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.2: OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td class="name"> owl:AllDifferent owl:AllDisjointClasses owl:AllDisjointProperties owl:allValuesFrom owl:annotatedProperty owl:annotatedSource owl:annotatedTarget owl:Annotation owl:AnnotationProperty owl:assertionProperty owl:AsymmetricProperty owl:Axiom owl:backwardCompatibleWith owl:bottomDataProperty owl:bottomObjectProperty owl:cardinality owl:Class owl:complementOf owl:DataRange owl:datatypeComplementOf owl:DatatypeProperty owl:deprecated owl:DeprecatedClass owl:DeprecatedProperty owl:differentFrom owl:disjointUnionOf owl:disjointWith owl:distinctMembers owl:equivalentClass owl:equivalentProperty owl:FunctionalProperty owl:hasKey owl:hasSelf owl:hasValue owl:imports owl:incompatibleWith owl:intersectionOf owl:InverseFunctionalProperty owl:inverseOf owl:IrreflexiveProperty owl:maxCardinality owl:maxQualifiedCardinality owl:members owl:minCardinality owl:minQualifiedCardinality owl:NamedIndividual owl:NegativePropertyAssertion owl:Nothing owl:ObjectProperty owl:onClass owl:onDataRange owl:onDatatype owl:oneOf owl:onProperty owl:onProperties owl:Ontology owl:OntologyProperty owl:priorVersion owl:propertyChainAxiom owl:propertyDisjointWith owl:qualifiedCardinality owl:ReflexiveProperty owl:Restriction owl:sameAs owl:someValuesFrom owl:sourceIndividual owl:SymmetricProperty owl:targetIndividual owl:targetValue owl:Thing owl:topDataProperty owl:topObjectProperty owl:TransitiveProperty owl:unionOf owl:versionInfo owl:versionIRI owl:withRestrictions
</td></tr>
</table>
</div>
<a name="Datatype_Names"></a><h3> <span class="mw-headline">3.3  Datatype Names </span></h3>
<p><a href="#table-vocab-datatypes" title="">Table 3.3</a> 
lists the IRIs of the <i>datatypes</i> used in the OWL 2 RDF-Based Semantics.
The datatype <span class="name">rdf:XMLLiteral</span> is described in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">Section 3.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
All other datatypes are described in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
The normative set of datatypes of the OWL 2 RDF-Based Semantics equals the set of datatypes
described in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p>
<div class="left" id="table-vocab-datatypes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.3: Datatypes of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td class="name"> xsd:anyURI xsd:base64Binary xsd:boolean xsd:byte xsd:dateTime xsd:dateTimeStamp xsd:decimal xsd:double xsd:float xsd:hexBinary xsd:int xsd:integer xsd:language xsd:long xsd:Name xsd:NCName xsd:negativeInteger xsd:NMTOKEN xsd:nonNegativeInteger xsd:nonPositiveInteger xsd:normalizedString rdf:PlainLiteral xsd:positiveInteger owl:rational owl:real xsd:short xsd:string xsd:token xsd:unsignedByte xsd:unsignedInt xsd:unsignedLong xsd:unsignedShort rdf:XMLLiteral
</td></tr>
</table>
</div>
<a name="Facet_Names"></a><h3> <span class="mw-headline">3.4  Facet Names </span></h3>
<p><a href="#table-vocab-facets" title="">Table 3.4</a> 
lists the IRIs of the <i>facets</i> used in the OWL 2 RDF-Based Semantics.
Each datatype listed in <a href="#Datatype_Names" title="">Section 3.3</a>
has a (possibly empty) set of constraining facets.
All facets are described in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
in the context of their respective datatypes.
The normative set of facets of the OWL 2 RDF-Based Semantics equals the set of facets
described in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p>
<div id="topic-vocab-facetexample"></div>
<p>In this specification,
facets are used for defining <i>datatype restrictions</i>
(see <a href="#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>).
For example,
to refer to the set of all strings of length 5
one can restrict
the datatype <span class="name">xsd:string</span> 
(<a href="#Datatype_Names" title="">Section 3.3</a>) 
by the facet <span class="name">xsd:length</span>
and the value 5.
</p>
<div class="left" id="table-vocab-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 3.4: Facets of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td class="name"> rdf:langRange xsd:length xsd:maxExclusive xsd:maxInclusive xsd:maxLength xsd:minExclusive xsd:minInclusive xsd:minLength xsd:pattern
</td></tr>
</table>
</div>
<a name="Interpretations"></a><h2> <span class="mw-headline">4  Interpretations </span></h2>
<p>The OWL 2 RDF-Based Semantics provides 
<i>vocabulary interpretations</i> and <i>vocabulary entailment</i>
(see <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#vocabulary_entail" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#vocabulary_entail">Section 2.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>])
for the
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a> 
and
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a> 
vocabularies 
and for the 
<a href="#Vocabulary" title="">OWL 2 RDF-Based vocabulary</a>.
This section defines
<i>OWL 2 RDF-Based datatype maps</i>
and
<i>OWL 2 RDF-Based interpretations</i>,
and specifies what 
<i>satisfaction</i> of ontologies, 
<i>consistency</i> and <i>entailment</i> 
means under the OWL 2 RDF-Based Semantics.
In addition,
the so called <i>"parts" of the universe</i> 
of an OWL 2 RDF-Based interpretation
are defined.
</p>
<a name="Datatype_Maps"></a><h3> <span class="mw-headline">4.1  Datatype Maps </span></h3>
<div id="topic-int-rdfdatatype"></div>
<p>According to 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatype" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatype">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
a <i><b>datatype</b></i> <i>d</i> has the following components:
</p>
<ul><li> LS(<i>d</i>), the <i>lexical space</i> of <i>d</i>, which is a set of <i>lexical forms</i>;
</li><li> VS(<i>d</i>), the <i>value space</i> of <i>d</i>, which is a set of <i>data values</i>;
</li><li> L2V(<i>d</i>), the <i>lexical-to-value mapping</i> of <i>d</i>, which maps lexical forms in LS(<i>d</i>) to data values in VS(<i>d</i>).
</li></ul>
<div id="topic-int-datavalueterm"></div>
<p><i>Terminological notes:</i>
The document at hand uses the term
<i>"data value"</i>
in accordance with the rest of the OWL 2 specification
(see 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
whereas the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">RDF Semantics specification</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
uses the term
<i>"datatype value"</i> instead.
Further, the names "LS" and "VS", 
which stand for the lexical space and the value space of a datatype, 
respectively,
are <i>not</i> used in the RDF Semantics specification,
but have been introduced here for easier reference. 
</p><p>In this document,
the basic definition of a datatype 
is extended to take <i>facets</i> into account.
See <a href="#Facet_Names" title="">Section 3.4</a> 
for information and an example on facets.
Note that
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
explicitly permits 
that semantic extensions 
may impose more elaborate datatyping conditions
than those listed above. 
</p>
<div id="topic-int-datatypewithfacets"></div>
<p>A <i><b>datatype with facets</b></i> <i>d</i>
is a datatype that has the following additional components:
</p>
<ul><li> FS(<i>d</i>), the <i>facet space</i> of <i>d</i>, which is a set of pairs of the form ( <i>F</i> , <i>v</i> ), where <i>F</i> is an IRI called the <i>constraining facet</i> and <i>v</i> is an arbitrary data value called the <i>constraining value</i>;
</li><li> F2V(<i>d</i>), the <i>facet-to-value mapping</i> of <i>d</i>, which maps each facet-value pair ( <i>F</i> , <i>v</i> ) in FS(<i>d</i>) to a subset of VS(<i>d</i>).
</li></ul>
<div id="topic-int-facetnature"></div>
<p>Note that
it is not further specified 
what the nature of the denotation of a facet IRI is,
i.e. it is only known that a facet IRI denotes some individual.
Semantic extensions 
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em>
impose further restrictions on the denotations of facets.
In fact,
<a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
will define additional restrictions on facets.
</p>
<div id="topic-int-facetvalueothertype"></div>
<p>Also note 
that for a datatype <i>d</i>
and a facet-value pair ( <i>F</i> , <i>v</i> ) in FS(<i>d</i>)
the value <i>v</i>
is not required
to be included in the value space VS(<i>d</i>) of <i>d</i> itself.
For example, 
the datatype <span class="name">xsd:string</span>
(<a href="#Datatype_Names" title="">Section 3.3</a>)
has the facet <span class="name">xsd:length</span>
(<a href="#Facet_Names" title="">Section 3.4</a>),
which takes nonnegative integers as its constraining values 
rather than strings.
</p>
<div id="topic-int-facetdatatypeassumption"></div>
<p>In this document,
it will always be assumed from now on that
any datatype <i>d</i> is a datatype with facets. 
If the facet space FS(<i>d</i>) of a datatype <i>d</i>
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(<i>d</i>) is the empty set.
Unless there is any risk of confusion,
the term <i>"datatype"</i> 
will always refer to a datatype with facets.
</p>
<div id="topic-int-rdfdatatypemap"></div>
<p><a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap">Section 5.1 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
further
defines a <i><b>datatype map</b></i> <i>D</i> to be 
a set of name-datatype pairs 
( <i>u</i> , <i>d</i> )
consisting of an IRI <i>u</i> and a datatype <i>d</i>,
such that no IRI appears twice in the set.
As a consequence of what has been said before,
in this document 
every datatype map <i>D</i> will entirely consist of datatypes with facets.
</p><p>The following definition specifies what an <i>OWL 2 RDF-Based datatype map</i> is.
</p>
<div id="def-owldatatypemap">
<p><b>Definition 4.1 (OWL 2 RDF-Based Datatype Map):</b>
A datatype map <i>D</i> 
is an <i>OWL 2 RDF-Based datatype map</i>,
if and only if 
for every datatype name <i>u</i> listed in <a href="#Datatype_Names" title="">Section 3.3</a>
and its respective set of constraining facets (<a href="#Facet_Names" title="">Section 3.4</a>)
there is 
a name-datatype pair ( <i>u</i>, <i>d</i> ) in <i>D</i>
with the specified
lexical space LS(<i>d</i>), 
value space VS(<i>d</i>),
lexical-to-value mapping L2V(<i>d</i>),
facet space FS(<i>d</i>) and
facet-to-value mapping F2V(<i>d</i>).
</p>
</div>
<div id="topic-int-facetclosedowlmap"></div>
<p>Note that <a href="#def-owldatatypemap" title="">Definition 4.1</a> 
does not prevent <i>additional</i> datatypes 
to be in an OWL 2 RDF-Based datatype map.
For the special case of
an OWL 2 RDF-Based datatype map <i>D</i>
that exclusively contains the datatypes listed in 
<a href="#Datatype_Names" title="">Section 3.3</a>,
it is ensured that
there are datatypes available for all the facet values,
i.e., 
for every name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i>
and for every facet-value pair
( <i>F</i> , <i>v</i> )
in FS(<i>d</i>)
there exists a name-datatype pair ( <i>u<sup>*</sup></i> , <i>d<sup>*</sup></i> ) in <i>D</i>
such that <i>v</i> is in VS(<i>d<sup>*</sup></i>).
</p>
<a name="Vocabulary_Interpretations"></a><h3> <span class="mw-headline">4.2  Vocabulary Interpretations </span></h3>
<div id="topic-int-rdfinterpretation"></div>
<p>From the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
let <i>V</i> be a set of literals and IRIs 
containing the RDF and RDFS vocabularies,
and let <i>D</i> be a datatype map according to 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDatatypeMap">Section 5.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
(and accordingly <a href="#Datatype_Maps" title="">Section 4.1</a>).
A <i><b>D-interpretation</b></i> <i>I</i> of <i>V</i> with respect to <i>D</i> is a tuple
</p>
<div class="indent">
<p><i>I</i> = ( IR , IP , IEXT , IS , IL , LV ) .
</p>
</div>
<p>IR is the <i>universe</i> of <i>I</i>,
i.e., a nonempty set
that contains at least 
the denotations of literals and IRIs in <i>V</i>.
IP is a subset of IR, 
the <i>properties</i> of <i>I</i>.
LV, 
the <i>data values</i> of <i>I</i>, 
is a subset of IR 
that contains at least the set of plain literals
(see <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal">Section 6.5 of RDF Concepts</a> 
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>])
in <i>V</i>, 
and
the value spaces of each datatype of <i>D</i>.
IEXT is used to associate properties with their <i>property extension</i>,
and is a mapping from IP to the powerset of IR &times; IR.
IS is a mapping from <i>IRIs</i> in <i>V</i> to their denotations in IR.
In particular,
IS(<i>u</i>) = <i>d</i> 
for any name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i>. 
IL is a mapping from <i>typed literals</i> 
"<i>s</i>"<span class="name">^^</span><i>u</i> 
in <i>V</i> 
to their denotations in IR,
where IL("<i>s</i>"<span class="name">^^</span><i>u</i>) = L2V(<i>d</i>)(<i>s</i>),
provided that <i>d</i> is a datatype of <i>D</i>, 
IS(<i>u</i>) = <i>d</i>, and
<i>s</i> is in the lexical space LS(<i>d</i>);
otherwise 
IL("<i>s</i>"<span class="name">^^</span><i>u</i>)
is not in LV.
</p>
<div id="topic-int-ifunction"></div>
<p><i>Convention:</i>
Following the practice introduced in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Section 1.4 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
for a given interpretation <i>I</i> of a vocabulary <i>V</i> 
the notation 
"<i>I</i>(<i>x</i>)" 
will be used 
instead of "IL(<i>x</i>)" and "IS(<i>x</i>)"
for the typed literals and IRIs <i>x</i> in <i>V</i>,
respectively.
</p>
<div id="topic-int-rdfconditions"></div>
<p>As detailed in the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
a D-interpretation has to meet all the semantic conditions
for <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">ground graphs</a> 
and <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">blank nodes</a>, 
those for <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF interpretations</a> and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS interpretations</a>,
and the
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">"general semantic conditions for datatypes"</a>.
</p><p>In this document,
the basic definition of a D-interpretation
is extended to take <i>facets</i> into account.
</p>
<div id="topic-int-facetinterpretation"></div>
<p>A <i><b>D-interpretation with facets</b></i> <i>I</i>
is a D-interpretation for a datatype map <i>D</i> 
consisting entirely of datatypes with facets
(<a href="#Datatype_Maps" title="">Section 4.1</a>),
where <i>I</i> meets the following additional semantic conditions:
for each name-datatype pair ( <i>u</i> , <i>d</i> ) in <i>D</i> 
and each facet-value pair ( <i>F</i> , <i>v</i> ) in the facet space FS(<i>d</i>)
</p>
<ul><li> <i>F</i> is in the vocabulary <i>V</i> of <i>I</i>;
</li><li> a name-datatype pair ( <i>u<sup>*</sup></i> , <i>d<sup>*</sup></i> ) exists in <i>D</i>, such that <i>v</i> is in the value space VS(<i>d<sup>*</sup></i>). 
</li></ul>
<div id="topic-int-facetinterpretassumption"></div>
<p>In this document,
it will always be assumed from now on that
any D-interpretation <i>I</i> 
is a D-interpretation with facets. 
Unless there is any risk of confusion,
the term <i>"D-interpretation"</i> 
will always refer to a D-interpretation with facets.
</p><p>The following definition specifies what an <i>OWL 2 RDF-Based interpretation</i> is.
</p>
<div id="def-owlinterpretation">
<p><b>Definition 4.2 (OWL 2 RDF-Based Interpretation):</b>
Let <i>D</i> be an OWL 2 RDF-Based datatype map,
and let <i>V</i> be a vocabulary
that includes 
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary 
together with all the datatype and facet names 
listed in <a href="#Vocabulary" title="">Section 3</a>.
An <i>OWL 2 RDF-Based interpretation</i>, 
<i>I</i> = ( IR , IP , IEXT , IS , IL , LV ), 
of <i>V</i> with respect to <i>D</i>
is a D-interpretation of <i>V</i> with respect to <i>D</i>
that meets all the extra semantic conditions 
given in <a href="#Semantic_Conditions" title="">Section 5</a>.
</p>
</div>
<a name="Satisfaction.2C_Consistency_and_Entailment"></a><h3> <span class="mw-headline">4.3  Satisfaction, Consistency and Entailment </span></h3>
<p>The following definitions specify 
what it means for an RDF graph to be <i>satisfied</i> 
by a given OWL 2 RDF-Based interpretation,
to be <i>consistent</i> 
under the OWL 2 RDF-Based Semantics,
and to <i>entail</i> another RDF graph.
</p>
<div id="topic-int-rdfsatisfaction"></div>
<p>The notion of <i>satisfaction</i> under the OWL 2 RDF-Based Semantics
is based on the notion of satisfaction for 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">D-interpretations</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defsatis" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defsatis">Simple interpretations</a>,
as defined in the RDF Semantics 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
In essence,
in order to satisfy an RDF graph,
an interpretation <i>I</i> has to satisfy all the triples in the graph,
i.e.,
for a triple "<i>s p o</i>"
it is necessary that the relationship
( <i>I</i>(<i>s</i>) , <i>I</i>(<i>o</i>) ) &isin; IEXT(<i>I</i>(<i>p</i>))
holds
(special treatment exists for blank nodes,
as detailed in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]).
In other words,
the given graph has to be compatible with 
the specific form of the IEXT mapping of <i>I</i>.
The distinguishing aspect of <i>OWL 2 RDF-Based satisfaction</i> is 
that an interpretation <i>I</i> needs to meet 
all the OWL 2 RDF-Based semantic conditions
(see <a href="#Semantic_Conditions" title="">Section 5</a>),
which have a constraining effect 
on the possible forms an IEXT mapping can have.
</p>
<div id="def-owlsatisfaction">
<p><b>Definition 4.3 (OWL 2 RDF-Based Satisfaction):</b>
Let <i>G</i> be an RDF graph,
let <i>D</i> be an OWL 2 RDF-Based datatype map,
let <i>V</i> be a vocabulary
that includes 
the RDF and RDFS vocabularies
and the OWL 2 RDF-Based vocabulary 
together with all the datatype and facet names listed in <a href="#Vocabulary" title="">Section 3</a>,
and let <i>I</i> be a D-interpretation of <i>V</i> with respect to <i>D</i>.
<i>I</i> <i>OWL 2 RDF-Based satisfies</i> <i>G</i> with respect to <i>V</i> and <i>D</i>
if and only if
<i>I</i> is an OWL 2 RDF-Based interpretation of <i>V</i> with respect to <i>D</i>
that
satisfies <i>G</i> 
as a <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">D-interpretation</a> of <i>V</i> with respect to <i>D</i>
according to the RDF Semantics
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p>
</div>
<div id="def-owlconsistency">
<p><b>Definition 4.4 (OWL 2 RDF-Based Consistency):</b>
Let <i>S</i> be a collection of RDF graphs,
and let <i>D</i> be an OWL 2 RDF-Based datatype map.
<i>S</i> is <i>OWL 2 RDF-Based consistent</i> with respect to <i>D</i>
if and only if
there is some OWL 2 RDF-Based interpretation <i>I</i> with respect to <i>D</i>
of some vocabulary <i>V</i> 
that includes
the RDF and RDFS vocabularies 
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in <a href="#Vocabulary" title="">Section 3</a>,
such that <i>I</i> OWL 2 RDF-Based satisfies all the RDF graphs in <i>S</i>
with respect to <i>V</i> and <i>D</i>.
</p>
</div>
<div id="def-owlentailment">
<p><b>Definition 4.5 (OWL 2 RDF-Based Entailment):</b>
Let <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i> be collections of RDF graphs,
and let <i>D</i> be an OWL 2 RDF-Based datatype map.
<i>S<sub>1</sub></i> <i>OWL 2 RDF-Based entails</i> <i>S<sub>2</sub></i> with respect to <i>D</i>
if and only if
for every OWL 2 RDF-Based interpretation <i>I</i> with respect to <i>D</i>
of any vocabulary <i>V</i> that includes
the RDF and RDFS vocabularies 
and the OWL 2 RDF-Based vocabulary
together with all the datatype and facet names listed in <a href="#Vocabulary" title="">Section 3</a>
the following holds:
If <i>I</i> 
OWL 2 RDF-Based satisfies all the RDF graphs in <i>S<sub>1</sub></i>
with respect to <i>V</i> and <i>D</i>,
then <i>I</i> 
OWL 2 RDF-Based satisfies all the RDF graphs in <i>S<sub>2</sub></i>
with respect to <i>V</i> and <i>D</i>.
</p>
</div>
<a name="Parts_of_the_Universe"></a><h3> <span class="mw-headline">4.4  Parts of the Universe </span></h3>
<p><a href="#table-int-parts" title="">Table 4.1</a> 
defines the <i>"parts"</i> of the universe
of a given OWL 2 RDF-Based interpretation <i>I</i>.
</p><p>The second column tells the <i>name</i> of the part.
The third column gives a <i>definition</i> of the part
in terms of the mapping IEXT of <i>I</i>,
and by referring to a particular term 
of the RDF, RDFS or OWL 2 RDF-Based vocabulary.
</p>
<div id="topic-int-partsdefexample"></div>
<p>As an example,
the part of all datatypes is named "IDC",
and it is defined as the set of all individuals <i>x</i>
for which the relationship
"( <i>x</i> , <i>I</i>(<span class="name">rdfs:Datatype</span>) ) 
&isin; 
IEXT(<i>I</i>(<span class="name">rdf:type</span>))"
holds.
According to the semantics of <span class="name">rdf:type</span>,
as defined in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
this means that the name "IDC" 
denotes the class extension 
(see <a href="#Class_Extensions" title="">Section 4.5</a>) 
of <i>I</i>(<span class="name">rdfs:Datatype</span>).
</p>
<div class="left" id="table-int-parts">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 4.1: Parts of the Universe</span>
</caption>
<tr>
<th style="text-align: center">
</th><th style="text-align: center"> Name of<br />Part <i>S</i>
</th><th style="text-align: center"> Definition of <i>S</i> as<br />{ <i>x</i> &isin; IR | ( <i>x</i> , <i>I</i>(<i>E</i>) ) &isin; IEXT(<i>I</i>(<span class="name">rdf:type</span>)) }<br />where IRI <i>E</i> is
</th></tr>
<tr>
<td> <span id="item-int-parts-individuals"></span>individuals
</td><td> IR
</td><td> <span class="name">rdfs:Resource</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-datavalues"></span>data values
</td><td> LV
</td><td> <span class="name">rdfs:Literal</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-ontologies"></span>ontologies
</td><td> IX
</td><td> <span class="name">owl:Ontology</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-classes"></span>classes
</td><td> IC
</td><td> <span class="name">rdfs:Class</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-datatypes"></span>datatypes
</td><td> IDC
</td><td> <span class="name">rdfs:Datatype</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-properties"></span>properties
</td><td> IP
</td><td> <span class="name">rdf:Property</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-dataproperties"></span>data properties
</td><td> IODP
</td><td> <span class="name">owl:DatatypeProperty</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-ontologyproperties"></span>ontology properties
</td><td> IOXP
</td><td> <span class="name">owl:OntologyProperty</span>
</td></tr>
<tr>
<td> <span id="item-int-parts-annotationproperties"></span>annotation properties
</td><td> IOAP
</td><td> <span class="name">owl:AnnotationProperty</span>
</td></tr>
</table>
</div>
<a name="Class_Extensions"></a><h3> <span class="mw-headline">4.5  Class Extensions </span></h3>
<p>The mapping ICEXT from IC to the powerset of IR,
which associates classes with their <i>class extension</i>,
is defined
for every <i>c</i> &isin; IC
as
</p>
<div class="indent">
<p>ICEXT(<i>c</i>) = { <i>x</i> &isin; IR | ( <i>x</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:type</span>)) } .
</p>
</div>
<a name="Semantic_Conditions"></a><h2> <span class="mw-headline">5  Semantic Conditions </span></h2>
<p>This section defines the semantic conditions of the OWL 2 RDF-Based 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 RDF-Based Semantics
is the combination of the semantic conditions presented here
and the semantic conditions 
for 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Simple Entailment</a>, 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">RDF</a>, 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">RDFS</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">D-Entailment</a>,
as specified in
the RDF Semantics specification
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>All semantic conditions in this section
are defined with respect to an interpretation <i>I</i>.
<a href="#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>
specifies semantic conditions for the different parts of the universe
of the interpretation being considered
(compare <a href="#Parts_of_the_Universe" title="">Section 4.4</a>).
<a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a> 
and 
<a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a> 
list semantic conditions for the classes and the properties of the OWL 2 RDF-Based vocabulary.
In the rest of this section, 
the OWL 2 RDF-Based semantic conditions 
for the different language constructs of OWL 2
are specified.
</p>
<div id="topic-semcond-conventions"></div>
<p><b>Conventions used in this Section</b>
</p>
<div id="topic-semcond-convention-iff"></div>
<p><i>iff:</i>
Throughout this section
the term "iff" is used as a shortform for "if and only if".
</p>
<div id="topic-semcond-convention-andcomma"></div>
<p><i>Conjunctive commas:</i>
A comma
("<span class="name">,</span>") 
separating two assertions in a semantic condition, as in
"<i>c</i> &isin; IC , <i>p</i> &isin; IP",
is read as a logical <i>"and"</i>.
Further,
a comma separating two variables,
as in
"<i>c</i>, <i>d</i> &isin; IC",
is used for abbreviating two comma separated assertions,
"<i>c</i> &isin; IC , <i>d</i> &isin; IC"
in this example.
</p>
<div id="topic-semcond-convention-varscope"></div>
<p><i>Unscoped variables:</i>
If no explicit scope is given for a variable "<i>x</i>", 
as in "&forall; <i>x</i>&nbsp;: &hellip;" or "{ <i>x</i> | &hellip; }",
then "<i>x</i>" is unconstrained, 
which means <i>x</i> &isin; IR,
i.e. "<i>x</i>" denotes an arbitrary individual in the universe.
</p>
<div id="topic-semcond-convention-setcard"></div>
<p><i>Set cardinality:</i>
For a set <i>S</i>, 
an expression of the form "#<i>S</i>" means the number of elements in <i>S</i>.
</p>
<div id="topic-semcond-convention-seq"></div>
<p><i>Sequence expressions:</i>
An expression of the form 
"<i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; <i>S</i>"
means that "<i>s</i>" represents an RDF list of <i>n</i> &ge; 0 
individuals <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i>,
all of them being members of the set <i>S</i>.
Precisely,
<i>s</i> = <i>I</i>(<span class="name">rdf:nil</span>) for <i>n</i> = 0;
and for <i>n</i> &gt; 0
there exist 
<i>z<sub>1</sub></i> &isin; IR , &hellip; , <i>z<sub>n</sub></i> &isin; IR, 
such that
</p>
<div class="indent">
<p><i>s</i> = <i>z<sub>1</sub></i> ,<br />
<i>a<sub>1</sub></i> &isin; <i>S</i> ,
( <i>z<sub>1</sub></i> , <i>a<sub>1</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:first</span>)) , 
( <i>z<sub>1</sub></i> , <i>z<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) ,<br />
&hellip; ,<br />
<i>a<sub>n</sub></i> &isin; <i>S</i>,
( <i>z<sub>n</sub></i> , <i>a<sub>n</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:first</span>)) ,
( <i>z<sub>n</sub></i> , <i>I</i>(<span class="name">rdf:nil</span>) ) &isin; IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) .
</p>
</div>
<p>Note, as mentioned in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#collections" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#collections">Section 3.3.3 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
there are no semantic constraints that enforce "well-formed" sequence structures.
So, for example,
it is possible for a sequence head <i>s</i> to refer to more than one sequence.
</p>
<div id="topic-semcond-convention-setnames"></div>
<p><i>Set names:</i>
The following names are used as convenient abbreviations for certain sets:
</p>
<ul><li> ISEQ: The set of all sequences. This set equals the class extension of <span class="name">rdf:List</span>, i.e., ISEQ&nbsp;:= ICEXT(<i>I</i>(<span class="name">rdf:List</span>)). 
</li><li> INNI: The set of all nonnegative integers. This set equals the value space of the datatype <span class="name">xsd:nonNegativeInteger</span>, i.e., INNI&nbsp;:= ICEXT(<i>I</i>(<span class="name">xsd:nonNegativeInteger</span>)), but is also subsumed by the value spaces of other numerical datatypes, such as <span class="name">xsd:integer</span>.
</li></ul>
<div id="topic-semcond-conditionform"></div>
<p><b>Notes on the Form of Semantic Conditions (Informative)</b>
</p>
<div id="topic-semcond-conditionform-correspondence"></div>
<p>One design goal of OWL 2
was to ensure an appropriate degree of alignment
between the OWL 2 RDF-Based Semantics and the 
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/" title="Direct Semantics">OWL 2 Direct Semantics</a>
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
under the different constraints the two semantics have to meet.
The way this semantic alignment is described
is via the <i>OWL 2 correspondence theorem</i>
in <a href="#Correspondence_Theorem" title="">Section 7.2</a>.
For this theorem to hold,
the semantic conditions
that treat the RDF encoding
of OWL 2 <i>axioms</i> 
(compare <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Axioms" title="Mapping to RDF Graphs">Section 3.2.5 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
and 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Axioms" title="Syntax">Section 9 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
such as
<a href="#Semantic_Conditions_for_Inverse_Properties" title="">inverse property axioms</a>,
must have the form of "iff" ("if-and-only-if") conditions.
This means that these semantic conditions
completely determine the semantics 
of the encoding of these constructs. 
On the other hand,
the RDF encoding
of OWL 2 <i>expressions</i>
(compare <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Section 3.2.4 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
and
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Property_Expressions" title="Syntax">Sections 6 &ndash; 8 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]),
such as
<a href="#Semantic_Conditions_for_Property_Restrictions" title="">property restrictions</a>,
are treated by "if-then" 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 RDF-Based Semantics 
for these language constructs.
