discontinued / server · Files

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      width="72" /></a> 
  <h1 id="htitle">RDF Semantics </h1>
  <h2>W3C Recommendation 10 February 2004</h2>


      

  <dl>
    <dt >This Version:</dt>
    <dd><a
        href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">http://www.w3.org/TR/2004/REC-rdf-mt-20040210/</a></dd>
    <dt>Latest Version:</dt>
    <dd><a
        href="http://www.w3.org/TR/rdf-mt/">http://www.w3.org/TR/rdf-mt/</a></dd>
    <dt>Previous Version:</dt>
    <dd><a href="http://www.w3.org/TR/2003/PR-rdf-mt-20031215/">http://www.w3.org/TR/2003/PR-rdf-mt-20031215/</a></dd>
    <dt>Editor:</dt>
    <dd><a href="http://www.ihmc.us/users/user.php?UserID=42">Patrick Hayes</a> (IHMC)&lt;
	<a href="mailto:phayes@ihmc.us">phayes@ihmc.us</a>&gt;</dd>
    <dt>Series Editor</dt>
    <dd><a href="http://www-uk.hpl.hp.com/people/bwm/">Brian McBride</a> (Hewlett 
      Packard Labs)&lt;<a
        href="mailto:bwm@hplb.hpl.hp.com">bwm@hplb.hpl.hp.com</a>&gt;</dd>
  </dl>

      






<p>Please refer to the <a
href="http://www.w3.org/2001/sw/RDFCore/errata#rdf-mt"><strong>errata</strong></a>
for this document, which may include some normative corrections.</p>

<p>See also <a href="http://www.w3.org/2001/sw/RDFCore/translation/rdf-mt">translations</a>.</p>

<p class="copyright"><a
      href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a>
      &#xa9; 2004 <a href="http://www.w3.org/"><acronym
      title="World Wide Web Consortium">W3C</acronym></a><sup>&#xae;</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>, <a
      href="http://www.w3.org/Consortium/Legal/copyright-documents">document
      use</a> and <a
      href="http://www.w3.org/Consortium/Legal/copyright-software">software
      licensing</a> rules apply.</p>
  </div>
    <hr />
  <h2 id="abstract">Abstract</h2>
<p>This is a specification of a precise semantics, <span >and 
  corresponding complete systems of inference rules,</span> for the Resource Description 
  Framework (RDF) and RDF Schema (RDFS).</p>

<div class="status">

<h2 class="nonum">
<a id="status" name="status">Status of this Document</a>
</h2>


<!-- Start Status-Of-This-Document Text -->

<p>This document has been reviewed by W3C Members and other interested
parties, and it has been endorsed by the Director as a <a
href="http://www.w3.org/2003/06/Process-20030618/tr.html#RecsW3C">W3C
Recommendation</a>.  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.</p>

<p>This is one document in a <a
href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Introduction">set
of six</a> (<a
href="http://www.w3.org/TR/2004/REC-rdf-primer-20040210/">Primer</a>,
<a
href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/">Concepts</a>,
<a
href="http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/">Syntax</a>,
<a
href="http://www.w3.org/TR/2004/REC-rdf-mt-20040210/">Semantics</a>,
<a
href="http://www.w3.org/TR/2004/REC-rdf-schema-20040210/">Vocabulary</a>,
and <a
href="http://www.w3.org/TR/2004/REC-rdf-testcases-20040210/">Test
Cases</a>) intended to jointly replace the original Resource
Description Framework specifications, <a
href="http://www.w3.org/TR/1999/REC-rdf-syntax-19990222/">RDF Model and Syntax (1999
Recommendation)</a> and <a
href="http://www.w3.org/TR/2000/CR-rdf-schema-20000327/">RDF Schema
(2000 Candidate Recommendation)</a>.  It has been developed by the <a
href="http://www.w3.org/2001/sw/RDFCore/">RDF Core Working Group</a>
as part of the <a href="http://www.w3.org/2001/sw/">W3C Semantic Web
Activity</a> (<a href="http://www.w3.org/2001/sw/Activity">Activity
Statement</a>, <a
href="http://www.w3.org/2002/11/swv2/charters/RDFCoreWGCharter">Group
Charter</a>) for publication on 10 February 2004.
</p>

<p>Changes to this document since the <a
href="http://www.w3.org/TR/2003/PR-rdf-mt-20031215/"
shape="rect">Proposed Recommendation Working Draft</a> are detailed in
the <a href="#changes" shape="rect">change log</a>.  </p>

<p> The public is invited to send comments to <a
href="mailto:www-rdf-comments@w3.org">www-rdf-comments@w3.org</a> (<a
href="http://lists.w3.org/Archives/Public/www-rdf-comments/">archive</a>)
and to participate in general discussion of related technology on <a
href="mailto:www-rdf-interest@w3.org"
shape="rect">www-rdf-interest@w3.org</a> (<a
href="http://lists.w3.org/Archives/Public/www-rdf-interest/"
shape="rect">archive</a>).  </p>

<p>A list of <a href="http://www.w3.org/2001/sw/RDFCore/impls">
implementations</a> is available.</p>

<p>The W3C maintains a list of <a href="http://www.w3.org/2001/sw/RDFCore/ipr-statements" 
rel="disclosure">any patent disclosures related to this work</a>.</p>

<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> 

<!-- End Status-Of-This-Document Text -->

</div>


<h2 id="contents">Table of Contents</h2>
<p class="toc"><a href="#prelim">0. Introduction</a><a href="#intro"> </a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#intro">0.1 Specifying a formal semantics: scope and limitations</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#graphsyntax">0.2 Graph Syntax</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#graphdefs">0.3 Graph Definitions</a><br />
  <a href="#sinterp">1. Interpretations </a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#technote">1.1 Technical Note (Informative)</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#urisandlit">1.2 URI references, Resources and Literals</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#interp">1.3 Interpretations</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#gddenot">1.4 Denotations of Ground Graphs</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#unlabel">1.5 Blank nodes as Existential variables</a><br />
  <a href="#entail">2. Simple Entailment between RDF graphs </a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#vocabulary_entail">2.1 Vocabulary interpretations and vocabulary entailment</a><br />
  <a href="#InterpVocab">3. Interpreting the RDF vocabulary </a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#RDFINTERP">3.1 RDF Interpretations</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#rdf_entail">3.2 RDF Entailment</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#ReifAndCont">3.3 Reification, Containers, Collections and rdf:value</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#Reif">3.3.1 Reification</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#Containers">3.3.2 RDF Containers</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#collections">3.3.3 RDF Collections</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#rdfValue">3.3.4 rdf:value</a><br />
  <a href="#rdfs_interp">4. Interpreting the RDFS Vocabulary</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#RDFSINTERP">4.1 RDFS Interpretations</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#ExtensionalDomRang">4.2 Extensional Semantic Conditions (Informative)</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#literalnote">4.3 A Note on rdfs:Literal</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#rdfs_entailment">4.4 RDFS Entailment</a><br />
  <a href="#dtype_interp">5. Interpreting Datatypes</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#DTYPEINTERP">5.1 Datatyped Interpretations</a><br />
  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<a href="#D_entailment">5.2 D-Entailment</a><br />
  <a href="#MonSemExt">6. Monotonicity of Semantic Extensions</a><br />
  <a href="#rules">7. Entailment Rules (Informative)</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#simpleRules">7.1 Simple Entailment Rules</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#RDFRules">7.2 RDF Entailment Rules</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#RDFSRules">7.3 RDFS Entailment Rules</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;<a href="#RDFSExtRules">7.3.1 Extensional Entailment Rules</a><br />
  &nbsp;&nbsp; &nbsp;&nbsp;<a href="#DtypeRules">7.4 Datatype Entailment Rules</a><br />
  <a href="#prf">Appendix A. Proofs of lemmas (Informative)</a><br />
  <a href="#gloss">Appendix B. Glossary (Informative)</a><br />
  <a href="#ack">Appendix C. Acknowledgements</a><br />
  <a href="#refs">References</a><br />
  <a href="#change">Appendix D. Change Log (Informative)</a></p>
<h2><a name="prelim" id="prelim">0. Introduction</a> </h2>

    <h3><a name="intro" id="intro">0.1 Specifying a formal semantics:
    scope and limitations</a></h3>

    
<p>RDF is an assertional language intended to be used to express <a
    href="#glossProposition" class="termref">propositions</a> using precise formal 
  vocabularies, particularly those specified using RDFS [<cite><a href="#ref-rdf-vocabulary">RDF-VOCABULARY</a></cite>], 
  for access and use over the World Wide Web, and is intended to provide a basic 
  foundation for more advanced assertional languages with a similar purpose. The 
  overall design goals emphasise generality and precision in expressing propositions 
  about any topic, rather than conformity to any particular processing model: 
  see the <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/">RDF Concepts document</a> [<cite><a
    href="#ref-rdf-concepts">RDF-CONCEPTS</a></cite>] for more discussion. </p>
<p>Exactly what is considered to be the 'meaning' of an <a
    href="#glossAssertion" class="termref">assertion</a> in RDF or RDFS in some 
  broad sense may depend on many factors, including social conventions, comments 
  in natural language or links to other content-bearing documents. Much of this 
  meaning will be inaccessible to machine processing and is mentioned here only 
  to emphasize that the <a href="#glossFormal"
    class="termref">formal</a> <a href="#glossSemantic"
    class="termref">semantics</a> described in this document is not intended to 
  provide a full analysis of 'meaning' in this broad sense; that would be a large 
  research topic. The semantics given here restricts itself to a <a href="#glossFormal"
    class="termref">formal</a> notion of meaning which could be characterized 
  as the part that is common to all other accounts of meaning, and can be captured 
  in mechanical <a
    href="#glossInference" class="termref">inference</a> rules. </p>
<p>This document uses a basic technique called <a href="#glossModeltheory"
    class="termref">model theory</a> for specifying the semantics of a formal 
  language. Readers unfamiliar with model theory may find the <a href="#gloss">glossary</a> 
  in appendix B helpful; throughout the text, uses of terms in a technical sense 
  are linked to their glossary definitions. Model theory assumes that the language 
  refers to a '<a href="#glossWorld"
    class="termref">world</a>', and describes the minimal conditions that a world 
  must satisfy in order to assign an appropriate meaning for every expression 
  in the language. A particular <a
    class="termref" href="#glossWorld" >world</a> is called an <i><a
    href="#glossInterpretation" class="termref">interpretation</a>,</i> so that 
  <a href="#glossModeltheory" class="termref">model theory</a> might be better 
  called 'interpretation theory'. The idea is to provide an abstract, mathematical 
  account of the properties that any such interpretation must have, making as 
  few assumptions as possible about its actual nature or intrinsic structure, 
  thereby retaining as much generality as possible. The chief utility of a formal 
  semantic theory is not to provide any deep analysis of the nature of the things 
  being described by the language or to suggest any particular processing model, 
  but rather to provide a technical way to determine when inference processes 
  are <a href="#glossValid"
    class="termref">valid</a>, i.e. when they preserve truth. This provides the 
  maximal freedom for implementations while preserving a globally coherent notion 
  of meaning.</p>
<p>Model theory tries to be <a href="#glossMetaphysical"
    class="termref">metaphysically</a> and <a href="#glossOntological"
    class="termref">ontologically</a> neutral. It is typically couched in the 
  language of set theory simply because that is the normal language of mathematics 
  - for example, this semantics assumes that names denote things in a <i>set</i> 
  IR called the '<a
    href="#glossUniverse" class="termref">universe</a>' - but the use of set-theoretic 
  language here is not supposed to imply that the things in the universe are set-theoretic 
  in nature. Model theory is usually most relevant to implementation via the notion 
  of <a
    href="#glossEntail" class="termref">entailment</a>, described later, which 
  makes it possible to define <a href="#glossValid"
    class="termref">valid</a> <a href="#glossInference"
    class="termref">inference</a> rules.</p>
<p>An alternative way to specify a semantics is to give a translation from RDF 
  into a formal logic with a <a href="#glossModeltheory" class="termref">model 
  theory</a> already attached, as it were. This 'axiomatic semantics' approach 
  has been suggested and used previously with various alternative versions of 
  the target logical language [<cite><a
    href="#ref-ConKla">Conen&amp;Klapsing</a></cite>] [<cite><a
    href="#ref-MarSaa">Marchiori&amp;Saarela</a></cite>] [<cite><a
    href="#ref-daml-axiomat">McGuinness&amp;al</a></cite>]. Such a translation 
  for RDF and RDFS is also given in the L<sub>base</sub> specification [<cite><a href="#ref-Lbase">LBASE</a></cite>]. 
  The axiomatic semantics style has some advantages for machine processing and 
  may be more readable, but in the event that any axiomatic semantics fails to 
  conform to the model-theoretic semantics described in this document, the model 
  theory should be taken as normative. </p>
<p>There are several aspects of meaning in RDF which are ignored by this semantics; 
  in particular, it treats URI references as simple names, ignoring aspects of 
  meaning encoded in particular URI forms [<cite><a href="#ref-2369">RFC 2396</a></cite>] 
  and does not provide any analysis of time-varying data or of changes to URI 
  references. It does not provide any analysis of <a href="#glossIndexical"
    class="termref">indexical</a> uses of URI references, for example to mean 
  'this document'. Some parts of the RDF and RDFS vocabularies are not assigned 
  any formal meaning, and in some cases, notably the reification and container 
  vocabularies, it assigns less meaning than one might expect. These cases are 
  noted in the text and the limitations discussed in more detail. RDF is an assertional 
  <a href="#glossLogic"
    class="termref">logic</a>, in which each triple expresses a simple <a href="#glossProposition" class="termref">proposition</a>. 
  This imposes a fairly strict <a href="#glossMonotonic"
    class="termref">monotonic</a> discipline on the language, so that it cannot 
  express closed-world assumptions, local default preferences, and several other 
  commonly used <a
    href="#glossNonmonotonic" class="termref">non-monotonic</a> constructs.</p>