</p>
<div id="topic-semcond-conditionform-axiomexpressions"></div>
<p>Special cases are
the semantic conditions for
<a href="#Semantic_Conditions_for_Boolean_Connectives" title="">Boolean connectives</a>
of classes 
and for
<a href="#Semantic_Conditions_for_Enumerations" title="">enumerations</a>.
These language constructs build OWL 2 expressions.
But for backward compatibility reasons
there is also RDF encoding of <i>axioms</i> 
based on the vocabulary for these language constructs
(see Table 18 in <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Axioms" title="Mapping to RDF Graphs">Section 3.2.5 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
For example, an RDF expression of the form
</p>
<div class="indent">
<p><span class="name">ex:c<sub>1</sub> owl:unionOf ( ex:c<sub>2</sub> ex:c<sub>3</sub> ) .</span>
</p>
</div> 
<p>is mapped by the reverse RDF mapping
to an OWL 2 axiom
that states the equivalence of the class denoted by
<span class="name">ex:c<sub>1</sub></span>
with the union of the classes denoted by
<span class="name">ex:c<sub>2</sub></span>
and
<span class="name">ex:c<sub>3</sub></span>.
In order to ensure that the
<a href="#Correspondence_Theorem" title="">correspondence theorem</a> 
holds,
and in accordance with the original 
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2">OWL 1 RDF-Compatible Semantics specification</a>
[<cite><a href="#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>],
the semantic conditions for the mentioned language constructs are therefore
"iff" conditions.
</p>
<div id="topic-semcond-conditionform-multitripleaxiom"></div>
<p>Further,
special treatment exists for OWL 2 axioms
that have a <i>multi-triple representation</i> in RDF,
where the different triples share a common <i>"root node"</i>,
such as the blank node 
"<span class="name">_:x</span>" 
in the following example:
</p>
<div class="indent">
<p><span class="name">_:x rdf:type owl:AllDisjointClasses .</span><br />
<span class="name">_:x owl:members ( ex:c<sub>1</sub> ex:c<sub>2</sub> ) .</span>
</p>
</div>
<p>In essence, 
the semantic conditions for the encoding of these language constructs 
are "iff" conditions,
as usual for axioms.
However, 
in order to cope with the specific syntactic aspect of a "root node",
the "iff" conditions of these language constructs have been split into two "if-then" conditions,
where the "if-then" condition representing the right-to-left direction
contains an additional premise 
having the form 
"&exist; <i>z</i> &isin; IR". 
The purpose of this premise is to ensure the existence of an individual
that is needed to satisfy the root node
under the OWL 2 RDF-Based semantics.
The language constructs in question are
<i>n-ary disjointness axioms</i> 
in <a href="#Semantic_Conditions_for_N-ary_Disjointness" title="">Section 5.10</a>,
and 
<i>negative property assertions</i> 
in <a href="#Semantic_Conditions_for_Negative_Property_Assertions" title="">Section 5.15</a>.
</p>
<div id="topic-semcond-conditionform-deducedrhs"></div>
<p>The "if-then" 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 
<span class="name">owl:someValuesFrom</span> restrictions in 
<a href="#Semantic_Conditions_for_Property_Restrictions" title="">Section 5.6</a>
does not list the statement 
"<i>x</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>))"
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 
<i>vocabulary classes</i> and <i>vocabulary properties</i>
in
<a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a> 
and
<a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>,
respectively,
occasionally in connection with the semantic conditions 
for the <i>parts of the universe</i>
in 
<a href="#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a>.
In the example above,
the omitted consequence can be obtained
from the third column of the entry for 
<span class="name">owl:someValuesFrom</span>
in the table in
<a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>,
which determines that 
IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) 
&sube; 
ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IC. 
</p>
<a name="Semantic_Conditions_for_the_Parts_of_the_Universe"></a><h3> <span class="mw-headline">5.1  Semantic Conditions for the Parts of the Universe </span></h3>
<p><a href="#table-semcond-parts" title="">Table 5.1</a> 
lists the semantic conditions 
for the parts of the universe
of the OWL 2 RDF-Based interpretation being considered.
Additional semantic conditions affecting these parts
are given in <a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>.
</p><p>The first column tells the <i>name</i> of the part,
as defined in
<a href="#Parts_of_the_Universe" title="">Section 4.4</a>.
The second column defines
certain <i>conditions</i> on the part.
In most cases,
the column specifies for the part 
by which other part it is subsumed,
and thus the position of the part 
in the "parts hierarchy" of the universe
is narrowed down.
The third column provides further 
<i>information about the instances</i> 
of those parts
that consist of classes or properties.
In general,
if the part consists of classes,
then for the class extensions of the member classes
is specified by which part of the universe they are subsumed.
If the part consists of properties,
then the domains and ranges of the member properties are determined. 
</p>
<div class="left" id="table-semcond-parts">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.1: Semantic Conditions for the Parts of the Universe</span>
</caption>
<tr>
<th style="text-align: center"> Name of<br />Part <i>S</i>
</th><th style="text-align: center"> Conditions on <i>S</i>
</th><th style="text-align: center"> Conditions on<br />Instances <i>x</i> of <i>S</i>
</th></tr>
<tr>
<td> <span id="item-semcond-parts-individuals"></span>IR
</td><td> <i>S</i> &ne; &empty;
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-datavalues"></span>LV
</td><td> <i>S</i> &sube; IR
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-ontologies"></span>IX
</td><td> <i>S</i> &sube; IR
</td><td>
</td></tr>
<tr>
<td> <span id="item-semcond-parts-classes"></span>IC
</td><td> <i>S</i> &sube; IR
</td><td> ICEXT(<i>x</i>) &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-parts-datatypes"></span>IDC
</td><td> <i>S</i> &sube; IC
</td><td> ICEXT(<i>x</i>) &sube; LV
</td></tr>
<tr>
<td> <span id="item-semcond-parts-properties"></span>IP
</td><td> <i>S</i> &sube; IR
</td><td> IEXT(<i>x</i>) &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-parts-dataproperties"></span>IODP
</td><td> <i>S</i> &sube; IP
</td><td> IEXT(<i>x</i>) &sube; IR &times; LV
</td></tr>
<tr>
<td> <span id="item-semcond-parts-ontologyproperties"></span>IOXP
</td><td> <i>S</i> &sube; IP
</td><td> IEXT(<i>x</i>) &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-parts-annotationproperties"></span>IOAP
</td><td> <i>S</i> &sube; IP
</td><td> IEXT(<i>x</i>) &sube; IR &times; IR
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_Vocabulary_Classes"></a><h3> <span class="mw-headline">5.2  Semantic Conditions for the Vocabulary Classes </span></h3>
<p><a href="#table-semcond-classes" title="">Table 5.2</a> 
lists the semantic conditions for the classes 
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those classes
with IRIs in the RDF and RDFS vocabularies
that represent
parts of the universe
of the OWL 2 RDF-Based interpretation being considered
(<a href="#Parts_of_the_Universe" title="">Section 4.4</a>).
The semantic conditions for the remaining classes
with names in the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a>
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a> vocabularies
can be found in the RDF Semantics specification
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>The first column tells the <i>IRI</i> of the class.
The second column defines
of what particular <i>kind</i> 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
(<a href="#Parts_of_the_Universe" title="">Section 4.4</a>)
it is <i>subsumed</i>:
from an entry of the form
"ICEXT(<i>I</i>(<i>C</i>))&nbsp;&sube;&nbsp;<i>S</i>",
for a class IRI <i>C</i>
and a set <i>S</i>,
and given an RDF triple of the form
"<i>u</i>&nbsp;<span class="name">rdf:type</span>&nbsp;<i>C</i>",
one can deduce 
that the relationship
"<i>I</i>(<i>u</i>)&nbsp;&isin;&nbsp;<i>S</i>"
holds.
Note that some entries are of the form
"ICEXT(<i>I</i>(<i>C</i>))&nbsp;=&nbsp;<i>S</i>",
which means that the class extension is exactly specified to be that set.
See <a href="#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a> 
for further semantic conditions 
on those classes that represent <i>parts</i>.
</p>
<div id="topic-semcond-classes-datatypes"></div>
<p>Not included in this table are the <i>datatypes</i> of the OWL 2 RDF-Based Semantics 
with IRIs listed in <a href="#Datatype_Names" title="">Section 3.3</a>.
For each such datatype IRI <i>E</i>, 
the following semantic conditions hold
(as a consequence of 
the fact that <i>E</i> is a member of the datatype map
of every OWL 2 RDF-Based interpretation
according to 
<a href="#def-owlinterpretation" title="">Definition 4.2</a>,
and by the "general semantic conditions for datatypes"
listed in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#defDinterp">Section 5.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]):
</p>
<ul><li> <i>I</i>(<i>E</i>) &isin; IDC
</li><li> ICEXT(<i>I</i>(<i>E</i>)) &sube; LV
</li></ul>
<div class="left" id="table-semcond-classes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.2: Semantic Conditions for the Vocabulary Classes</span>
</caption>
<tr>
<th style="text-align: center"> IRI <i>E</i>
</th><th style="text-align: center"> <i>I</i>(<i>E</i>)
</th><th style="text-align: center"> ICEXT(<i>I</i>(<i>E</i>))
</th></tr>
<tr>
<td> <span id="item-semcond-classes-alldifferent"></span><span class="name">owl:AllDifferent</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-alldisjointclasses"></span><span class="name">owl:AllDisjointClasses</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-alldisjointproperties"></span><span class="name">owl:AllDisjointProperties</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-annotation"></span><span class="name">owl:Annotation</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-annotationproperty"></span><span class="name">owl:AnnotationProperty</span>
</td><td> &isin; IC
</td><td> = IOAP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-asymmetricproperty"></span><span class="name">owl:AsymmetricProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-axiom"></span><span class="name">owl:Axiom</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-rdfsclass"></span><span class="name">rdfs:Class</span>
</td><td> &isin; IC
</td><td> = IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-owlclass"></span><span class="name">owl:Class</span>
</td><td> &isin; IC
</td><td> = IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-datarange"></span><span class="name">owl:DataRange</span>
</td><td> &isin; IC
</td><td> = IDC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-datatype"></span><span class="name">rdfs:Datatype</span>
</td><td> &isin; IC
</td><td> = IDC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-dataproperty"></span><span class="name">owl:DatatypeProperty</span>
</td><td> &isin; IC
</td><td> = IODP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-deprecatedclass"></span><span class="name">owl:DeprecatedClass</span>
</td><td> &isin; IC
</td><td> &sube; IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-deprecatedproperty"></span><span class="name">owl:DeprecatedProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-functionalproperty"></span><span class="name">owl:FunctionalProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-inversefunctionalproperty"></span><span class="name">owl:InverseFunctionalProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-irreflexiveproperty"></span><span class="name">owl:IrreflexiveProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-literal"></span><span class="name">rdfs:Literal</span>
</td><td> &isin; IDC
</td><td> = LV
</td></tr>
<tr>
<td> <span id="item-semcond-classes-namedindividual"></span><span class="name">owl:NamedIndividual</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-negativepropertyassertion"></span><span class="name">owl:NegativePropertyAssertion</span>
</td><td> &isin; IC
</td><td> &sube; IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-nothing"></span><span class="name">owl:Nothing</span>
</td><td> &isin; IC
</td><td> = &empty;
</td></tr>
<tr>
<td> <span id="item-semcond-classes-objectproperty"></span><span class="name">owl:ObjectProperty</span>
</td><td> &isin; IC
</td><td> = IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-ontology"></span><span class="name">owl:Ontology</span>
</td><td> &isin; IC
</td><td> = IX
</td></tr>
<tr>
<td> <span id="item-semcond-classes-ontologyproperty"></span><span class="name">owl:OntologyProperty</span>
</td><td> &isin; IC
</td><td> = IOXP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-property"></span><span class="name">rdf:Property</span>
</td><td> &isin; IC
</td><td> = IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-reflexiveproperty"></span><span class="name">owl:ReflexiveProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-resource"></span><span class="name">rdfs:Resource</span>
</td><td> &isin; IC
</td><td> = IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-restriction"></span><span class="name">owl:Restriction</span>
</td><td> &isin; IC
</td><td> &sube; IC
</td></tr>
<tr>
<td> <span id="item-semcond-classes-symmetricproperty"></span><span class="name">owl:SymmetricProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
<tr>
<td> <span id="item-semcond-classes-thing"></span><span class="name">owl:Thing</span>
</td><td> &isin; IC
</td><td> = IR
</td></tr>
<tr>
<td> <span id="item-semcond-classes-transitiveproperty"></span><span class="name">owl:TransitiveProperty</span>
</td><td> &isin; IC
</td><td> &sube; IP
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_Vocabulary_Properties"></a><h3> <span class="mw-headline">5.3  Semantic Conditions for the Vocabulary Properties </span></h3>
<p><a href="#table-semcond-properties" title="">Table 5.3</a> 
lists the semantic conditions for the properties
that have IRIs in the OWL 2 RDF-Based vocabulary.
In addition,
the table contains all those properties
with IRIs in the RDFS vocabulary
that are specified to be annotation properties 
under the OWL 2 RDF-Based Semantics.
The semantic conditions for the remaining properties
with names in the
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFINTERP">RDF</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFSINTERP">RDFS</a> 
vocabularies
can be found in the RDF Semantics specification
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p><p>The first column tells the <i>IRI</i> of the property.
The second column defines
of what particular <i>kind</i> 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 <i>domain and range</i> of the property:
from an entry of the form
"IEXT(<i>I</i>(<i>p</i>))&nbsp;&sube;&nbsp;<i>S<sub>1</sub></i>&nbsp;&times;&nbsp;<i>S<sub>2</sub></i>",
for a property IRI <i>p</i>
and sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
and given an RDF triple
"<i>s</i>&nbsp;<i>p</i>&nbsp;<i>o</i>",
one can deduce the relationships
"<i>I</i>(<i>s</i>)&nbsp;&isin;&nbsp;<i>S<sub>1</sub></i>"
and 
"<i>I</i>(<i>o</i>)&nbsp;&isin;&nbsp;<i>S<sub>2</sub></i>".
Note that some entries are of the form
"IEXT(<i>I</i>(<i>p</i>))&nbsp;=&nbsp;<i>S<sub>1</sub></i>&nbsp;&times;&nbsp;<i>S<sub>2</sub></i>",
which means that the property extension is exactly specified
to be the Cartesian product of the two sets.
</p>
<div id="topic-semcond-properties-facets"></div>
<p>Not included in this table are the <i>facets</i> of the OWL 2 RDF-Based Semantics
with IRIs 
listed in <a href="#Facet_Names" title="">Section 3.4</a>,
which are used to specify datatype restrictions
(see <a href="#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>).
For each such facet IRI <i>E</i>,
the following semantic conditions
<i>extend</i>
the basic semantics specification
that has been given for 
<i>datatypes with facets</i>
in <a href="#Datatype_Maps" title="">Section 4.1</a>:
</p>
<ul><li> <i>I</i>(<i>E</i>) &isin; IODP
</li><li> IEXT(<i>I</i>(<i>E</i>)) &sube; IR &times; LV
</li></ul>
<div id="topic-semcond-properties-narydatatype"></div>
<p>Implementations are <i>not</i> required 
to support the semantic condition for 
<span class="name">owl:onProperties</span>, 
but 
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> 
support it 
in order to realize 
<i>n-ary dataranges</i> with arity &ge; 2
(see 
Sections
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Ranges" title="Syntax">7</a>
and
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for further information).
</p>
<div id="topic-semcond-properties-informativenotes"></div>
<p><i>Informative notes:</i>
</p><p><span class="name">owl:topObjectProperty</span> 
relates every two individuals in the universe with each other.
Likewise, <span class="name">owl:topDataProperty</span>
relates every individual with every data value.
Further,
<span class="name">owl:bottomObjectProperty</span> 
and 
<span class="name">owl:bottomDataProperty</span>
stand both for the <i>empty</i> relationship.
</p><p>The ranges of the properties
<span class="name">owl:deprecated</span> and <span class="name">owl:hasSelf</span>
are not restricted in any form,
and, in particular, 
they are not restricted to Boolean values.
The actual object values of these properties
do not have any intended meaning,
but could as well have been defined to be of any other value.
Therefore, the semantics given here are of a form
that the values can be arbitrarily chosen
without leading to any nontrivial semantic conclusions.
It is, however, recommended to still use an object literal of the form
<span class="name">"true"^^xsd:boolean</span>
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 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Section 3.2.4 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
for <span class="name">owl:hasSelf</span>,
and 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for <span class="name">owl:deprecated</span>).
</p><p>The range of the property
<span class="name">owl:annotatedProperty</span> 
is unrestricted,
i.e. it is <i>not</i> specified as the set of properties.
Annotations are meant to be "semantically weak",
i.e. their formal meaning should not significantly exceed 
that originating from the RDF Semantics specification.
</p><p>Several properties,
such as <span class="name">owl:priorVersion</span>,
have been specified as both ontology properties and annotation properties,
in order to be in line with both 
the original 
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#5.2">OWL 1 RDF-Compatible Semantics specification</a>
[<cite><a href="#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>]
and
the rest of the OWL 2 specification
(see <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
</p>
<div class="left" id="table-semcond-properties">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.3: Semantic Conditions for the Vocabulary Properties</span>
</caption>

<tr>
<th style="text-align: center"> IRI <i>E</i>
</th><th style="text-align: center"> <i>I</i>(<i>E</i>)
</th><th style="text-align: center"> IEXT(<i>I</i>(<i>E</i>))
</th></tr>
<tr>
<td> <span id="item-semcond-properties-allvaluesfrom"></span><span class="name">owl:allValuesFrom</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedproperty"></span><span class="name">owl:annotatedProperty</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedsource"></span><span class="name">owl:annotatedSource</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-annotatedtarget"></span><span class="name">owl:annotatedTarget</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-assertionproperty"></span><span class="name">owl:assertionProperty</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) &times; IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-backwardcompatiblewith"></span><span class="name">owl:backwardCompatibleWith</span>
</td><td> &isin; IOXP ,<br />&isin; IOAP
</td><td> &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-bottomdataproperty"></span><span class="name">owl:bottomDataProperty</span>
</td><td> &isin; IODP
</td><td> = &empty;
</td></tr>
<tr>
<td> <span id="item-semcond-properties-bottomobjectproperty"></span><span class="name">owl:bottomObjectProperty</span>
</td><td> &isin; IP
</td><td> = &empty;
</td></tr>
<tr>
<td> <span id="item-semcond-properties-cardinality"></span><span class="name">owl:cardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-comment"></span><span class="name">rdfs:comment</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-complementof"></span><span class="name">owl:complementOf</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-datatypecomplementof"></span><span class="name">owl:datatypeComplementOf</span>
</td><td> &isin; IP
</td><td> &sube; IDC &times; IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-deprecated"></span><span class="name">owl:deprecated</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-differentfrom"></span><span class="name">owl:differentFrom</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-disjointunionof"></span><span class="name">owl:disjointUnionOf</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-disjointwith"></span><span class="name">owl:disjointWith</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-distinctmembers"></span><span class="name">owl:distinctMembers</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-equivalentclass"></span><span class="name">owl:equivalentClass</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-equivalentproperty"></span><span class="name">owl:equivalentProperty</span>
</td><td> &isin; IP
</td><td> &sube; IP &times; IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-haskey"></span><span class="name">owl:hasKey</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-hasself"></span><span class="name">owl:hasSelf</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-hasvalue"></span><span class="name">owl:hasValue</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-imports"></span><span class="name">owl:imports</span>
</td><td> &isin; IOXP
</td><td> &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-incompatiblewith"></span><span class="name">owl:incompatibleWith</span>
</td><td> &isin; IOXP ,<br />&isin; IOAP
</td><td> &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-intersectionof"></span><span class="name">owl:intersectionOf</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-inverseof"></span><span class="name">owl:inverseOf</span>
</td><td> &isin; IP
</td><td> &sube; IP &times; IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-isdefinedby"></span><span class="name">rdfs:isDefinedBy</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-label"></span><span class="name">rdfs:label</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-maxcardinality"></span><span class="name">owl:maxCardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-maxqualifiedcardinality"></span><span class="name">owl:maxQualifiedCardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-members"></span><span class="name">owl:members</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-mincardinality"></span><span class="name">owl:minCardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-minqualifiedcardinality"></span><span class="name">owl:minQualifiedCardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onclass"></span><span class="name">owl:onClass</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-ondatarange"></span><span class="name">owl:onDataRange</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-ondatatype"></span><span class="name">owl:onDatatype</span>
</td><td> &isin; IP
</td><td> &sube; IDC &times; IDC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-oneof"></span><span class="name">owl:oneOf</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onproperty"></span><span class="name">owl:onProperty</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-onproperties"></span><span class="name">owl:onProperties</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-priorversion"></span><span class="name">owl:priorVersion</span>
</td><td> &isin; IOXP ,<br />&isin; IOAP
</td><td> &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-propertychainaxiom"></span><span class="name">owl:propertyChainAxiom</span>
</td><td> &isin; IP
</td><td> &sube; IP &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-propertydisjointwith"></span><span class="name">owl:propertyDisjointWith</span>
</td><td> &isin; IP
</td><td> &sube; IP &times; IP
</td></tr>
<tr>
<td> <span id="item-semcond-properties-qualifiedcardinality"></span><span class="name">owl:qualifiedCardinality</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; INNI
</td></tr>
<tr>
<td> <span id="item-semcond-properties-sameas"></span><span class="name">owl:sameAs</span>
</td><td> &isin; IP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-seealso"></span><span class="name">rdfs:seeAlso</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-somevaluesfrom"></span><span class="name">owl:someValuesFrom</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:Restriction</span>)) &times; IC
</td></tr>
<tr>
<td> <span id="item-semcond-properties-sourceindividual"></span><span class="name">owl:sourceIndividual</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-targetindividual"></span><span class="name">owl:targetIndividual</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-targetvalue"></span><span class="name">owl:targetValue</span>
</td><td> &isin; IP
</td><td> &sube; ICEXT(<i>I</i>(<span class="name">owl:NegativePropertyAssertion</span>)) &times; LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-topdataproperty"></span><span class="name">owl:topDataProperty</span>
</td><td> &isin; IODP
</td><td> = IR &times; LV
</td></tr>
<tr>
<td> <span id="item-semcond-properties-topobjectproperty"></span><span class="name">owl:topObjectProperty</span>
</td><td> &isin; IP
</td><td> = IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-unionof"></span><span class="name">owl:unionOf</span>
</td><td> &isin; IP
</td><td> &sube; IC &times; ISEQ
</td></tr>
<tr>
<td> <span id="item-semcond-properties-versioninfo"></span><span class="name">owl:versionInfo</span>
</td><td> &isin; IOAP
</td><td> &sube; IR &times; IR
</td></tr>
<tr>
<td> <span id="item-semcond-properties-versioniri"></span><span class="name">owl:versionIRI</span>
</td><td> &isin; IOXP
</td><td> &sube; IX &times; IX
</td></tr>
<tr>
<td> <span id="item-semcond-properties-withrestrictions"></span><span class="name">owl:withRestrictions</span>
</td><td> &isin; IP
</td><td> &sube; IDC &times; ISEQ
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Boolean_Connectives"></a><h3> <span class="mw-headline">5.4  Semantic Conditions for Boolean Connectives </span></h3>
<p><a href="#table-semcond-booleans" title="">Table 5.4</a>
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.
</p>
<div id="topic-semcond-booleans-informativenotes"></div>
<p><i>Informative notes:</i>
Of the three pairs of semantic conditions in the table
every first is an "iff" condition,
since the corresponding OWL 2 language constructs
are both
class expressions and axioms.
In contrast,
the semantic condition on datatype complements
is an "if-then" condition,
since it only corresponds to a datarange expression.
See the
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
For the remaining semantic conditions
that treat the cases of intersections and unions of datatypes
it is sufficient to have "if-then" conditions,
since stronger "iff" conditions would be redundant
due to the more general "iff" conditions 
that already exist for classes. 
Note that the datatype related semantic conditions
do not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class
to be an instance of class <span class="name">rdfs:Datatype</span>.
</p>
<div class="left" id="table-semcond-booleans">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.4: Semantic Conditions for Boolean Connectives</span>
</caption>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-booleans-intersectionof-main"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC ,<br />ICEXT(<i>z</i>) = ICEXT(<i>c<sub>1</sub></i>) &cap; &hellip; &cap; ICEXT(<i>c<sub>n</sub></i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-intersectionof-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>d<sub>1</sub></i> , &hellip; , <i>d<sub>n</sub></i> &isin; IDC , <i>n</i> &ge; 1 ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td><td colspan="2"> <i>z</i> &isin; IDC
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-booleans-unionof-main"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC ,<br />ICEXT(<i>z</i>) = ICEXT(<i>c<sub>1</sub></i>) &cup; &hellip; &cup; ICEXT(<i>c<sub>n</sub></i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-unionof-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>d<sub>1</sub></i> , &hellip; , <i>d<sub>n</sub></i> &isin; IDC , <i>n</i> &ge; 1 ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td><td colspan="2"> <i>z</i> &isin; IDC
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<td> <span id="item-semcond-booleans-complementof"></span>( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:complementOf</span>))
</td><th colspan="2" style="text-align: center"> iff
</th><td> <i>z</i> , <i>c</i> &isin; IC ,<br />ICEXT(<i>z</i>) = IR \ ICEXT(<i>c</i>)
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-booleans-datatypecomplementof"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> ( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:datatypeComplementOf</span>))
</td><td colspan="2"> ICEXT(<i>z</i>) = LV \ ICEXT(<i>d</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Enumerations"></a><h3> <span class="mw-headline">5.5  Semantic Conditions for Enumerations </span></h3>
<p><a href="#table-semcond-enums" title="">Table 5.5</a> 
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.
</p>
<div id="topic-semcond-enums-informativenotes"></div>
<p><i>Informative notes:</i>
The first semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is both a class expression and an axiom.
See the
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
For the remaining semantic condition
that treats the case of enumerations of data values
it is sufficient to have an "if-then" condition,
since a stronger "iff" condition would be redundant
due to the more general "iff" condition 
that already exists for individuals.
Note that the data value related semantic condition
does not apply to empty sets,
but one can still receive a datatype from an empty set
by explicitly asserting the resulting class 
to be an instance of class <span class="name">rdfs:Datatype</span>.
</p>
<div class="left" id="table-semcond-enums">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.5: Semantic Conditions for Enumerations</span>
</caption>
<tr>
<th colspan="4" style="text-align: center"> <span id="item-semcond-enums-main"></span>if <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td><th colspan="2" rowspan="1" style="text-align: center"> iff
</th><td> <i>z</i> &isin; IC ,<br />ICEXT(<i>z</i>) = { <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> }
</td></tr>
<tr>
<td colspan="4">
</td></tr>
<tr>
<th colspan="2" style="text-align: center"> <span id="item-semcond-enums-data"></span>if
</th><th colspan="2" style="text-align: center"> then
</th></tr>
<tr>
<td colspan="2"> <i>s</i> sequence of <i>v<sub>1</sub></i> , &hellip; , <i>v<sub>n</sub></i> &isin; LV , <i>n</i> &ge; 1 ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td><td colspan="2"> <i>z</i> &isin; IDC
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Property_Restrictions"></a><h3> <span class="mw-headline">5.6  Semantic Conditions for Property Restrictions </span></h3>
<p><a href="#table-semcond-restrictions" title="">Table 5.6</a> 
lists the semantic conditions for property restrictions.
</p><p><i>Value restrictions</i> require that 
some or all of the values of a certain property 
must be instances of a given class or data range,
or that the property has a specifically defined value.
By placing a <i>self restriction</i> on some given property 
one only considers those individuals
that are reflexively related to themselves via this property. 
<i>Cardinality restrictions</i> determine 
how often a certain property is allowed 
to be applied to a given individual.
<i>Qualified cardinality restrictions</i> 
are more specific than cardinality restrictions
in that they determine the quantity of a property application 
with respect to a particular class or data range
from which the property values are taken.
</p>
<div id="topic-semcond-restrictions-narydatatype"></div>
<p>Implementations are <i>not</i> required 
to support the semantic conditions for 
<span class="name">owl:onProperties</span>, 
but 
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> 
support them 
in order to realize 
<i>n-ary dataranges</i> with arity &ge; 2
(see 
Sections
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Ranges" title="Syntax">7</a>
and
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for further information).
</p>
<div id="topic-semcond-restrictions-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "if-then" conditions,
since the corresponding OWL 2 language constructs
are class expressions.
The "if-then" 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 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
Note that the semantic condition for <i>self restrictions</i>
does not constrain the right hand side of 
a <span class="name">owl:hasSelf</span> assertion 
to be the Boolean value <span class="name">"true"^^xsd:boolean</span>.
See <a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a> for an explanation.
</p>
<div class="left" id="table-semcond-restrictions">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.6: Semantic Conditions for Property Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <span id="item-semcond-restrictions-somevaluesfrom"></span>( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | &exist; <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-somevaluesfrom-nary"></span><i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IR , <i>n</i> &ge; 1 ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td><td> <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | &exist; <i>y<sub>1</sub></i> , &hellip; , <i>y<sub>n</sub></i>&nbsp;: ( <i>x</i> , <i>y<sub>k</sub></i> ) &isin; IEXT(<i>p<sub>k</sub></i>) for each 1 &le; <i>k</i> &le; <i>n</i> and ( <i>y<sub>1</sub></i> , &hellip; , <i>y<sub>n</sub></i> ) &isin; ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-allvaluesfrom"></span>( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | &forall; <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies <i>y</i> &isin; ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-allvaluesfrom-nary"></span><i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IR , <i>n</i> &ge; 1 ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td><td> <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | &forall; <i>y<sub>1</sub></i> , &hellip; , <i>y<sub>n</sub></i>&nbsp;: ( <i>x</i> , <i>y<sub>k</sub></i> ) &isin; IEXT(<i>p<sub>k</sub></i>) for each 1 &le; <i>k</i> &le; <i>n</i> implies ( <i>y<sub>1</sub></i> , &hellip; , <i>y<sub>n</sub></i> ) &isin; ICEXT(<i>c</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-hasvalue"></span>( <i>z</i> , <i>a</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:hasValue</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ( <i>x</i> , <i>a</i> ) &isin; IEXT(<i>p</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-hasself"></span>( <i>z</i> , <i>v</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:hasSelf</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | ( <i>x</i> , <i>x</i> ) &isin; IEXT(<i>p</i>) }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-mincardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) } &ge; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxcardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) } &le; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-cardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:cardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) } = <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-minqualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>c</i>) } &ge; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-minqualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> &isin; IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> &isin; LV | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>d</i>) } &ge; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxqualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>c</i>) } &le; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-maxqualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> &isin; IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> &isin; LV | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>d</i>) } &le; <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-qualifiedcardinality"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>))
</td><td> ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>c</i>) } = <i>n</i> }
</td></tr>
<tr>
<td> <span id="item-semcond-restrictions-qualifiedcardinality-data"></span>( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>))
</td><td> <i>p</i> &isin; IODP ,<br />ICEXT(<i>z</i>) = { <i>x</i> | #{ <i>y</i> &isin; LV | ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and <i>y</i> &isin; ICEXT(<i>d</i>) } = <i>n</i> }
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Datatype_Restrictions"></a><h3> <span class="mw-headline">5.7  Semantic Conditions for Datatype Restrictions </span></h3>
<p><a href="#table-semcond-facets" title="">Table 5.7</a> 
lists the semantic conditions for datatype restrictions,
which are used to define sub datatypes of existing datatypes
by restricting the original datatype 
by means of a set of facet-value pairs.
See <a href="#Facet_Names" title="">Section 3.4</a> 
for information and an example on constraining facets.
</p>
<div id="topic-semcond-facets-empty"></div>
<p>Certain special cases exist:
If no facet-value pair is applied to a given datatype, 
then the resulting datatype will be equivalent to the original datatype.
Further,
if a facet-value 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 facet-value pairs
will result in an inconsistent ontology, 
if applied to a datatype with an empty facet space.
</p>
<div id="topic-semcond-facets-facetspace"></div>
<p>The set <b>IFS</b>
is defined by 
IFS(<i>d</i>)&nbsp;:= { ( <i>I</i>(<i>F</i>) , <i>v</i> ) | ( <i>F</i> , <i>v</i> ) &isin; FS(<i>d</i>) } ,
where 
<i>d</i> is a datatype,
<i>F</i> is the IRI of a constraining facet, 
and <i>v</i> is a constraining value of the facet.
This set corresponds to the facet space FS(<i>d</i>), 
as defined in <a href="#Datatype_Maps" title="">Section 4.1</a>,
but rather consists of 
pairs of the <i>denotation</i> of a facet and a value. 
</p>
<div id="topic-semcond-facets-facetmapping"></div>
<p>The mapping <b>IF2V</b> 
is defined by
IF2V(<i>d</i>)(( <i>I</i>(<i>F</i>) , <i>v</i> ))&nbsp;:= F2V(<i>d</i>)(( <i>F</i> , <i>v</i> )) , 
where
<i>d</i> is a datatype,
<i>F</i> is the IRI of a constraining facet, 
and <i>v</i> is a constraining value of the facet.
This mapping corresponds to the facet-to-value mapping F2V(<i>d</i>), 
as defined in <a href="#Datatype_Maps" title="">Section 4.1</a>,
resulting in the same subsets of the value space VS(<i>d</i>), 
but rather applies to 
pairs of the <i>denotation</i> of a facet and a value.
</p>
<div id="topic-semcond-facets-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "if-then" condition,
since the corresponding OWL 2 language construct
is a datarange expression.
The "if-then" 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
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.7: Semantic Conditions for Datatype Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-facets"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>z</i><sub>1</sub><i> , &hellip; , </i>z<i><sub>n</sub></i> &isin; IR ,<br /><i>f<sub>1</sub></i> , &hellip; , <i>f<sub>n</sub></i> &isin; IP ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDatatype</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:withRestrictions</span>)) ,<br />( <i>z<sub>1</sub></i> , <i>v<sub>1</sub></i> ) &isin; IEXT(<i>f<sub>1</sub></i>) , &hellip; , ( <i>z<sub>n</sub></i> , <i>v<sub>n</sub></i> ) &isin; IEXT(<i>f<sub>n</sub></i>)
</td><td> <i>f<sub>1</sub></i> , &hellip; , <i>f<sub>n</sub></i> &isin; IODP ,<br /><i>v<sub>1</sub></i> , &hellip; , <i>v<sub>n</sub></i> &isin; LV ,<br />( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> ) , &hellip; , ( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ) &isin; IFS(<i>d</i>) ,<br />ICEXT(<i>z</i>) = ICEXT(<i>d</i>) &cap; IF2V(<i>d</i>)(( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> )) &cap; &hellip; &cap; IF2V(<i>d</i>)(( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ))
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_the_RDFS_Vocabulary"></a><h3> <span class="mw-headline">5.8  Semantic Conditions for the RDFS Vocabulary </span></h3>
<p><a href="#table-semcond-rdfs" title="">Table 5.8</a> 
<i>extends</i> the RDFS semantic conditions
for subclass axioms, subproperty axioms, domain axioms and range axioms.
The semantic conditions provided here are "iff" conditions,
while the original semantic conditions,
as specified in 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">Section 4.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
are weaker "if-then" conditions.
Only the additional semantic conditions are given here 
and the other conditions of RDF and RDFS 
are retained.
</p>
<div id="topic-semcond-rdfs-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-rdfs">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.8: Semantic Conditions for the RDFS Vocabulary</span>
</caption>
<tr>
<td> <span id="item-semcond-rdfs-subclassof"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdfs:subClassOf</span>))
</td><th rowspan="4" style="text-align: center"> iff
</th><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> &isin; IC ,<br />ICEXT(<i>c<sub>1</sub></i>) &sube; ICEXT(<i>c<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-subpropertyof"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdfs:subPropertyOf</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>1</sub></i>) &sube; IEXT(<i>p<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-domain"></span>( <i>p</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">rdfs:domain</span>))
</td><td> <i>p</i> &isin; IP , <i>c</i> &isin; IC ,<br />&forall; <i>x</i> , <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies <i>x</i> &isin; ICEXT(<i>c</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-rdfs-range"></span>( <i>p</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">rdfs:range</span>))
</td><td> <i>p</i> &isin; IP , <i>c</i> &isin; IC ,<br />&forall; <i>x</i> , <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies <i>y</i> &isin; ICEXT(<i>c</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Equivalence_and_Disjointness"></a><h3> <span class="mw-headline">5.9  Semantic Conditions for Equivalence and Disjointness </span></h3>
<p><a href="#table-semcond-eqdis" title="">Table 5.9</a> 
lists the semantic conditions for specifying 
that two individuals are equal or different from each other,
and that either two classes or two properties 
are equivalent or disjoint with each other,
respectively.
The
property <span class="name">owl:equivalentClass</span>
is also used to formulate <i>datatype definitions</i>
(see <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Definitions" title="Syntax">Section 9.4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for information about datatype definitions).
In addition,
the table treats disjoint union axioms.
</p>
<div id="topic-semcond-eqdis-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-eqdis">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.9: Semantic Conditions for Equivalence and Disjointness</span>
</caption>
<tr>
<td> <span id="item-semcond-eqdis-sameas"></span>( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:sameAs</span>))
</td><th rowspan="6" style="text-align: center"> iff
</th><td> <i>a<sub>1</sub></i> = <i>a<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-differentfrom"></span>( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:differentFrom</span>))
</td><td> <i>a<sub>1</sub></i> &ne; <i>a<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-equivalentclass"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:equivalentClass</span>))
</td><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> &isin; IC ,<br />ICEXT(<i>c<sub>1</sub></i>) = ICEXT(<i>c<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-disjointwith"></span>( <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:disjointWith</span>))
</td><td> <i>c<sub>1</sub></i> , <i>c<sub>2</sub></i> &isin; IC ,<br />ICEXT(<i>c<sub>1</sub></i>) &cap; ICEXT(<i>c<sub>2</sub></i>) = &empty;
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-equivalentproperty"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:equivalentProperty</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>1</sub></i>) = IEXT(<i>p<sub>2</sub></i>)
</td></tr>
<tr>
<td> <span id="item-semcond-eqdis-propertydisjointwith"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:propertyDisjointWith</span>))
</td><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>1</sub></i>) &cap; IEXT(<i>p<sub>2</sub></i>) = &empty;
</td></tr>
<tr>
<td colspan="3">
</td></tr>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-eqdis-disjointunionof"></span>if <i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>c</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:disjointUnionOf</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>c</i> , <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC ,<br />ICEXT(<i>c</i>) = ICEXT(<i>c<sub>1</sub></i>) &cup; &hellip; &cup; ICEXT(<i>c<sub>n</sub></i>) ,<br />ICEXT(<i>c<sub>j</sub></i>) &cap; ICEXT(<i>c<sub>k</sub></i>) = &empty; for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_N-ary_Disjointness"></a><h3> <span class="mw-headline">5.10  Semantic Conditions for N-ary Disjointness </span></h3>
<p><a href="#table-semcond-ndis" title="">Table 5.10</a> 
lists the semantic conditions for specifying 
n-ary diversity and disjointness axioms,
i.e. that several given individuals 
are mutually different from each other,
and that several given classes or properties 
are mutually disjoint with each other,
respectively.
</p>
<div id="topic-semcond-ndis-variants"></div>
<p>Note that there are two alternative ways 
to specify <span class="name">owl:AllDifferent</span> axioms,
by using either the property
<span class="name">owl:members</span>
that is used for all other constructs, too,
or by applying the legacy property
<span class="name">owl:distinctMembers</span>.
Both variants have an equivalent formal meaning.
</p>
<div id="topic-semcond-ndis-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However, 
there are actually <i>two</i> semantic conditions for each language construct
due to the multi-triple RDF encoding of these language constructs.
The "if-then" 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 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-ndis">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.10: Semantic Conditions for N-ary Disjointness</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-members-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR ,<br /><i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>a<sub>j</sub></i> &ne; <i>a<sub>k</sub></i> for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-members-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR ,<br /><i>a<sub>j</sub></i> &ne; <i>a<sub>k</sub></i> for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td><td> <i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-distinctmembers-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR ,<br /><i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:distinctMembers</span>))
</td><td> <i>a<sub>j</sub></i> &ne; <i>a<sub>k</sub></i> for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldifferent-distinctmembers-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR ,<br /><i>a<sub>j</sub></i> &ne; <i>a<sub>k</sub></i> for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td><td> <i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:distinctMembers</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointclasses-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IR ,<br /><i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDisjointClasses</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC ,<br />ICEXT(<i>c<sub>j</sub></i>) &cap; ICEXT(<i>c<sub>k</sub></i>) = &empty; for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointclasses-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC ,<br />ICEXT(<i>c<sub>j</sub></i>) &cap; ICEXT(<i>c<sub>k</sub></i>) = &empty; for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td><td> <i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDisjointClasses</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointproperties-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IR ,<br /><i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDisjointProperties</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td><td> <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>j</sub></i>) &cap; IEXT(<i>p<sub>k</sub></i>) = &empty; for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-eqdis-alldisjointproperties-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>j</sub></i>) &cap; IEXT(<i>p<sub>k</sub></i>) = &empty; for each 1 &le; <i>j</i> &le; <i>n</i> and each 1 &le; <i>k</i> &le; <i>n</i> such that <i>j</i> &ne; <i>k</i>
</td><td> <i>z</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AllDisjointProperties</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:members</span>))
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Sub_Property_Chains"></a><h3> <span class="mw-headline">5.11  Semantic Conditions for Sub Property Chains </span></h3>
<p><a href="#table-semcond-chains" title="">Table 5.11</a> 
lists the semantic conditions for sub property chains,
which allow for specifying complex property subsumption axioms.
</p><p>As an example,
one can define a sub property chain axiom
that specifies
the chain consisting of the property extensions 
of properties 
<span class="name">ex:hasFather</span>
and
<span class="name">ex:hasBrother</span>
to be a sub relation of 
the extension of the property
<span class="name">ex:hasUncle</span>.
</p>
<div id="topic-semcond-chains-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
The semantics has been specified in a way
such that a sub property chain axiom can be satisfied
without requiring the existence of a property
that has the property chain as its property extension.
</p>
<div class="left" id="table-semcond-chains">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.11: Semantic Conditions for Sub Property Chains</span>
</caption>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-chains"></span>if <i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>p</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:propertyChainAxiom</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>p</i> &isin; IP ,<br /><i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />&forall; <i>y<sub>0</sub></i> , &hellip; , <i>y<sub>n</sub></i>&nbsp;: ( <i>y<sub>0</sub></i> , <i>y<sub>1</sub></i> ) &isin; IEXT(<i>p<sub>1</sub></i>) and &hellip; and ( <i>y<sub>n-1</sub></i> , <i>y<sub>n</sub></i> ) &isin; IEXT(<i>p<sub>n</sub></i>) implies ( <i>y<sub>0</sub></i> , <i>y<sub>n</sub></i> ) &isin; IEXT(<i>p</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Inverse_Properties"></a><h3> <span class="mw-headline">5.12  Semantic Conditions for Inverse Properties </span></h3>
<p><a href="#table-semcond-inverses" title="">Table 5.12</a> 
lists the semantic conditions for inverse property axioms.
The inverse of a given property
is the corresponding property
with subject and object swapped 
for each property assertion built from it.
</p>
<div id="topic-semcond-inverses-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-inverses">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.12: Semantic Conditions for Inverse Properties</span>
</caption>
<tr>
<td> <span id="item-semcond-inverses"></span>( <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:inverseOf</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>p<sub>1</sub></i> , <i>p<sub>2</sub></i> &isin; IP ,<br />IEXT(<i>p<sub>1</sub></i>) = { ( <i>x</i> , <i>y</i> ) | ( <i>y</i> , <i>x</i> ) &isin; IEXT(<i>p<sub>2</sub></i>) }
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Property_Characteristics"></a><h3> <span class="mw-headline">5.13  Semantic Conditions for Property Characteristics </span></h3>
<p><a href="#table-semcond-characteristics" title="">Table 5.13</a> 
lists the semantic conditions for property characteristics.
</p><p>If a property is <i>functional</i>,
then at most one distinct value can be assigned 
to any given individual 
via this property.
An <i>inverse functional</i> property can be regarded as a "key" property,
i.e. no two different individuals 
can be assigned the same value 
via this property.
A <i>reflexive</i> property relates every individual in the universe to itself,
whereas an <i>irreflexive</i> property does not relate any individual with itself.
If two individuals are related by a <i>symmetric</i> property,
then this property also relates them reversely,
while this is never the case for an <i>asymmetric</i> property.
A <i>transitive</i> property 
that relates an individual <i>a</i> with an individual <i>b</i>, 
and the latter with an individual <i>c</i>,
also relates <i>a</i> with <i>c</i>.
</p>
<div id="topic-semcond-characteristics-informativenotes"></div>
<p><i>Informative notes:</i>
All the semantic conditions are "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
See the 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-characteristics">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.13: Semantic Conditions for Property Characteristics</span>
</caption>
<tr>
<td> <span id="item-semcond-characteristics-functionalproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:FunctionalProperty</span>))
</td><th rowspan="7" style="text-align: center"> iff
</th><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i> , <i>y<sub>1</sub></i> , <i>y<sub>2</sub></i>&nbsp;: ( <i>x</i> , <i>y<sub>1</sub></i> ) &isin; IEXT(<i>p</i>) and ( <i>x</i> , <i>y<sub>2</sub></i> ) &isin; IEXT(<i>p</i>) implies <i>y<sub>1</sub></i> = <i>y<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-inversefunctionalproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:InverseFunctionalProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x<sub>1</sub></i> , <i>x<sub>2</sub></i> , <i>y</i>&nbsp;: ( <i>x<sub>1</sub></i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and ( <i>x<sub>2</sub></i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies <i>x<sub>1</sub></i> = <i>x<sub>2</sub></i>
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-reflexiveproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:ReflexiveProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i>&nbsp;: ( <i>x</i> , <i>x</i> ) &isin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-irreflexiveproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:IrreflexiveProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i>&nbsp;: ( <i>x</i> , <i>x</i> ) &notin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-symmetricproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:SymmetricProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i> , <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies ( <i>y</i> , <i>x</i> ) &isin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-asymmetricproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:AsymmetricProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i> , <i>y</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) implies ( <i>y</i> , <i>x</i> ) &notin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td> <span id="item-semcond-characteristics-transitiveproperty"></span><i>p</i> &isin; ICEXT(<i>I</i>(<span class="name">owl:TransitiveProperty</span>))
</td><td> <i>p</i> &isin; IP ,<br />&forall; <i>x</i> , <i>y</i> , <i>z</i>&nbsp;: ( <i>x</i> , <i>y</i> ) &isin; IEXT(<i>p</i>) and ( <i>y</i> , <i>z</i> ) &isin; IEXT(<i>p</i>) implies ( <i>x</i> , <i>z</i> ) &isin; IEXT(<i>p</i>)
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Keys"></a><h3> <span class="mw-headline">5.14  Semantic Conditions for Keys </span></h3>
<p><a href="#table-semcond-keys" title="">Table 5.14</a> 
lists the semantic conditions for Keys.
</p><p>Keys provide an alternative to inverse functional properties 
(see <a href="#Semantic_Conditions_for_Property_Characteristics" title="">Section 5.13</a>).
They allow for defining a property as a key local to a given class:
the specified property
will have the features of a key
only for individuals being instances of the class,
and no assumption is made
about individuals 
for which membership of the class cannot be entailed.
Further,
it is possible to define "compound keys",
i.e. several properties can be combined into a single key 
applicable to composite values.
Note that 
keys are not functional by default
under the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-semcond-keys-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic condition is an "iff" condition,
since the corresponding OWL 2 language construct
is an axiom.
See the 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-keys">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.14: Semantic Conditions for Keys</span>
</caption>
<tr>
<th colspan="3" style="text-align: center"> <span id="item-semcond-keys"></span>if <i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IR then
</th></tr>
<tr>
<td> ( <i>c</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:hasKey</span>))
</td><th rowspan="1" style="text-align: center"> iff
</th><td> <i>c</i> &isin; IC ,<br /><i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP ,<br />&forall; <i>x</i> , <i>y</i> , <i>z<sub>1</sub></i> , &hellip; , <i>z<sub>n</sub></i>&nbsp;:<br />&nbsp;&nbsp;&nbsp;if <i>x</i> &isin; ICEXT(<i>c</i>) and <i>y</i> &isin; ICEXT(<i>c</i>) and<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;( <i>x</i> , <i>z<sub>k</sub></i> ) &isin; IEXT(<i>p<sub>k</sub></i>) and ( <i>y</i> , <i>z<sub>k</sub></i> ) &isin; IEXT(<i>p<sub>k</sub></i>) for each 1 &le; <i>k</i> &le; <i>n</i><br />&nbsp;&nbsp;&nbsp;then <i>x</i> = <i>y</i>
</td></tr>
</table>
</div>
<a name="Semantic_Conditions_for_Negative_Property_Assertions"></a><h3> <span class="mw-headline">5.15  Semantic Conditions for Negative Property Assertions </span></h3>
<p><a href="#table-semcond-negassert" title="">Table 5.15</a> 
lists the semantic conditions for negative property assertions.
They allow to state that 
two given individuals are <i>not</i> related by a given property.
</p>
<div id="topic-semcond-negassert-datavariant"></div>
<p>The second form based on <span class="name">owl:targetValue</span> 
is more specific than the first form based on <span class="name">owl:targetIndividual</span>
in that the second form is restricted 
to the case of negative <i>data</i> property assertions.
Note that the second form 
will coerce the target value of a negative property assertion
into a data value,
due to the range defined for the property 
<span class="name">owl:targetValue</span> 
in 
<a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>.
</p>
<div id="topic-semcond-negassert-informativenotes"></div>
<p><i>Informative notes:</i>
The semantic conditions essentially represent "iff" conditions,
since the corresponding OWL 2 language constructs
are axioms.
However, 
there are actually <i>two</i> semantic conditions for each language construct,
due to the multi-triple RDF encoding of these language constructs.
The "if-then" 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 
<a href="#topic-semcond-conditionform" title=""><i>notes on the form of semantic conditions</i></a>
for further information.
</p>
<div class="left" id="table-semcond-negassert">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 5.15: Semantic Conditions for Negative Property Assertions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-object-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>a<sub>1</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>a<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:targetIndividual</span>))
</td><td> ( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) &notin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-object-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>a<sub>1</sub></i> &isin; IR ,<br /><i>p</i> &isin; IP ,<br /><i>a<sub>2</sub></i> &isin; IR ,<br />( <i>a<sub>1</sub></i> , <i>a<sub>2</sub></i> ) &notin; IEXT(<i>p</i>)
</td><td> ( <i>z</i> , <i>a<sub>1</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>a<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:targetIndividual</span>))
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-data-fw"></span>if
</th><th style="text-align: center"> then
</th></tr>
<tr>
<td> ( <i>z</i> , <i>a</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>v</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:targetValue</span>))
</td><td> <i>p</i> &isin; IODP ,<br />( <i>a</i> , <i>v</i> ) &notin; IEXT(<i>p</i>)
</td></tr>
<tr>
<td colspan="2">
</td></tr>
<tr>
<th style="text-align: center"> <span id="item-semcond-negassert-data-bw"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>a</i> &isin; IR ,<br /><i>p</i> &isin; IODP ,<br /><i>v</i> &isin; LV ,<br />( <i>a</i> , <i>v</i> ) &notin; IEXT(<i>p</i>)
</td><td> ( <i>z</i> , <i>a</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:sourceIndividual</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:assertionProperty</span>)) ,<br />( <i>z</i> , <i>v</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:targetValue</span>))
</td></tr>
</table>
</div>
<a name="Appendix:_Axiomatic_Triples_.28Informative.29"></a><h2> <span class="mw-headline">6  Appendix: Axiomatic Triples (Informative) </span></h2>
<p>The RDF Semantics specification
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]
defines so called <i>"axiomatic triples"</i>
as part of the semantics of 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples">RDF</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples">RDFS</a>.
Unlike the RDF Semantics,
the OWL 2 RDF-Based Semantics does not normatively specify any axiomatic triples,
since one cannot expect to find a set of RDF triples
that fully captures all "axiomatic aspects"
of the OWL 2 RDF-Based Semantics.
Furthermore,
axiomatic triples for the OWL 2 RDF-Based Semantics could,
in principle,
contain arbitrarily complex class expressions,
e.g. the union of several classes,
and by this it becomes nonobvious 
which of several possible nonequivalent sets of axiomatic triples 
should be selected.
However,
the OWL 2 RDF-Based Semantics includes many semantic conditions 
that can in a sense be regarded as being "axiomatic",
and thus can be considered a replacement for the missing axiomatic triples.
After an overview on axiomatic triples for RDF and RDFS 
in <a href="#Axiomatic_Triples_in_RDF" title="">Section 6.1</a>, 
Sections <a href="#Axiomatic_Triples_for_the_Vocabulary_Classes" title="">6.2</a> 
and 
<a href="#Axiomatic_Triples_for_the_Vocabulary_Properties" title="">6.3</a>
will discuss how the "axiomatic" semantic conditions 
of the OWL 2 RDF-Based Semantics
relate to axiomatic triples.
Based on this discussion,
an explicit example set of axiomatic triples
that is compatible with the OWL 2 RDF-Based Semantics
will be provided in
<a href="#A_Set_of_Axiomatic_Triples" title="">Section 6.4</a>.
</p>
<a name="Axiomatic_Triples_in_RDF"></a><h3> <span class="mw-headline">6.1  Axiomatic Triples in RDF </span></h3>
<p>In 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDF_axiomatic_triples">RDF</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#RDFS_axiomatic_triples">RDFS</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
axiomatic triples are used
to provide basic meaning 
for all the vocabulary terms 
of the two languages.
This formal meaning is independent of any given RDF graph,
and it even holds for vocabulary terms,
which do not occur in a graph 
that is interpreted by an RDF or RDFS interpretation.
As a consequence,
all the axiomatic triples of RDF and RDFS 
are entailed by the <i>empty</i> graph,
when being interpreted under the semantics of RDF or RDFS, 
respectively.
</p>
<div id="topic-axiomatic-rdf-examples"></div>
<p>Examples of RDF and RDFS axiomatic triples are:
</p>
<div class="indent">
<p>(1) <span class="name">rdf:type rdf:type rdf:Property .</span><br />
(2) <span class="name">rdf:type rdfs:domain rdfs:Resource .</span><br />
(3) <span class="name">rdf:type rdfs:range rdfs:Class .</span><br />
(4) <span class="name">rdfs:Datatype rdfs:subClassOf rdfs:Class .</span><br />
(5) <span class="name">rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .</span>
</p>
</div>
<p>As shown by these examples,
axiomatic triples are typically used by the RDF Semantics specification
to determine the part of the universe
to which the denotation of a vocabulary term belongs (1).
In the case of a property,
the domain (2) and range (3) is specified as well.
Also, in some cases,
hierarchical relationships 
between classes (4) or properties (5) of the vocabulary
are determined.
</p>
<div id="topic-axiomatic-rdf-semcond"></div>
<p>Under the OWL 2 RDF-Based Semantics,
all the axiomatic triples of RDF and RDFS
could, in principle, be replaced by
"axiomatic" semantic conditions
that have neither premises nor bound variables.
By applying the <i>RDFS semantic conditions</i> 
given in <a href="#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>,
the example axiomatic triples (1) &ndash; (5) 
can be equivalently restated as:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">rdf:type</span>) &isin; ICEXT(<i>I</i>(<span class="name">rdf:Property</span>)) ,<br /> 
IEXT(<i>I</i>(<span class="name">rdf:type</span>)) &sube; ICEXT(<i>I</i>(<span class="name">rdfs:Resource</span>)) &times; ICEXT(<i>I</i>(<span class="name">rdfs:Class</span>)) ,<br />
ICEXT(<i>I</i>(<span class="name">rdfs:Datatype</span>)) &sube; ICEXT(<i>I</i>(<span class="name">rdfs:Class</span>)) ,<br />
IEXT(<i>I</i>(<span class="name">rdfs:isDefinedBy</span>)) &sube; IEXT(<i>I</i>(<span class="name">rdfs:seeAlso</span>)) .
</p>
</div>
<div id="topic-axiomatic-rdf-simple"></div>
<p>All the axiomatic triples of RDF and RDFS 
can be considered <i>"simple"</i> 
in the sense that 
they have in their object position
only single terms 
from the RDF and RDFS vocabularies,
and no complex class or property expressions 
appear there.
</p>
<a name="Axiomatic_Triples_for_the_Vocabulary_Classes"></a><h3> <span class="mw-headline">6.2  Axiomatic Triples for the Vocabulary Classes </span></h3>
<p>The semantic conditions for <i>vocabulary classes</i>
in <a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
can be considered as corresponding to
a set of axiomatic triples 
for the classes in the vocabulary of the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-axiomatic-classes-secondcoldef"></div>
<p>First, 
for each IRI <i>E</i> 
occurring in the first column of <a href="#table-semcond-classes" title="">Table 5.2</a>,
if the <i>second</i> column contains an entry 
of the form 
"<i>I</i>(<i>E</i>) &isin; <i>S</i>" 
for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form 
"<i>E</i> <span class="name">rdf:type</span> <i>C</i>", 
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In the table, <i>S</i> will always be either 
the <a href="#Parts_of_the_Universe" title="">part</a> IC of all classes, 
or some sub part of IC.
Hence, in a corresponding RDF triple the IRI <i>C</i> will be
one of 
"<span class="name">rdfs:Class</span>", 
"<span class="name">owl:Class</span>" 
(<i>S</i>=IC in both cases) 
or "<span class="name">rdfs:Datatype</span>" (<i>S</i>=IDC).
</p>
<div id="topic-axiomatic-classes-secondcolexample"></div>
<p>For example,
for the IRI "<span class="name">owl:FunctionalProperty</span>",
the semantic condition
</p>
<div class="indent">
<p><i>I</i>(<span class="name">owl:FunctionalProperty</span>) &isin; IC
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:FunctionalProperty rdf:type rdfs:Class .</span>
</p>
</div>
<div id="topic-axiomatic-classes-thirdcoldef"></div>
<p>Further, 
for each IRI <i>E</i> in the first column of the table, 
if the <i>third</i> column contains an entry 
of the form 
"ICEXT(<i>I</i>(<i>E</i>)) &sube; <i>S</i>" 
(or "ICEXT(<i>I</i>(<i>E</i>)) = <i>S</i>") 
for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form
"<i>E</i> <span class="name">rdfs:subClassOf</span> <i>C</i>" 
(or additionally "<i>C</i> <span class="name">rdfs:subClassOf</span> <i>E</i>"), 
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In each case,
<i>S</i> will be one of 
the <a href="#Parts_of_the_Universe" title="">parts of the universe</a> of <i>I</i>.