    
<p><a id="DefSemanticExtension" name="DefSemanticExtension"></a> Particular uses 
  of RDF, including as a basis for more expressive languages such as DAML+OIL 
  [<cite><a href="#ref-daml">DAML</a></cite>] and OWL [<cite><a
    href="#ref-owl">OWL</a></cite>], may impose further semantic conditions in 
  addition to those described here, and such extra semantic conditions can also 
  be imposed on the meanings of terms in particular RDF vocabularies.<a name="RDFSemanticExtension" id="RDFSemanticExtension"></a> 
  Extensions or dialects of RDF which are obtained by imposing such extra semantic 
  conditions may be referred to as <i>semantic extensions</i> of RDF. Semantic 
  extensions of RDF are constrained in this recommendation using the keywords 
  <strong title="MUST in RFC 2119 context" class="RFC2119">MUST</strong> , <strong title="MUST NOT in RFC 2119 context" class="RFC2119">MUST 
  NOT</strong>, <strong title="SHOULD in RFC 2119 context" class="RFC2119">SHOULD</strong> 
  and <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> of 
  [<cite><a href="#ref-2119">RFC 2119</a></cite>]. Semantic extensions of RDF 
  <strong title="MUST in RFC 2119 context" class="RFC2119">MUST</strong> conform 
  to the semantic conditions for simple interpretations described in <a href="#interp">sections 
  1.3</a> <a href="#gddenot"> and 1.4</a> <a href="#unlabel">and 1.5</a> and those 
  for RDF interpretations described in <a href="#RDFINTERP">section 3.1</a> of 
  this document. Any name for entailment in a semantic extension <strong title="SHOULD in RFC 2119 context" class="RFC2119">SHOULD</strong> 
  be indicated by the use of a <a href="#vocabulary_entail" class="termref">vocabulary 
  entailment</a> term. The semantic conditions imposed on an RDF semantic extension 
  <strong title="MUST in RFC 2119 context" class="RFC2119">MUST</strong> define 
  a notion of <a href="#vocabulary_entail"  class="termref">vocabulary entailment</a> 
  which is <a href="#glossValid"
    class="termref">valid</a> according to the model-theoretic semantics described 
  in the normative parts of this document; except that if the semantic extension 
  is defined on some syntactically restricted subset of <a href="#defgraph" class="termref">RDF 
  graphs</a>, then the semantic conditions need only apply to this subset. Specifications 
  of such syntactically restricted semantic extensions <strong title="MUST in RFC 2119 context" class="RFC2119">MUST</strong> 
  include a specification of their syntactic conditions which are sufficient to 
  enable software to distinguish unambiguously those <a href="#defgraph" class="termref">RDF 
  graphs</a> to which the extended semantic conditions apply. Applications based 
  on such syntactically restricted semantic extensions <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> 
  treat <a href="#defgraph" class="termref">RDF graphs</a> which do not conform 
  to the required syntactic restrictions as syntax errors.</p>
<p>An example of a semantic extension of RDF is RDF Schema [<cite><a href="#ref-rdf-vocabulary">RDF-VOCABULARY</a></cite>], 
  abbreviated as RDFS, the semantics of which are defined in later parts of this 
  document. RDF Schema imposes no extra syntactic restrictions.</p>

    <h3><a name="graphsyntax" id="graphsyntax">0.2 Graph
    Syntax</a></h3>

    
<p>Any semantic theory must be attached to a syntax. This semantics is defined 
  as a mapping on the <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax">abstract 
  syntax</a> of RDF described in the RDF concepts and abstract syntax document 
  [<cite><a href="#ref-rdf-concepts">RDF-CONCEPTS</a></cite>]. This document uses 
  the following terminology defined there: <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-URIref"><em>URI 
  reference</em></a>, <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal"><em> 
  literal</em></a>, <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal"><em>plain 
  literal</em></a>, <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal"><em>typed 
  literal</em></a>, <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#dfn-rdf-XMLLiteral"><em>XML 
  literal</em></a>, <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#XMLLiteral-value-space"> 
  <em>XML value,</em></a> <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-rdf-graph"><em>node</em></a>, 
  <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-blank-nodes"><em>blank 
  node</em></a>, <a
    href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#xtocid103646"><em>triple</em></a> 
  <span >and <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-syntax"><em>RDF 
  graph</em></a></span>. Throughout this document we use the term 'character string' 
  or 'string' to refer to a sequence of Unicode characters, and 'language tag' 
  in the sense of RFC 3066, c.f. <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal">section 
  6.5</a> in [<cite><a href="#ref-rdf-concepts">RDF-CONCEPTS</a></cite>]. Note 
  that strings in an RDF graph <strong title="SHOULD in RFC 2119 context" class="RFC2119">SHOULD</strong> 
  be in Normal Form C. </p>
<p >This document uses the <a
    href="http://www.w3.org/TR/rdf-testcases/#ntriples">N-Triples</a> syntax described 
  in the RDF test cases document [<cite><a href="#ref-rdf-tests">RDF-TESTS</a></cite>] 
  to describe <a href="#defgraph" class="termref">RDF graphs</a>. This notation 
  uses a <a
    href="http://www.w3.org/TR/rdf-testcases/#bNode">node identifier</a> (nodeID) 
  convention to indicate blank nodes in the triples of a graph. <a id="nodeIDnote"
    name="nodeIDnote"></a> While node identifiers such as '<code>_:xxx</code>' 
  serve to identify blank nodes in the surface syntax, these expressions are not 
  considered to be the label of the graph node they identify; they are not names, 
  and do not occur in the actual graph. In particular, the <a href="#defgraph" class="termref">RDF 
  graphs</a> described by two <a
    href="http://www.w3.org/TR/rdf-testcases/#ntriples">N-Triples documents</a> 
  which differ only by re-naming their node identifiers will be understood to 
  be <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-graph-equality">equivalent</a> 
  .<span > <a name="lcc7R1" id="lcc7R1"></a>This re-naming convention should be 
  understood as applying only to whole documents, since re-naming the node identifiers 
  in part of a document may result in a document describing a different <a href="#defgraph" class="termref">RDF 
  graph</a>. </span></p>
    <p>The N-Triples syntax requires that URI references be given in full,
    enclosed in angle brackets. In the interests of brevity,  the
    imaginary URI scheme 'ex:' is used to provide illustrative examples. To
    obtain a more realistic view of the normal appearance of the
    N-Triples syntax, the reader should imagine this replaced with
    something like '<code>http://www.example.org/rdf/mt/artificial-example/</code>'.
    The QName prefixes
    <code>rdf:</code>, <code>rdfs:</code> and <code>xsd:</code> are defined
    as follows:</p>

    <p>Prefix <code>rdf:</code> namespace URI:
    <code>http://www.w3.org/1999/02/22-rdf-syntax-ns#</code></p>

    <p>Prefix <code>rdfs:</code> namespace URI:
    <code>http://www.w3.org/2000/01/rdf-schema#</code></p>

    <p>Prefix <code>xsd:</code> namespace URI:
    <code>http://www.w3.org/2001/XMLSchema#</code></p>

    <p>Since QName syntax is not legal N-Triples syntax, and in the
    interests of brevity and readability, examples use the convention
    whereby a QName is used without surrounding angle brackets to
    indicate the corresponding URI reference enclosed in angle brackets, e.g.
    the triple</p>

    <p><code>&lt;ex:a&gt; rdf:type rdfs:Class .</code></p>

    <p>should be read as an abbreviation for the N-Triples syntax</p>

    <p><code>&lt;ex:a&gt;
    &lt;http://www.w3.org/1999/02/22-rdf-syntax-ns#type&gt;
    &lt;http://www.w3.org/2000/01/rdf-schema#Class&gt; .</code></p>
<p >In stating general semantic conditions, single characters 
  or character sequences without a colon indicate an arbitrary name, blank node, 
  character string and so on. The exact meaning will be specified in context.</p>
    <h3><a name="graphdefs" id="graphdefs">0.3 Graph
    Definitions</a></h3>

    
<p><a name="defgraph" id="defgraph">An <span > <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#dfn-rdf-graph"><em>RDF 
  graph</em></a></span>, or simply a <em>graph</em>, is a set of RDF triples.</a></p>
<p><a name="defsubg" id="defsubg">A <i>subgraph</i> of an RDF graph is a subset 
  of the triples in the graph.</a> A triple is identified with the singleton set 
  containing it, so that each triple in a graph is considered to be a subgraph. 
  A <em>proper</em> subgraph is a proper subset of the triples in the graph. </p>

    
<p><a name="defgd" id="defgd">A <em>ground</em> RDF graph is one with no blank 
  nodes.</a></p>

    
<p><a name="defname" id="defname">A <em>name</em> is a URI reference or a literal.</a> 
  These are the expressions that need to be assigned a meaning by an <a href="#glossInterpretation"
    class="termref">interpretation</a>. Note that a typed literal <span >comprises</span> 
  two <a href="#defname"  class="termref">name</a>s: itself and its internal type 
  URI reference. </p>
<p><a name="defvocab" id="defvocab"></a> A set of <a href="#defname"  class="termref">name</a>s 
  is referred to as a <i>vocabulary</i>. The vocabulary <em>of</em> a graph is 
  the set of names which occur as the subject, predicate or object of any triple 
  in the graph. Note that URI references which occur only inside typed literals 
  are not required to be in the vocabulary of the graph.</p>
<p><a name="definst" id="definst"> Suppose that M is a mapping from a set of blank 
  nodes to some set of literals, blank nodes and URI references; then any graph obtained 
  from a graph G by replacing some or all of the blank nodes N in G by M(N) is 
  an <em>instance</em> of G.</a> Note that any graph is an instance of itself, 
  <span >an instance of an instance of G is an instance of G,</span> 
  and if H is an instance of G then every triple in H is an instance of some triple 
  in G.</p>
<p><a name="definstvoc" id="definstvoc">An instance <i>with respect to a vocabulary</i> 
  V </a>is an <a href="#definst" class="termref">instance</a> in which all the 
  <a href="#defname" class="termref">name</a>s in the instance that were substituted 
  for blank nodes in the original are <a href="#defname" class="termref">name</a>s 
  from V.</p>
<p><a name="defpropinst" id="defpropinst">A <i>proper</i> instance</a> of a graph 
  is an instance in which a blank node has been replaced by a name, or two blank 
  nodes in the graph have been mapped into the same node in the instance. </p>
<p >Any instance of a graph in which a blank node is mapped to a new blank node 
  not in the original graph is an instance of the original and also has it as 
  an instance, and this process can be iterated so that any 1:1 mapping between 
  blank nodes defines an instance of a graph which has the original graph as an 
  instance. Two such graphs, each an instance of the other but neither a proper 
  instance, which differ only in the identity of their blank nodes, are considered 
  to be <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-graph-equality">equivalent</a>. 
  We will treat such equivalent graphs as identical; this allows us to ignore 
  some issues which arise from 're-naming' nodeIDs, and is in conformance with 
  the <a href="#nodeIDnote" class="termref">convention</a> that blank nodes have 
  no label. Equivalent graphs are mutual instances with an invertible instance 
  mapping.</p>
<p ><span ><a id="deflean"
    name="deflean">An RDF graph is <em>lean</em> if it has no instance which is 
  a proper subgraph of the graph.</a> Non-lean graphs have internal redundancy 
  and express the same content as their lean subgraphs. For example, the graph</span></p>
<p ><code>&lt;ex:a&gt; &lt;ex:p&gt; _:x .<br />
  _:y &lt;ex:p&gt; _:x .</code></p>
<p >is not <a
      href="#deflean" class="termref">lean</a>, but</p>
<p ><code>&lt;ex:a&gt; &lt;ex:p&gt; _:x .<br />
  _:x &lt;ex:p&gt; _:x .</code></p>
<p >is <a
      href="#deflean" class="termref">lean</a>. </p>
<p> <a name="defmerge" id="defmerge"></a> A 
  <em>merge</em> of a set of RDF graphs is defined as follows. If the graphs in 
  the set have no blank nodes in common, then the union of the graphs is a merge; 
  if they do share blank nodes, then it is the union of a set of graphs that is 
  obtained by replacing the graphs in the set by equivalent graphs that share 
  no blank nodes. This is often described by saying that the blank nodes have 
  been 'standardized apart'. It 
  is easy to see that any two merges are equivalent, so we will refer to <em>the</em> 
  merge, following the convention on equivalent graphs. 
  Using the convention on equivalent graphs and identity, any graph in the original 
  set is considered to be a subgraph of the merge. </p>
<p>One does not, in general, obtain the merge of a set of graphs by concatenating 
  their corresponding <a
    href="http://www.w3.org/TR/rdf-testcases/#ntriples">N-Triples</a> documents 
  and constructing the graph described by the merged document. If some of the 
  documents use the same node identifiers, the merged document will describe a 
  graph in which some of the blank nodes have been 'accidentally' identified. 
  To merge <a
    href="http://www.w3.org/TR/rdf-testcases/#ntriples">N-Triples</a> documents 
  it is necessary to check if the same nodeID is used in two or more documents, 
  and to replace it with a distinct nodeID in each of them, before merging the 
  documents. Similar cautions apply to merging graphs described by RDF/XML documents 
  which contain nodeIDs, see <a
      href="http://www.w3.org/TR/2002/WD-rdf-syntax-grammar-20021108/">RDF/XML 
  Syntax Specification (Revised)</a> [<cite><a href="#ref-rdf-syntax">RDF-SYNTAX</a></cite>].</p>
<h2><a id="sinterp" name="sinterp"> 1. Interpretations</a> </h2>

    
<h3><a name="technote" id="technote">1.1 Technical note (Informative)</a></h3>
<p>RDF does not impose any logical restrictions on the domains and ranges of properties; 
  in particular, a property may be applied to itself. When <a href="#glossClass" class="termref">classes</a> 
  are introduced in RDFS, they may contain themselves. Such 'membership loops' 
  might seem to violate the axiom of foundation, one of the axioms of standard 
  (Zermelo-Fraenkel) set theory, which forbids infinitely descending chains of 
  membership. However, the semantic model given here distinguishes properties 
  and classes considered as objects from their <a id="defexten"
    name="defexten"><i>extensions</i> - the sets of object-value pairs which satisfy 
  the property, or things that are 'in' the class</a> - thereby allowing the extension 
  of a property or class to contain the property or class itself without violating 
  the axiom of foundation. In particular, this use of a class extension mapping 
  allows classes to contain themselves. For example, it is quite OK for (the extension 
  of) a 'universal' class to contain the class itself as a member, a convention 
  that is often adopted at the top of a classification hierarchy. (If an extension 
  contained itself then the axiom would be violated, but that case never arises.) 
  The technique is described more fully in [<cite><a
    href="#ref-HayMen">Hayes&amp;Menzel</a></cite>].</p>
<p>In this respect, RDFS differs from many conventional ontology frameworks such 
  as UML which assume a more structured hierarchy of individuals, sets of individuals, 
  etc., or which draw a sharp distinction between data and meta-data. However, 
  while RDFS does not assume the existence of such structure, it does not prohibit 
  it. RDF allows membership loops, but it does not mandate their use for all parts 
  of a user vocabulary. If this aspect of RDFS is found worrying, then it is possible 
  to restrict oneself to a subset of RDF graphs which do not contain any such 
  'loops' of class membership or property application while retaining much of 
  the expressive power of RDFS for many practical purposes, <span >and semantic 
  extensions may impose syntactic conditions which forbid such looped constructions. 
  </span></p>
<p>The use of the explicit extension mapping also makes it possible for two properties 
  to have exactly the same values, or two classes to contain the same instances, 
  and still be distinct entities. This means that RDFS classes can be considered 
  to be rather more than simple sets; they can be thought of as 'classifications' 
  or 'concepts' which have a robust notion of identity which goes beyond a simple 
  <a href="#glossExtensional"
    class="termref">extensional</a> correspondence. This property of the <a href="#glossModeltheory" class="termref">model 
  theory</a> has significant consequences in more expressive languages built on 
  top of RDF, such as OWL [<cite><a
    href="#ref-owl">OWL</a></cite>], which are capable of expressing identity 
  between properties and classes directly. This '<a href="#glossIntensional" class="termref">intensional</a>' 
  nature of classes and properties is sometimes claimed to be a useful property 
  of a descriptive language, but a full discussion of this issue is beyond the 
  scope of this document.</p>
  
 <p>Notice that the question of whether or not a class contains itself as a member is quite different 
 from the question of whether or not it is a subclass of itself. All classes are subclasses of themselves.</p>
	
<p>Readers who are familiar with conventional logical semantics may find it useful 
  to think of RDF as a version of existential binary relational logic in which 
  relations are first-class entities in the universe of quantification. Such a 
  logic can be obtained by encoding the relational atom R(a,b) into a conventional 
  logical syntax, using a notional three-place relation Triple(a,R,b); the basic 
  semantics described here can be reconstructed from this intuition by defining 
  the extension of y as the set {&lt;x,z&gt; : Triple(x,y,z)} and noting that 
  this would be precisely the denotation of R in the conventional Tarskian model 
  theory of the original form R(a,b) of the relational atom. This construction 
  can also be traced in the semantics of the L<sub>base</sub> axiomatic description 
  [<cite><a href="#ref-Lbase">LBASE</a></cite>].</p>

   

    <h3><a name="urisandlit" id="urisandlit">1.2 URI references, Resources and
    Literals</a>.</h3>

    <p>This document does  not take any position  on the way that URI references
      may be composed from other expressions, e.g. from relative URIs or QNames;
    the semantics simply assumes that such lexical issues have been
    resolved in some way that is globally coherent, so that a single
    URI reference can be taken to have the same meaning wherever it occurs.
    Similarly, the semantics has no special provision for tracking
    temporal changes. It assumes, implicitly, that URI references have the
    same meaning <em>whenever</em> they occur. To provide an adequate
    semantics which would be sensitive to temporal changes is a
    research problem which is beyond the scope of this document.</p>

    <p>The semantics does not assume any particular relationship
    between the denotation of a URI reference and a document or Web
    resource which can be retrieved by using that URI reference in an HTTP
    transfer protocol, or any entity which is considered to be the
    source of such documents. Such a requirement could be added as a
    semantic extension, but the formal semantics described here makes
    no assumptions about any connection between the denotations of
    URI references and the uses of those URI references in other protocols.</p>
<p >The semantics treats all RDF <a href="#defname" class="termref">name</a>s 
  as expressions which denote. The things denoted are called 'resources', following 
  [<cite><a
    href="#ref-2369">RFC 2396</a></cite>], but no assumptions are made here about 
  the nature of <a href="#glossResource"
    class="termref">resources</a>; 'resource' is treated here as synonymous with 
  'entity',<span > i.e. as a generic term for anything in the 
  universe of discourse. </span> </p>
<p >The different syntactic forms of <a href="#defname" class="termref">name</a>s 
  are treated in particular ways. URI references are treated simply as logical 
  constants. Plain literals are considered to denote themselves, so have a fixed 
  meaning. The denotation of a typed literal is the value mapped from its enclosed 
  character string by the datatype associated with its enclosed type. RDF assigns 
  a particular meaning to literals typed with <code>rdf:XMLLiteral</code>, <a href="#defXMLdatatype" class="termref">described 
  </a> in <a href="#InterpVocab" class="termref">section 3</a>. </p>
    <h3><a name="interp" id="interp">1.3 Interpretations</a></h3>

    
<p>The basic intuition of model-theoretic semantics is that asserting a sentence 
  makes a claim about the <a href="#glossWorld"
    class="termref">world</a>: it is another way of saying that the world is, 
  in fact, so arranged as to be an <a
    href="#glossInterpretation" class="termref">interpretation</a> which makes 
  the sentence true. In other words, an assertion amounts to stating a <i>constraint</i> 
  on the <i>possible</i> ways the world might be. Notice that there is no presumption 
  here that any assertion contains enough information to specify a single unique 
  interpretation. It is usually impossible to assert enough in any language to 
  completely constrain the interpretations to a single possible world, so there 
  is no such thing as 'the' unique interpretation of an RDF graph. 
  In general, the larger an RDF graph is - the more it says about the world - 
  then the smaller the set of interpretations that an <a href="#glossAssertion"
    class="termref">assertion</a> of the graph allows to be true - the fewer the 
  ways the world could be, while making the asserted graph true of it.</p>

    <p>The following definition of an interpretation is couched in
    mathematical language, but what it amounts to intuitively is that
    an interpretation provides just enough information about a possible
    way the world might be - a 'possible world' - in order to fix the
    truth-value (true or false) of any ground RDF triple. It does this
    by specifying for each URI reference, what it is supposed to be a name of;
    and also, if it is used to indicate a property, what values that
    property has for each thing in the <a href="#glossUniverse"
    class="termref">universe</a>; and if it is used to indicate a
    <a href="#defDatatype" class="termref">datatype</a>, that the <a href="#defDatatype" class="termref">datatype</a> defines a mapping between
    lexical forms and datatype values. This is just enough information
    to fix the truth-value of any <a href="#defgd"
    class="termref">ground</a> triple, and hence any ground RDF
    graph. (Non-ground
    graphs are considered in the following section.) Note that if any of
    this information were omitted, it would be possible for some <a
    href="#glossWellformed" class="termref">well-formed</a> triple to
    be left without a determinate value; and also that any other
    information - such as the exact nature of the things in the <a
    href="#glossUniverse" class="termref">universe</a> - would,
    regardless of its intrinsic interest, be irrelevant to the actual
    truth-values of any triple.</p>

    
<p>All interpretations will be relative to a set of <a href="#defname" class="termref">name</a>s, 
  called the vocabulary of the interpretation; so that one should speak, strictly, 
  of an interpretation of an RDF vocabulary, rather than of RDF itself. Some interpretations 
  may assign special meanings to the symbols in a particular vocabulary. Interpretations 
  which share the special meaning of a particular vocabulary will be named for 
  that vocabulary, e.g. '<a href="#rdfinterpdef" class="termref">rdf-interpretation</a>s', 
  '<a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s', etc. An 
  interpretation with no particular extra conditions on a vocabulary (including 
  the RDF vocabulary itself) will be called a <i>simple</i> interpretation, or 
  simply an interpretation. </p>

    
<p>RDF uses several forms of literal. The chief semantic characteristic of literals 
  is that their meaning is largely determined by the form of the string they contain. 
  Plain literals, without an embedded type URI reference, are always interpreted 
  as referring to themselves: either a character string or a pair consisting of 
  a character string and a <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Graph-Literal" class="termref">language 
  tag</a>; in either case, the character string is referred to as the &quot;literal 
  character string&quot;. In the case of typed literals, however, the full specification 
  of the meaning depends on being able to access <a href="#defDatatype" class="termref">datatype</a> 
  information which is external to RDF itself. A full discussion of the meaning 
  of typed literals is described in <a href="#dtype_interp">section 5</a> , where 
  a special notion of datatype interpretation is introduced. Each interpretation 
  defines a mapping IL from typed literals to their interpretations. Stronger 
  conditions on IL will be defined as the notion of 'interpretation' is extended 
  in later sections. </p>
<p>Throughout this document, precise semantic conditions will be set out in tables 
  which state semantic conditions, tables containing true assertions and <a href="#glossValid"
    class="termref">valid</a> inference rules, and tables listing syntax, which 
  are distinguished by background color. These tables, taken together, amount 
  to a formal summary of the entire semantics. Note that the semantics of RDF 
  does not depend on that of RDFS. The full semantics of RDF is defined in sections 
  <a href="#sinterp">1</a> and <a href="#InterpVocab">3</a> ; the full semantics 
  of RDFS in sections <a href="#sinterp">1</a>, <a href="#InterpVocab">3</a> and 
  <a href="#rdfs_interp">4</a>.</p>
  <div class="title">Definition of a simple interpretation.</div>
<table border="1" summary="definition of a simple interpretation">
  <tr>
        <td class="semantictable"><p>A <i>simple</i> <em>interpretation</em> I of a vocabulary V is
            defined by:</p>
          <p>1. A non-empty set IR of resources, called the domain or universe
            of I.</p>
      <p>2. <span >A set IP, called the set of properties of I.</span></p>
          <p>3. A mapping IEXT from IP into the powerset of IR x IR i.e. the
            set of sets of pairs &lt;x,y&gt; with x and y in IR .</p>
      <p>4. A mapping IS from URI references in V into (IR union IP)</p>
      <p>5. A mapping IL from typed literals in V into IR.</p>
      <p >6. A distinguished subset LV of IR, called the set of literal values, 
        which contains all the plain literals in V</p></td>
  </tr>
  </table>
<p>IEXT(x), called
      the <a href="#defexten" class="termref"><i>extension</i></a> of x, is a
      set of pairs which identify the arguments for which the property is true,
      that is, a binary
      relational extension.
      This trick of distinguishing a relation as an object from its
      relational extension allows a property to occur in its own
      extension, as <a href="#technote" class="termref">noted earlier</a>.</p>
<p>The assumption that LV is a subset of IR amounts to saying that literal values 
  are thought of as real entities that 'exist'. This amounts to saying that literal 
  values are resources. However, this does not imply that literals should be identified 
  with URI references. Note that LV may contain other items in addition to plain 
  literals. There is a technical reason why the range of IL is IR rather than 
  restricted to LV. When interpretations take account of <a href="#defDatatype" class="termref">datatype</a> 
  information, it is syntactically possible for a typed literal to be internally 
  inconsistent, and such ill-typed literals are required to denote a <em>non</em>-literal 
  value, as <a href="#illformedliteral" class="termref">explained</a> in <a href="#dtype_interp" class="termref">section 
  5</a>.</p>
<p>The next sections define how an interpretation of a vocabulary determines the 
  truth-values of any RDF graph, by a recursive definition of the denotation - 
  the semantic "value" - of any RDF expression in terms of those of its immediate 
  subexpressions. These apply to all subsequent semantic extensions. RDF has two 
  kinds of denotation: <a href="#defname" class="termref">name</a>s denote things 
  in the universe, and sets of triples denote truth-values. </p>
    <h3><a name="gddenot" id="gddenot">1.4 Denotations of Ground
    Graphs</a></h3>