</p>
<div id="topic-axiomatic-classes-thirdcolexample"></div>
<p>For example,
the semantic condition
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">owl:FunctionalProperty</span>)) &sube; IP
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:FunctionalProperty rdfs:subClassOf rdf:Property .</span>
</p>
</div>
<div id="topic-axiomatic-classes-partstab"></div>
<p>In addition, 
the semantic conditions for the 
<i>parts of the universe</i> 
in <a href="#table-semcond-parts" title="">Table 5.1</a>
of <a href="#Semantic_Conditions_for_the_Parts_of_the_Universe" title="">Section 5.1</a> 
have to be taken into account.
In particular, 
if an entry in the <i>second</i> column of <a href="#table-semcond-parts" title="">Table 5.1</a>
is of the form
"<i>S<sub>1</sub></i> &sube; <i>S<sub>2</sub></i>" 
for some sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
then this corresponds to an RDF triple
of the form
"<i>C<sub>1</sub></i> <span class="name">owl:subClassOf</span> <i>C<sub>2</sub></i>",
where 
<i>C<sub>1</sub></i> and <i>C<sub>2</sub></i> 
are the IRIs of vocabulary classes with 
ICEXT(<i>I</i>(<i>C<sub>1</sub></i>)) = <i>S<sub>1</sub></i>
and
ICEXT(<i>I</i>(<i>C<sub>2</sub></i>)) = <i>S<sub>2</sub></i>,
respectively,
according to 
<a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>.
</p>
<div id="topic-axiomatic-classes-datatypes"></div>
<p><a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>
also specifies semantic conditions 
for all the <i>datatypes</i> of the OWL 2 RDF-Based Semantics,
as listed in <a href="#Datatype_Names" title="">Section 3.3</a>. 
For each datatype IRI <i>E</i>,
such as <i>E</i>&nbsp;:= "<span class="name">xsd:string</span>",
for the semantic conditions
"<i>I</i>(<i>E</i>) &isin; IDC" 
and
"ICEXT(<i>I</i>(<i>E</i>)) &sube; LV"
the corresponding axiomatic triples are of the form
</p>
<div class="indent">
<p><i>E</i> <span class="name">rdf:type rdfs:Datatype .</span><br />
<i>E</i> <span class="name">rdfs:subClassOf rdfs:Literal .</span>
</p>
</div>
<div id="topic-axiomatic-classes-simple"></div>
<p>In analogy to 
<a href="#Axiomatic_Triples_in_RDF" title="">Section 6.1</a> 
for the RDF axiomatic triples, 
all the axiomatic triples for the vocabulary classes
(including datatypes)
can be considered <i>"simple"</i>
in the sense that
they will have in their object position
only single terms 
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(<a href="#Vocabulary_Terms" title="">Section 3.2</a>).
</p>
<div id="topic-axiomatic-classes-redundant"></div>
<p>Note that some of the axiomatic triples obtained in this way
already follow from the semantics of 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfinterpdef">RDF</a> 
and 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfsinterpdef">RDFS</a>,
as defined in 
the RDF Semantics [<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
</p>
<a name="Axiomatic_Triples_for_the_Vocabulary_Properties"></a><h3> <span class="mw-headline">6.3  Axiomatic Triples for the Vocabulary Properties </span></h3>
<p>The semantic conditions for <i>vocabulary properties</i>
in <a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a> 
can be considered as corresponding to
a set of axiomatic triples
for the properties in the vocabulary of the OWL 2 RDF-Based Semantics.
</p>
<div id="topic-axiomatic-properties-secondcoldef"></div>
<p>First,
for each IRI <i>E</i> 
occurring in the first column of <a href="#table-semcond-properties" title="">Table 5.3</a>,
if the <i>second</i> column contains an entry
of the form 
"<i>I</i>(<i>E</i>) &isin; <i>S</i>" for some set <i>S</i>,
then this entry corresponds to an RDF triple of the form 
"<i>E</i> <span class="name">rdf:type</span> <i>C</i>",
where <i>C</i> is the IRI of a vocabulary class with ICEXT(<i>I</i>(<i>C</i>)) = <i>S</i>.
In the table, 
<i>S</i> will always be either 
the <a href="#Parts_of_the_Universe" title="">part</a> IP of all properties, 
or some sub part of IP.
Hence, in a corresponding RDF triple the IRI <i>C</i> will be
one of 
"<span class="name">rdf:Property</span>", 
"<span class="name">owl:ObjectProperty</span>", 
(<i>S</i>=IP in both cases), 
"<span class="name">owl:DatatypeProperty</span>" (<i>S</i>=IODP), 
"<span class="name">owl:OntologyProperty</span>" (<i>S</i>=IOXP)
or "<span class="name">owl:AnnotationProperty</span>" (<i>S</i>=IOAP). 
</p>
<div id="topic-axiomatic-properties-secondcolexample"></div>
<p>For example,
for the IRI "<span class="name">owl:disjointWith</span>",
the semantic condition
</p>
<div class="indent">
<p><i>I</i>(<span class="name">owl:disjointWith</span>) &isin; IP
</p>
</div>
<p>has the corresponding axiomatic triple
</p>
<div class="indent">
<p><span class="name">owl:disjointWith rdf:type rdf:Property .</span>
</p>
</div>
<div id="topic-axiomatic-properties-thirdcoldef"></div>
<p>Further, 
for each IRI <i>E</i> in the first column of the table,
if the <i>third</i> column contains an entry
of the form 
"IEXT(<i>I</i>(<i>E</i>)) &sube; <i>S<sub>1</sub></i> &times; <i>S<sub>2</sub></i>" 
for some sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>,
then this entry corresponds to RDF triples of the form
"<i>E</i> <span class="name">rdfs:domain</span> <i>C<sub>1</sub></i>" 
and 
"<i>E</i> <span class="name">rdfs:range</span> <i>C<sub>2</sub></i>",
where <i>C<sub>1</sub></i> and <i>C<sub>2</sub></i> 
are the IRIs of vocabulary classes with 
ICEXT(<i>I</i>(<i>C<sub>1</sub></i>)) = <i>S<sub>1</sub></i>
and
ICEXT(<i>I</i>(<i>C<sub>2</sub></i>)) = <i>S<sub>2</sub></i>,
respectively.
Note that the sets <i>S<sub>1</sub></i> and <i>S<sub>2</sub></i>
do <i>not always</i> correspond 
to any of the <a href="#Parts_of_the_Universe" title="">parts of the universe</a> of <i>I</i>.
</p>
<div id="topic-axiomatic-properties-thirdcolexample"></div>
<p>For example,
the semantic condition
</p>
<div class="indent">
<p>IEXT(<i>I</i>(<span class="name">owl:disjointWith</span>)) &sube; IC &times; IC
</p>
</div>
<p>has the corresponding axiomatic triples
</p>
<div class="indent">
<p><span class="name">owl:disjointWith rdfs:domain owl:Class .</span><br />
<span class="name">owl:disjointWith rdfs:range owl:Class .</span>
</p>
</div>
<div id="topic-axiomatic-properties-thirdcolexeption"></div>
<p>Exceptions are the semantic conditions
"IEXT(<i>I</i>(<span class="name">owl:topObjectProperty</span>)) = IR &times; IR"
and
"IEXT(<i>I</i>(<span class="name">owl:topDataProperty</span>)) = IR &times; LV",
since the <i>exactly</i> 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
<span class="name">owl:sameAs</span>
are equal to those for
<span class="name">owl:topObjectProperty</span>,
but the property extension of 
<span class="name">owl:sameAs</span>
is different from the property extension of 
<span class="name">owl:topObjectProperty</span>.
</p>
<div id="topic-axiomatic-properties-facets"></div>
<p><a href="#Semantic_Conditions_for_the_Vocabulary_Properties" title="">Section 5.3</a>
also specifies semantic conditions
for all the <i>facets</i> of the OWL 2 RDF-Based Semantics,
as listed in <a href="#Facet_Names" title="">Section 3.4</a>.
For each facet IRI <i>E</i>,
such as <i>E</i>&nbsp;:= "<span class="name">xsd:length</span>",
for the semantic conditions
"<i>I</i>(<i>E</i>) &isin; IODP"
and
"IEXT(<i>I</i>(<i>E</i>)) &sube; IR &times; LV"
the corresponding axiomatic triples are of the form
</p>
<div class="indent">
<p><i>E</i> <span class="name">rdf:type owl:DatatypeProperty .</span><br />
<i>E</i> <span class="name">rdfs:domain rdfs:Resource .</span><br />
<i>E</i> <span class="name">rdfs:range rdfs:Literal .</span><br />
</p>
</div>
<div id="topic-axiomatic-properties-simple"></div>
<p>In analogy to 
<a href="#Axiomatic_Triples_in_RDF" title="">Section 6.1</a> 
for the RDF axiomatic triples, 
all the axiomatic triples for the vocabulary properties 
(including facets)
can be considered <i>"simple"</i>
in the sense that
they will have in their object position
only single terms 
from the RDF, RDFS and OWL 2 RDF-Based vocabularies
(<a href="#Vocabulary_Terms" title="">Section 3.2</a>).
</p>
<a name="A_Set_of_Axiomatic_Triples"></a><h3> <span class="mw-headline">6.4  A Set of Axiomatic Triples </span></h3>
<p>This section provides a concrete example set of axiomatic triples
based on the discussion in the Sections
<a href="#Axiomatic_Triples_for_the_Vocabulary_Classes" title="">6.2</a>
and
<a href="#Axiomatic_Triples_for_the_Vocabulary_Properties" title="">6.3</a>.
The axiomatic triples are grouped by different tables
for the <a href="#table-axiomatic-classes" title="">classes</a> 
and the <a href="#table-axiomatic-properties" title="">properties</a> 
of the OWL 2 RDF-Based vocabulary,
for the <a href="#table-axiomatic-datatypes" title="">datatypes</a> 
and the <a href="#table-axiomatic-facets" title="">facets</a> 
of the OWL 2 RDF-Based Semantics,
and for some of the 
<a href="#table-axiomatic-rdfs" title="">classes and properties of the RDFS vocabulary</a>.
Note that this set of axiomatic triples 
is not meant to be free of redundancy.
</p>
<div class="left" id="table-axiomatic-classes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.1: Axiomatic Triples for the Classes of the OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-classes-alldifferent">owl:AllDifferent rdf:type rdfs:Class .<br />owl:AllDifferent rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-alldisjointclasses">owl:AllDisjointClasses rdf:type rdfs:Class .<br />owl:AllDisjointClasses rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-alldisjointproperties">owl:AllDisjointProperties rdf:type rdfs:Class .<br />owl:AllDisjointProperties rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-annotation">owl:Annotation rdf:type rdfs:Class .<br />owl:Annotation rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-annotationproperty">owl:AnnotationProperty rdf:type rdfs:Class .<br />owl:AnnotationProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-asymmetricproperty">owl:AsymmetricProperty rdf:type rdfs:Class .<br />owl:AsymmetricProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-axiom">owl:Axiom rdf:type rdfs:Class .<br />owl:Axiom rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-class">owl:Class rdf:type rdfs:Class .<br />owl:Class rdfs:subClassOf rdfs:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-datarange">owl:DataRange rdf:type rdfs:Class .<br />owl:DataRange rdfs:subClassOf rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-datatypeproperty">owl:DatatypeProperty rdf:type rdfs:Class .<br />owl:DatatypeProperty rdfs:subClassOf rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-deprecatedclass">owl:DeprecatedClass rdf:type rdfs:Class .<br />owl:DeprecatedClass rdfs:subClassOf rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-deprecatedproperty">owl:DeprecatedProperty rdf:type rdfs:Class .<br />owl:DeprecatedProperty rdfs:subClassOf rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-functionalproperty">owl:FunctionalProperty rdf:type rdfs:Class .<br />owl:FunctionalProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-inversefunctionalproperty">owl:InverseFunctionalProperty rdf:type rdfs:Class .<br />owl:InverseFunctionalProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-irreflexiveproperty">owl:IrreflexiveProperty rdf:type rdfs:Class .<br />owl:IrreflexiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-namedindividual">owl:NamedIndividual rdf:type rdfs:Class .<br />owl:NamedIndividual rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-negativepropertyassertion">owl:NegativePropertyAssertion rdf:type rdfs:Class .<br />owl:NegativePropertyAssertion rdfs:subClassOf rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-nothing">owl:Nothing rdf:type owl:Class .<br />owl:Nothing rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-objectproperty">owl:ObjectProperty rdf:type rdfs:Class .<br />owl:ObjectProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-ontology">owl:Ontology rdf:type rdfs:Class .<br />owl:Ontology rdfs:subClassOf rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-ontologyproperty">owl:OntologyProperty rdf:type rdfs:Class .<br />owl:OntologyProperty rdfs:subClassOf rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-reflexiveproperty">owl:ReflexiveProperty rdf:type rdfs:Class .<br />owl:ReflexiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-restriction">owl:Restriction rdf:type rdfs:Class .<br />owl:Restriction rdfs:subClassOf owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-symmetricproperty">owl:SymmetricProperty rdf:type rdfs:Class .<br />owl:SymmetricProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-classes-thing">owl:Thing rdf:type owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-classes-transitiveproperty">owl:TransitiveProperty rdf:type rdfs:Class .<br />owl:TransitiveProperty rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-properties">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.2: Axiomatic Triples for the Properties of the OWL 2 RDF-Based Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-properties-allvaluesfrom">owl:allValuesFrom rdf:type rdf:Property .<br />owl:allValuesFrom rdfs:domain owl:Restriction .<br />owl:allValuesFrom rdfs:range rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-annotatedproperty">owl:annotatedProperty rdf:type rdf:Property .<br />owl:annotatedProperty rdfs:domain rdfs:Resource .<br />owl:annotatedProperty rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-annotatedsource">owl:annotatedSource rdf:type rdf:Property .<br />owl:annotatedSource rdfs:domain rdfs:Resource .<br />owl:annotatedSource rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-annotatedtarget">owl:annotatedTarget rdf:type rdf:Property .<br />owl:annotatedTarget rdfs:domain rdfs:Resource .<br />owl:annotatedTarget rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-assertionproperty">owl:assertionProperty rdf:type rdf:Property .<br />owl:assertionProperty rdfs:domain owl:NegativePropertyAssertion .<br />owl:assertionProperty rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-backwardcompatiblewith">owl:backwardCompatibleWith rdf:type owl:AnnotationProperty .<br />owl:backwardCompatibleWith rdf:type owl:OntologyProperty .<br />owl:backwardCompatibleWith rdfs:domain owl:Ontology .<br />owl:backwardCompatibleWith rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-bottomdataproperty">owl:bottomDataProperty rdf:type owl:DatatypeProperty .<br />owl:bottomDataProperty rdfs:domain owl:Thing .<br />owl:bottomDataProperty rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-bottomobjectproperty">owl:bottomObjectProperty rdf:type owl:ObjectProperty .<br />owl:bottomObjectProperty rdfs:domain owl:Thing .<br />owl:bottomObjectProperty rdfs:range owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-cardinality">owl:cardinality rdf:type rdf:Property .<br />owl:cardinality rdfs:domain owl:Restriction .<br />owl:cardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-complementof">owl:complementOf rdf:type rdf:Property .<br />owl:complementOf rdfs:domain owl:Class .<br />owl:complementOf rdfs:range owl:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-datatypecomplementof">owl:datatypeComplementOf rdf:type rdf:Property .<br />owl:datatypeComplementOf rdfs:domain rdfs:Datatype .<br />owl:datatypeComplementOf rdfs:range rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-deprecated">owl:deprecated rdf:type owl:AnnotationProperty .<br />owl:deprecated rdfs:domain rdfs:Resource .<br />owl:deprecated rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-differentfrom">owl:differentFrom rdf:type rdf:Property .<br />owl:differentFrom rdfs:domain owl:Thing .<br />owl:differentFrom rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-disjointunionof">owl:disjointUnionOf rdf:type rdf:Property .<br />owl:disjointUnionOf rdfs:domain owl:Class .<br />owl:disjointUnionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-disjointwith">owl:disjointWith rdf:type rdf:Property .<br />owl:disjointWith rdfs:domain owl:Class .<br />owl:disjointWith rdfs:range owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-distinctmembers">owl:distinctMembers rdf:type rdf:Property .<br />owl:distinctMembers rdfs:domain owl:AllDifferent .<br />owl:distinctMembers rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-equivalentclass">owl:equivalentClass rdf:type rdf:Property .<br />owl:equivalentClass rdfs:domain rdfs:Class .<br />owl:equivalentClass rdfs:range rdfs:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-equivalentproperty">owl:equivalentProperty rdf:type rdf:Property .<br />owl:equivalentProperty rdfs:domain rdf:Property .<br />owl:equivalentProperty rdfs:range rdf:Property .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-haskey">owl:hasKey rdf:type rdf:Property .<br />owl:hasKey rdfs:domain owl:Class .<br />owl:hasKey rdfs:range rdf:List .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-hasself">owl:hasSelf rdf:type rdf:Property .<br />owl:hasSelf rdfs:domain owl:Restriction .<br />owl:hasSelf rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-hasvalue">owl:hasValue rdf:type rdf:Property .<br />owl:hasValue rdfs:domain owl:Restriction .<br />owl:hasValue rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-imports">owl:imports rdf:type owl:OntologyProperty .<br />owl:imports rdfs:domain owl:Ontology .<br />owl:imports rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-incompatiblewith">owl:incompatibleWith rdf:type owl:AnnotationProperty .<br />owl:incompatibleWith rdf:type owl:OntologyProperty .<br />owl:incompatibleWith rdfs:domain owl:Ontology .<br />owl:incompatibleWith rdfs:range owl:Ontology .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-intersectionof">owl:intersectionOf rdf:type rdf:Property .<br />owl:intersectionOf rdfs:domain rdfs:Class .<br />owl:intersectionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-inverseof">owl:inverseOf rdf:type rdf:Property .<br />owl:inverseOf rdfs:domain owl:ObjectProperty .<br />owl:inverseOf rdfs:range owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-maxcardinality">owl:maxCardinality rdf:type rdf:Property .<br />owl:maxCardinality rdfs:domain owl:Restriction .<br />owl:maxCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-maxqualifiedcardinality">owl:maxQualifiedCardinality rdf:type rdf:Property .<br />owl:maxQualifiedCardinality rdfs:domain owl:Restriction .<br />owl:maxQualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-members">owl:members rdf:type rdf:Property .<br />owl:members rdfs:domain rdfs:Resource .<br />owl:members rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-mincardinality">owl:minCardinality rdf:type rdf:Property .<br />owl:minCardinality rdfs:domain owl:Restriction .<br />owl:minCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-minqualifiedcardinality">owl:minQualifiedCardinality rdf:type rdf:Property .<br />owl:minQualifiedCardinality rdfs:domain owl:Restriction .<br />owl:minQualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-onclass">owl:onClass rdf:type rdf:Property .<br />owl:onClass rdfs:domain owl:Restriction .<br />owl:onClass rdfs:range owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-ondatarange">owl:onDataRange rdf:type rdf:Property .<br />owl:onDataRange rdfs:domain owl:Restriction .<br />owl:onDataRange rdfs:range rdfs:Datatype .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-ondatatype">owl:onDatatype rdf:type rdf:Property .<br />owl:onDatatype rdfs:domain rdfs:Datatype .<br />owl:onDatatype rdfs:range rdfs:Datatype .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-oneof">owl:oneOf rdf:type rdf:Property .<br />owl:oneOf rdfs:domain rdfs:Class .<br />owl:oneOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-onproperty">owl:onProperty rdf:type rdf:Property .<br />owl:onProperty rdfs:domain owl:Restriction .<br />owl:onProperty rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-onproperties">owl:onProperties rdf:type rdf:Property .<br />owl:onProperties rdfs:domain owl:Restriction .<br />owl:onProperties rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-priorversion">owl:priorVersion rdf:type owl:AnnotationProperty .<br />owl:priorVersion rdf:type owl:OntologyProperty .<br />owl:priorVersion rdfs:domain owl:Ontology .<br />owl:priorVersion rdfs:range owl:Ontology .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-propertychainaxiom">owl:propertyChainAxiom rdf:type rdf:Property .<br />owl:propertyChainAxiom rdfs:domain owl:ObjectProperty .<br />owl:propertyChainAxiom rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-propertydisjointwith">owl:propertyDisjointWith rdf:type rdf:Property .<br />owl:propertyDisjointWith rdfs:domain rdf:Property .<br />owl:propertyDisjointWith rdfs:range rdf:Property .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-qualifiedcardinality">owl:qualifiedCardinality rdf:type rdf:Property .<br />owl:qualifiedCardinality rdfs:domain owl:Restriction .<br />owl:qualifiedCardinality rdfs:range xsd:nonNegativeInteger .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-sameas">owl:sameAs rdf:type rdf:Property .<br />owl:sameAs rdfs:domain owl:Thing .<br />owl:sameAs rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-somevaluesfrom">owl:someValuesFrom rdf:type rdf:Property .<br />owl:someValuesFrom rdfs:domain owl:Restriction .<br />owl:someValuesFrom rdfs:range rdfs:Class .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-sourceindividual">owl:sourceIndividual rdf:type rdf:Property .<br />owl:sourceIndividual rdfs:domain owl:NegativePropertyAssertion .<br />owl:sourceIndividual rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-targetindividual">owl:targetIndividual rdf:type rdf:Property .<br />owl:targetIndividual rdfs:domain owl:NegativePropertyAssertion .<br />owl:targetIndividual rdfs:range owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-targetvalue">owl:targetValue rdf:type rdf:Property .<br />owl:targetValue rdfs:domain owl:NegativePropertyAssertion .<br />owl:targetValue rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-topdataproperty">owl:topDataProperty rdf:type owl:DatatypeProperty .<br />owl:topDataProperty rdfs:domain owl:Thing .<br />owl:topDataProperty rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-topobjectproperty">owl:topObjectProperty rdf:type rdf:ObjectProperty .<br />owl:topObjectProperty rdfs:domain owl:Thing .<br />owl:topObjectProperty rdfs:range owl:Thing .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-unionof">owl:unionOf rdf:type rdf:Property .<br />owl:unionOf rdfs:domain rdfs:Class .<br />owl:unionOf rdfs:range rdf:List .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-versioninfo">owl:versionInfo rdf:type owl:AnnotationProperty .<br />owl:versionInfo rdfs:domain rdfs:Resource .<br />owl:versionInfo rdfs:range rdfs:Resource .<br /></span>
</td><td> <span class="name" id="item-axiomatic-properties-versioniri">owl:versionIRI rdf:type owl:OntologyProperty .<br />owl:versionIRI rdfs:domain owl:Ontology .<br />owl:versionIRI rdfs:range owl:Ontology .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-properties-withrestrictions">owl:withRestrictions rdf:type rdf:Property .<br />owl:withRestrictions rdfs:domain rdfs:Datatype .<br />owl:withRestrictions rdfs:range rdf:List .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-datatypes">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.3: Axiomatic Triples for the Datatypes of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-anyuri">xsd:anyURI rdf:type rdfs:Datatype .<br />xsd:anyURI rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-base64binary">xsd:base64Binary rdf:type rdfs:Datatype .<br />xsd:base64Binary rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-boolean">xsd:boolean rdf:type rdfs:Datatype .<br />xsd:boolean rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-byte">xsd:byte rdf:type rdfs:Datatype .<br />xsd:byte rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-datetime">xsd:dateTime rdf:type rdfs:Datatype .<br />xsd:dateTime rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-datetimestamp">xsd:dateTimeStamp rdf:type rdfs:Datatype .<br />xsd:dateTimeStamp rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-decimal">xsd:decimal rdf:type rdfs:Datatype .<br />xsd:decimal rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-double">xsd:double rdf:type rdfs:Datatype .<br />xsd:double rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-float">xsd:float rdf:type rdfs:Datatype .<br />xsd:float rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-hexbinary">xsd:hexBinary rdf:type rdfs:Datatype .<br />xsd:hexBinary rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-int">xsd:int rdf:type rdfs:Datatype .<br />xsd:int rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-integer">xsd:integer rdf:type rdfs:Datatype .<br />xsd:integer rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-language">xsd:language rdf:type rdfs:Datatype .<br />xsd:language rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-long">xsd:long rdf:type rdfs:Datatype .<br />xsd:long rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-name">xsd:Name rdf:type rdfs:Datatype .<br />xsd:Name rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-ncname">xsd:NCName rdf:type rdfs:Datatype .<br />xsd:NCName rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-negativeinteger">xsd:negativeInteger rdf:type rdfs:Datatype .<br />xsd:negativeInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-nmtoken">xsd:NMTOKEN rdf:type rdfs:Datatype .<br />xsd:NMTOKEN rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-nonnegativeinteger">xsd:nonNegativeInteger rdf:type rdfs:Datatype .<br />xsd:nonNegativeInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-nonpositiveinteger">xsd:nonPositiveInteger rdf:type rdfs:Datatype .<br />xsd:nonPositiveInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-normalizedstring">xsd:normalizedString rdf:type rdfs:Datatype .<br />xsd:normalizedString rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-plainliteral">rdf:PlainLiteral rdf:type rdfs:Datatype .<br />rdf:PlainLiteral rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-positiveinteger">xsd:positiveInteger rdf:type rdfs:Datatype .<br />xsd:positiveInteger rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-rational">owl:rational rdf:type rdfs:Datatype .<br />owl:rational rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-real">owl:real rdf:type rdfs:Datatype .<br />owl:real rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-short">xsd:short rdf:type rdfs:Datatype .<br />xsd:short rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-string">xsd:string rdf:type rdfs:Datatype .<br />xsd:string rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-token">xsd:token rdf:type rdfs:Datatype .<br />xsd:token rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-unsignedbyte">xsd:unsignedByte rdf:type rdfs:Datatype .<br />xsd:unsignedByte rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-unsignedint">xsd:unsignedInt rdf:type rdfs:Datatype .<br />xsd:unsignedInt rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-unsignedlong">xsd:unsignedLong rdf:type rdfs:Datatype .<br />xsd:unsignedLong rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-datatypes-unsignedshort">xsd:unsignedShort rdf:type rdfs:Datatype .<br />xsd:unsignedShort rdfs:subClassOf rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-datatypes-xmlliteral">rdf:XMLLiteral rdf:type rdfs:Datatype .<br />rdf:XMLLiteral rdfs:subClassOf rdfs:Literal .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<div class="left" id="table-axiomatic-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.4: Axiomatic Triples for the Facets of the OWL 2 RDF-Based Semantics</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-facets-langrange">rdf:langRange rdf:type owl:DatatypeProperty .<br />rdf:langRange rdfs:domain rdfs:Resource .<br />rdf:langRange rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-length">xsd:length rdf:type owl:DatatypeProperty .<br />xsd:length rdfs:domain rdfs:Resource .<br />xsd:length rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-maxexclusive">xsd:maxExclusive rdf:type owl:DatatypeProperty .<br />xsd:maxExclusive rdfs:domain rdfs:Resource .<br />xsd:maxExclusive rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-maxinclusive">xsd:maxInclusive rdf:type owl:DatatypeProperty .<br />xsd:maxInclusive rdfs:domain rdfs:Resource .<br />xsd:maxInclusive rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-maxlength">xsd:maxLength rdf:type owl:DatatypeProperty .<br />xsd:maxLength rdfs:domain rdfs:Resource .<br />xsd:maxLength rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-minexclusive">xsd:minExclusive rdf:type owl:DatatypeProperty .<br />xsd:minExclusive rdfs:domain rdfs:Resource .<br />xsd:minExclusive rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-mininclusive">xsd:minInclusive rdf:type owl:DatatypeProperty .<br />xsd:minInclusive rdfs:domain rdfs:Resource .<br />xsd:minInclusive rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-facets-minlength">xsd:minLength rdf:type owl:DatatypeProperty .<br />xsd:minLength rdfs:domain rdfs:Resource .<br />xsd:minLength rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-facets-pattern">xsd:pattern rdf:type owl:DatatypeProperty .<br />xsd:pattern rdfs:domain rdfs:Resource .<br />xsd:pattern rdfs:range rdfs:Literal .<br /></span>
</td><td>
</td></tr></table>
</div>
<div class="left" id="table-axiomatic-rdfs">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 6.5: Additional Axiomatic Triples for Classes and Properties of the RDFS Vocabulary</span>
</caption>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-class">rdfs:Class rdfs:subClassOf owl:Class .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-comment">rdfs:comment rdf:type owl:AnnotationProperty .<br />rdfs:comment rdfs:domain rdfs:Resource .<br />rdfs:comment rdfs:range rdfs:Literal .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-datatype">rdfs:Datatype rdfs:subClassOf owl:DataRange .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-isdefinedby">rdfs:isDefinedBy rdf:type owl:AnnotationProperty .<br />rdfs:isDefinedBy rdfs:domain rdfs:Resource .<br />rdfs:isDefinedBy rdfs:range rdfs:Resource .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-label">rdfs:label rdf:type owl:AnnotationProperty .<br />rdfs:label rdfs:domain rdfs:Resource .<br />rdfs:label rdfs:range rdfs:Literal .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-literal">rdfs:Literal rdf:type rdfs:Datatype .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-property">rdf:Property rdfs:subClassOf owl:ObjectProperty .<br /></span>
</td><td> <span class="name" id="item-axiomatic-rdfs-resource">rdfs:Resource rdfs:subClassOf owl:Thing .<br /></span>
</td></tr>
<tr>
<td> <span class="name" id="item-axiomatic-rdfs-seealso">rdfs:seeAlso rdf:type owl:AnnotationProperty .<br />rdfs:seeAlso rdfs:domain rdfs:Resource .<br />rdfs:seeAlso rdfs:range rdfs:Resource .<br /></span>
</td><td>
</td></tr>
</table>
</div>
<a name="Appendix:_Relationship_to_the_Direct_Semantics_.28Informative.29"></a><h2> <span class="mw-headline">7  Appendix: Relationship to the Direct Semantics (Informative) </span></h2>
<p>This section compares
the OWL 2 RDF-Based Semantics
with the 
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/" title="Direct Semantics"><i>OWL 2 Direct Semantics</i></a>
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
While 
the OWL 2 RDF-Based Semantics is based on the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
the OWL 2 Direct Semantics
is a <i>description logic</i> style semantics.