    
<p>The denotation of a ground RDF graph in I is given recursively by the following 
  rules, which extend the interpretation mapping I from <a href="#defname" class="termref">name</a>s 
  to ground graphs. These rules (and extensions of them given later) work by defining 
  the denotation of any piece of RDF syntax E in terms of the denotations of the 
  immediate syntactic constituents of E, hence allowing the denotation of any 
  piece of RDF to be determined by a kind of syntactic recursion.</p>
<p>In this table, and throughout this document, the equality sign = indicates 
  identity and angle brackets &lt;x,y&gt; are used to indicate an ordered pair 
  of x and y. RDF graph syntax is indicated using the notational conventions of 
  the <a
    href="http://www.w3.org/TR/rdf-testcases/#ntriples">N-Triples</a> syntax described 
  in the RDF test cases document [<cite><a href="#ref-rdf-tests">RDF-TESTS</a></cite>]: 
  literal strings are encloded within double quote marks, language tags indicated 
  by the use of the <code>@</code> sign, and triples terminate with a 'code dot' 
  <code>.</code> . </p>

  <div class="title">Semantic conditions for ground graphs.</div>
  <table cellpadding="5" border="2" summary="semantic conditions for RDF graphs">
        <tbody>
          <tr>
            
      <td  class="semantictable">if E is a plain literal &quot;aaa&quot; in V 
        then I(E) = aaa</td>
          </tr>
          <tr>
            
      <td class="semantictable">if E is a plain literal &quot;aaa&quot;<code>@</code>ttt 
        in V then I(E) = &lt;aaa, ttt&gt;</td>
          </tr>

          <tr>
            
        
      <td class="semantictable">if E is a typed literal in V then I(E) = IL(E)</td>
          </tr>

          <tr>
            
      <td class="semantictable">if E is a URI reference in V then I(E) = IS(E)</td>
          </tr>

          <tr>
            
        <td class="semantictable"><p>if E is a ground triple s p o<code>.</code> 
          then I(E) = true if </p>
        <p>s, p and o are in V, <span >I(p) is in IP and</span> &lt;I(s),I(o)&gt; 
          is in IEXT(I(p))</p>
          <p>otherwise I(E)= false.</p></td>
          </tr>

          <tr>
            <td class="semantictable">if E is a ground RDF graph then I(E) = false if I(E') =
            false for some triple E' in E, otherwise I(E) =true.</td>
          </tr>
        </tbody>
  </table>

<p>If the vocabulary of an RDF graph contains names that are not in the vocabulary 
  of an interpretation I - that is, if I simply does not give a semantic value 
  to some <a href="#defname" class="termref">name</a> that is used in the graph 
  - then these truth-conditions will always yield the value false for some triple 
  in the graph, and hence for the graph itself. Turned around, this means that 
  any assertion of a graph implicitly asserts that all the <a href="#defname" class="termref">name</a>s 
  in the graph actually refer to something in the world. The final condition implies 
  that an empty graph (an empty set of triples) is trivially true.</p>
<p>Note that the denotation of plain literals is always in LV; <span >and that 
  those of the subject and object of any true triple must be in IR; so any URI 
  reference which occurs in a graph both as a predicate and as a subject or object 
  must denote something in the intersection of IR and IP in any interpretation 
  which satisfies the graph.</span></p>

    
<p>As an illustrative example, the following is a small interpretation for the 
  artificial vocabulary {<code>ex:a</code>,<code> ex:b</code>,<code> ex:c</code>, 
  <code>&quot;whatever&quot;</code>, <code>&quot;whatever&quot;^^ex:b</code>}. 
  Integers are used to indicate the non-literal 'things' in the universe. This 
  is not meant to imply that interpretations should be interpreted as being about 
  arithmetic, but more to emphasize that the exact nature of the things in the 
  universe is irrelevant. <span >LV can be any set satisfying the semantic conditions.</span> 
  (In this and subsequent examples the greater-than and less-than symbols are 
  used in several ways: following mathematical usage to indicate abstract pairs 
  and n-tuples; following N-Triples syntax to enclose URI references, and also 
  as arrowheads when indicating mappings.)</p>
<p>IR = LV union{1, 2}</p>
<p>IP={1}</p>

    <p>IEXT: 1<code>=&gt;</code>{&lt;1,2&gt;,&lt;2,1&gt;}</p>
<p>IS: <code>ex:a=&gt;</code>1, <code>ex:b=&gt;</code>1, <code>ex:c=&gt;</code>2</p>
<p>IL: <code>&quot;whatever&quot;^^ex:b</code> <code>=&gt;</code>2 </p>

    <div class="c1">
      <p><img src="RDFSemanticsFigure1.jpg"
      alt="A drawing of the domains and mappings described in the text"
       /><br />
       <b>Figure 1</b>: An example of an interpretation. Note, this is
      not a picture of an RDF graph.<br />
       The figure does not show the infinite number of members of
      LV.</p>
    </div>
<p>This interpretation makes these triples true:</p>

    <p><code>&nbsp;&nbsp;&lt;ex:a&gt; &lt;ex:b&gt; &lt;ex:c&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:c&gt; &lt;ex:a&gt; &lt;ex:a&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:c&gt; &lt;ex:b&gt; &lt;ex:a&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:a&gt; &lt;ex:b&gt;
    "whatever"^^&lt;ex:b&gt; .</code></p>

    <p>For example, I(<code>&lt;ex:a&gt; &lt;ex:b&gt; &lt;ex:c&gt;
    .</code>) = true if
    &lt;I(<code>ex:a</code>),I(<code>ex:c</code>)&gt; is in
    IEXT(I(<code>&lt;ex:b&gt;</code>)), i.e. if &lt;1,2&gt; is in
    IEXT(1), which is {&lt;1,2&gt;,&lt;2,1&gt;} and so does contain
    &lt;1,2&gt; and so I(<code>&lt;ex:a &lt;ex:b&gt; ex:c&gt;</code>)
    is true.</p>
<p>The truth of the fourth <span >triple</span> is a consequence of the rather 
  idiosyncratic interpretation chosen here for typed literals.</p>
<p>In this interpretation IP is a subset of IR; this will be typical of RDF semantic 
  interpretations, but is not required. </p>

    <p>It makes these triples false:</p>

    <p><code>&nbsp;&nbsp;&lt;ex:a&gt; &lt;ex:c&gt; &lt;ex:b&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:a&gt; &lt;ex:b&gt; &lt;ex:b&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:c&gt; &lt;ex:a&gt; &lt;ex:c&gt;
    .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:a&gt; &lt;ex:b&gt; "whatever"
    .</code></p>

    
<p>For example, I(<code>&lt;ex:a&gt; &lt;ex:c&gt; &lt;ex:b&gt; .</code>) = true 
  if &lt;I(<code>ex:a</code>), I(<code>&lt;ex:b&gt;</code>)&gt;, i.e.&lt;1,1&gt;, 
  is in IEXT(I(<code>ex:c</code>)); but I(<code>ex:c</code>)=2 which is not in 
  IP, so IEXT is not defined on 2, so the condition fails and I(<code>&lt;ex:a&gt; 
  &lt;ex:c&gt; &lt;ex:b&gt; .</code>) = false.</p>
<p>It also makes all triples containing a plain literal false, since the property 
  extension does not have any pairs containing a plain literal.</p>
<p>To emphasize; this is only one possible interpretation of this vocabulary; 
  there are (infinitely) many others. For example, if this interpretation were 
  modified by attaching the property extension to 2 instead of 1, none of the 
  above triples would be true.</p>
<p>This example illustrates that any interpretation which maps any URI reference 
  which occurs in the predicate position of a triple in a graph to something not 
  in IP will make the graph false. </p>

    <h3><a name="unlabel" id="unlabel">1.5. Blank Nodes as Existential
    Variables</a></h3>

    
<p>Blank nodes are treated as simply indicating the existence of a thing, without 
  using, or saying anything about, the name of that thing. (This is not the same 
  as assuming that the blank node indicates an 'unknown' URI reference; for example, 
  it does not assume that there is any URI reference which refers to the thing. 
  The discussion of <a
    href="#glossSkolemization" class="termref">Skolemization</a> in <a href="#prf">appendix 
  A</a> is relevant to this point.)</p>
<p>An interpretation can specify the truth-value of a graph containing blank nodes. 
  This will require some definitions, as the theory so far provides no meaning 
  for blank nodes. Suppose I is an interpretation and A is a mapping from some 
  set of blank nodes to the universe IR of I, and define I+A to be an extended 
  interpretation which is like I except that it uses A to give the interpretation 
  of blank nodes. Define blank(E) to be the set of blank nodes in E. Then the 
  above rules can be extended to include the two new cases that are introduced 
  when blank nodes occur in the graph:</p>

    <div  class="title">Semantic conditions for blank nodes.</div>
      <table cellpadding="5" border="2" summary="Semantic conditions for blank nodes">
        <tbody>
          <tr>
            
      <td class="semantictable">If E is a blank node and A(E) is defined then 
        [I+A](E) = A(E)</td>
          </tr>

          <tr>
            <td class="semantictable">If E is an RDF graph then I(E) = true if [I+A'](E) =
            true for some mapping A' from blank(E) to IR, otherwise
            I(E)= false.</td>
          </tr>
        </tbody>
      </table>
<p class="termref">Notice that this does not change the definition of an interpretation; 
  it still consists of the same values IR, IP, IEXT, IS, LV and IL. It simply 
  extends the rules for defining denotations under an interpretation, so that 
  the same interpretation that provides a truth-value for ground graphs also assigns 
  truth-values to graphs with blank nodes, even though it provides no denotation 
  for the blank nodes themselves. Notice also that the blank nodes themselves 
  are perfectly well-defined entities; they differ from other nodes only in not 
  being assigned a denotation by an interpretation, reflecting the intuition that 
  they have no 'global' meaning (i.e. outside the graph in which they occur).</p>

    
<p><a name="bnodeExample" id="bnodeExample"></a>For example, the graph defined 
  by the following triples is false in the interpretation shown in figure 1:</p>

    <p>&nbsp;<code>&nbsp;_:xxx &lt;ex:a&gt; &lt;ex:b&gt; .</code></p>

    <p><code>&nbsp;&nbsp;&lt;ex:c&gt; &lt;ex:b&gt; _:xxx .</code></p>

    <p>since if A' maps the blank node to 1 then the first triple is
    false in I+A', and if it maps it to 2 then the second triple is
    false.</p>

    <p>Note that each of these triples, if thought of as a single
    graph, would be true in I, but the whole graph is not; and that if
    a different nodeID were used in the two triples, indicating that
    the RDF graph had two blank nodes instead of one, then A' could map
    one node to 2 and the other to 1, and the resulting graph would be
    true under the interpretation I.</p>
    
<p>This effectively treats all blank nodes as having the same meaning as existentially 
  quantified variables in the RDF graph in which they occur, and which have the 
  scope of the entire graph. In terms of the N-Triples syntax, this amounts to 
  the convention that would place the quantifiers just outside, or at the outer 
  edge of, the N-Triples document corresponding to the graph. This in turn means 
  that there is a subtle but important distinction in meaning between the operation 
  of forming the union of two graphs and that of forming the <a href="#defmerge" class="termref">merge</a>. 
  The simple union of two graphs corresponds to the conjunction ( 'and' ) of all 
  the triples in the graphs, maintaining the identity of any blank nodes which 
  occur in both graphs. This is appropriate when the information in the graphs 
  comes from a single source, or where one is derived from the other by means 
  of some <a href="#glossValid"
    class="termref">valid</a> inference process, as for example when applying 
  an inference rule to add a triple to a graph. Merging two graphs treats <span >the 
  blank nodes in each graph as being existentially quantified in that graph</span>, 
  so that no blank node from one graph is allowed to stray into the scope of the 
  other graph's surrounding quantifier. This is appropriate when the graphs come 
  from different sources and there is no justification for assuming that a blank 
  node in one refers to the same entity as any blank node in the other. </p>
<h2><a name="entail" id="entail">2. Simple Entailment between RDF graphs</a> </h2>
<p>Following conventional terminology, I <a
    id="defsatis" name="defsatis"></a><i>satisfies</i> E if I(E)=true, and a set 
  S of RDF graphs <a id="defentail"
    name="defentail"></a><em>(simply)</em> <a href="#glossEntail"
    class="termref"><em>entails</em></a> <span >a graph</span> E if every interpretation 
  which satisfies every member of S also satisfies E. In later sections these 
  notions will be adapted to other classes of interpretations, but throughout 
  this section 'entailment' should be interpreted as meaning simple entailment. 
</p>

    <p>Entailment is the key idea which connects model-theoretic
    semantics to real-world applications. As noted earlier, making an
    assertion amounts to claiming that the world is an interpretation
    which assigns the value true to the assertion. If A entails B, then
    any interpretation that makes A true also makes B true, so that an
    assertion of A already contains the same "meaning" as an assertion
    of B; one could say that the meaning of B is somehow contained in,
    or subsumed by, that of A. If A and B entail each other, then they
    both "mean" the same thing, in the sense that asserting either of
    them makes the same claim about the world. The interest of this
    observation arises most vividly when A and B are different
    expressions, since then the relation of entailment is exactly the
    appropriate semantic license to justify an application inferring or
    generating one of them from the other. Through the notions of
    satisfaction, entailment and validity, formal semantics gives a
    rigorous definition to a notion of "meaning" that can be related
    directly to computable methods of determining whether or not
    meaning is preserved by some transformation on a representation of
    knowledge.</p>

    <p><a id="defvalid" name="defvalid">Any process which constructs a
    graph E from some other graph(s) S is said to be <em>(simply)
    valid</em> if S entails E </a><span >in every case</span>,
    otherwise <em>invalid.</em> Note
    that being an invalid process does not mean that the conclusion is
    false, and being <a href="#glossValid"
    class="termref">valid</a> does not guarantee truth. However, validity
    represents the best guarantee that any assertional language can
    offer: if given true inputs, it will never draw a false conclusion
    from them.</p>
<p>This section gives a few basic results about simple entailment and <a href="#glossValid"
    class="termref">valid</a> <a href="#glossInference"
    class="termref">inference</a>. Simple entailment can be recognized by relatively 
  simple syntactic comparisons. The two basic forms of simply valid inference 
  in RDF are, in logical terms, the inference from (P and Q) to P, and the inference 
  from foo(baz) to (exists (?x) foo(?x)).</p>

    
<p>These results apply only to simple entailment, not to the extended notions 
  of entailment introduced in later sections. Proofs, all of which are straightforward, 
  are given in <a href="#prf" class="termref">appendix A</a>, which also describes some other 
  properties of entailment which may be of interest.</p>
<p><strong>Empty Graph Lemma.</strong> The empty set of triples is entailed by 
  any graph, and does not entail any graph except itself. <a href="#emptygraphlemmaprf" class="termref">[Proof]</a></p>
<p><a name="subglem" id="subglem"><strong>Subgraph Lemma.</strong></a> A graph 
  entails all its <a
    href="#defsubg" class="termref">subgraphs</a>. <a href="#subglemprf" class="termref">[Proof]</a></p>
<p><a name="instlem" id="instlem"><strong>Instance Lemma.</strong></a> A graph 
  is entailed by any of its <a
    href="#definst" class="termref">instances</a>.<a href="#instlemprf" class="termref">[Proof]</a></p>
<p >The relationship between merging and entailment is simple, 
  and obvious from the definitions:</p>
<p><a name="mergelem" id="mergelem"><strong>Merging lemma.</strong></a> The merge 
  of a set S of RDF graphs is entailed by S, and entails every member of S. <a href="#mergelemprf" class="termref">[Proof]</a></p>

    <p>This means that a set of graphs can be treated as <a
    href="#glossEquivalent" class="termref">equivalent</a> to its
    merge, i.e. a single graph, as far as the <a
    href="#glossModeltheory" class="termref">model theory</a> is
    concerned. This can
    be used to simplify the terminology somewhat: for example, the definition
    of
    S entails E, above, can be paraphrased by saying that S entails E when every
    interpretation which satisfies S also
    satisfies E.</p>
    <p >The <a href="#bnodeExample" >example</a> given in section 1.5 shows  that it is not the case, in general,
      that the simple union of a set of graphs is entailed by the set.</p>
    <p>The main result for simple RDF inference is:</p>
<p><a name="interplemma" id="interplemma"><strong>Interpolation Lemma.</strong> 
  S entails a graph E if and only if a subgraph of S is an instance of E.</a><a href="#interplemmaprf" class="termref">[Proof]</a></p>
<p>The interpolation lemma completely characterizes simple RDF entailment in syntactic 
  terms. To tell whether a set of RDF graphs entails another, <span >check that 
  there is some instance of the entailed graph which is a subset of the merge 
  of the original set of graphs</span>. Of course, there is no need to actually 
  construct the merge. If working backwards from the <a
    href="#glossConsequent" class="termref">consequent E</a>, an efficient technique 
  would be to treat blank nodes as variables in a process of subgraph-matching, 
  allowing them to bind to 'matching' <a href="#defname" class="termref">name</a>s 
  in the <a href="#glossAntecedent"
    class="termref">antecedent</a> graph(s) in S, i.e. those which may entail 
  the <a href="#glossConsequent"
    class="termref">consequent</a> graph. The interpolation lemma shows that this 
  process is <a href="#glossValid"
    class="termref">valid</a>, and is also <a href="#glossComplete"
    class="termref">complete</a> if the subgraph-matching algorithm is. The existence 
  of <a href="#glossComplete"
    class="termref">complete</a> subgraph-checking algorithms also shows that 
  RDF entailment is decidable, i.e. there is a terminating algorithm which will 
  determine for any finite set S and any graph E, whether or not S entails E.</p>
<p>Such a variable-binding process would only be appropriate when applied to the 
  <em>conclusion</em> of a proposed entailment. This corresponds to using the 
  document as a goal or a query, in contrast to asserting it, i.e. claiming it 
  to be true. If an RDF document is asserted, then it would be invalid to bind 
  new values to any of its blank nodes, since the resulting graph might not be 
  entailed by the assertion.</p>
<p>The interpolation lemma has an immediate consequence a criterion for non-entailment:</p>
<p><a name="Anonlem1" id="Anonlem1"><strong>Anonymity lemma.</strong></a> Suppose 
  E is a <a href="#deflean"
    class="termref">lean</a> graph and E' is a proper instance of E. Then E does 
  not entail E'.<a href="#Anonlem1prf" class="termref">[Proof]</a></p>
<p>Note again that this applies only to simple entailment, not to the vocabulary 
  entailment relationships defined in rest of the document.</p>
    
<p>Several basic properties of entailment follow directly from the above definitions 
  and results but are stated here for completeness sake:</p>
<p><strong><a name="monotonicitylemma" id="monotonicitylemma"></a>Monotonicity 
  Lemma</strong>. Suppose S is a subgraph of S' and S entails E. Then S' entails 
  E.<a href="#monotonicitylemmaprf" class="termref">[Proof]</a></p>
<p>The property of finite expressions always being derivable from a finite set 
  of antecedents is called <em>compactness</em>. Semantic theories which support 
  non-compact notions of entailment do not have corresponding computable inference 
  systems.</p>
<p><strong><a name="compactlemma" id="compactlemma"></a>Compactness Lemma</strong>. 
  If S entails E and E is a finite graph, then some finite subset S' of S entails 
  E.</p>
<h3><a name="vocabulary_entail" id="vocabulary_entail"></a><span >2.1 
  Vocabulary interpretations and vocabulary entailment</span></h3>
<p >Simple interpretations and simple entailment capture the semantics of RDF 
  graphs when no attention is paid to the particular meaning of any of the names 
  in the graph. To obtain the full meaning of an RDF graph written using a particular 
  vocabulary, it is usually necessary to add further semantic conditions which 
  attach stronger meanings to particular URI references and typed literals in 
  the graph. Interpretations which are required to satisfy extra semantic conditions 
  on a particular vocabulary will be generically referred to as <em>vocabulary 
  interpretations</em>. Vocabulary entailment means entailment with respect to 
  such vocabulary interpretations. These stronger notions of interpretation and 
  entailment are indicated by the use of a namespace prefix, so that we will refer 
  to <em>rdf-entailment</em>, <em>rdfs-entailment</em> and so on in what follows. 
  In each case, the vocabulary whose meaning is being restricted, and the exact 
  conditions associated with that vocabulary, are spelled out in detail. </p>
<h2><a name="InterpVocab" id="InterpVocab">3. Interpreting the RDF Vocabulary</a> </h2>
<h3><a name="RDFINTERP" id="RDFINTERP">3.1 RDF Interpretations</a></h3>

    <p ><a name="defRDFV" id="defRDFV"></a>The <em>RDF vocabulary, </em>rdfV, is a 
  set of URI references in the <code>rdf:</code> namespace:</p>

    <div class="c1">
      
  <table  border="1" summary="rdf vocabulary">
    <tbody>
          <tr>
            <td class="othertable"><strong>RDF  vocabulary</strong></td>
          </tr>

          <tr>
            
        <td class="othertable">&nbsp;<code>rdf:type</code> &nbsp;&nbsp;<code>rdf:Property 
          rdf:XMLLiteral rdf:nil rdf:List rdf:Statement rdf:subject rdf:predicate 
          rdf:object rdf:first rdf:rest rdf:Seq rdf:Bag rdf:Alt rdf:_1 rdf:_2 
          ... rdf:value</code></td>
          </tr>
    </tbody>
      </table>
  <p><a name="defXMLdatatype" id="defXMLdatatype"></a><a href="#rdfinterpdef" class="termref">rdf-interpretation</a>s 
    impose extra semantic conditions on <a href="#defRDFV" class="termref">rdfV</a> 
    and on typed literals with the type <code>rdf:XMLLiteral</code>, which is 
    referred to as the RDF built-in datatype. This datatype is <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-XMLLiteral">fully 
    described</a> in the <i><a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/">RDF 
    Concepts and Abstract Syntax</a></i> document [<cite><a href="#ref-rdf-concepts">RDF-CONCEPTS</a></cite>]. 
    Any character string <em>sss</em> which satisfies the conditions for being 
    in the <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#XMLLiteral-lexical-space"> 
    lexical space of <code>rdf:XMLLiteral</code></a> will be called a <em>well-typed 
    XML literal string</em>. The corresponding value will be called the <em>XML 
    value of</em> the literal. <span >Note that the XML values of well-typed XML 
    literals are in precise 1:1 correspondence with the XML literal strings of 
    such literals, but are not themselves character strings</span>. An XML literal 
    whose literal string is well-typed will be called a <em>well-typed XML literal</em>; 
    other XML literals will be called <em>ill-typed</em>. </p>
</div>

    
<p> <a id="rdfinterpdef" name="rdfinterpdef"></a>An <i>rdf-interpretation</i> 
  of a vocabulary V is a simple interpretation I <span >of (V union <a href="#defRDFV" class="termref">rdfV</a>)</span> 
  which satisfies the extra conditions described in the following table<span > 
  and all the triples in the subsequent table. These triples are called the <em>rdf 
  axiomatic triples</em>. </span></p>
   
<div class="title">RDF semantic conditions.</div> 
<table  border="1" summary="RDF semantic conditions">
  <tbody>
    <tr> 
      <td class="semantictable"><p><a name="rdfsemcond1" id="rdfsemcond1"></a>x is 
        in IP if and only if &lt;x, I(<code>rdf:Property</code>)&gt; is in IEXT(I(<code>rdf:type</code>))</p></td>
    </tr>
    <tr> 
      <td class="semantictable"><p><a name="rdfsemcond2" id="rdfsemcond2"></a>If 
          <span ><code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code></span> 
          is in V and xxx is a well-typed XML literal string, then </p>
        <p>IL<span >(<code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code>) 
          is the XML <span >value</span> of xxx;</span><br />
          <span >IL(<code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code>) 
          is in LV;<br />
          IEXT(I(<code>rdf:type</code>)) contains &lt;IL(<code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code>), 
          I(<code>rdf:XMLLiteral</code>)&gt;</span></p>
      </td>
    </tr>
    <tr> 
      <td class="semantictable"><p><span > <a name="rdfsemcond3" id="rdfsemcond3"></a>If 
          <span ><code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code></span> 
          is in V and xxx is an ill-typed XML literal string, then</span></p>
        <p><span >IL(<code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code>) 
          is not in LV;<br />
          IEXT(I(<code>rdf:type</code>)) does not contain &lt;IL(<code>&quot;</code>xxx<code>&quot;^^rdf:XMLLiteral</code>), 
          I(<code>rdf:XMLLiteral</code>)&gt;.</span> </p></td>
    </tr>
  </tbody>
</table>
    

  
<p>
  The <a href="#rdfsemcond1" class="termref">first condition</a> could be regarded as defining IP to be the set of 
  resources in the universe of the interpretation which have the value I(<code>rdf:Property</code>) 
  of the property I(<code>rdf:type</code>). Such subsets of the universe will 
  be central in interpretations of RDFS. Note that this condition requires 
  IP to be a subset of IR. The <a href="#rdfsemcond3" class="termref">third condition</a> requires that ill-typed XML literals 
  denote something other than a literal value: this will be the standard way of 
  handling ill-formed typed literals. 
   
  </p>


 
 <div class="title">RDF axiomatic triples.</div> 

  <table  border="1" summary="RDF axiomatic triples">
    <tr>
      <td class="ruletable"><a name="RDF_axiomatic_triples" id="RDF_axiomatic_triples"> </a><code>rdf:type rdf:type rdf:Property .<br />
        rdf:subject rdf:type rdf:Property .<br />
        rdf:predicate rdf:type rdf:Property .<br />
        rdf:object rdf:type rdf:Property .<br />
        rdf:first rdf:type rdf:Property .<br />
        rdf:rest rdf:type rdf:Property .<br />
        rdf:value rdf:type rdf:Property .<br />
        rdf:_1 rdf:type rdf:Property .<br />
        rdf:_2 rdf:type rdf:Property .<br />
        ... <br />
        rdf:nil rdf:type rdf:List .</code></td>
    </tr>
  </table>
  <p >The <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s described
    in <a href="#rdfs_interp">section 4</a> below assign further semantic 
    conditions (range and domain conditions) to the properties used in the RDF 
    vocabulary, and other semantic extensions <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> 
    impose further conditions so as to further restrict their meanings, provided 
    that such conditions<strong title="MUST in RFC 2119 context" class="RFC2119"> 
    MUST</strong> be compatible with the conditions described in this section.</p>

    <p>For example, the following rdf-interpretation extends the simple interpretation 
  in figure 1 to the case <span >where V contains <a href="#defRDFV" class="termref">rdfV</a></span>. 
  For simplicity, we ignore XML literals in this example.</p>
    
<p>IR = <span >LV union </span>{1, 2, T , P}</p>
<p>IP = {1, T}</p>

    <p>IEXT: 1<code>=&gt;</code>{&lt;1,2&gt;,&lt;2,1&gt;},
    T<code>=&gt;</code>{&lt;1,P&gt;,&lt;T,P&gt;}</p>