Several fundamental differences
exist between the two semantics,
but 
there is also a strong relationship 
basically stating that the OWL 2 RDF-Based Semantics is able
to reflect all logical conclusions 
of the OWL 2 Direct Semantics.
This means that the OWL 2 Direct Semantics
can
in a sense
be regarded as a semantics subset of the OWL 2 RDF-Based Semantics.
</p><p>Technically,
the comparison will be performed
by comparing the sets of <i>entailments</i>
that hold for each of the two semantics, 
respectively.
The definition of an <i><b>OWL 2 RDF-Based entailment</b></i> 
was given in 
<a href="#Satisfaction.2C_Consistency_and_Entailment" title="">Section 4.3</a> 
of this document,
while the definition of an <i><b>OWL 2 Direct entailment</b></i>
is provided in
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#Inference_Problems" title="Direct Semantics">Section 2.5 of the OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
In both cases,
entailments are defined for pairs of ontologies,
and such an ordered pair of two ontologies will be called an 
<i><b>entailment query</b></i>
in this section.
</p><p>Comparing the two semantics by means of entailments
will only be meaningful
if the entailment queries
allow for applying
both 
the OWL 2 RDF-Based 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 left-hand side and right-hand side ontologies 
are both
<i><b>OWL 2 DL ontologies in RDF graph form</b></i>.
These are RDF graphs that,
by applying the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs"><i>reverse RDF mapping</i></a>
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>],
can be transformed
into corresponding 
<i><b>OWL 2 DL ontologies in Functional Syntax form</b></i>
according to the <i>functional style syntax</i> defined in the 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>],
and which must further meet
all the <i>restrictions on OWL 2 DL ontologies</i>
that are specified in
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
In fact,
these restrictions must be <i>mutually met</i> 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
<i><b>OWL 2 DL entailment query</b></i>.
</p><p>Ideally,
the relationship between 
the OWL 2 RDF-Based 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 RDF-Based entailment.
However,
this desirable relationship 
cannot hold in general
due to a variety of differences
that exist between 
the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
as demonstrated in
<a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>.
</p><p>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
<i>OWL 2 correspondence theorem</i>,
which will be presented in 
<a href="#Correspondence_Theorem" title="">Section 7.2</a>.
The 
<i>proof</i> of this theorem,
given in <a href="#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>, 
will further demonstrate
that such a substitute OWL 2 DL entailment query
can always be algorithmically constructed
by means of simple syntactic transformations.
</p>
<a name="Example_on_Semantic_Differences"></a><h3> <span class="mw-headline">7.1  Example on Semantic Differences </span></h3>
<p>This section will show
that differences exist
between
the OWL 2 RDF-Based 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 RDF-Based 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 RDF-Based entailment,
while being semantically unchanged 
compared to the original entailment query
under the OWL 2 Direct Semantics.
This example will motivate
the <i>OWL 2 correspondence theorem</i>
in <a href="#Correspondence_Theorem" title="">Section 7.2</a>
and its proof 
in <a href="#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>.
</p><p>The example entailment query consists of the following 
pair 
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
of RDF graphs:
</p>
<div id="topic-correspondence-diff-example-rdf"></div>
<div class="indent">
<p><i>G<sub>1</sub><sup>*</sup></i>&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">ex:o1 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c1 rdfs:subClassOf ex:c2 .</span>
</p>
</div>
</div>
<div class="indent">
<p><i>G<sub>2</sub><sup>*</sup></i>&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">ex:o2 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
(6) <span class="name">_:x rdf:type owl:Class .</span><br />
(7) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span><br />
(8) <span class="name">ex:c3 rdfs:label "c3" .</span>
</p>
</div>
</div>
<p>Both <i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i> 
are
OWL 2 DL ontologies in RDF graph form
and can therefore be mapped by the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
to the following two OWL 2 DL ontologies in Functional Syntax form
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>):
</p>
<div id="topic-correspondence-diff-example-fs"></div>
<div class="indent">
<p>F(<i>G<sub>1</sub><sup>*</sup></i>)&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o1</span><br />
(2) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) &nbsp;&nbsp;&nbsp; <span class="name">SubClassOf( ex:c1 ex:c2 )</span><br />
(5) <span class="name">)</span>
</p>
</div>
</div>
<div class="indent">
<p>F(<i>G<sub>2</sub><sup>*</sup></i>)&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o2</span><br />
(2) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) &nbsp;&nbsp;&nbsp; <span class="name">SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(6) &nbsp;&nbsp;&nbsp; <span class="name">AnnotationAssertion( rdfs:label ex:c3 "c3" )</span><br />
(7) <span class="name">)</span>
</p>
</div>
</div>
<p>Note that
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
mutually meet the restrictions on OWL 2 DL ontologies
as specified in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
For example,
none of the IRIs being declared as a class in F(<i>G<sub>1</sub><sup>*</sup></i>)
is declared as a datatype in F(<i>G<sub>2</sub><sup>*</sup></i>),
since this would not be allowed for an OWL 2 DL entailment query.
</p><p>It follows that
F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>).
To show this,
only the axioms 
(4) of F(<i>G<sub>1</sub><sup>*</sup></i>)
and
(5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
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 
<i>G<sub>1</sub><sup>*</sup></i> 
does <i>not</i> OWL 2 RDF-Based entail 
<i>G<sub>2</sub><sup>*</sup></i>,
for reasons discussed in detail now.
</p>
<div id="topic-correspondence-diff-reason-annotations"></div>
<p><b><i>Reason 1: An Annotation in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology F(<i>G<sub>2</sub><sup>*</sup></i>)
contains an annotation (6).
The OWL 2 Direct Semantics
does not give a formal meaning to
annotations.
In contrast, 
under the OWL 2 RDF-Based Semantics 
<i>every</i> RDF triple occurring in an RDF graph 
has a formal meaning,
including
the corresponding annotation triple (8) in <i>G<sub>2</sub><sup>*</sup></i>.
Since this annotation triple
only occurs in <i>G<sub>2</sub><sup>*</sup></i> 
but not in <i>G<sub>1</sub><sup>*</sup></i>,
there will exist OWL 2 RDF-Based interpretations
that satisfy <i>G<sub>1</sub><sup>*</sup></i>
without satisfying triple (8) of <i>G<sub>2</sub><sup>*</sup></i>.
Hence,
<i>G<sub>1</sub><sup>*</sup></i>
does <i>not</i> OWL 2 RDF-Based entail <i>G<sub>2</sub><sup>*</sup></i>.
</p><p><i><b>Resolution of Reason 1.</b></i> 
The annotation triple (8) in <i>G<sub>2</sub><sup>*</sup></i>
will be removed,
which will avoid requiring 
OWL 2 RDF-Based interpretations to interpret this triple.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since annotations are strictly optional in OWL 2 DL ontologies
and may therefore be omitted.
Also, this operation will not change the formal meaning of the ontologies 
under the OWL 2 Direct Semantics,
since annotations do not have a formal meaning under this semantics. 
</p>
<div id="topic-correspondence-diff-reason-declarations"></div>
<p><b><i>Reason 2: An Entity Declaration exclusively in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology F(<i>G<sub>2</sub><sup>*</sup></i>)
contains an entity declaration for the class IRI
<span class="name">ex:c3</span> (4),
for which there is no corresponding entity declaration 
in F(<i>G<sub>1</sub><sup>*</sup></i>).
The OWL 2 Direct Semantics does not give a formal meaning to
entity declarations,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding declaration statement (4) in <i>G<sub>2</sub><sup>*</sup></i>.
The consequences are analog to those described for reason 1.
</p><p><i><b>Resolution of Reason 2.</b></i> 
The declaration statement (4) in <i>G<sub>2</sub><sup>*</sup></i>
will be copied to <i>G<sub>1</sub><sup>*</sup></i>.
An OWL 2 RDF-Based interpretation 
that satisfies the modified graph <i>G<sub>1</sub><sup>*</sup></i>
will then also satisfy the declaration statement.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the copied declaration statement is not in conflict 
with any of the other entity declarations 
in <i>G<sub>1</sub><sup>*</sup></i>.
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.
</p>
<div id="topic-correspondence-diff-reason-headers"></div>
<p><b><i>Reason 3: Different Ontology IRIs in F(</i>G<sub>1</sub><sup>*</sup><i>) and F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
The ontology IRIs for the two ontologies,
given by (1) in F(<i>G<sub>1</sub><sup>*</sup></i>) 
and by (1) in F(<i>G<sub>2</sub><sup>*</sup></i>),
differ from each other.
The OWL 2 Direct Semantics does not give a formal meaning to 
ontology headers,
while the OWL 2 RDF-Based Semantics gives a formal meaning
to the corresponding header triples
(1) in <i>G<sub>1</sub><sup>*</sup></i>
and
(1) in <i>G<sub>2</sub><sup>*</sup></i>.
Since these header triples differ from each other,
the consequences are analog to those described for reason 1.
</p><p><i><b>Resolution of Reason 3.</b></i> 
The IRI
in the subject position of the header triple (1)
in <i>G<sub>2</sub><sup>*</sup></i> 
is changed into a blank node.
Due to the existential semantics of blank nodes under the OWL 2 RDF-Based Semantics
the resulting triple will then be entailed
by triple (1)
in <i>G<sub>1</sub><sup>*</sup></i>.
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 <i>G<sub>2</sub><sup>*</sup></i>,
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.
</p>
<div id="topic-correspondence-diff-reason-expressions"></div>
<p><b><i>Reason 4: A Class Expression in F(</i>G<sub>2</sub><sup>*</sup><i>).</i></b>
Axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
contains a class expression
that represents the union of the two classes
denoted by
<span class="name">ex:c2</span>
and
<span class="name">ex:c3</span>.
Within <i>G<sub>2</sub><sup>*</sup></i>,
this class expression is represented 
by the triples (6) and (7),
both having the blank node 
"<span class="name">_:x</span>" 
in their respective subject position.
The way the OWL 2 RDF-Based Semantics interprets these two triples
differs from the way
the OWL 2 Direct Semantics treats the class expression
in axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>).
</p><p>The OWL 2 Direct Semantics treats classes as <i>sets</i>, 
i.e. subsets of the universe.
Thus,
the IRIs
<span class="name">ex:c2</span>
and
<span class="name">ex:c3</span>
in F(<i>G<sub>2</sub><sup>*</sup></i>)
denote two sets,
and the class expression
in axiom (5) of F(<i>G<sub>2</sub><sup>*</sup></i>)
therefore represents the set
that consists of the union of these two sets.
</p><p>The OWL 2 RDF-Based Semantics,
on the other hand,
treats classes as <i>individuals</i>,
i.e. members of the universe.
While every class under the OWL 2 RDF-Based 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 RDF-Based Semantics,
just as for the OWL 2 Direct Semantics, 
that the union of their class extensions will always exist
as a subset of the universe.
However,
there is no guarantee
that there will also exist 
an individual in the universe
that has this set union as its class extension.
</p><p>Under the OWL 2 RDF-Based Semantics,
triple (7) of <i>G<sub>2</sub><sup>*</sup></i>
essentially claims that a class exists 
being the union of two other classes. 
But since 
the existence of such a union class
is not ensured by <i>G<sub>1</sub><sup>*</sup></i>,
there will be OWL 2 RDF-Based interpretations
that satisfy <i>G<sub>1</sub><sup>*</sup></i>
without satisfying
triple (7) of <i>G<sub>2</sub><sup>*</sup></i>.
Hence,
<i>G<sub>1</sub><sup>*</sup></i>
does <i>not</i>
OWL 2 RDF-Based entail
<i>G<sub>2</sub><sup>*</sup></i>.
</p><p><i><b>Resolution of Reason 4.</b></i>
The triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i>
are copied to <i>G<sub>1</sub><sup>*</sup></i>
together with the new triple
"<span class="name">_:x owl:equivalentClass _:x</span>".
In addition,
for the IRI
<span class="name">ex:c3</span>,
which only occurs in the union class expression
but not in <i>G<sub>1</sub><sup>*</sup></i>,
an entity declaration is added
to <i>G<sub>1</sub><sup>*</sup></i>
by the resolution of reason 2.
If an OWL 2 RDF-Based interpretation satisfies the modified graph <i>G<sub>1</sub><sup>*</sup></i>,
then the triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i>
will now be satisfied.
The changed RDF graphs will still be
OWL 2 DL ontologies in RDF graph form,
since the whole set of added triples 
validly encodes an OWL 2 axiom,
and since none of the restrictions on OWL 2 DL ontologies is hurt.
Also, this operation will not change 
the formal meaning of the ontologies 
under the OWL 2 Direct Semantics,
since the added equivalence axiom 
is a tautology under this semantics.
</p><p>Note that it would have been an error 
to simply copy the 
triples (6) and (7) of <i>G<sub>2</sub><sup>*</sup></i> 
to <i>G<sub>1</sub><sup>*</sup></i>,
without also adding the new triple
"<span class="name">_:x owl:equivalentClass _:x</span>".
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 <i>arbitrary</i> 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 <i>tautological</i> axiom
that contains the original class expression
would have been sufficient for this purpose as well.
</p>
<div id="topic-correspondence-diff-resolution-complete"></div>
<p><i><b>Complete Resolution: The Transformed Entailment Query.</b></i>
</p><p>Combining the resolutions of all the above reasons
leads to the following new pair of RDF graphs
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ):
</p>
<div id="topic-correspondence-diff-resolution-rdf"></div>
<div class="indent">
<p><i>G<sub>1</sub></i>&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">ex:o1 rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf ex:c2 .</span><br />
(6) <span class="name">_:x owl:equivalentClass _:x .</span><br />
(7) <span class="name">_:x rdf:type owl:Class .</span><br />
(8) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<div class="indent">
<p><i>G<sub>2</sub></i>&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">_:o rdf:type owl:Ontology .</span><br />
(2) <span class="name">ex:c1 rdf:type owl:Class .</span><br />
(3) <span class="name">ex:c2 rdf:type owl:Class .</span><br />
(4) <span class="name">ex:c3 rdf:type owl:Class .</span><br />
(5) <span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
(6) <span class="name">_:x rdf:type owl:Class .</span><br />
(7) <span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<div id="topic-correspondence-diff-resolution-reiteration"></div>
<p>The following list reiterates the changes compared to the original RDF graphs
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>:
</p>
<ul><li> <i><b>Resolution of Reason 1 (Annotation):</b></i> Triple (8) in <i>G<sub>2</sub><sup>*</sup></i> has been removed, i.e. there is no corresponding annotation triple in <i>G<sub>2</sub></i>.
</li><li> <i><b>Resolution of Reason 2 (Entity Declaration):</b></i> Triple (4) in <i>G<sub>2</sub><sup>*</sup></i> has been copied to <i>G<sub>1</sub><sup>*</sup></i>, becoming triple (4) in <i>G<sub>1</sub></i>.
</li><li> <i><b>Resolution of Reason 3 (Ontology IRIs):</b></i> The IRI in the subject position of triple (1) in <i>G<sub>2</sub><sup>*</sup></i> has been changed into a blank node, becoming triple (1) in <i>G<sub>2</sub></i>.
</li><li> <i><b>Resolution of Reason 4 (Class Expression):</b></i> Triples (6) and (7) in <i>G<sub>2</sub><sup>*</sup></i> have been copied to <i>G<sub>1</sub><sup>*</sup></i> together with the new triple "<span class="name">_:x owl:equivalentClass _:x</span>", becoming triples (6), (7) and (8) in <i>G<sub>1</sub></i>. 
</li></ul>
<p><i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are again
OWL 2 DL ontologies in RDF graph form
and can be mapped to the following
OWL 2 DL ontologies in Functional Syntax form
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>),
which again mutually meet the restrictions on OWL 2 DL ontologies:
</p>
<div id="topic-correspondence-diff-resolution-fs"></div>
<div class="indent">
<p>F(<i>G<sub>1</sub></i>)&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">Ontology( ex:o1</span><br />
(2) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) &nbsp;&nbsp;&nbsp; <span class="name">SubClassOf( ex:c1 ex:c2 )</span><br />
(6) &nbsp;&nbsp;&nbsp; <span class="name">EquivalentClasses( ObjectUnionOf( ex:c2 ex:c3 ) ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(7) <span class="name">)</span>
</p>
</div>
</div>
<div class="indent">
<p>F(<i>G<sub>2</sub></i>)&nbsp;:
</p>
<div class="indent">
<p>(1) <span class="name">Ontology(</span><br />
(2) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c1 ) )</span><br />
(3) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c2 ) )</span><br />
(4) &nbsp;&nbsp;&nbsp; <span class="name">Declaration( Class( ex:c3 ) )</span><br />
(5) &nbsp;&nbsp;&nbsp; <span class="name">SubClassOf( ex:c1 ObjectUnionOf( ex:c2 ex:c3 ) )</span><br />
(6) <span class="name">)</span>
</p>
</div>
</div>
<p>As said earlier,
all the applied changes 
preserve the formal meaning 
of the original OWL 2 DL ontologies
under the OWL 2 Direct Semantics.
Hence,
it is still the case
that 
F(<i>G<sub>1</sub></i>)
OWL 2 Direct entails
F(<i>G<sub>2</sub></i>).
However, 
due to the syntactic transformation
the situation has changed for the OWL 2 RDF-Based Semantics:
it is now possible to show,
by following the lines of argumentation 
for the resolutions of the different reasons given above,
that <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>
as well.
</p>
<a name="Correspondence_Theorem"></a><h3> <span class="mw-headline">7.2  Correspondence Theorem </span></h3>
<p>This section presents the <i>OWL 2 correspondence theorem</i>,
which compares the semantic expressivity of 
the OWL 2 RDF-Based Semantics 
with that of 
the OWL 2 Direct Semantics.
The theorem basically states that 
the OWL 2 RDF-Based 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 <i>entailment</i>.
</p><p>However,
as discussed in
<a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>,
there exist semantic differences 
between the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics,
which do not allow for stating
that <i>any</i> OWL 2 DL entailment query 
that is an OWL 2 Direct entailment
will always also be an 
OWL 2 RDF-Based entailment. 
Nevertheless,
it can still be ensured that
any given OWL 2 DL entailment query
can be <i>substituted</i> 
by another OWL 2 DL entailment query
in a way 
that for the substitute entailment query 
the desired relationship will really hold,
while preserving the formal meaning 
compared to the original entailment query
under the OWL 2 Direct Semantics.
</p><p>In fact, 
the theorem only makes the seemingly weak assertion 
that such a substitute entailment query 
will always <i>exist</i>.
But the actual 
<i>proof for the theorem</i> 
in <a href="#Proof_for_the_Correspondence_Theorem" title="">Section 7.3</a>
will be more concrete
in that it will substitute each given OWL 2 DL entailment query
with a variant
that can be algorithmically constructed
by applying a set of simple syntactic transformations
to the original entailment query.
One can get an idea of how this works
from <a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>. 
</p>
<div id="topic-correspondence-datatypemaps"></div>
<p><i><b>Technical Note on Corresponding Datatype Maps.</b></i>
A distinction exists 
between the format of
an <i>OWL 2 RDF-Based datatype map</i>,
as defined by <a href="#def-owldatatypemap" title="">Definition 4.1</a>,
and the format of an <i>OWL 2 Direct datatype map</i>,
as defined in
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
It is, however, possible to translate 
between
an OWL 2 RDF-Based datatype map <i>D</i>
and
the corresponding OWL 2 Direct datatype map F(<i>D</i>)
in the following way:
</p><p>For an <a href="#def-owldatatypemap" title="">OWL 2 RDF-Based datatype map</a> <i>D</i>,
the <i>corresponding OWL 2 Direct datatype map</i>
F(<i>D</i>)&nbsp;:= ( 
<i>N<sub>DT</sub></i> , 
<i>N<sub>LS</sub></i> , 
<i>N<sub>FS</sub></i> , 
<i>&sdot;&nbsp;<sup>DT</sup></i> , 
<i>&sdot;&nbsp;<sup>LS</sup></i> , 
<i>&sdot;&nbsp;<sup>FS</sup></i> 
)
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
is given by
</p>
<ul><li> <i>Datatype Names:</i> <i>N<sub>DT</sub></i> is defined as the set of all IRIs <i>u</i>, for which there is a datatype <i>d</i>, such that ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li><li> <i>Lexical Space:</i> For each datatype name <i>u</i> &isin; <i>N<sub>DT</sub></i>, set <i>N<sub>LS</sub>(u)</i>&nbsp;:= LS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li><li> <i>Facet Space:</i> For each datatype name <i>u</i> &isin; <i>N<sub>DT</sub></i>, set <i>N<sub>FS</sub>(u)</i>&nbsp;:= FS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li><li> <i>Value Space:</i> For each datatype name <i>u</i> &isin; <i>N<sub>DT</sub></i>, set <i>(u)&nbsp;<sup>DT</sup></i>&nbsp;:= VS(<i>d</i>), where ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li><li> <i>Lexical-to-Value Mapping:</i> For each datatype name <i>u</i> &isin; <i>N<sub>DT</sub></i> and each lexical form <i>a</i> &isin; <i>N<sub>LS</sub>(u)</i>, set ( <i>a</i> , <i>u</i> )&nbsp;<sup><i>LS</i></sup>&nbsp;:= L2V(<i>d</i>)(<i>a</i>), where ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li><li> <i>Facet-to-Value Mapping:</i> For each datatype name <i>u</i> &isin; <i>N<sub>DT</sub></i> and each facet-value pair ( <i>F</i> , <i>v</i> ) &isin; <i>N<sub>FS</sub>(u)</i>, set ( <i>F</i> , <i>v</i> )&nbsp;<sup><i>FS</i></sup>&nbsp;:= F2V(<i>d</i>)(( <i>F</i> , <i>v</i> )), where ( <i>u</i> , <i>d</i> ) &isin; <i>D</i>.
</li></ul>
<div id="thm-correspondence">
<p><b>Theorem 7.1 (OWL 2 Correspondence Theorem):</b>
</p><p>Let <i>D</i> be an OWL 2 RDF-Based datatype map
according to <a href="#def-owldatatypemap" title="">Definition 4.1</a>,
with F(<i>D</i>) 
being the
OWL 2 Direct datatype map 
according to
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
that corresponds to <i>D</i> according to the
<a href="#topic-correspondence-datatypemaps" title=""><i>technical note on corresponding datatype maps</i></a>.
Let 
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i> 
be RDF graphs
that are
OWL 2 DL ontologies in RDF graph form,
with
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
being the 
OWL 2 DL ontologies in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">Functional Syntax</a> 
form
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">the reverse RDF mapping</a>
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
to
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
Let
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
mutually meet 
the restrictions on OWL 2 DL ontologies
as specified in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p><p>Then, 
there exist RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
that are
OWL 2 DL ontologies in RDF graph form,
such that all the following relationships hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the
OWL 2 DL ontologies in Functional Syntax form
that result from applying the reverse RDF mapping
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively:
</p>
<ol><li> F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>) mutually meet the restrictions on OWL 2 DL ontologies.
</li><li> F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>1</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>1</sub></i>) with respect to <i>F(D)</i>.
</li><li> F(<i>G<sub>2</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>2</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>.
</li><li> If F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>, then <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i> with respect to <i>D</i>.
</li></ol>
</div>
<a name="Proof_for_the_Correspondence_Theorem"></a><h3> <span class="mw-headline">7.3  Proof for the Correspondence Theorem </span></h3>
<p>This is the sketch of a proof for 
<a href="#thm-correspondence" title=""><i>Theorem 7.1 (OWL 2 Correspondence Theorem)</i></a>
in 
<a href="#Correspondence_Theorem" title="">Section 7.2</a>.
The proof sketch provides
the basic line of argumentation for showing the theorem.
However, 
for complexity reasons,
some technical aspects of the theorem are only coarsely treated,
and the proof sketch also refrains 
from considering the full amount of OWL 2 language constructs.
For certain steps of the proof
there are example calculations
that focus only on a small fraction of language constructs,
but which can be taken as a hint 
on how a complete proof 
taking into account every feature of the OWL 2 RDF-Based Semantics 
could be constructed in principle. 
A complete proof could make use of the observation 
that the definitions of the OWL 2 Direct Semantics 
and the OWL 2 RDF-Based Semantics, 
despite their technical differences 
as outlined in <a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>, 
are closely aligned with respect to the different language constructs of OWL 2. 
</p><p>The proof sketch will make use of an approach 
that will be called <i>"balancing"</i> throughout this section,
and which will now be introduced.
The basic idea is to substitute 
the original pair of RDF graphs in an OWL 2 DL entailment query
by another entailment query
having the same semantic characteristics 
under the OWL 2 Direct Semantics,
but for which the technical differences 
between the two semantics specifications
have no relevant consequences
under the OWL 2 RDF-Based Semantics anymore.
A concrete example 
for the application of this approach
was given in <a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>.
</p>
<div id="def-balanced">
<p><b>Definition (Balanced):</b>
A pair of RDF graphs
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is called
<i>balanced</i>,
if and only if
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> 
are OWL 2 DL ontologies in RDF graph form,
such that all the following conditions hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the 
OWL 2 DL ontologies in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">Functional Syntax</a> 
form
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively:
</p>
<ol><li> F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>) mutually meet the restrictions on OWL 2 DL ontologies as specified in <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> [<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</li><li> Nodes in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>:
<ol><li> for every IRI <i>u</i> occurring in <i>G<sub>1</sub></i> or <i>G<sub>2</sub></i> that corresponds to a non-built-in entity in F(<i>G<sub>1</sub></i>) or F(<i>G<sub>2</sub></i>), respectively, the graph contains, for every entity type <i>T</i> of <i>u</i>, a declaration statement of the form "<i>u</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI corresponding to <i>T</i> (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 7 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] and <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Entity_Declarations_and_Typing" title="Syntax">Section 5.8 of the OWL 2 Structural Specification</a> [<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]);
</li><li> every plain or typed literal occurring in <i>G<sub>2</sub></i> also occurs in <i>G<sub>1</sub></i> (see <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Datatype_Maps" title="Syntax">Section 4 of the OWL 2 Structural Specification</a> [<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]). 
</li></ol>
</li><li> <i>G<sub>2</sub></i> contains exactly one ontology header consisting of a single RDF triple of the form "<i>x</i> <span class="name">rdf:type owl:Ontology</span>", where <i>x</i> is either a blank node or, if an ontology IRI is used in <i>G<sub>1</sub></i>, may alternatively equal that ontology IRI (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 4 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li><li> <i>G<sub>2</sub></i> does <i>not</i> contain:
<ol><li> the RDF encoding of an annotation (see Sections <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Annotations" title="Mapping to RDF Graphs">3.2.2</a> and <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Ontology_Annotations" title="Mapping to RDF Graphs">3.2.3</a>, and <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 17</a> in the OWL 2 RDF Mapping [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> a statement with an ontology property such as "<span class="name">owl:imports</span>";
</li><li> a deprecation statement based on "<span class="name">owl:DeprecatedClass</span>", "<span class="name">owl:DeprecatedProperty</span>" and "<span class="name">owl:deprecated</span>" (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 16 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> an annotation property axiom based on "<span class="name">rdfs:subClassOf</span>", "<span class="name">rdfs:domain</span>" and "<span class="name">rdfs:range</span>" (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Axioms" title="Mapping to RDF Graphs">Table 16 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li></ol>
</li><li> Any of the following sub graphs of <i>G<sub>2</sub></i> is also a sub graph of <i>G<sub>1</sub></i>:
<ol><li> the RDF encoding of an entity declaration (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_the_Ontology_Header_and_Declarations" title="Mapping to RDF Graphs">Table 7 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a property expression (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Table 11 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a class expression (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Tables 13 and 15 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> the RDF encoding of a data range expression (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Tables 12 and 14 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]);
</li><li> an RDF sequence (see <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">Table 3 in the OWL 2 RDF Mapping</a> [<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]).
</li></ol>
</li></ol>
</div>
<div id="thm-balancing">
<p><b>Balancing Lemma:</b>
An algorithm exists
that terminates on every valid input
and that has the following input/output behavior:
</p><p>The <i>valid input</i> of the algorithm 
is given by
all the pairs of RDF graphs
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> ),
where
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
are OWL 2 DL ontologies in RDF graph form,
with
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
being the 
OWL 2 DL ontologies in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">Functional Syntax</a> 
form
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]
that result from applying the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a>
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>] 
to
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
Further,
F(<i>G<sub>1</sub><sup>*</sup></i>) and F(<i>G<sub>2</sub><sup>*</sup></i>)
have to mutually meet 
the restrictions on OWL 2 DL ontologies
as specified in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</p><p>For a valid input,
the <i>output</i> of the algorithm
is a pair of RDF graphs 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ),
where
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form,
such that 
for any OWL 2 RDF-Based datatype map <i>D</i>
according to <a href="#def-owldatatypemap" title="">Definition 4.1</a>
all the following relationships hold,
with
F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
being the 
OWL 2 DL ontologies in Functional Syntax form
that result from applying the reverse RDF mapping
to
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>,
respectively,
and 
with F(<i>D</i>)
being the 
OWL 2 Direct datatype map 
according to
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_datatype_map" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>]
that corresponds to <i>D</i> according to the 
<a href="#topic-correspondence-datatypemaps" title=""><i>technical note on corresponding datatype maps</i></a>
in <a href="#Correspondence_Theorem" title="">Section 7.2</a>:
</p>
<ol><li> The pair ( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) is <a href="#def-balanced" title=""><i>balanced</i></a>.