    <p>IS: <code>ex:a=&gt;</code>1, <code>ex:b=&gt;</code>1,
    <code>ex:c=&gt;</code> 2, <code>rdf:type=&gt;</code>T,
    <code>rdf:Property=&gt;</code>P, <code>rdf:nil=&gt;</code>1,
    <code>rdf:List=&gt;</code>P, &nbsp;<code>rdf:Statement=&gt;</code>P<code>,
    rdf:subject=&gt;</code>1<code>, rdf:predicate=&gt;</code>1<code>, rdf:object=&gt;</code>1<code>,
    rdf:first=&gt;</code>1<code>, rdf:rest=&gt;</code>1<code>, rdf:Seq=&gt;</code>P<code>,
    rdf:Bag=&gt;</code>P<code>, rdf:Alt=&gt;</code>P<code>, rdf:_1, rdf:_2, ...
    =&gt;</code>1</p>
    <div class="c1">
      <p><img src="RDFMTFigure2.jpg"
      alt="A drawing of the domains and mappings described in the text" />
      <br />
       <b>Figure 2</b>: An rdf-interpretation.</p>
    </div>

    <p>This is not the smallest rdf-interpretation which extends the
    earlier example, since one could have made
     IEXT(T) be
    {&lt;1,2&gt;,&lt;T,2&gt;}, and managed without having P in the
    universe. In general, a given entity in an interpretation may play
    several 'roles' at the same time, as long as this can be done
    without violating any of the required semantic conditions. The
    above interpretation identifies properties with lists, for example;
    of course, other interpretations might not make such an
    identification.</p>
<p>Every <a href="#rdfinterpdef" class="termref">rdf-interpretation</a> is also a simple interpretation. The 'extra' structure 
  does not prevent it acting in the simpler role.</p>
<h3><a name="rdf_entail" id="rdf_entail"></a>3.2. RDF entailment</h3>
<p>S <i>rdf-entails</i> E when every rdf-interpretation which satisfies every 
  member of S also satisfies E. This follows the wording of the definition of 
  <a href="#defentail" class="termref">simple entailment</a> in <a href="#entail" > 
  Section 2</a>, but refers only to <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>s 
  instead of all simple interpretations. <span >Rdf-entailment is an example of 
  <a href="#vocabulary_entail" class="termref">vocabulary entailment</a>. </span></p>
<p>It is easy to see that the lemmas in <a href="#entail" > Section 2</a> do not all apply to rdf-entailment: 
  for example, the triple</p>
<p><code>rdf:type rdf:type rdf:Property .</code></p>
<p>is true in every <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>, so is rdf-entailed by the empty graph, 
  contradicting the interpolation lemma for rdf-entailment. <a href="#RDFRules" > Section 7.2</a> describes 
  exact conditions for detecting RDF entailment.</p>
<h3><a name="ReifAndCont" id="ReifAndCont">3.3. Reification, Containers</a>, Collections 
  and rdf:value</h3>
<p>The RDF semantic conditions impose significant formal constraints on the meaning 
  only of the central RDF vocabulary, so the notions of rdf-entailment and <a href="#rdfinterpdef" class="termref">rdf-interpretation</a> 
  apply to the remainder of the vocabulary without further change. This includes 
  vocabulary which is intended for use in describing containers and bounded collections, 
  and a reification vocabulary to enable an RDF graph to describe, as well as 
  exhibit, triples. In this section we review the intended meanings of this vocabulary, 
  and note some intuitive consequences which are not supported by the formal <a
    href="#glossModeltheory" class="termref">model theory</a>. Semantic extensions 
  <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> limit the 
  formal interpretations of these vocabularies to conform to these intended meanings.</p>
<p>The omission of these conditions from the formal semantics is a design decision 
  to accomodate variations in existing RDF usage and to make it easier to implement 
  processes to check formal RDF entailment. For example, implementations may decide 
  to use special procedural techniques to implement the RDF collection vocabulary.</p>
    
<h4><a name="Reif" id="Reif">3.3.1 Reification</a></h4>

    <div class="c1">  
      <table  border="1" summary="reification vocabulary">
        <tbody>
          <tr>
            <td class="othertable"><strong>RDF reification vocabulary</strong></td>
          </tr>
          <tr>
            <td class="othertable"><code>rdf:Statement rdf:subject rdf:predicate
                rdf:object</code></td>
          </tr>
        </tbody>
      </table>
    </div>

    <p>Semantic extensions <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> limit the interpretation of these so
    that a triple of the form</p>

    <p>aaa <code>rdf:type rdf:Statement .</code></p>

    <p>is true in I just when I(aaa) is a <a href="#glossToken"
    class="termref">token</a> of an RDF triple in some RDF document,
    and the three properties, when applied to such a denoted triple,
    have the same values as the respective components of that
    triple.</p>

    <p>This may be illustrated by considering the following two RDF
    graphs, the first of which consists of a single triple.</p>

    <p><code>&lt;ex:a&gt; &lt;ex:b&gt; &lt;ex:c&gt; .</code></p>

    <p>and</p>

    <p><code>_:xxx rdf:type rdf:Statement .<br />
     _:xxx rdf:subject &lt;ex:a&gt; .<br />
     _:xxx rdf:predicate &lt;ex:b&gt; .<br />
     _:xxx rdf:object &lt;ex:c&gt; .</code></p>

    <p>The second graph is called a <i><a href="#glossReify"
    class="termref">reification</a></i> of the triple in the first
    graph, and the node which is intended to refer to the first triple
    - the blank node in the second graph - is called, rather
    confusingly, a <em>reified triple</em>. (This can be a blank node
    or a URI reference.) In the intended interpretation of the reification
    vocabulary, the second graph would be made true in an
    interpretation I by interpreting the reified triple to refer to a
    token of the triple in the first graph in some concrete RDF
    document, considering that token to be valid RDF syntax, and then
    using I to interpret the syntactic triple which the token
    instantiates, so that the subject, predicate and object of that
    triple are interpreted in the same way in the reification as in the
    triple described by the reification. This could be stated formally
    as follows: &lt;x,y&gt; is in IEXT(I(<code>rdf:subject</code>))
    just when x is a token of an RDF triple of the form</p>

    <p>aaa bbb ccc .</p>

    <p>and y is I(aaa); similarly for predicate and object. Notice that
    the value of the <code>rdf:subject</code> property is not the
    subject URI reference itself but its interpretation, and so this condition
    involves a two-stage interpretation process: one has to interpret
    the reified node - the subject of the triples in the reification -
    to refer to another triple, then treat that triple as RDF syntax
    and apply the interpretation mapping again to get to the referent
    of its subject. This requires triple tokens to exist as first-class
    entities in the universe IR of an interpretation. In sum: the
    meaning of the reification is that a document exists containing a
    triple token which means whatever the first graph means.<span >Note
    that this way of understanding the reification vocabulary does not interpret
    reification as a form of quotation. Rather, the reification describes the
    relationship between a token of a triple and the resources that triple refers
    to. The reification can be read intuitively as saying &quot;'this piece of
    RDF talks about these things&quot; rather than &quot;this piece of RDF has
    this form&quot;.</span></p>

    <p>The semantic extension described here requires
    the reified triple that the reification describes -
    I(<code>_:xxx</code>) in the above example - to be a    particular token
    or instance of a triple in a (real
    or notional) RDF document, rather than an 'abstract' triple
    considered as a grammatical form. There could be several such
    entities which have the same subject, predicate and object
    properties. Although a graph is defined as a set of triples,
    several such tokens with the same triple structure might occur in
    different documents. Thus, it would be meaningful to claim that the
    blank node in the second graph above does not refer to the triple
    in the first graph, but to some other triple with the same
    structure. This particular interpretation of reification was chosen
    on the basis of use cases where properties such as dates of
    composition or provenance information have been applied to the
    reified triple, which are meaningful only when thought of as
    referring to a particular instance or token of a triple. </p>
<p>Although RDF applications may use reification to refer to triple tokens in 
  RDF documents, the connection between the document and its reification must 
  be maintained by some means external to <span >the RDF graph 
  syntax. (In the RDF/XML syntax described in <i><a
      href="http://www.w3.org/TR/2003/WD-rdf-syntax-grammar-20030123/">RDF/XML 
  Syntax Specification (Revised)</a></i> [<a href="#ref-rdf-syntax">RDF-SYNTAX</a>], 
  the rdf:ID attribute can be used in the description of a triple to create a 
  reification of that triple in which the reified triple is a URI constructed 
  from the baseURI of the XML document and the value of rdf:ID as a fragment.)</span> 
  Since an assertion of a reification of a triple does not implicitly assert the 
  triple itself, this means that there are <em>no</em> entailment relationships 
  which hold between a triple and a reification of it. Thus the reification vocabulary 
  has no effective semantic constraints on it, other than those that apply to 
  an <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>. </p>

    <p>A reification of a triple does not entail the triple, and is not
    entailed by it. (The
    reification only says that the triple token exists and what it is about,
      not that it is true. The second non-entailment is a consequence of the
      fact
    that asserting a triple does not automatically assert that any
    triple tokens exist in the universe being described by the triple.
    For example, the triple might be part of an ontology describing
    animals, which could be satisfied by an interpretation in which the
    universe contained only animals, and in which a reification of it was therefore
      false.)</p>

    <p>Since the relation between triples and reifications of triples
    in any RDF graph or graphs need not be one-to-one, asserting a
    property about some entity described by a reification need not
    entail that the same property holds of another such entity, even if
    it has the same components. For example,</p>

    <p><code>_:xxx rdf:type rdf:Statement .<br />
     _:xxx rdf:subject &lt;ex:subject&gt; .<br />
     _:xxx rdf:predicate &lt;ex:predicate&gt; .<br />
     _:xxx rdf:object &lt;ex:object&gt; .<br />
     _:yyy rdf:type rdf:Statement .<br />
     _:yyy rdf:subject &lt;ex:subject&gt; .<br />
     _:yyy rdf:predicate &lt;ex:predicate&gt; .<br />
     _:yyy rdf:object &lt;ex:object&gt; .<br />
     _:xxx &lt;ex:property&gt; &lt;ex:foo&gt; .</code></p>

    <p>does not entail</p>

    <p><code>_:yyy &lt;ex:property&gt; &lt;ex:foo&gt; .</code></p>

    
<h4><a name="Containers" id="Containers">3.3.2 RDF containers</a></h4>

    <table border="1" summary="container vocabulary">
      <tbody>
        <tr>
          <td class="othertable"><strong>RDF Container Vocabulary</strong></td>
        </tr>
        <tr>
          <td class="othertable"><code>rdf:Seq rdf:Bag rdf:Alt rdf:_1 rdf:_2
              ...</code></td>
        </tr>
      </tbody>
    </table>
    <p>RDF provides vocabularies for describing three classes of
    containers. Containers have a type, and their members can
    be enumerated by using a fixed set of <em>container membership
    properties</em>. These properties are indexed by integers to
    provide a way to distinguish the members from each other, but these
    indices should not necessarily be thought of as defining an
    ordering of the container itself; some containers are considered to be unordered.</p>

    <p>The RDFS vocabulary, described below, adds a generic membership
    property which holds regardless of position, and classes containing
    all the containers and all the membership properties.</p>

  <p>One should understand this RDF vocabulary as <em>describing</em>
    containers, rather than as a vocabulary for constructing them, as
    would typically be supplied by a programming language. On this
    view, the actual containers are entities in the semantic universe,
    and RDF graphs which use the vocabulary simply provide very basic
    information about these entities, enabling an RDF graph to
    characterize the container type and give partial information about
    the members of a container. Since the RDF container vocabulary is
    so limited, many 'natural' assumptions concerning RDF containers
    are not formally sanctioned by the RDF <a href="#glossModeltheory"
    class="termref">model theory</a>. This should not be taken as
    meaning that these assumptions are false, but only that RDF does
    not formally entail that they must be true.</p>

    <p>There are no special semantic conditions on the container
    vocabulary: the only 'structure' which RDF presumes its containers
    to have is what can be inferred from the use of this vocabulary and
    the <span >general RDF</span> semantic conditions. <span >In
    general, this amounts to knowing the type of a container, and having a partial
    enumeration
    of the items in the container.</span> The intended mode of use is that things
    of type <code>rdf:Bag</code>
    are considered to be unordered but to allow duplicates; things of
    type <code>rdf:Seq</code> are considered to be ordered, and things
    of type <code>rdf:Alt</code> are considered to represent a
    collection of alternatives, possibly with a preference ordering.
    The ordering of items in an ordered container is intended to be
    indicated by the numerical ordering of the container membership
    properties, <span >which are assumed to be single-valued</span>.
    However, these informal interpretations are not reflected in any formal RDF
    entailments.</p>

    
<p>RDF does not support any entailments which could arise from <span >enumerating 
  the elements of an <code>rdf:Bag</code> in a different order</span>. For example,</p>

    <p><code>_:xxx rdf:type rdf:Bag .<br />
     _:xxx rdf:_1 &lt;ex:a&gt; .<br />
     _:xxx rdf:_2 &lt;ex:b&gt; .</code></p>

    <p>does not entail</p>

    <p><code>_:xxx rdf:_1 &lt;ex:b&gt; .<br />
     _:xxx rdf:_2 &lt;ex:a&gt; .</code></p>

    <p>Notice that if this conclusion were <a href="#glossValid"
    class="termref">valid</a>, then the result of
    conjoining it to the original graph would also be a <a href="#glossValid"
    class="termref">valid</a>
    entailment, which would assert that both elements were in both
    positions. This is a consequence of the fact that RDF is a purely
    assertional language.</p>

    <p>There is no assumption that a property of a container applies to
    any of the elements of the container, or vice versa. </p>
    <p>There is no formal requirement that
      the three container classes are disjoint, so that for example
      something can be asserted to be both an <code>rdf:Bag</code> and an <code>rdf:Seq</code>.
      There is no assumption that containers are gap-free, so that for example</p>
    <p><code>_:xxx rdf:type rdf:Seq.<br />
     _:xxx rdf:_1 &lt;ex:a&gt; .<br />
     _:xxx rdf:_3 &lt;ex:c&gt; .</code></p>

    <p>does not entail</p>

    <p><code>_:xxx rdf:_2 _:yyy .</code></p>

    <p>There is no way in RDF to 'close' a container, i.e. to assert
    that it contains only a fixed number of members. This is a
    reflection of the fact that it is always consistent to add a triple
    to a graph asserting a membership property of any container. And
    finally, there is no built-in assumption that an RDF container has
    only finitely many members.</p>

    
<h4><a name="collections" id="collections"></a>3.3.3 RDF collections</h4>

    <table  border="1" summary="collection vocabulary">
      <tbody>
        <tr>
          <td class="othertable"><strong>RDF Collection Vocabulary</strong></td>
        </tr>
        <tr>
          <td class="othertable"><code>rdf:List rdf:first rdf:rest rdf:nil</code></td>
        </tr>
      </tbody>
    </table>
    <p>RDF provides a vocabulary for describing collections, i.e.'list
    structures', in terms of head-tail links. Collections differ from
    containers in allowing branching structure and in having an
    explicit terminator, allowing applications to determine the exact
    set of items in the collection.</p>

  
<p>As with containers, no special semantic conditions are imposed on this vocabulary 
  other than the type of <code>rdf:nil</code> being <code>rdf:List</code>. It 
  is intended for use typically in a context where a container is described using 
  blank nodes to connect a 'well-formed' sequence of items, each described by 
  two triples of the form<code><br />
  <br />
  _:c1 rdf:first aaa .<br />
  _:c1 rdf:rest _:c2</code></p>

    
<p>where the final item is indicated by the use of <code>rdf:nil</code> as the 
  value of the property <code>rdf:rest</code>. In a familiar convention, <code>rdf:nil</code> 
  can be thought of as the empty collection. Any such graph amounts to an assertion 
  that the collection exists, and since the members of the collection can be determined 
  by inspection, this is often sufficient to enable applications to determine 
  what is meant. Note however that the semantics does not require any collections 
  to exist other than those mentioned explicitly in a graph (and the empty collection). 
  For example, the existence of a collection containing two items does not automatically 
  guarantee that the similar collection with the items permuted also exists:<code><br />
  <br />
  _:c1 rdf:first &lt;ex:aaa&gt; .<br />
  _:c1 rdf:rest _:c2 .<br />
  <span > _:c2 rdf:first</span> &lt;ex:bbb&gt; .<br />
  _:c2 rdf:rest rdf:nil . </code></p>
  <p>does not entail</p>
  
<p><code>_:c3 rdf:first &lt;ex:bbb&gt; .<br />
  _:c3 rdf:rest _:c4 .<br />
  <span >_:c4 rdf:first</span> &lt;ex:aaa&gt; .<br />
     _:c4 rdf:rest rdf:nil .
    </code></p>

    <p>Also, RDF imposes no '<a href="#glossWellformed"
    class="termref">well-formedness</a>' conditions on the use of this
    vocabulary, so that it is possible to write RDF graphs which assert
    the existence of highly peculiar objects such as lists with forked
    or non-list tails, or multiple heads:</p>
  
<p><code>_:666 rdf:first &lt;ex:aaa&gt; .<br />
     _:666 rdf:first &lt;ex:bbb&gt; .<br />
     _:666 rdf:rest &lt;ex:ccc&gt; .<br />
  _:666 rdf:rest rdf:nil . </code></p>

    
<p>It is also possible to write a set of triples which underspecify a collection 
  by failing to specify its <code>rdf:rest</code> property value.</p>

    
<p>Semantic extensions <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> 
  place extra syntactic well-formedness restrictions on the use of this vocabulary 
  in order to rule out such graphs. They <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> 
  exclude interpretations of the collection vocabulary which violate the convention 
  that the subject of a 'linked' collection of two-triple items of the form described 
  above, ending with an item ending with <code>rdf:nil</code>, denotes a totally 
  ordered sequence whose members are the denotations of the <code>rdf:first</code> 
  values of the items, in the order got by tracing the <code>rdf:rest</code> properties 
  from the subject to <code>rdf:nil</code>. This permits sequences which contain 
  other sequences.</p>
<p >Note that the RDFS semantic conditions, described below, require that any 
  subject of the <code>rdf:first</code> property, and any subject or object of 
  the <code>rdf:rest</code> property, be of <code>rdf:type rdf:List</code>. </p>

    
<h4><a name="rdfValue" id="rdfValue"></a>3.3.4 rdf:value</h4>
<p>The intended use for <code>rdf:value</code> is <a href="http://www.w3.org/TR/rdf-primer/#rdfvalue">explained 
  intuitively</a> in the RDF Primer
  document [<cite><a href="#ref-rdf-primer">RDF-PRIMER</a></cite>]. It is typically 
  used to identify a 'primary' or 'main' value of a property which has several 
  values, or has as its value a complex entity with several facets or properties 
  of its own.</p>
<p>Since the range of possible uses for <code>rdf:value</code> is so wide, it 
  is difficult to give a precise statement which covers all the intended meanings 
  or use cases. Users are cautioned, therefore, that <span >the 
  meaning of <code>rdf:value</code> may vary from application to application</span>. 
  In practice, the intended meaning is often clear from the context, but may be 
  lost when graphs are merged or when conclusions are inferred.</p>

    
<h2><a name="rdfs_interp" id="rdfs_interp">4. Interpreting the RDFS Vocabulary</a></h2>

    
<h3><a name="RDFSINTERP" id="RDFSINTERP">4.1 RDFS Interpretations</a></h3>

<p>RDF Schema [<cite><a href="#ref-rdf-vocabulary">RDF-VOCABULARY</a></cite>] 
  extends RDF to include a larger <a id="defRDFSV" name="defRDFSV"></a>vocabulary 
  rdfsV with more complex semantic constraints:</p>

    <div class="c1">
      <table border="1" summary="RDFS vocabulary">
        <tbody>
          <tr>
            <td class="othertable"><strong>RDFS vocabulary</strong></td>
          </tr>

          <tr>
            <td class="othertable"><code>rdfs:domain rdfs:range rdfs:Resource rdfs:Literal
            rdfs:Datatype rdfs:Class rdfs:subClassOf rdfs:subPropertyOf
            rdfs:member rdfs:Container rdfs:ContainerMembershipProperty
            rdfs:comment rdfs:seeAlso rdfs:isDefinedBy
            rdfs:label</code></td>
          </tr>
        </tbody>
      </table>
    </div>
<p>(<code>rdfs:comment</code>,<code> rdfs:seeAlso</code>, <code>rdfs:isDefinedBy</code> 
  and <code>rdfs:label</code> are included here because some constraints which 
  apply to their use can be stated using <code>rdfs:domain</code>,<code> rdfs:range</code> 
  and <code>rdfs:subPropertyOf</code>. Other than this, the formal semantics does 
  not assign them any particular meanings.)</p>
<p>Although not strictly necessary, it is convenient to state the RDFS semantics 
  in terms of a new semantic construct, a '<a
    href="#glossClass" class="termref">class</a>', i.e. a resource which represents 
  a set of things in the universe which all have that class as the value of their 
  <code>rdf:type</code> property. Classes are defined to be things of type <code>rdfs:Class</code>, 
  and <span >the set of all classes in an interpretation will be called IC</span>. 
  The semantic conditions are stated in terms of a mapping ICEXT (for the <em>C</em>lass 
  <em>Ext</em>ension in I) from IC to the set of subsets of IR. The meanings of 
  ICEXT and IC in a <a href="#rdfinterpdef" class="termref">rdf-interpretation</a> 
  of the RDFS vocabulary are completely defined by the first two conditions in 
  the table of RDFS semantic conditions, below. Notice that a class may have an 
  empty class extension; that (as <a class="termref" href="#technote">noted</a> 
  earlier) two different class entities could have the same class extension; and 
  that the class extension of <code>rdfs:Class</code> contains the class <code>rdfs:Class</code>. 
</p>
    
<p><a id="rdfsinterpdef" name="rdfsinterpdef"></a>An <i>rdfs-interpretation</i> 
  of V is an <a href="#rdfinterpdef" class="termref">rdf-interpretation</a> I 
  <span >of (V union </span><a href="#defRDFV" class="termref">rdfV</a> union 
  <a href="#defRDFSV" class="termref">rdfsV)</a> which satisfies the following 
  semantic conditions and all the triples in the subsequent table, called the<em> 
  RDFS axiomatic triples</em>.</p>
  
<div class="title">RDFS semantic conditions.</div>
  <table  border="1">
    <tr> 
      
    <td class="semantictable"> <p><a name="rdfssemcond1" id="rdfssemcond1"></a>x 
        is in ICEXT(y) if and only if &lt;x,y&gt; is in IEXT(I(<code>rdf:type</code>))</p>
        <p>IC = ICEXT(I(<code>rdfs:Class</code>))</p>
        <p>IR = ICEXT(I(<code>rdfs:Resource</code>))</p>
        <p >LV = ICEXT(I(<code>rdfs:Literal</code>)) </p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"> <p><a name="rdfssemcond2" id="rdfssemcond2"></a>If 
        &lt;x,y&gt; is in IEXT(I(<code>rdfs:domain</code>)) and &lt;u,v&gt; is 
        in IEXT(x) then u is in ICEXT(y)</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"> <p><a name="rdfssemcond3" id="rdfssemcond3"></a>If 
        &lt;x,y&gt; is in IEXT(I(<code>rdfs:range</code>)) and &lt;u,v&gt; is 
        in IEXT(x) then v is in ICEXT(y)</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"><p><a name="rdfssemcond4" id="rdfssemcond4"></a>IEXT(I(<code>rdfs:subPropertyOf</code>)) 
      is transitive and reflexive on IP</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"> <p><a name="rdfssemcond5" id="rdfssemcond5"></a>If 
        &lt;x,y&gt; is in IEXT(I(<code>rdfs:subPropertyOf</code>)) then x and 
        y are in IP and IEXT(x) is a subset of IEXT(y)</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"><p><a name="rdfssemcond6" id="rdfssemcond6"></a>If 
        x is in IC then &lt;x, I(<code>rdfs:Resource</code>)&gt; is in IEXT(I(<code>rdfs:subClassOf</code>))</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"> <p><a name="rdfssemcond7" id="rdfssemcond7"></a>If 
        &lt;x,y&gt; is in IEXT(I(<code>rdfs:subClassOf</code>)) then x and y are 
        in IC and ICEXT(x) is a subset of ICEXT(y)</p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"><p><a name="rdfssemcond8" id="rdfssemcond8"></a>IEXT(I(<code>rdfs:subClassOf</code>)) 
      is transitive and reflexive on IC</p></td>
    </tr>
    <tr> 
      <td class="semantictable"><p><a name="rdfssemcond9" id="rdfssemcond9"></a>If 
        x is in ICEXT(I(<code>rdfs:ContainerMembershipProperty</code>)) then:<br />
        &lt; x, I(<code>rdfs:member</code>)&gt; is in IEXT(I(<code>rdfs:subPropertyOf</code>))<br />
      </p></td>
    </tr>
    <tr> 
      
    <td class="semantictable"><p><a name="rdfssemcond10" id="rdfssemcond10"></a>If 
        x is in ICEXT(I(<code>rdfs:Datatype</code>)) then <span >&lt;x, 
        I(<code>rdfs:Literal</code>)&gt; is in IEXT(I(<code>rdfs:subClassOf</code>))</span></p></td>
    </tr>
  </table>
 
    <p><a id="RDFS_axiomatic_triples" name="RDFS_axiomatic_triples">  </a>
	</p>
	  <div class="title">RDFS axiomatic triples.</div>
  <table  border="1" summary="RDFS axioms">
        