</li><li> F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>1</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>1</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>1</sub></i>) with respect to <i>F(D)</i>.
</li><li> F(<i>G<sub>2</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub><sup>*</sup></i>) with respect to <i>F(D)</i>, and F(<i>G<sub>2</sub><sup>*</sup></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) with respect to <i>F(D)</i>.
</li></ol>
</div>
<div id="topic-correspondence-proof-balancing"></div>
<p><i><b>Proof for the Balancing Lemma:</b></i> 
</p><p>Let the graph pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
be a valid input.
The resulting RDF graphs 
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> 
are constructed as follows,
starting from copies of 
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>,
respectively.
</p><p>Since the initial versions of 
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> 
are OWL 2 DL ontologies in RDF graph form,
the <i>canonical parsing process</i> (CP)
for computing the reverse RDF mapping,
as described in 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Ontologies" title="Syntax">Section 3 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>], 
can be applied.
Based on CP, it is possible to identify within these graphs
</p>
<ul><li> all entity types for every non-built-in IRI,
</li><li> all blank nodes that correspond to anonymous individuals, and
</li><li> all sub graphs that correspond to OWL 2 language constructs (ontology headers, declarations, expressions, axioms and annotations) as described in the <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/" title="Syntax">OWL 2 Structural Specification</a> [<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>].
</li></ul>
<p>Based on this observation, the following steps are performed:
</p>
<ol><li> Consistently substitute all blank nodes in <i>G<sub>2</sub></i> such that <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> have no common blank nodes.
</li><li> Apply CP to <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> (without changing these graphs) to identify the entity types of the IRIs, the anonymous individuals, and the sub graphs encoding OWL 2 language constructs.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i>: remove <i>g</i> from <i>G<sub>2</sub></i>, if <i>g</i> is the RDF encoding of 
<ul><li> an <i>annotation</i>, or
</li><li> a <i>deprecation statement</i>, or
</li><li> an <i>annotation property axiom</i>.
</li></ul>
</li><li> For the sub graph <i>g</i> of <i>G<sub>2</sub></i> corresponding to the <i>ontology header</i> in F(<i>G<sub>2</sub></i>): substitute <i>g</i> in <i>G<sub>2</sub></i> by a triple of the form "<i>x</i> <span class="name">rdf:type owl:Ontology</span>", where <i>x</i> is a new blank node not yet used in <i>G<sub>2</sub></i>.
</li><li> For each non-built-in IRI <i>u</i> in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> and for each entity type <i>T</i> of <i>u</i> identified by CP: add to <i>G<sub>1</sub></i> or <i>G<sub>2</sub></i>, respectively, the RDF triple "<i>u</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI corresponding to <i>T</i>.
</li><li> For each plain or typed literal <i>L</i> in <i>G<sub>2</sub></i>: add to <i>G<sub>1</sub></i> the RDF triple "<i>o</i> <span class="name">rdfs:comment</span> <i>L</i>", where <i>o</i> is the IRI or blank node of the ontology header triple "<i>o</i> <span class="name">rdf:type</span> <span class="name">owl:Ontology</span>" in <i>G<sub>1</sub></i>.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of an <i>entity declaration</i>: add <i>g</i> to <i>G<sub>1</sub></i>.
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>property expression</i> with root blank node <i>x</i>: add <i>g</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>x</i> <span class="name">owl:equivalentProperty</span> <i>x</i>".
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>class expression</i> with root blank node <i>x</i>: add <i>g</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>x</i> <span class="name">owl:equivalentClass</span> <i>x</i>".
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is the RDF encoding of a <i>data range expression</i> with root blank node <i>x</i>:
<ul><li> If <i>g</i> is part of a <i>data property restriction expression</i>, then nothing needs to be done, since the comprising restriction expression is covered by the treatment of class expressions, and therefore <i>g</i> occurs in <i>G<sub>1</sub></i> as well.
</li><li> Otherwise, add a declaration triple to <i>G<sub>1</sub></i> for a new data property <i>p</i> that does not yet occur in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>. Then, the RDF encoding <i>r</i> of a <i>universal data property restriction expression</i> on property <i>p</i> is created for <i>g</i>. Let <i>r</i> have the new root blank node <i>y</i>. Add <i>r</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>y</i> <span class="name">owl:equivalentClass</span> <i>y</i>".
</li></ul>
</li><li> For each sub graph <i>g</i> of <i>G<sub>2</sub></i> that is an RDF sequence with root blank node <i>x</i>, which does not occur in the RDF encoding of language constructs already treated by one of the earlier steps, i.e. <i>g</i> is part of the encoding of an axiom: create the RDF encoding <i>r</i> of an enumeration class expression with a new root blank node <i>y</i> having the main RDF triple "<i>y</i> <span class="name">owl:oneOf</span> <i>x</i>". Then, add <i>r</i> to <i>G<sub>1</sub></i> together with the RDF triple "<i>y</i> <span class="name">owl:equivalentClass</span> <i>y</i>". Additionally, for every IRI <i>u</i> being a member of the RDF sequence, add to <i>G<sub>1</sub></i> a typing triple "<i>u</i> <span class="name">rdf:type owl:NamedIndividual</span>". If one of the sequence members is a blank node <i>z</i> that is the root node of some property expression or class expression <i>e</i>, then select a new IRI <i>w</i> not yet occurring in <i>G<sub>1</sub></i>, consistently replace <i>z</i> by <i>w</i> everywhere in <i>r</i>, add to <i>G<sub>1</sub></i> the triple "<i>w</i> <span class="name">owl:equivalentProperty</span> <i>z</i>" or "<i>w</i> <span class="name">owl:equivalentClass</span> <i>z</i>", respectively, and add to <i>G<sub>1</sub></i> the two triples "<i>w</i> <span class="name">rdf:type owl:NamedIndividual</span>" and "<i>w</i> <span class="name">rdf:type</span> <i>t</i>", where <i>t</i> is the vocabulary class IRI that represents the appropriate entity type of the expression <i>e</i>. No further treatment of <i>e</i> is needed, since <i>e</i> is treated by the earlier steps covering expressions.
</li></ol>
<p>In the following it is shown that all the claims of the balancing lemma hold.
</p><p><i><b>A: Existence of a Terminating Algorithm.</b></i>
An algorithm <i>exists</i>
for mapping 
the input graph pair
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
to the output graph pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ),
since CP (applied in step 2)
is described in the form of an algorithm
in the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>],
and since all other steps
can obviously be performed algorithmically.
The algorithm <i>terminates</i>,
since CP terminates 
on arbitrary input graphs,
and since all other steps
can obviously be executed in finite time.
</p><p><i><b>B: The Resulting RDF Graphs are OWL 2 DL Ontologies.</b></i>
The RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
are OWL 2 DL ontologies in RDF graph form
that mutually meet 
the restrictions on OWL 2 DL ontologies,
since the original RDF graphs 
<i>G<sub>1</sub><sup>*</sup></i> and <i>G<sub>2</sub><sup>*</sup></i>
have this feature,
and since each of the steps described above
transforms a pair of RDF graphs with this feature
again into a pair of RDF graphs with this feature,
for the following reasons:
</p>
<ul><li> The consistent substitution of blank nodes in step 1 does not change the structure of an OWL 2 DL ontology.
</li><li> The application of CP in step 2 does not change the graphs.
</li><li> Annotations, deprecation statements and annotation property axioms are optional information in an OWL 2 DL ontology and can therefore be omitted in step 3.
</li><li> The ontology header of an OWL 2 DL ontology does neither require the existence of an ontology IRI nor of any ontology properties, and so the substitution of the ontology header in step 4 is a valid operation. 
</li><li> If an entity has some particular entity type for which there is no explicitly given entity declaration, then the entity declaration may be added, as done in step 5.
</li><li> It is allowed to add arbitrary annotations to the ontology header of an OWL 2 DL ontology, as done in step 6.
</li><li> Entity declarations may be copied from <i>G<sub>2</sub></i> to <i>G<sub>1</sub></i> in step 7 without conflict, since the original ontologies have been assumed to mutually meet the restrictions on OWL 2 DL ontologies regarding different entity declarations for the same IRI (e.g. that one IRI must not be the name of both an object property and a data property).
</li><li> Adding to <i>G<sub>1</sub></i> an axiom that claims equivalence of some property expression (step 8) or class expression (step 9) with itself, where the expression already occurs in <i>G<sub>2</sub></i>, is an allowed operation, since the original ontologies are assumed to mutually meet the restrictions on OWL 2 DL ontologies concerning property and class expressions, and since no syntactic restrictions exist on this specific use of equivalence axioms.
</li><li> For the case of data ranges (step 10) it is sufficient to note that placing universal property restrictions on arbitrary (simple or complex) property expressions is allowed in OWL 2 DL. The rest of the argumentation follows the lines of the treatment of class expressions in step 9. 
</li><li> For the treatment of RDF sequences in step 11: First, the enumeration class expressions being constructed from the RDF sequences are syntactically valid in OWL 2 DL, since all enumerated entries are IRIs by construction. Second, there is no restriction in OWL 2 DL disallowing axioms that claim equivalence of enumeration class expressions with themselves. Third, punning in OWL 2 DL allows a given non-built-in IRI of any entity type to be additionally declared as a named individual. Forth, there is no OWL 2 DL restriction forbidding to add an entity declaration for a new (i.e. not elsewhere used) IRI and to assert the denotation of this new IRI to be equivalent to some existing property or class expression. Hence, the resulting ontologies still mutually meet all syntactic restrictions on OWL 2 DL ontologies.
</li></ul>
<p><i><b>C: The Resulting Pair of RDF Graphs is Balanced.</b></i>
All the conditions of <a href="#def-balanced" title=""><i>balanced</i></a> pairs of RDF graphs
are met by the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
for the following reasons: 
</p>
<ul><li> Condition 1: It has already been shown in paragraph <i>B</i> that <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> mutually meet the restrictions on OWL 2 DL ontologies.
</li><li> Conditions 2.1 and 2.2 on nodes in <i>G<sub>1</sub></i> and <i>G<sub>2</sub></i> are met by steps 5 and 6, respectively. 
</li><li> Condition 3 on ontology headers in <i>G<sub>2</sub></i> is satisfied by step 4, always applying an anonymous ontology header.
</li><li> Conditions 4.1, 4.3 and 4.4 on annotations, deprecation statements and annotation property axioms in <i>G<sub>2</sub></i>, respectively, are all satisfied by step 3. 
</li><li> Condition 4.2 on statements with ontology properties is implicitly satisfied by step 4, since the substitution of the ontology header in <i>G<sub>2</sub></i> removes all existing statements with ontology properties.
</li><li> Condition 5.1 on entity declarations in <i>G<sub>2</sub></i> being reflected in <i>G<sub>1</sub></i> is satisfied by step 7. 
</li><li> Conditions 5.2, 5.3 and 5.4 on property, class and data range expressions in <i>G<sub>2</sub></i>, respectively, being reflected in <i>G<sub>1</sub></i> are met by steps 8, 9 and 10, respectively. 
</li><li> Condition 5.5 on RDF sequences in <i>G<sub>2</sub></i> being reflected in <i>G<sub>1</sub></i> is satisfied by step 11.
</li></ul>
<p><i><b>D: The Resulting Ontologies are semantically equivalent with the Original Ontologies under the OWL 2 Direct Semantics.</b></i> 
F(<i>G<sub>1</sub></i>) is semantically equivalent with F(<i>G<sub>1</sub><sup>*</sup></i>),
since F(<i>G<sub>1</sub></i>) differs from F(<i>G<sub>1</sub><sup>*</sup></i>) only by (potentially):
</p>
<ul><li> additional entity declarations (steps 5, 7 and 11), which have no formal meaning under the OWL 2 Direct Semantics;
</li><li> additional annotations (step 6), which have no formal meaning;
</li><li> additional tautological axioms (steps 8, 9, 10 and 11), which do not change the formal meaning; 
</li></ul>
<p>F(<i>G<sub>2</sub></i>) is semantically equivalent with F(<i>G<sub>2</sub><sup>*</sup></i>),
since F(<i>G<sub>2</sub></i>) differs from F(<i>G<sub>2</sub><sup>*</sup></i>) only by (potentially):
</p>
<ul><li> differently labeled anonymous individuals (step 1), by which the formal meaning under the OWL 2 Direct Semantics keeps unchanged, since anonymous individuals are existentially interpreted;
</li><li> missing annotations, deprecation statements and annotation property axioms (step 3), which have no formal meaning; 
</li><li> a modified ontology header (step 4), which has no formal meaning; 
</li><li> additional entity declarations (step 5), which have no formal meaning.
</li></ul>
<p><i><b>End of Proof for the Balancing Lemma.</b></i>
</p><p>In the following,
the correspondence theorem will be proven. 
</p>
<div id="topic-correspondence-proof-main"></div>
<p>Assume that the premises of the correspondence theorem are true
for a given pair 
( <i>G<sub>1</sub><sup>*</sup></i> , <i>G<sub>2</sub><sup>*</sup></i> )
of RDF graphs. 
This allows for applying the 
<a href="#thm-balancing" title=""><i>balancing lemma</i></a>,
which provides the existence of corresponding RDF graphs
<i>G<sub>1</sub></i> and <i>G<sub>2</sub></i>
that are OWL 2 DL ontologies in RDF graph form,
and which meet the 
definition of <a href="#def-balanced" title=""><i>balanced</i></a> graph pairs.
Let F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>)
be the corresponding OWL 2 DL ontologies in Functional Syntax form.
Then,
the claimed relationship 1 of the correspondence theorem
follows directly from relationship 1 of the <a href="#thm-balancing" title=""><i>balancing lemma</i></a> 
and from condition 1 of the definition of <a href="#def-balanced" title="">balanced</a> graph pairs.
Further, 
the claimed relationships 2 and 3 of the correspondence theorem
follow directly from the relationships 2 and 3 of the <a href="#thm-balancing" title=""><i>balancing lemma</i></a>, 
respectively.
</p><p>The rest of this proof will treat 
the claimed relationship 4 of the correspondence theorem,
which states that 
if F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>) 
with respect to <i>F(D)</i>, 
then <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i> 
with respect to <i>D</i>.
For this to see,
an arbitrary OWL 2 RDF-Based interpretation <i>I</i> will be selected 
that OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>. 
For <i>I</i>,
a closely corresponding OWL 2 Direct interpretation <i>J</i> 
will be constructed,
and it will then be shown 
that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>).
Since it was assumed that
F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>),
it will follow that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>2</sub></i>).
Based on this result, it will then be possible to show 
that <i>I</i> also OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
Since <i>I</i> was arbitrarily selected,
this will mean 
that <i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>.
</p>
<div id="topic-correspondence-proof-step1"></div>
<p><i><b>Step 1: Selection of a Pair of Corresponding Interpretations.</b></i> 
</p><p>Let
F(<i>G<sub>1</sub></i>) OWL 2 Direct entail F(<i>G<sub>2</sub></i>) w.r.t. F(<i>D</i>),
and let <i>I</i> be an OWL 2 RDF-Based interpretation
of a vocabulary <i>V<sup>I</sup></i> w.r.t. <i>D</i>,
such that
<i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
</p><p>Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) 
is <a href="#def-balanced" title=""><i>balanced</i></a>,
there exist <i>entity declarations</i>
in F(<i>G<sub>1</sub></i>)
for each entity type
of every non-built-in IRI
occurring in <i>G<sub>1</sub></i>:
For each entity declaration
of the form
"<span class="name">Declaration</span>(<i>T</i>(<i>u</i>))"
in F(<i>G<sub>1</sub></i>),
such that <i>T</i> is the entity type for some IRI <i>u</i>,
a typing triple 
of the form
"<i>u</i> <span class="name">rdf:type</span> <i>t</i>"
exists in <i>G<sub>1</sub></i>,
where <i>t</i> is the vocabulary class IRI 
representing the part of the universe of <i>I</i>
that corresponds to <i>T</i>.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
all these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
and thus all non-built-in IRIs in <i>G<sub>1</sub></i>
are instances of all their declared parts of the universe of <i>I</i>.
</p><p>The vocabulary
<i>V<sup>J</sup></i>&nbsp;:= (
<i>V<sup>J</sup><sub>C</sub></i> , 
<i>V<sup>J</sup><sub>OP</sub></i> , 
<i>V<sup>J</sup><sub>DP</sub></i> , 
<i>V<sup>J</sup><sub>I</sub></i> , 
<i>V<sup>J</sup><sub>DT</sub></i> , 
<i>V<sup>J</sup><sub>LT</sub></i> , 
<i>V<sup>J</sup><sub>FA</sub></i>
)
of the OWL 2 Direct interpretation <i>J</i> w.r.t. the datatype map <i>F(D)</i> is now constructed as follows.
</p>
<ul><li> The set <i>V<sup>J</sup><sub>C</sub></i> of classes contains all IRIs in <i>V<sup>I</sup></i> that are declared as classes in F(<i>G<sub>1</sub></i>), together with all the required class names listed in <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>OP</sub></i> of object properties contains all IRIs in <i>V<sup>I</sup></i> that are declared as object properties in F(<i>G<sub>1</sub></i>), together with all the required object property names listed in <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>DP</sub></i> of data properties contains all IRIs in <i>V<sup>I</sup></i> that are declared as data properties in F(<i>G<sub>1</sub></i>), together with all the required data property names listed in <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
</li><li> The set <i>V<sup>J</sup><sub>I</sub></i> of individuals contains all IRIs in <i>V<sup>I</sup></i> that are declared as named individuals in F(<i>G<sub>1</sub></i>), and additionally all anonymous individuals occurring in F(<i>G<sub>1</sub></i>) and F(<i>G<sub>2</sub></i>).
</li><li> The set <i>V<sup>J</sup><sub>DT</sub></i> of datatypes is defined according to <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>), together with all other IRIs in <i>V<sup>I</sup></i> that are declared as datatypes in F(<i>G<sub>1</sub></i>).
</li><li> The set <i>V<sup>J</sup><sub>LT</sub></i> of literals is defined according to <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>).
</li><li> The set <i>V<sup>J</sup><sub>FA</sub></i> of facet-literal pairs is defined according to <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#def_vocabulary" title="Direct Semantics">Section 2.1 of the OWL 2 Direct Semantics</a> [<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>] w.r.t. the datatype map F(<i>D</i>).
</li></ul>
<p>The OWL 2 Direct interpretation
<i>J</i>&nbsp;:= ( 
<i>&Delta;<sub>I</sub></i> , 
<i>&Delta;<sub>D</sub></i> , 
<i>&sdot;&nbsp;<sup>C</sup></i> , 
<i>&sdot;&nbsp;<sup>OP</sup></i> , 
<i>&sdot;&nbsp;<sup>DP</sup></i> , 
<i>&sdot;&nbsp;<sup>I</sup></i> , 
<i>&sdot;&nbsp;<sup>DT</sup></i> , 
<i>&sdot;&nbsp;<sup>LT</sup></i> , 
<i>&sdot;&nbsp;<sup>FA</sup></i> 
) 
is now defined as follows.
The object and data domains of <i>J</i> are identified 
with the universe IR and the set of data values LV of <i>I</i>, 
respectively,
i.e.,
<i>&Delta;<sub>I</sub></i>&nbsp;:= IR and
<i>&Delta;<sub>D</sub></i>&nbsp;:= LV.
The class interpretation function <i>&sdot;&nbsp;<sup>C</sup></i>,
the object property interpretation function <i>&sdot;&nbsp;<sup>OP</sup></i>,
the data property interpretation function <i>&sdot;&nbsp;<sup>DP</sup></i>, 
the datatype interpretation function <i>&sdot;&nbsp;<sup>DT</sup></i>,
the literal interpretation function <i>&sdot;&nbsp;<sup>LT</sup></i>, and 
the facet interpretation function <i>&sdot;&nbsp;<sup>FA</sup></i>
are defined according to
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/#Interpretations" title="Direct Semantics">Section 2.2 of the OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>].
Specifically,
for every non-built-in IRI <i>u</i> 
occurring in F(<i>G<sub>1</sub></i>):
</p>
<ul><li> If <i>u</i> is declared as a class, then set <i>u<sup>C</sup></i>&nbsp;:= ICEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:Class</span>", i.e., <i>I</i>(<i>u</i>) &isin; IC.
</li><li> If <i>u</i> is declared as an object property, then set <i>u<sup>OP</sup></i>&nbsp;:= IEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:ObjectProperty</span>", i.e., <i>I</i>(<i>u</i>) &isin; IP.
</li><li> If <i>u</i> is declared as a data property, then set <i>u<sup>DP</sup></i>&nbsp;:= IEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:DatatypeProperty</span>", i.e., <i>I</i>(<i>u</i>) &isin; IODP.
</li><li> If <i>u</i> is declared as a named individual, then set <i>u<sup>I</sup></i>&nbsp;:= <i>I</i>(<i>u</i>), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type owl:NamedIndividual</span>", i.e., <i>I</i>(<i>u</i>) &isin; IR.
</li><li> If <i>u</i> is declared as a datatype, then set <i>u<sup>DT</sup></i>&nbsp;:= ICEXT(<i>I</i>(<i>u</i>)), since <i>G<sub>1</sub></i> contains the triple "<i>u</i> <span class="name">rdf:type rdfs:Datatype</span>", i.e., <i>I</i>(<i>u</i>) &isin; IDC.
</li></ul>
<p><i>Notes:</i>
</p>
<ul><li> A <i>literal</i> occurring in <i>G<sub>1</sub></i> is mapped by the reverse RDF mapping to the same literal in F(<i>G<sub>1</sub></i>), and the formal meaning of a well-formed literal is analog for both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics.
</li><li> A <i>blank node</i> <i>b</i> occurring in <i>G<sub>1</sub></i> that represents an <i>anonymous individual</i> is written as the same blank node <i>b</i> in F(<i>G<sub>1</sub></i>). Both the OWL 2 RDF-Based Semantics and the OWL 2 Direct Semantics treat anonymous individuals in an analog way as <i>existential variables</i> defined locally to a given ontology, i.e. some individual <i>x</i> exists in the universe to which all occurrences of <i>b</i> in the ontology can be mapped (see <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 in the RDF Semantics</a> [<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>] for the precise definition on how blank nodes are treated under the OWL 2 RDF-Based Semantics). Hence, the same mapping from <i>b</i> to <i>x</i> can be used with both <i>I</i> and <i>J</i>.
</li><li> <i>G<sub>1</sub></i> may also contain declarations for <i>annotation properties</i>. Since annotation properties have no formal meaning under the OWL 2 Direct Semantics, the OWL 2 Direct interpretation <i>J</i> does not treat them.
</li><li> With the above definition it is possible for <i>J</i> to have a <i>nonseparated vocabulary</i> according to <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Metamodeling" title="Syntax">Section 5.9 of the OWL 2 Structural Specification</a> [<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]. Since <i>G<sub>1</sub></i> is an OWL 2 DL ontology in RDF graph form, it is allowed that the same IRI <i>u</i> may be declared as one or more of an individual name, either a class name or a datatype name, and either an object property name or a data property name. For the OWL 2 RDF-Based interpretation <i>I</i>, the IRI <i>u</i> will always denote the same individual in the universe IR, where <i>I</i>(<i>u</i>) may additionally have a class extension or a property extension, or both. For the OWL 2 Direct interpretation <i>J</i>, however, <i>u</i> will denote as an individual name an element of <i>&Delta;<sub>I</sub></i>, as a class name a subset of <i>&Delta;<sub>I</sub></i>, as a datatype name a subset of <i>&Delta;<sub>D</sub></i>, as an object property name a subset of <i>&Delta;<sub>I</sub></i> &times; <i>&Delta;<sub>I</sub></i>, and as a data property name a subset of <i>&Delta;<sub>I</sub></i> &times; <i>&Delta;<sub>D</sub></i>.
</li></ul>
<div id="topic-correspondence-proof-step2"></div>
<p><b><i>Step 2: Satisfaction of F(</i>G<sub>1</sub><i>) by the OWL 2 Direct Interpretation.</i></b>
</p><p>Based on the premise that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
it has to be shown that <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>).
For this to hold,
it will be sufficient that
<i>J</i> OWL 2 Direct satisfies every axiom <i>A</i> occurring in F(<i>G<sub>1</sub></i>).
Let <i>g<sub>A</sub></i> be the sub graph of <i>G<sub>1</sub></i>
that is mapped to <i>A</i> by the reverse RDF mapping.
The basic idea can roughly be described as follows:
</p><p>Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions 
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the axiom <i>A</i>.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of <i>A</i>, 
i.e. the graph <i>g<sub>A</sub></i>, 
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition). 
Now,
all the RDF triples in <i>g<sub>A</sub></i> 
are OWL 2 RDF-Based satisfied by <i>I</i>, 
since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
Hence,
the antecedent of the semantic condition becomes true, 
and therefore its consequent becomes true as well.
This will reveal a certain semantic relationship
that 
according to <i>I</i> 
holds between the denotations of the 
IRIs, literals and anonymous individuals 
occurring in <i>g<sub>A</sub></i>,
which, 
roughly speaking,
expresses the meaning of the OWL 2 axiom <i>A</i>.
Because of the close semantic correspondence 
of the OWL 2 Direct interpretation <i>J</i> to <i>I</i>,
the analog semantic relationship holds 
according to <i>J</i>
between the denotations of the
IRIs, literals and anonymous individuals 
occurring in <i>A</i>.
This semantic relationship
turns out to be compatible
with the formal meaning of the axiom <i>A</i>
as specified by the OWL 2 Direct Semantics,
i.e. <i>J</i> satisfies <i>A</i>.
</p><p>This basic idea is now demonstrated in more detail
for a single example axiom <i>A</i> in F(<i>G<sub>1</sub></i>),
which can be taken as a hint on 
how a complete proof 
taking into account every feature of the OWL 2 RDF-Based Semantics
could be constructed in principle.
</p>
<div class="anexample" id="topic-correspondence-proof-example1"> 
<p>Let <i>A</i> be the following OWL 2 axiom in F(<i>G<sub>1</sub></i>):
</p>
<div class="indent">
<p><i>A</i>&nbsp;: <span class="name">SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))</span>
</p>
</div>
<p>and let <i>g<sub>A</sub></i> be the corresponding sub graph in <i>G<sub>1</sub></i>
that is being mapped to <i>A</i> via the reverse RDF mapping, 
namely
</p>
<div class="indent">
<p><i>g<sub>A</sub></i>&nbsp;:
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
<span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<p>Since the pair ( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>1</sub></i> contains the typing triples
</p>
<div class="indent">
<p><span class="name">ex:c1 rdf:type owl:Class .</span><br />
<span class="name">ex:c2 rdf:type owl:Class .</span><br />
<span class="name">ex:c3 rdf:type owl:Class .</span>
</p>
</div>
<p>that correspond to class entity declarations in F(<i>G<sub>1</sub></i>) for the IRIs
"<span class="name">ex:c1</span>", 
"<span class="name">ex:c2</span>", and 
"<span class="name">ex:c3</span>",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
since it has been postulated 
that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>.
Hence,
by applying the semantics of <span class="name">rdf:type</span>
(see
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]), 
all the IRIs denote classes, precisely:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) &isin; IC ,<br />
<i>I</i>(<span class="name">ex:c2</span>) &isin; IC , and<br />
<i>I</i>(<span class="name">ex:c3</span>) &isin; IC .
</p>
</div>
<p>Since <i>I</i> is an OWL 2 RDF-Based interpretation,
it meets all the OWL 2 RDF-Based semantic conditions,
and since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
all the triples in <i>g<sub>A</sub></i> are OWL 2 RDF-Based satisfied.
This meets the left-to-right directions of the semantic conditions
for subclass axioms 
("<span class="name">rdfs:subClassOf</span>",
see <a href="#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>)
and union class expressions
("<span class="name">owl:unionOf</span>",
see <a href="#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>),
which results in the following semantic relationship 
that holds between the extensions of the classes above
according to <i>I</i>: 
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">ex:c1</span>)) 
&sube; 
ICEXT(<i>I</i>(<span class="name">ex:c2</span>)) 
&cup; 
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) . 
</p>
</div>
<p>By applying the definition of <i>J</i>,
one can conclude 
that the following semantic relationship
holds between the denotations of the class names occurring in <i>A</i>
according to <i>J</i>:
</p>
<div class="indent">
<p>(<span class="name">ex:c1</span>)&nbsp;<sup><i>C</i></sup> 
&sube; 
(<span class="name">ex:c2</span>)&nbsp;<sup><i>C</i></sup> 
&cup; 
(<span class="name">ex:c3</span>)&nbsp;<sup><i>C</i></sup> . 
</p>
</div>
<p>This semantic relationship is compatible 
with the formal meaning of the axiom <i>A</i> 
under the OWL 2 Direct Semantics.
Hence, <i>J</i> OWL 2 Direct satisfies <i>A</i>.
</p>
</div>
<p>Since <i>J</i> OWL 2 Direct satisfies F(<i>G<sub>1</sub></i>),
and since it has been postulated that
F(<i>G<sub>1</sub></i>) OWL 2 Direct entails F(<i>G<sub>2</sub></i>), 
it follows that
<i>J</i> OWL 2 Direct satisfies F(<i>G<sub>2</sub></i>).
</p>
<div id="topic-correspondence-proof-step3"></div>
<p><b><i>Step 3: Satisfaction of </i>G<sub>2</sub><i> by the OWL 2 RDF-Based Interpretation.</i></b>
</p><p>The last step will be
to show that <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
For this to hold,
<i>I</i> needs to OWL 2 RDF-Based satisfy 
every triple occurring in <i>G<sub>2</sub></i>.
The basic idea can roughly be described as follows:
</p><p><i>First:</i> 
According to the <i>"semantic conditions for ground graphs"</i>
in <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#gddenot">Section 1.4 of the RDF Semantics specification</a>
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>],
all the IRIs and literals used in RDF triples in <i>G<sub>2</sub></i>
need to be in the vocabulary <i>V<sup>I</sup></i> of <i>I</i>.
This is true for the following reason:
Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="#def-balanced" title=""><i>balanced</i></a>,
all IRIs and literals occurring in <i>G<sub>2</sub></i> 
do also occur in <i>G<sub>1</sub></i>.
Since <i>I</i> satisfies <i>G<sub>1</sub></i>,
all IRIs and literals in <i>G<sub>1</sub></i>,
including those in <i>G<sub>2</sub></i>,
are contained in <i>V<sup>I</sup></i> 
due to the semantic conditions for ground graphs.
</p><p><i>Second:</i>
If a set of RDF triples encodes an OWL 2 language construct
that is not interpreted by the OWL 2 Direct Semantics,
such as annotations,
then <i>G<sub>2</sub></i> should contain such a set of RDF triples
only if they are also included in <i>G<sub>1</sub></i>.
The reason is
that with such triples 
there will, in general, exist OWL 2 RDF-Based interpretations
only satisfying the graph <i>G<sub>1</sub></i> but not <i>G<sub>2</sub></i>,
which will render the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
into a nonentailment
(an exception are RDF triples
that are true
under every OWL 2 RDF-Based interpretation).
Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>2</sub></i> will not contain the RDF encoding for any 
<i>annotations</i>, 
statements with <i>ontology properties</i>, 
<i>deprecation</i> statements or 
<i>annotation property axioms</i>.
Hence,
there are no corresponding RDF triples that need to be satisfied by <i>I</i>.
</p><p><i>Third:</i>
Since <i>G<sub>2</sub></i> is an OWL 2 DL ontology in RDF graph form,
the graph is partitioned by the 
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Mapping_from_RDF_Graphs_to_the_Structural_Specification" title="Mapping to RDF Graphs">reverse RDF mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]
into sub graphs corresponding to
either <i>ontology headers</i>,
<i>entity declarations</i>
or <i>axioms</i>,
where axioms may further consist of different kinds of <i>expressions</i>,
such as Boolean class expressions.
It has to be shown that all the triples in each such sub graph 
are OWL 2 RDF-Based satisfied by <i>I</i>. 
</p><p><i>For ontology headers:</i>
Let <i>A</i> be the ontology header of F(<i>G<sub>2</sub></i>)
and let <i>g<sub>A</sub></i> be the corresponding sub graph of <i>G<sub>2</sub></i>.
Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>g<sub>A</sub></i> is encoded as a single RDF triple of the form
"<i>x</i> <span class="name">rdf:type owl:Ontology</span>",
where <i>x</i> is either an IRI or a blank node.
Since <i>G<sub>1</sub></i> is an OWL 2 DL ontology in RDF graph form,
<i>G<sub>1</sub></i> also contains the encoding of an ontology header
including a triple <i>g<sub>1</sub></i> of the form
"<i>y</i> <span class="name">rdf:type owl:Ontology</span>",
where <i>y</i> is either an IRI or a blank node.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>g<sub>1</sub></i> is satisfied by <i>I</i>.
If both <i>y</i> and <i>x</i> are IRIs,
then, due to balancing, 
<i>x</i> equals <i>y</i>,
and therefore <i>g<sub>A</sub></i> equals <i>g<sub>1</sub></i>,
i.e. <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
Otherwise, 
balancing forces <i>x</i> to be a blank node,
i.e. <i>x</i> is treated as an existential variable 
under the OWL 2 RDF-Based Semantics
according to the
<i><a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">"semantic conditions for blank nodes"</a></i> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
From this observation,
and from the premise that <i>I</i> satisfies <i>g<sub>1</sub></i>,
it follows that <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
</p><p><i>For entity declarations:</i>
Let <i>A</i> be an entity declaration in F(<i>G<sub>2</sub></i>),
and let <i>g<sub>A</sub> be the corresponding sub graph of </i>G<sub>2</sub><i>.</i>
Since the pair
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>A</i> occurs in F(<i>G<sub>1</sub></i>),
and hence <i>g<sub>A</sub></i> is a sub graph of <i>G<sub>1</sub></i>.
Since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>I</i> OWL 2 RDF-Based satisfies <i>g<sub>A</sub></i>.
</p><p><i>For axioms:</i>
Let <i>A</i> be an axiom in F(<i>G<sub>2</sub></i>),
and let <i>g<sub>A</sub></i> be the corresponding sub graph of <i>G<sub>2</sub></i>.
Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions 
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond 
to the definition of the interpretation of the axiom <i>A</i>.
In particular,
the consequent of one of these semantic conditions
corresponds to the RDF-encoding of <i>A</i>, 
i.e. the graph <i>g<sub>A</sub></i>,
except for declaration typing triples,
for which satisfaction has already been shown
(in the case of an "if-and-only-if" semantic condition
this will generally be the right-to-left direction of that condition).
Hence, 
in order to show that <i>g<sub>A</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>,
it will be sufficient to show 
that the antecedent of this semantic condition is true.
In general,
several requirements have to be met to ensure this:
</p><p><i>Requirement 1:</i>
The denotations of all the non-built-in IRIs in <i>g<sub>A</sub></i> 
have to be contained in the appropriate part of the universe of <i>I</i>.
This can be shown as follows.
For every non-built-in IRI <i>u</i> occurring in <i>g<sub>A</sub></i>,
<i>u</i> also occurs in <i>A</i>.
Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> )
is <a href="#def-balanced" title=""><i>balanced</i></a>,
there are entity declarations in F(<i>G<sub>2</sub></i>)
for all the entity types of <i>u</i>,
each being of the form 
<i>D</i>&nbsp;:= "<span class="name">Declaration</span>(<i>T</i>(<i>u</i>))"
for some entity type <i>T</i>.
From the reverse RDF mapping follows
that for each such declaration <i>D</i>
a typing triple <i>d</i> exists in <i>G<sub>2</sub></i>,
being of the form <i>d</i>&nbsp;:= "<i>u</i> <span class="name">rdf:type</span> <i>t</i>",
where <i>t</i> is the vocabulary class IRI 
representing the part of the universe of <i>I</i>
that corresponds to the entity type <i>T</i>.
It has already been shown that,
for <i>D</i> being an entity declaration in F(<i>G<sub>2</sub></i>)
and <i>d</i> being the corresponding sub graph in <i>G<sub>2</sub></i>,
<i>I</i> OWL 2 RDF-Based satisfies <i>d</i>.
Hence, <i>I</i>(<i>u</i>) is an individual 
contained in the appropriate part of the universe.
</p><p><i>Requirement 2:</i>
For every expression <i>E</i> occurring in <i>A</i>,
with the RDF encoding <i>g<sub>E</sub></i> in <i>g<sub>A</sub></i>,
an individual has to exist in the universe of <i>I</i>
that appropriately represents the denotation of <i>E</i>.
Since <i>I</i> is an OWL 2 RDF-Based interpretation,
all the OWL 2 RDF-Based semantic conditions are met by <i>I</i>.
Due to the close alignment between the definitions 
in the OWL 2 RDF-Based Semantics
and the OWL 2 Direct Semantics,
OWL 2 RDF-Based semantic conditions exist
that semantically correspond
to the definition of the interpretation of the expression <i>E</i>.
In particular,
the antecedent of one of these semantic conditions
will become true,
if the RDF-encoding of <i>E</i>, 
i.e. the graph <i>g<sub>E</sub></i>, 
is satisfied
(in the case of an "if-and-only-if" semantic condition
this will generally be the left-to-right direction of that condition). 
Now, 
since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) 
is <a href="#def-balanced" title=""><i>balanced</i></a>, 
<i>g<sub>E</sub></i> also occurs in <i>G<sub>1</sub></i>. 
So, 
since <i>I</i> OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>, 
<i>g<sub>E</sub></i> is OWL 2 RDF-Based satisfied by <i>I</i>.
Hence,
the antecedent of the semantic condition becomes true,
and therefore its consequent becomes true as well.
This will result in the existence of an individual with the required properties,
when taking into account existential blank node semantics. 
</p><p><i>Requirement 3:</i>
A semantic relationship
has to hold
between the denotations of the 
IRIs, literals and anonymous individuals 
occurring in <i>g<sub>A</sub></i>
with respect to <i>I</i>,
which, 
roughly speaking,
expresses the meaning of the OWL 2 axiom <i>A</i>. 
This is the case for the following reasons:
First,
the literals and anonymous individuals 
occurring in <i>A</i> and <i>g<sub>A</sub></i>, respectively,
are interpreted in an analog way 
under the OWL 2 Direct Semantics and the OWL 2 RDF-Based Semantics.
Second, 
it was assumed that the OWL 2 Direct interpretation <i>J</i> 
OWL 2 Direct satisfies <i>A</i>,
and therefore a semantic relationship 
with the desired properties
holds with respect to <i>J</i>.
Third,
<i>J</i> has been defined in close correspondence to <i>I</i>,
so that for the semantic relationship expressed by <i>J</i>
an analog semantic relationship holds with respect to <i>I</i>.
</p><p>This basic idea is now demonstrated in more detail
for a single example axiom <i>A</i> in F(<i>G<sub>2</sub></i>),
which can be taken as a hint on 
how a complete proof 
taking into account every feature of the OWL 2 RDF-Based Semantics 
could be constructed in principle.
</p>
<div class="anexample" id="topic-correspondence-proof-example2">
<p>Let <i>A</i> be the following OWL 2 axiom in F(<i>G<sub>2</sub></i>):
</p>
<div class="indent">
<p><i>A</i>&nbsp;: <span class="name">SubClassOf(ex:c1 ObjectUnionOf(ex:c2 ex:c3))</span>
</p>
</div>
<p>and let <i>g<sub>A</sub></i> be the corresponding sub graph in <i>G<sub>2</sub></i>
that is being mapped to <i>A</i> via the reverse RDF mapping, 
namely
</p>
<div class="indent">
<p><i>g<sub>A</sub></i>&nbsp;:
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span><br />
<span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( ex:c2 ex:c3 ) .</span>
</p>
</div>
</div>
<p><i>First</i>,
since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) 
is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>G<sub>2</sub></i> contains the typing triples 
</p>
<div class="indent">
<p><span class="name">ex:c1 rdf:type owl:Class .</span><br />
<span class="name">ex:c2 rdf:type owl:Class .</span><br />
<span class="name">ex:c3 rdf:type owl:Class .</span>
</p>
</div>
<p>that correspond to class entity declarations in F(<i>G<sub>2</sub></i>) for the IRIs
"<span class="name">ex:c1</span>", 
"<span class="name">ex:c2</span>", and 
"<span class="name">ex:c3</span>",
respectively.
All these declaration typing triples are OWL 2 RDF-Based satisfied by <i>I</i>,
since due to balancing
the typing triples exist in <i>G<sub>1</sub></i> as well,
and since it has been postulated 
that <i>I</i> OWL 2 RDF-Based satisfies all triples in <i>G<sub>1</sub></i>.
Hence,
by applying the semantics of <span class="name">rdf:type</span>
(see
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#rdfssemcond1">Section 4.1 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]),
all the IRIs denote classes, 
and therefore the denotations of the IRIs 
are included in the appropriate part of the universe of <i>I</i>, 
precisely:
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) &isin; IC ,<br />
<i>I</i>(<span class="name">ex:c2</span>) &isin; IC , and<br /> 
<i>I</i>(<span class="name">ex:c3</span>) &isin; IC .
</p>
</div>
<p><i>Second</i>,
<i>g<sub>A</sub></i> contains the sub graph <i>g<sub>E</sub></i>,
given by 
</p>
<div class="indent">
<p><i>g<sub>E</sub></i>&nbsp;:<br />
</p>
<div class="indent">
<p><span class="name">_:x rdf:type owl:Class .</span><br />
<span class="name">_:x owl:unionOf ( c2 c3 ) .</span>
</p>
</div>
</div>
<p>which corresponds to the union class expression <i>E</i> in <i>A</i>, 
given by
</p>
<div class="indent">
<p><i>E</i>&nbsp;: <span class="name">ObjectUnionOf(ex:c2 ex:c3)</span>
</p>
</div>
<p>Since the pair 
( <i>G<sub>1</sub></i> , <i>G<sub>2</sub></i> ) 
is <a href="#def-balanced" title=""><i>balanced</i></a>,
<i>g<sub>E</sub></i> occurs as a sub graph of <i>G<sub>1</sub></i> as well.
<i>g<sub>E</sub></i> contains blank nodes
and,
since <i>I</i> satisfies <i>G<sub>1</sub></i>,
the semantic conditions for RDF graphs with blank nodes apply 
(see 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#unlabel">Section 1.5 of the RDF Semantics</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>]).
This provides the existence of
a mapping <i>B</i> from blank(<i>g<sub>E</sub></i>) to IR, 
where blank(<i>g<sub>E</sub></i>) is 
the set of all blank nodes occurring in <i>g<sub>E</sub></i>.
It follows that 
the extended interpretation <i>I</i>+<i>B</i> 
OWL 2 RDF-Based satisfies all the triples in <i>g<sub>E</sub></i>.
Further,
since <i>I</i> is an OWL 2 RDF-Based interpretation,
<i>I</i> meets all the OWL 2 RDF-Based semantic conditions.
Thus, the left-to-right direction 
of the semantic condition for union class expressions
("<span class="name">owl:unionOf</span>",
see <a href="#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a>)
applies, providing:
</p>
<div class="indent">
<p>[<i>I</i>+<i>B</i>](<span class="name">_:x</span>) &isin; IC ,<br />
ICEXT([<i>I</i>+<i>B</i>](<span class="name">_:x</span>)) 
=
ICEXT(<i>I</i>(<span class="name">ex:c2</span>)) 
&cup; 
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) .
</p>
</div>
<p><i>Third</i>,
since the OWL 2 Direct interpretation <i>J</i> OWL 2 Direct satisfies <i>A</i>,
the following semantic relationship
holds between the denotations of the class names in <i>A</i> 
according to <i>J</i>:
</p>
<div class="indent">
<p>(<span class="name">ex:c1</span>)&nbsp;<sup><i>C</i></sup> 
&sube; 
(<span class="name">ex:c2</span>)&nbsp;<sup><i>C</i></sup> 
&cup; 
(<span class="name">ex:c3</span>)&nbsp;<sup><i>C</i></sup> .
</p>
</div>
<p>By applying the definition of
the OWL 2 Direct interpretation <i>J</i>, 
one can conclude that the following semantic relationship
holds between the extensions of the classes above
according to <i>I</i>:
</p>
<div class="indent">
<p>ICEXT(<i>I</i>(<span class="name">ex:c1</span>)) 
&sube; 
ICEXT(<i>I</i>(<span class="name">ex:c2</span>)) 
&cup; 
ICEXT(<i>I</i>(<span class="name">ex:c3</span>)) .
</p>
</div>
<p><i>Finally</i>,
combining all intermediate results gives
</p>
<div class="indent">
<p><i>I</i>(<span class="name">ex:c1</span>) &isin; IC ,<br />
[<i>I</i>+<i>B</i>](<span class="name">_:x</span>) &isin; IC ,<br />
ICEXT(<i>I</i>(<span class="name">ex:c1</span>)) 
&sube; 
ICEXT([<i>I</i>+<i>B</i>](<span class="name">_:x</span>)) .
</p>
</div>
<p>Therefore, all the premises are met
to apply the right-to-left direction of the semantic condition for subclass axioms
("<span class="name">rdfs:subClassOf</span>",
see <a href="#Semantic_Conditions_for_the_RDFS_Vocabulary" title="">Section 5.8</a>),
which results in
</p>
<div class="indent">
<p>( <i>I</i>(<span class="name">ex:cl</span>) , [<i>I</i>+<i>B</i>](<span class="name">_:x</span>) )
&isin;
IEXT(<i>I</i>(<span class="name">rdfs:subClassOf</span>)) .
</p>
</div>
<p>So,
the remaining triple
</p>
<div class="indent">
<p><span class="name">ex:c1 rdfs:subClassOf _:x .</span>
</p>
</div>
<p>in <i>g<sub>A</sub></i> 
is OWL 2 RDF-Based satisfied by <i>I</i>+<i>B</i>,
where "<span class="name">_:x</span>" is 
the root blank node of the union class expression <i>g<sub>E</sub></i>.
Hence, 
w.r.t. existential blank node semantics,
<i>I</i> OWL 2 RDF-Based satisfies all the triples in <i>g<sub>A</sub></i>.
</p>
</div>
<p>To conclude,
for any OWL 2 RDF-Based interpretation <i>I</i>
that OWL 2 RDF-Based satisfies <i>G<sub>1</sub></i>,
<i>I</i> also OWL 2 RDF-Based satisfies <i>G<sub>2</sub></i>.
Hence,
<i>G<sub>1</sub></i> OWL 2 RDF-Based entails <i>G<sub>2</sub></i>,
and therefore relationship 4 of the correspondence theorem holds. 
<i><b>Q.E.D.</b></i>
</p>
<a name="Appendix:_Comprehension_Conditions_.28Informative.29"></a><h2> <span class="mw-headline">8  Appendix: Comprehension Conditions (Informative) </span></h2>
<p>The <a href="#thm-correspondence" title="">correspondence theorem</a> 
in <a href="#Correspondence_Theorem" title="">Section 7.2</a>
shows
that it is possible for the OWL 2 RDF-Based Semantics
to reflect all the entailments of the 
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/" title="Direct Semantics">OWL 2 Direct Semantics</a>
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>],
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 RDF-Based Semantics,
for the technical reasons discussed in 
<a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>.
It seems natural to ask for similar entailments
even 
when an entailment query
does not consist of OWL 2 DL ontologies in RDF graph form.
However,
the correspondence theorem does not apply to such cases,
and thus the OWL 2 Direct Semantics cannot be taken 
as a reference frame
for "desirable" and "useful" entailments,
or for when a graph transformation 
can be considered "harmless" or not.
</p><p>As discussed in 
<a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>,
a core obstacle for the correspondence theorem to hold 
was the RDF encoding of OWL 2 expressions, 
such as union class expressions,
when they appear on the right hand side of an entailment query.
Under the OWL 2 RDF-Based Semantics
it is not generally ensured that an individual exists,
which represents the denotation of such an expression.
The <i>"comprehension conditions"</i> defined in this section
are additional semantic conditions
that provide the necessary individuals
for <i>every</i> sequence, class and property expression.
By this,
the combination
of the normative semantic conditions of the OWL 2 RDF-Based Semantics
(<a href="#Semantic_Conditions" title="">Section 5</a>) 
and the comprehension conditions
can be regarded to "simulate" the semantic expressivity 
of the OWL 2 Direct Semantics
on entailment queries consisting of <i>arbitrary</i> RDF graphs.
</p><p>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
<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/119" title="http://www.w3.org/2007/OWL/tracker/issues/119">lead to formal inconsistency</a>.
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 <i>unrestricted</i> entailment queries
in order to receive additional entailments under the OWL 2 RDF-Based Semantics.
Such a concrete method is, however, 
outside the scope of this specification.
</p><p><i>Note:</i>
The <a href="#topic-semcond-conventions" title="">conventions</a>
in the introduction of
<a href="#Semantic_Conditions" title="">Section 5 ("Semantic Conditions")</a>
apply to the current section as well.
</p>
<a name="Comprehension_Conditions_for_Sequences"></a><h3> <span class="mw-headline">8.1  Comprehension Conditions for Sequences </span></h3>
<p><a href="#table-comprehension-lists" title="">Table 8.1</a> 
lists the comprehension conditions for sequences,
i.e. RDF lists.
These comprehension conditions provide the existence 
of sequences 
built from any finite combination of individuals
contained in the universe.
</p>
<div class="left" id="table-comprehension-lists">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.1: Comprehension Conditions for Sequences</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-lists"></span>if
</th><th style="text-align: center"> then exists z<sub>1</sub> , &hellip; , z<sub>n</sub> &isin; IR
</th></tr>
<tr>
<td> <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR
</td><td> ( <i>z<sub>1</sub></i> , <i>a<sub>1</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:first</span>)) , ( <i>z<sub>1</sub></i> , <i>z<sub>2</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:rest</span>)) , &hellip; ,<br />( <i>z<sub>n</sub></i> , <i>a<sub>n</sub></i> ) &isin; IEXT(<i>I</i>(<span class="name">rdf:first</span>)) , ( <i>z<sub>n</sub></i> , <i>I</i>(<span class="name">rdf:nil</span>) ) &isin; IEXT(<i>I</i>(<span class="name">rdf:rest</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Boolean_Connectives"></a><h3> <span class="mw-headline">8.2  Comprehension Conditions for Boolean Connectives </span></h3>
<p><a href="#table-comprehension-booleans" title="">Table 8.2</a> 
lists the comprehension conditions for 
Boolean connectives
(see <a href="#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a> 
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of complements for any class and datatype,
and of intersections and unions
built from any finite set of classes
contained in the universe.
</p>
<div class="left" id="table-comprehension-booleans">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.2: Comprehension Conditions for Boolean Connectives</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <span id="item-comprehension-booleans-intersectionof"></span><i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC
</td><td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:intersectionOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-unionof"></span><i>s</i> sequence of <i>c<sub>1</sub></i> , &hellip; , <i>c<sub>n</sub></i> &isin; IC
</td><td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:unionOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-complementof"></span><i>c</i> &isin; IC
</td><td> ( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:complementOf</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-booleans-datatypecomplement"></span><i>d</i> &isin; IDC
</td><td> ( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:datatypeComplementOf</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Enumerations"></a><h3> <span class="mw-headline">8.3  Comprehension Conditions for Enumerations </span></h3>
<p><a href="#table-comprehension-enums" title="">Table 8.3</a> 
lists the comprehension conditions for 
enumerations
(see <a href="#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a> 
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of enumeration classes
built from any finite set of individuals
contained in the universe.
</p>
<div class="left" id="table-comprehension-enums">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.3: Comprehension Conditions for Enumerations</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-enums"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>s</i> sequence of <i>a<sub>1</sub></i> , &hellip; , <i>a<sub>n</sub></i> &isin; IR
</td><td> ( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:oneOf</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Property_Restrictions"></a><h3> <span class="mw-headline">8.4  Comprehension Conditions for Property Restrictions </span></h3>
<p><a href="#table-comprehension-restrictions" title="">Table 8.4</a> 
lists the comprehension conditions for 
property restrictions
(see <a href="#Semantic_Conditions_for_Property_Restrictions" title="">Section 5.6</a> 
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of cardinality restrictions
on any property and for any nonnegative integer,
as well as value restrictions
on any property and on any class
contained in the universe.
</p>
<div id="topic-comprehension-restrictions-self"></div>
<p>Note that the comprehension conditions for self restrictions 
constrains the right hand side of 
the produced <span class="name">owl:hasSelf</span> assertions
to be the Boolean value 
<span class="name">"true"^^xsd:boolean</span>.
This is in accordance with
<a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/#Parsing_of_Expressions" title="Mapping to RDF Graphs">Table 13 in Section 3.2.4 of the OWL 2 RDF Mapping</a> 
[<cite><a href="#ref-owl-2-rdf-mapping" title="">OWL 2 RDF Mapping</a></cite>]. 
</p>
<div id="topic-comprehension-restrictions-narydatatype"></div>
<p>Implementations are <i>not</i> required 
to support the comprehension conditions for 
<span class="name">owl:onProperties</span>, 
but 
<em class="RFC2119" title="MAY in RFC 2119 context">MAY</em> 
support them 
in order to realize 
<i>n-ary dataranges</i> with arity &ge; 2
(see 
Sections
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Ranges" title="Syntax">7</a>
and
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Data_Property_Restrictions" title="Syntax">8.4</a>
of the OWL 2 Structural Specification 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>] 
for further information).
</p>
<div class="left" id="table-comprehension-restrictions">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.4: Comprehension Conditions for Property Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <span id="item-comprehension-restrictions-somevaluesfrom"></span><i>c</i> &isin; IC ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-somevaluesfrom-nary"></span><i>c</i> &isin; IC ,<br /><i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP , <i>n</i> &ge; 1
</td><td> ( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:someValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-allvaluesfrom"></span><i>c</i> &isin; IC ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-allvaluesfrom-nary"></span><i>c</i> &isin; IC ,<br /><i>s</i> sequence of <i>p<sub>1</sub></i> , &hellip; , <i>p<sub>n</sub></i> &isin; IP , <i>n</i> &ge; 1
</td><td> ( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:allValuesFrom</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperties</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-hasvalue"></span><i>a</i> &isin; IR ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>a</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:hasValue</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-hasself"></span><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>I</i>(<span class="name">"true"^^xsd:boolean</span>) ) &isin; IEXT(<i>I</i>(<span class="name">owl:hasSelf</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-mincardinality"></span><i>n</i> &isin; INNI ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxcardinality"></span><i>n</i> &isin; INNI ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxCardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-cardinality"></span><i>n</i> &isin; INNI ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:cardinality</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-minqualifiedcardinality"></span><i>n</i> &isin; INNI ,<br /><i>c</i> &isin; IC ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-minqualifiedcardinality-data"></span><i>n</i> &isin; INNI ,<br /><i>d</i> &isin; IDC ,<br /><i>p</i> &isin; IODP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:minQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxqualifiedcardinality"></span><i>n</i> &isin; INNI ,<br /><i>c</i> &isin; IC ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-maxqualifiedcardinality-data"></span><i>n</i> &isin; INNI ,<br /><i>d</i> &isin; IDC ,<br /><i>p</i> &isin; IODP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:maxQualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-qualifiedcardinality"></span><i>n</i> &isin; INNI ,<br /><i>c</i> &isin; IC ,<br /><i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>c</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onClass</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
<tr>
<td> <span id="item-comprehension-restrictions-qualifiedcardinality-data"></span><i>n</i> &isin; INNI ,<br /><i>d</i> &isin; IDC ,<br /><i>p</i> &isin; IODP
</td><td> ( <i>z</i> , <i>n</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:qualifiedCardinality</span>)) ,<br />( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDataRange</span>)) ,<br />( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onProperty</span>))
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Datatype_Restrictions"></a><h3> <span class="mw-headline">8.5  Comprehension Conditions for Datatype Restrictions </span></h3>
<p><a href="#table-comprehension-facets" title="">Table 8.5</a> 
lists the comprehension conditions for 
datatype restrictions
(see <a href="#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a> 
for the corresponding semantic conditions).
These comprehension conditions provide the existence
of datatypes 
built from restricting any datatype 
contained in the universe
by any finite set of facet-value pairs
contained in the facet space
(see <a href="#Datatype_Maps" title="">Section 4.1</a>)
of the original datatype.
</p><p>The set IFS is defined in 
<a href="#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>.
</p>
<div class="left" id="table-comprehension-facets">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.5: Comprehension Conditions for Datatype Restrictions</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-facets"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR , <i>s</i> sequence of <i>z<sub>1</sub></i> , &hellip; , <i>z<sub>n</sub></i> &isin; IR
</th></tr>
<tr>
<td> <i>d</i> &isin; IDC ,<br /><i>f<sub>1</sub></i> , &hellip; , <i>f<sub>n</sub></i> &isin; IODP ,<br /><i>v<sub>1</sub></i> , &hellip; , <i>v<sub>n</sub></i> &isin; LV ,<br />( <i>f<sub>1</sub></i> , <i>v<sub>1</sub></i> ) , &hellip; , ( <i>f<sub>n</sub></i> , <i>v<sub>n</sub></i> ) &isin; IFS(<i>d</i>)
</td><td> ( <i>z</i> , <i>d</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:onDatatype</span>)) ,<br />( <i>z</i> , <i>s</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:withRestrictions</span>)) ,<br />( <i>z<sub>1</sub></i> , <i>v<sub>1</sub></i> ) &isin; IEXT(<i>f<sub>1</sub></i>) , &hellip; , ( <i>z<sub>n</sub></i> , <i>v<sub>n</sub></i> ) &isin; IEXT(<i>f<sub>n</sub></i>)
</td></tr>
</table>
</div>
<a name="Comprehension_Conditions_for_Inverse_Properties"></a><h3> <span class="mw-headline">8.6  Comprehension Conditions for Inverse Properties </span></h3>
<p><a href="#table-comprehension-inverses" title="">Table 8.6</a> 
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.
</p><p>Inverse property expressions can be used 
to build axioms with anonymous inverse properties, 
such as in the graph
</p>
<div class="indent">
<p><span class="name">_:x owl:inverseOf ex:p .</span><br />
<span class="name">_:x rdfs:subPropertyOf owl:topObjectProperty .</span>
</p>
</div>
<p>Note that,
to some extent,
the OWL 2 RDF-Based Semantics already covers the use of inverse property expressions
by means of the semantic conditions of inverse property axioms
(see <a href="#Semantic_Conditions_for_Inverse_Properties" title="">Section 5.12</a>),
since these semantic conditions also apply to an existential variable
on the left hand side of an inverse property axiom.
Nevertheless, 
not all relevant cases are covered by this semantic condition.
For example,
one might expect the above example graph 
to be generally true.
However,
the normative semantic conditions
do not permit this conclusion,
since it is not ensured that 
for every property <i>p</i>
there is an individual in the universe
with a property extension being inverse to that of <i>p</i>.
</p>
<div class="left" id="table-comprehension-inverses">
<table border="2" cellpadding="5" style="text-align: left">
<caption> <span class="caption">Table 8.6: Comprehension Conditions for Inverse Properties</span>
</caption>
<tr>
<th style="text-align: center"> <span id="item-comprehension-inverses"></span>if
</th><th style="text-align: center"> then exists <i>z</i> &isin; IR
</th></tr>
<tr>
<td> <i>p</i> &isin; IP
</td><td> ( <i>z</i> , <i>p</i> ) &isin; IEXT(<i>I</i>(<span class="name">owl:inverseOf</span>))
</td></tr>
</table>
</div>
<a name="Appendix:_Changes_from_OWL_1_.28Informative.29"></a><h2> <span class="mw-headline">9  Appendix: Changes from OWL 1 (Informative) </span></h2>
<p>This section lists relevant differences 
between the OWL 2 RDF-Based Semantics and the original specification of the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html"><i>OWL 1 RDF-Compatible Semantics</i></a>
[<cite><a href="#ref-owl-1-rdf-semantics" title="">OWL 1 RDF-Compatible Semantics</a></cite>].
Significant effort has been spent
in keeping the design of the OWL 2 RDF-Based Semantics 
as close as possible 
to that of the OWL 1 RDF-Compatible Semantics.
While this aim was achieved to a large degree, 
the OWL 2 RDF-Based Semantics actually deviates from its predecessor in several aspects.
In most cases this is because of serious technical problems 
that would have arisen 
from a conservative 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DefSemanticExtension">semantic extension</a>.
Not listed are
the new language constructs and the new datatypes of OWL 2.
</p>
<div id="topic-languagechanges-markers"></div>
<p>The following markers are used:
</p>
<ul><li> <b>[DEV]</b>: a deviation from OWL 1 that breaks backward compatibility
</li><li> <b>[EXT]</b>: a backward compatible extension to OWL 1
</li><li> <b>[NOM]</b>: a change of the nomenclature originally used in OWL 1
</li><li> <b>[DPR]</b>: a feature of OWL 1 that has been deprecated as of OWL 2
</li></ul>
<div id="topic-languagechanges-graphsyntax"></div>
<p><b>Generalized Graph Syntax [EXT].</b>
The OWL 2 RDF-Based Semantics 
allows RDF graphs to contain
<a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt"><i>IRIs</i></a> 
[<cite><a href="#ref-rfc-3987" title="">RFC 3987</a></cite>]
(see <a href="#Syntax" title="">Section 2.1</a>),
whereas the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs with 
<a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt"><i>URIs</i></a> 
[<cite><a href="#ref-rfc-2396" title="">RFC 2396</a></cite>].
This change is in accordance with the rest of the OWL 2 specification
(see 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#IRIs" title="Syntax">Section 2.4 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
In addition,
the OWL 2 RDF-Based Semantics
is now explicitly allowed to 
be applied to RDF graphs containing
<i>"generalized" RDF triples</i>,
i.e. triples that can consist of 
IRIs, literals or blank nodes 
in all three positions
(<a href="#Syntax" title="">Section 2.1</a>),
although implementations are not required to support this.
In contrast,
the OWL 1 RDF-Compatible Semantics was restricted to RDF graphs 
conforming to the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax">RDF Concepts specification</a>
[<cite><a href="#ref-rdf-concepts" title="">RDF Concepts</a></cite>].
These limitations of the OWL 1 RDF-Compatible Semantics 
were actually inherited from the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#graphsyntax">RDF Semantics specification</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>].
The relaxations are intended to warrant interoperability
with existing and future technologies and tools.
Both changes are compatible with OWL 1,
since all RDF graphs that were legal under the OWL 1 RDF-Compatible Semantics 
are still legal under the OWL 2 RDF-Based Semantics. 
</p>
<div id="topic-languagechanges-facets"></div>
<p><b>Facets for Datatypes [EXT].</b>
The basic definitions of a <i>datatype</i> and a <i>D-interpretation</i>,
as defined by the RDF Semantics specification 
and as applied by the OWL 1 RDF-Compatible Semantics,
have been extended
to take into account <i>constraining facets</i>
(see <a href="#Interpretations" title="">Section 4</a>),
in order to allow for <i>datatype restrictions</i>
as specified in <a href="#Semantic_Conditions_for_Datatype_Restrictions" title="">Section 5.7</a>.
This change is compatible with OWL 1,
since <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#DTYPEINTERP">Section 5.1</a> 
of the RDF Semantics specification
explicitly allows for extending the minimal datatype definition provided there.
</p>
<div id="topic-languagechanges-correspondence"></div>
<p><b>Correspondence Theorem and Comprehension Conditions [DEV].</b> 
The semantic conditions of the OWL 1 RDF-Compatible Semantics included 
a set of so called 
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#comprehension_principles" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#comprehension_principles"><i>"comprehension conditions"</i></a>,
which allowed to prove the original
<a class="external text" href="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#theorem-2" title="http://www.w3.org/TR/2004/REC-owl-semantics-20040210/rdfs.html#theorem-2"><i>"correspondence theorem"</i></a>
stating that every entailment of OWL 1 DL was also an entailment of OWL 1 Full.
The document at hand adds comprehension conditions 
for the new language constructs of OWL 2
(see <a href="#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a>).
However,
the comprehension conditions
are <i>not</i> a normative aspect of the OWL 2 RDF-Based Semantics
anymore.
It has turned out 
that combining the comprehension conditions 
with the normative set of semantic conditions in 
<a href="#Semantic_Conditions" title="">Section 5</a>
would lead to formal inconsistency of the resulting semantics
(<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/119" title="http://www.w3.org/2007/OWL/tracker/issues/119">Issue 119</a>).
In addition,
it became clear that
a correspondence theorem along the lines of the original theorem
would not work for the relationship between the
OWL 2 RDF-Based Semantics and the 
<a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/" title="Direct Semantics">OWL 2 Direct Semantics</a> 
[<cite><a href="#ref-owl-2-direct-semantics" title="">OWL 2 Direct Semantics</a></cite>],
since it is not possible to "balance" the differences between
the two semantics
solely by means of additional comprehension conditions
(see <a href="#Example_on_Semantic_Differences" title="">Section 7.1</a>).
Consequently,
the correspondence theorem 
of the OWL 2 RDF-Based Semantics
(<a href="#Correspondence_Theorem" title="">Section 7.2</a>)
follows an alternative approach 
that replaces the use of the comprehension conditions
and can be seen as a technical refinement
of an idea 
originally discussed by the WebOnt Working Group
(<a class="external text" href="http://lists.w3.org/Archives/Public/www-webont-wg/2002Mar/0179.html" title="http://lists.w3.org/Archives/Public/www-webont-wg/2002Mar/0179.html">email</a>). 
This change is an <i>incompatible deviation</i> from OWL 1,
since certain aspects of the originally normative definition of the semantics 
have been removed.
</p>
<div id="topic-languagechanges-arglists"></div>
<p><b>Flawed Semantics of Language Constructs with Argument Lists [DEV].</b> 
In the OWL 1 RDF-Compatible Semantics,
the semantic conditions for 
unions, intersections and enumerations of classes 
were defined in a flawed form, 
which lead to formal inconsistency of the semantics
(<a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/120" title="http://www.w3.org/2007/OWL/tracker/issues/120">Issue 120</a>; 
see also this
<a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=BrokenOwl1FullFeaturesWithArgumentLists&amp;oldid=26002" title="http://www.w3.org/2007/OWL/wiki/index.php?title=BrokenOwl1FullFeaturesWithArgumentLists&amp;oldid=26002">unofficial problem description</a>). 
The affected semantic conditions have been revised; 
see 
<a href="#Semantic_Conditions_for_Boolean_Connectives" title="">Section 5.4</a> 
and 
<a href="#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>.
This change is an <i>incompatible deviation</i> from OWL 1,
since the semantics has formally been weakened
in order to eliminate a source of inconsistency.
</p>
<div id="topic-languagechanges-ndis"></div>
<p><b>Incomplete Semantics of <span class="name">owl:AllDifferent</span> [EXT].</b> 
The OWL 1 RDF-Compatible Semantics missed a certain semantic condition 
for axioms based on the vocabulary term "<span class="name">owl:AllDifferent</span>"
(see also this 
<a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=MissingOwl1FullAllDifferentSemanticCondition&amp;oldid=26003" title="http://www.w3.org/2007/OWL/wiki/index.php?title=MissingOwl1FullAllDifferentSemanticCondition&amp;oldid=26003">unofficial problem description</a>). 
The missing semantic condition 
has been added to the OWL 2 RDF-Based Semantics
(see <a href="#Semantic_Conditions_for_N-ary_Disjointness" title="">Section 5.10</a>).
This change is compatible with OWL 1,
since the semantics has been conservatively extended.
</p>
<div id="topic-languagechanges-datarange"></div>
<p><b>Aligned Semantics of <span class="name">owl:DataRange</span> and <span class="name">rdfs:Datatype</span> [EXT].</b>
The class 
<span class="name">owl:DataRange</span>
has been made an <i>equivalent class</i> 
to <span class="name">rdfs:Datatype</span>
(see <a href="#Semantic_Conditions_for_the_Vocabulary_Classes" title="">Section 5.2</a>).
The main purpose for this change was 
to allow for the deprecation of the term 
<span class="name">owl:DataRange</span> 
in favor of <span class="name">rdfs:Datatype</span>.
This change is compatible with OWL 1
according to an analysis
of the relationship between the two classes
in the OWL 1 RDF-Compatible Semantics
(<a class="external text" href="http://lists.w3.org/Archives/Public/public-owl-wg/2008Jan/0229.html" title="http://lists.w3.org/Archives/Public/public-owl-wg/2008Jan/0229.html">email</a>).
</p>
<div id="topic-languagechanges-ontoprop"></div>
<p><b>Ontology Properties as Annotation Properties [EXT].</b>
Several properties
that have been ontology properties in OWL 1,
such as <span class="name">owl:priorVersion</span>,
have now been specified 
as both ontology properties and annotation properties,
in order to be in line 
with the rest of the OWL 2 specification
(see 
<a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Annotation_Properties" title="Syntax">Section 5.5 of the OWL 2 Structural Specification</a> 
[<cite><a href="#ref-owl-2-specification" title="">OWL 2 Specification</a></cite>]).
This change is compatible with OWL 1,
since the semantics has been conservatively extended:
all the ontology properties of OWL 1 are still ontology properties in OWL 2.
</p>
<div id="topic-languagechanges-dataenums"></div>
<p><b>Nonempty Data Value Enumerations [DEV].</b> 
The semantic condition for enumerations of data values 
in <a href="#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>
is now restricted to <i>nonempty</i> sets of data values. 
This prevents the class <span class="name">owl:Nothing</span> 
from unintentionally becoming an instance 
of the class <span class="name">rdfs:Datatype</span>,
as analyzed in
(<a class="external text" href="http://lists.w3.org/Archives/Public/public-webont-comments/2008May/0001.html" title="http://lists.w3.org/Archives/Public/public-webont-comments/2008May/0001.html">email</a>). 
This restriction of the semantics 
is an <i>incompatible deviation</i> from OWL 1.
Note, however,
that it is still possible
to define a datatype as an empty enumeration of data values,
as explained in <a href="#Semantic_Conditions_for_Enumerations" title="">Section 5.5</a>.
</p>
<div id="topic-languagechanges-nameing"></div>
<p><b>Terminological Clarifications [NOM].</b>
This document uses the term <i>"OWL 2 RDF-Based Semantics"</i>
to refer to the specified semantics only.
According to <a href="#Syntax" title="">Section 2.1</a>,
the term <i>"OWL 2 Full"</i>
refers to the language
that is determined
by the set of all RDF graphs
(also called <i>"OWL 2 Full ontologies"</i>)
being interpreted using the OWL 2 RDF-Based Semantics.
OWL 1 has not been particularly clear on this distinction.
Where the OWL 1 RDF-Compatible Semantics specification talked about
<i>"OWL Full interpretations"</i>,
<i>"OWL Full satisfaction"</i>,
<i>"OWL Full consistency"</i>
and
<i>"OWL Full entailment"</i>,
the OWL 2 RDF-Based Semantics Specification talks 
in <a href="#Interpretations" title="">Section 4</a>
about
<i>"OWL 2 RDF-Based interpretations"</i>,
<i>"OWL 2 RDF-Based satisfaction"</i>,
<i>"OWL 2 RDF-Based consistency"</i>
and
<i>"OWL 2 RDF-Based entailment"</i>,
respectively,
since these terms are primarily meant to be related to 
the semantics 
rather than the whole language.
</p>
<div id="topic-languagechanges-abbreviations"></div>
<p><b>Modified Abbreviations [NOM].</b>
The names 
"R<sub>I</sub>", "P<sub>I</sub>", "C<sub>I</sub>", 
"EXT<sub>I</sub>", "CEXT<sub>I</sub>", 
"S<sub>I</sub>", "L<sub>I</sub>" and "LV<sub>I</sub>",
which have been used in the
OWL 1 RDF-Compatible Semantics specification, 
have been replaced by the corresponding names 
defined in the 
<a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">RDF Semantics document</a> 
[<cite><a href="#ref-rdf-semantics" title="">RDF Semantics</a></cite>], 
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>I</i>", 
following the conventions in the RDF Semantics document.
These changes are intended to support 
the use of the OWL 2 RDF-Based Semantics document 
as an incremental extension 
of the RDF Semantics document. 
Names for the <a href="#Parts_of_the_Universe" title="">"parts of the universe"</a> 
that were exclusively used in the OWL 1 RDF-Compatible Semantics document, 
such as "IX" or "IODP", 
have not been changed.
Other abbreviations, 
such as "IAD" for the class extension of <span class="name">owl:AllDifferent</span>, 
have in general not been reused in the document at hand,
but the explicit nonabbreviated form,
such as 
"IEXT(<i>I</i>(<span class="name">owl:AllDifferent</span>))",
is used instead.
</p><p><b>Modified Tuple Notation Style [NOM].</b>
Tuples are written in the form
"( &hellip; )"
instead of "&lt; &hellip; &gt;",
as in the other OWL 2 documents.
</p>
<div id="topic-languagechanges-deprecated"></div>
<p><b>Deprecated Vocabulary Terms [DPR].</b> 
The following vocabulary terms have been deprecated as of OWL 2 
by the Working Group, 
and <em class="RFC2119" title="SHOULD NOT in RFC 2119 context">SHOULD NOT</em> be used 
in new ontologies anymore:
</p>
<ul><li> <span class="name">owl:DataRange</span> (per <a class="external text" href="http://www.w3.org/2007/OWL/wiki/Teleconference.2008.01.23/Minutes" title="http://www.w3.org/2007/OWL/wiki/Teleconference.2008.01.23/Minutes">resolution</a> of <a class="external text" href="http://www.w3.org/2007/OWL/tracker/issues/29" title="http://www.w3.org/2007/OWL/tracker/issues/29">Issue 29</a>)
</li></ul>
<div id="changelog">
<a name="Appendix:_Change_Log_.28Informative.29"></a><h2> <span class="mw-headline">10  Appendix: Change Log (Informative) </span></h2>
<a name="Changes_Since_Proposed_Recommendation"></a><h3> <span class="mw-headline">10.1  Changes Since Proposed Recommendation </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/" title="http://www.w3.org/TR/2009/PR-owl2-rdf-based-semantics-20090922/">Proposed Recommendation of 22 September, 2009</a>.
</p>
<ul><li> [editorial] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=25950&amp;oldid=25949" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=25950&amp;oldid=25949">Correction of grammar</a> (punctuation, word order, etc.), mainly in the <a href="#Introduction_.28Informative.29" title="">Introduction section</a>.
</li><li> [editorial] Updated and corrected several hyperlinks.
</li></ul>
<a name="Changes_Since_Candidate_Recommendation"></a><h3> <span class="mw-headline">10.2  Changes Since Candidate Recommendation </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/CR-owl2-rdf-based-semantics-20090611/" title="http://www.w3.org/TR/2009/CR-owl2-rdf-based-semantics-20090611/">Candidate Recommendation of 11 June, 2009</a>.
</p>
<ul><li> [resolution] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24829&amp;oldid=24826" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24829&amp;oldid=24826">Re-definition of several ontology properties</a> to be both ontology properties and annotation properties, in order to align the RDF-Based Semantics with the rest of the <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/#Annotation_Properties" title="Syntax">OWL 2 specification</a>, and in particular to avoid an equivocal definition of the <a href="http://www.w3.org/TR/2009/REC-owl2-profiles-20091027/#Reasoning_in_OWL_2_RL_and_RDF_Graphs_using_Rules" title="Profiles">OWL 2 RL/RDF rules</a> (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-07-15#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-07-15#resolution_2">WG resolution</a>).
</li><li> [correction] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24865&amp;oldid=24864" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24865&amp;oldid=24864">Correction of the type of facets</a>: Facets are intended to be data properties and have been used as such <a href="#table-semcond-facets" title="">elsewhere in the document</a>, but they were wrongly specified as unrestricted properties so far.
</li><li> [correction] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24758&amp;oldid=24757" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24758&amp;oldid=24757">Correction of a mismatch</a> between the <a href="#topic-int-rdfinterpretation" title="">definition of D-interpretations</a> in the document at hand and the RDF Semantics specification: according to the <a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/#interp">definition of simple interpretations</a>, LV contains all plain literals in the vocabulary <i>V</i>. The missing reference to "<i>V</i>" has been added.
</li><li> [nonnormative] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24755&amp;oldid=24670" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24755&amp;oldid=24670">Correction of an error</a> in the formulation of the <a href="#thm-correspondence" title="">correspondence theorem</a>.
</li><li> [nonnormative] The <a href="#Appendix:_Axiomatic_Triples_.28Informative.29" title="">section on Axiomatic Triples</a> has been extended by an explicit <a href="#A_Set_of_Axiomatic_Triples" title="">set of axiomatic triples</a>, based on the discussion in the rest of the section.
</li><li> [nonnormative] The <a href="#Appendix:_Axiomatic_Triples_.28Informative.29" title="">section on Axiomatic Triples</a> now explicitly mentions axiomatic triples for <a href="#topic-axiomatic-classes-datatypes" title="">datatypes</a> and <a href="#topic-axiomatic-properties-facets" title="">facets</a> corresponding to the semantic conditions for <a href="#topic-semcond-classes-datatypes" title="">datatypes</a> and <a href="#topic-semcond-properties-facets" title="">facets</a>, respectively. 
</li><li> [nonnormative] Refinement of the <a href="#Proof_for_the_Correspondence_Theorem" title="">proof for the correspondence theorem</a> and correction of several errors. Motivated by these changes, the <a href="#Example_on_Semantic_Differences" title="">example in Section 7.1</a> has been slightly revised as well.
</li><li> [editorial] Added a description and ALT-attribute text to <a href="#fig-partshierarchy" title="">Figure 1 on the parts hierarchy</a>. 
</li><li> [editorial] Distinction between <a href="#References" title="">normative and nonnormative references</a>, as in other OWL 2 documents.
</li><li> [editorial] Added some clarification to the <a href="#Introduction_.28Informative.29" title="">introduction section</a>. 
</li><li> [editorial] <a class="external text" href="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24766&amp;oldid=24765" title="http://www.w3.org/2007/OWL/wiki/index.php?title=RDF-Based_Semantics&amp;diff=24766&amp;oldid=24765">Removed a redundant conclusion</a> from the table presenting the <a href="#table-semcond-facets" title="">semantic conditions for datatype restrictions</a>, since this conclusion already follows from the <a href="#item-semcond-properties-ondatatype" title="">semantic conditions for the vocabulary properties</a>, and having the conclusion repeated would not match the <a href="#topic-semcond-conditionform-deducedrhs" title="">general approach</a> that is applied when presenting "if-then" semantic conditions in this document.
</li><li> [editorial] Reworded the description of the <a href="#topic-languagechanges-markers" title="">markers in the section on changes from OWL 1</a>, and added a marker "DPR" for the <a href="#topic-languagechanges-deprecated" title="">deprecated features</a>.
</li><li> [editorial] Changed the presentation style of references and citations to a form used in all OWL 2 documents.
</li><li> [editorial] Changed the presentation style for tuples from "&lang; &hellip; &rang;" to "( &hellip; )", to follow the conventions used in the other OWL 2 documents.
</li><li> [editorial] Numerous minor corrections and stylistic improvements.
</li></ul>
<a name="Changes_Since_Last_Call"></a><h3> <span class="mw-headline">10.3  Changes Since Last Call </span></h3>
<p>This section summarizes the changes to this document since the <a class="external text" href="http://www.w3.org/TR/2009/WD-owl2-rdf-based-semantics-20090421/" title="http://www.w3.org/TR/2009/WD-owl2-rdf-based-semantics-20090421/">Last Call Working Draft of 21 April, 2009</a>.
</p>
<ul><li> [resolution] Renamed the annotation vocabulary terms "<span class="name">owl:subject</span>", "<span class="name">owl:predicate</span>" and "<span class="name">owl:object</span>" to "<span class="name">owl:annotatedSource</span>", "<span class="name">owl:annotatedProperty</span>" and "<span class="name">owl:annotatedTarget</span>", respectively (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-05-20#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-05-20#resolution_2">WG resolution</a>).
</li><li> [resolution] Replaced the datatype "<span class="name">rdf:text</span>" by "<span class="name">rdf:PlainLiteral</span>" (per <a class="external text" href="http://www.w3.org/2007/OWL/meeting/2009-05-27#resolution_2" title="http://www.w3.org/2007/OWL/meeting/2009-05-27#resolution_2">WG resolution</a>).
</li><li> [resolution] Replaced the facet "<span class="name">rdf:langPattern</span>" by "<span class="name">rdf:langRange</span>", following the same replacement in the original <a class="external text" href="http://www.w3.org/TR/2009/WD-rdf-text-20090421/" title="http://www.w3.org/TR/2009/WD-rdf-text-20090421/">rdf:PlainLiteral specification</a>.
</li><li> [correction] Changed the range of the property "<span class="name">owl:annotatedProperty</span>" from IP to IR in order to avoid undesired semantic side effects from annotations. This was an oversight when the original semantic conditions for annotations of axioms and annotations were removed from the document.
</li><li> [nonnormative] The semantic conditions and comprehension conditions for the n-ary property restrictions have been changed to only cover property sequences of length greater than 0, since the meaning of an expression with an empty property set is not clear.
</li><li> [editorial] Explained the optional status of the semantic conditions concerned with the IRI "<span class="name">owl:onProperties</span>", in accordance with the rest of the OWL 2 specification.
</li><li> [editorial] Shortened and clarified some section titles, moved the section on <a href="#Semantic_Conditions_for_Sub_Property_Chains" title="">semantic conditions for sub property chains</a> within <a href="#Semantic_Conditions" title="">Section 5</a>, and aligned the entry order of all tables in <a href="#Appendix:_Comprehension_Conditions_.28Informative.29" title="">Section 8</a> with those in <a href="#Semantic_Conditions" title="">Section 5</a>.
</li><li> [editorial] Several clarifications, minor corrections and cosmetic changes.
</li></ul>
</div>
<a name="Acknowledgments"></a><h2> <span class="mw-headline">11  Acknowledgments </span></h2>
<p>The starting point for the development of OWL 2 was the <a class="external text" href="http://www.w3.org/Submission/2006/10/" title="http://www.w3.org/Submission/2006/10/">OWL1.1 member submission</a>, itself a result of user and developer feedback, and in particular of information gathered during the <a class="external text" href="http://www.webont.org/owled/" title="http://www.webont.org/owled/">OWL Experiences and Directions (OWLED) Workshop series</a>. The working group also considered <a class="external text" href="http://www.w3.org/2001/sw/WebOnt/webont-issues.html" title="http://www.w3.org/2001/sw/WebOnt/webont-issues.html">postponed issues</a> from the <a class="external text" href="http://www.w3.org/2004/OWL/" title="http://www.w3.org/2004/OWL/">WebOnt Working Group</a>.
</p><p>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. Patel-Schneider (Bell Labs Research, Alcatel-Lucent) and
Zhe Wu (Oracle Corporation)
for their thorough reviews.
</p><p>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 Bozen-Bolzano),
Bernardo Cuenca Grau (Oxford University Computing Laboratory),
Martin Dzbor (Open University),
Achille Fokoue (IBM Corporation),
Christine Golbreich (Universit&eacute; de Versailles St-Quentin and LIRMM),
Sandro Hawke (W3C/MIT),
Ivan Herman (W3C/ERCIM),
Rinke Hoekstra (University of Amsterdam),
Ian Horrocks (Oxford University Computing Laboratory),
Elisa Kendall (Sandpiper Software),
Markus Kr&ouml;tzsch (FZI),
Carsten Lutz (Universit&auml;t Bremen),
Deborah L. McGuinness (RPI),
Boris Motik (Oxford University Computing Laboratory),
Jeff Pan (University of Aberdeen),
Bijan Parsia (University of Manchester),
Peter F. Patel-Schneider (Bell Labs Research, Alcatel-Lucent),
Sebastian Rudolph (FZI),
Alan Ruttenberg (Science Commons),
Uli Sattler (University of Manchester),
Michael Schneider (FZI),
Mike Smith (Clark &amp; Parsia),
Evan Wallace (NIST),
Zhe Wu (Oracle Corporation), and
Antoine Zimmermann (DERI Galway).
We would also like to thank past members of the working group:
Jeremy Carroll,
Jim Hendler,
Vipul Kashyap.
</p>
<a name="References"></a><h2> <span class="mw-headline">12  References </span></h2>
<a name="Normative_References"></a><h3> <span class="mw-headline">12.1  Normative References </span></h3>
<dl><dt> <span id="ref-owl-2-specification">[OWL 2 Specification]</span>
</dt><dd><span><cite><a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/">OWL 2 Web Ontology Language: <span>Structural Specification and Functional-Style Syntax</span></a></cite> Boris Motik, Peter F. Patel-Schneider, Bijan Parsia, eds. W3C Recommendation, 27 October 2009, <a href="http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/">http://www.w3.org/TR/2009/REC-owl2-syntax-20091027/</a>.  Latest version available at <a href="http://www.w3.org/TR/owl2-syntax/">http://www.w3.org/TR/owl2-syntax/</a>.</span></dd><dt> <span id="ref-rdf-concepts">[RDF Concepts]</span>
</dt><dd> <cite><a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/">Resource Description Framework (RDF): Concepts and Abstract Syntax</a></cite>. Graham Klyne and Jeremy J. Carroll, eds. W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/.  Latest version available as http://www.w3.org/TR/rdf-concepts/.
</dd><dt> <span id="ref-rdf-semantics">[RDF Semantics]</span>
</dt><dd> <cite><a class="external text" href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/" title="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">RDF Semantics</a></cite>. Patrick Hayes, ed., W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-mt-20040210/.  Latest version available as http://www.w3.org/TR/rdf-mt/.
</dd><dt> <span id="ref-rfc-2119">[RFC 2119]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc2119.txt" title="http://www.ietf.org/rfc/rfc2119.txt">RFC 2119: Key words for use in RFCs to Indicate Requirement Levels</a></cite>. Network Working Group, S. Bradner. IETF, March 1997, http://www.ietf.org/rfc/rfc2119.txt
</dd><dt> <span id="ref-rfc-3987">[RFC 3987]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc3987.txt" title="http://www.ietf.org/rfc/rfc3987.txt">RFC 3987: Internationalized Resource Identifiers (IRIs)</a></cite>. M. Duerst and M. Suignard. IETF, January 2005, http://www.ietf.org/rfc/rfc3987.txt
</dd></dl>
<a name="Nonnormative_References"></a><h3> <span class="mw-headline">12.2  Nonnormative References </span></h3>
<dl><dt> <span id="ref-owl-2-direct-semantics">[OWL 2 Direct Semantics]</span>
</dt><dd><span><cite><a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/">OWL 2 Web Ontology Language: <span>Direct Semantics</span></a></cite> Boris Motik, Peter F. Patel-Schneider, Bernardo Cuenca Grau, eds. W3C Recommendation, 27 October 2009, <a href="http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/">http://www.w3.org/TR/2009/REC-owl2-direct-semantics-20091027/</a>.  Latest version available at <a href="http://www.w3.org/TR/owl2-direct-semantics/">http://www.w3.org/TR/owl2-direct-semantics/</a>.</span></dd><dt> <span id="ref-owl-2-rdf-mapping">[OWL 2 RDF Mapping]</span>
</dt><dd><span><cite><a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/">OWL 2 Web Ontology Language: <span>Mapping to RDF Graphs</span></a></cite> Peter F. Patel-Schneider, Boris Motik, eds. W3C Recommendation, 27 October 2009, <a href="http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/">http://www.w3.org/TR/2009/REC-owl2-mapping-to-rdf-20091027/</a>.  Latest version available at <a href="http://www.w3.org/TR/owl2-mapping-to-rdf/">http://www.w3.org/TR/owl2-mapping-to-rdf/</a>.</span></dd><dt> <span id="ref-owl-1-rdf-semantics">[OWL 1 RDF-Compatible Semantics]</span>
</dt><dd> <cite><a class="external text" href="http://www.w3.org/TR/owl-semantics/rdfs.html" title="http://www.w3.org/TR/owl-semantics/rdfs.html">OWL Web Ontology Language: Semantics and Abstract Syntax, Section 5. RDF-Compatible Model-Theoretic Semantics</a></cite>. Peter F. Patel-Schneider, Patrick Hayes, and Ian Horrocks, eds., W3C Recommendation, 10 February 2004.
</dd><dt> <span id="ref-rfc-2396">[RFC 2396]</span>
</dt><dd> <cite><a class="external text" href="http://www.ietf.org/rfc/rfc2396.txt" title="http://www.ietf.org/rfc/rfc2396.txt">RFC 2396 - Uniform Resource Identifiers (URI): Generic Syntax</a></cite>. T. Berners-Lee, R. Fielding, U.C. Irvine and L. Masinter. IETF, August 1998.
</dd></dl>


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