          <tr>
            
        
    <td class="ruletable"> <code>rdf:type rdfs:domain rdfs:Resource .<br />
      rdfs:domain rdfs:domain rdf:Property .<br />
      rdfs:range rdfs:domain rdf:Property .<br />
      rdfs:subPropertyOf rdfs:domain rdf:Property .<br />
      <a name="axtripleforproof1" id="axtripleforproof1"></a>rdfs:subClassOf rdfs:domain 
      rdfs:Class .<br />
      rdf:subject rdfs:domain rdf:Statement .<br />
      rdf:predicate rdfs:domain rdf:Statement .<br />
      rdf:object rdfs:domain rdf:Statement .<br />
      rdfs:member rdfs:domain rdfs:Resource . <br />
      rdf:first rdfs:domain rdf:List .<br />
      rdf:rest rdfs:domain rdf:List .<br />
      rdfs:seeAlso rdfs:domain rdfs:Resource .<br />
      rdfs:isDefinedBy rdfs:domain rdfs:Resource .<br />
      rdfs:comment rdfs:domain rdfs:Resource .<br />
      rdfs:label rdfs:domain rdfs:Resource .<br />
      rdf:value rdfs:domain rdfs:Resource .<br />
      <br />
      rdf:type rdfs:range rdfs:Class .<br />
      rdfs:domain rdfs:range rdfs:Class .<br />
      rdfs:range rdfs:range rdfs:Class .<br />
      rdfs:subPropertyOf rdfs:range rdf:Property .<br />
      <a name="axtripleforproof2" id="axtripleforproof2"></a>rdfs:subClassOf rdfs:range 
      rdfs:Class .<br />
      rdf:subject rdfs:range rdfs:Resource .<br />
      rdf:predicate rdfs:range rdfs:Resource .<br />
      rdf:object rdfs:range rdfs:Resource .<br />
      rdfs:member rdfs:range rdfs:Resource .<br />
      rdf:first rdfs:range rdfs:Resource .<br />
      rdf:rest rdfs:range rdf:List .<br />
      rdfs:seeAlso rdfs:range rdfs:Resource .<br />
      rdfs:isDefinedBy rdfs:range rdfs:Resource .<br />
      rdfs:comment rdfs:range rdfs:Literal .<br />
      rdfs:label rdfs:range rdfs:Literal .<br />
      rdf:value rdfs:range rdfs:Resource .<br />
      <br />
      rdf:Alt rdfs:subClassOf rdfs:Container .<br />
      rdf:Bag rdfs:subClassOf rdfs:Container .<br />
      rdf:Seq rdfs:subClassOf rdfs:Container .<br />
      rdfs:ContainerMembershipProperty rdfs:subClassOf rdf:Property .<br />
      <br />
      rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso .<br />
      <br />
      rdf:XMLLiteral rdf:type rdfs:Datatype .<br />
      rdf:XMLLiteral rdfs:subClassOf rdfs:Literal . <br />
      rdfs:Datatype rdfs:subClassOf rdfs:Class .<br />
      <br />
      rdf:_1 rdf:type rdfs:ContainerMembershipProperty .<br />
      <span >rdf:_1 rdfs:domain rdfs:Resource .<br />
      rdf:_1 rdfs:range rdfs:Resource .</span> <br />
      rdf:_2 rdf:type rdfs:ContainerMembershipProperty .<br />
      rdf:_2 rdfs:domain rdfs:Resource .<br />
      rdf:_2 rdfs:range rdfs:Resource . <br />
      ...</code> </td>
          </tr>
        
  </table>
    
<p><span >Since I is an <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>, the first condition implies that IP 
  = ICEXT(I(<code>rdf:Property</code>)).</span></p>
<p>These axioms and conditions have some redundancy: for example, all but one 
  of the RDF axiomatic triples can be derived from the RDFS axiomatic triples 
  and the semantic conditions on ICEXT,<code> rdfs:domain</code> and <code>rdfs:range</code>. 
  Other triples which must be true in all <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s 
  include the following:</p>
  <div class="title">Some triples which are rdfs-valid.</div>
<table  border="1">
  <tr>
    <td class="ruletable"><code>rdfs:Resource rdf:type rdfs:Class .<br />
      rdfs:Class rdf:type rdfs:Class .<br />
      rdfs:Literal rdf:type rdfs:Class .<br />
      rdf:XMLLiteral rdf:type rdfs:Class .<br />
      rdfs:Datatype rdf:type rdfs:Class .<br />
      rdf:Seq rdf:type rdfs:Class .<br />
      rdf:Bag rdf:type rdfs:Class .<br />
      rdf:Alt rdf:type rdfs:Class .<br />
      rdfs:Container rdf:type rdfs:Class .<br />
      rdf:List rdf:type rdfs:Class .<br />
      rdfs:ContainerMembershipProperty rdf:type rdfs:Class .<br />
      rdf:Property rdf:type rdfs:Class .<br />
      rdf:Statement rdf:type rdfs:Class .<br />
      <br />
      rdfs:domain rdf:type rdf:Property .<br />
      rdfs:range rdf:type rdf:Property .<br />
      rdfs:subPropertyOf rdf:type rdf:Property .<br />
      rdfs:subClassOf rdf:type rdf:Property .<br />
      rdfs:member rdf:type rdf:Property .<br />
      rdfs:seeAlso rdf:type rdf:Property .<br />
      rdfs:isDefinedBy rdf:type rdf:Property .<br />
      rdfs:comment rdf:type rdf:Property .<br />
      rdfs:label rdf:type rdf:Property .<br />
      </code><code></code></td>
  </tr>
</table>

<p><br />
  Note that <a href="#defDatatype" class="termref">datatype</a>s are allowed to 
  have class extensions, i.e. are considered to be classes, in RDFS. As illustrated 
  by the semantic condition on the class extension of <code>rdf:XMLLiteral</code>, 
  the members of a datatype class are the values of the <a href="#defDatatype" class="termref">datatype</a>. 
  This is explained in more detail in <a href="#dtype_interp" >section 5</a> below. 
  <span >The class <code>rdfs:Literal</code> contains all literal values; however, 
  typed literals whose strings do not conform to the lexical requirements of their 
  <a href="#defDatatype" class="termref">datatype</a> are required to have meanings 
  not in this class. The semantic conditions on <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>s 
  imply that ICEXT(I(<code>rdf:XMLLiteral</code>)) contains all XML values of 
  well-typed XML literals.</span></p>
<p >The conditions on <code>rdf:XMLLiteral</code> and <code>rdfs:range</code> 
  taken together make it possible to write a contradictory statement in RDFS, 
  by asserting that a property value must be in the class <code>rdf:XMLLiteral</code> 
  but applying this property with a value which is an ill-formed XML literal, 
  and therefore required to not be in that class: for example</p>
<p ><code>&lt;ex:a&gt; &lt;ex:p&gt; &quot;&lt;notLegalXML&quot;^^rdf:XMLLiteral 
  .<br />
  &lt;ex:p&gt; rdfs:range rdf:XMLLiteral .</code></p>
<p >cannot be true in any rdfs-interpretation; it is <em>rdfs-<a href="#glossInconsistent" class="termref">inconsistent</a></em>.</p>
<h3><a name="ExtensionalDomRang" id="ExtensionalDomRang"></a>4.2 Extensional Semantic 
  Conditions (Informative)</h3>
<p>The semantics given above is deliberately chosen to be the weakest 'reasonable' 
  interpretation of the RDFS vocabulary. Semantic extensions <strong title="MAY in RFC 2119 context" class="RFC2119">MAY</strong> 
  strengthen the range, domain, subclass and subproperty semantic conditions to 
  the following '<a class="termref" href="#glossExtensional">extensional</a>' 
  versions:</p>
  <div class="title">Extensional alternatives for some RDFS semantic conditions.</div>
<table summary="range and domain extension conditions"  border="1">
  <tr> 
    <td class="semantictable"> <p>&lt;x,y&gt; is in IEXT(I(<code>rdfs:subClassOf</code>)) 
        if and only if x and y are in IC and ICEXT(x) is a subset of ICEXT(y)</p></td>
  </tr>
  <tr> 
    <td class="semantictable"> <p>&lt;x,y&gt; is in IEXT(I(<code>rdfs:subPropertyOf</code>)) 
        if and only if x and y are in IP and IEXT(x) is a subset of IEXT(y)</p></td>
  </tr>
  <tr> 
    <td class="semantictable"> <p>&lt;x,y&gt; is in IEXT(I(<code>rdfs:range</code>)) 
        if and only if (if &lt;u,v&gt; is in IEXT(x) then v is in ICEXT(y))</p></td>
  </tr>
  <tr> 
    <td class="semantictable"> <p>&lt;x,y&gt; is in IEXT(I(<code>rdfs:domain</code>)) 
        if and only if (if &lt;u,v&gt; is in IEXT(x) then u is in ICEXT(y))</p></td>
  </tr>
</table>
<p>which would guarantee that the subproperty and subclass properties were transitive 
  and reflexive, but would also have further consequences. </p>
<p>These stronger conditions would be trivially satisfied when properties are 
  identified with property extensions, classes with class extensions, and <code>rdfs:SubClassOf</code> 
  understood to mean subset, and hence would be satisfied by an <a href="#glossExtensional" class="termref">extensional</a> semantics 
  for RDFS. In some ways the extensional versions provide a simpler semantics, 
  but they require more complex inference rules. The 'intensional' semantics described 
  in the main text provides for most common uses of subclass and subproperty assertions, 
  and allows for simpler implementations of a <a href="#glossComplete" class="termref"> 
  complete</a> set of RDFS entailment rules, described in <a href="#RDFSRules" class="termref"> section 7.3</a>.</p>
<h3><a name="literalnote" id="literalnote">4.3 A Note on rdfs:Literal</a> </h3>

    
<p>Although the semantic conditions on <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s include the intuitively 
  sensible condition that ICEXT(I(<code>rdfs:Literal</code>)) must be the set 
  LV, there is no way to impose this condition by any RDF assertion or inference 
  rule. This limitation is due to the fact that RDF does not allow literals to 
  occur in the subject position of a triple, so there are severe restrictions 
  on what can be said <i>about</i> literals in RDF. Similarly, while properties 
  may be asserted of the class <code>rdfs:Literal</code>, none of these can be 
  validly transferred to literals themselves.</p>

    <p>For example, a triple of the form</p>

    <p><code>&lt;ex:a&gt; rdf:type rdfs:Literal .</code></p>

    <p>is consistent even though '<code>ex:a</code>' is a URI reference rather
    than a literal. What it says is that I(<code>ex:a</code>) is a
    literal value, ie that the URI reference '<code>ex:a</code>'
    <i>denotes</i> a literal value. It does not specify exactly which
    literal value it denotes.</p>

    
<p>The semantic conditions guarantee that any triple containing a plain literal 
  object entails a similar triple with a blank node as object:</p>

    <p><code>&lt;ex:a&gt; &lt;ex:b&gt; "10"</code> .</p>

    <p>entails</p>

    
<p><code>&lt;ex:a&gt; &lt;ex:b&gt; _:xxx .</code></p>
<p>This means that the literal denotes an entity, which could therefore also be 
  named, at least in principle, by a URI reference.</p>
<h3><a name="rdfs_entailment" id="rdfs_entailment"></a>4.4 RDFS Entailment</h3>
<p>S <i>rdfs-entails</i> E when every <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a> 
  which satisfies every member of S also satisfies E. This follows the wording 
  of the definition of <a href="#defentail" class="termref">simple entailment</a> 
  in <a href="#entail" class="termref"> Section 2</a>, but refers only to <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s 
  instead of all simple interpretations. <span >Rdfs-entailment is an example 
  of <a href="#vocabulary_entail" class="termref">vocabulary entailment</a>.</span> 
</p>
<p> Since every <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a> is an <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>, if S rdfs-entails 
  E then it rdf-entails E; but rdfs-entailment is stronger than rdf-entailment. 
  Even the empty graph has a large number of rdfs-entailments which are not rdf-entailments, 
  for example all triples of the form </p>
<p>xxx <code>rdf:type rdfs:Resource .</code></p>
<p>are true in all <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s 
  of any vocabulary containing the URI reference xxx.</p>
<p >An rdfs-inconsistent graph rdfs-entails any graph, by the 
  definition of entailment; such 'trivial entailments' by an inconsistent set 
  are not usually considered useful inferences to draw in practice, however. </p>
    
<h2><a name="dtype_interp" id="dtype_interp">5. Interpreting Datatypes</a></h2>
<h3><a name="DTYPEINTERP" id="DTYPEINTERP">5.1 Datatyped Interpretations</a></h3>
    
<p >RDF provides for the use of externally defined <a href="#defDatatype" class="termref">datatype</a>s 
  identified by a particular URI reference. In the interests of generality, RDF imposes 
  minimal conditions on a datatype. It also includes a single built-in datatype 
  <code>rdf:XMLLiteral.</code></p>
    <p>This semantics for datatypes is minimal. It makes no provision for associating
      a datatype with a property so that it applies to all values of the property,
       and does not provide any way of explicitly asserting that
      a blank node denotes a particular datatype value. Semantic
      extensions and future versions of RDF may impose more elaborate datatyping
      conditions. Semantic extensions may also refer to other kinds of information
      about a datatype, such as orderings of the value space.</p>
<p > <a name="defDatatype" id="defDatatype"></a>A <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#section-Datatypes">datatype</a> 
  is an entity characterized by a set of character strings called <em>lexical 
  forms</em> and a mapping from that set to a set of <em>values</em>. Exactly 
  how these sets <span >and mappings</span> are defined is a matter external to 
  RDF. </p>
    
<p >Formally, a datatype d is defined by three items:</p>
    
<p >1. a non-empty set of character strings called the <em>lexical space</em> 
  of d;</p>
    
<p >2. a non-empty set called the <em>value space</em> of d;</p>
<p >3. a mapping from the lexical space of d to the value space of d, called the 
  <em>lexical-to-value mapping</em> of d.</p>
<p  >The lexical-to-value mapping of a datatype d is written as 
  L2V(d).</p>
    <p>In stating the semantics we assume that interpretations are relativized
      to a particular set of datatypes each of which is identified by a 
      URI reference. </p>
<p><a name="defDatatypeMap" id="defDatatypeMap"></a>Formally, let D be a set of 
  pairs consisting of a URI reference and a <a href="#defDatatype" class="termref">datatype</a> 
  such that no URI reference appears twice in the set, so that D can be regarded 
  as a function from a set of URI references to a set of datatypes: call this 
  a <em>datatype map</em>. (The particular URI references must be mentioned explicitly 
  in order to ensure that interpretations conform to any naming conventions imposed 
  by the external authority responsible for defining the datatypes.) Every <a href="#defDatatypeMap" class="termref">datatype 
  map</a> is understood to contain &lt;<code>rdf:XMLLiteral</code>, x&gt; where 
  x is the built-in XML Literal datatype whose lexical and value spaces and lexical-to-value 
  mapping are <span ><a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/#dfn-rdf-XMLLiteral">defined</a> 
  in the <a href="http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/"> RDF Concepts and Abstract 
  Syntax</a> document [<cite><a href="#ref-rdf-concepts">RDF-CONCEPTS</a></cite>]. 
  </span></p>
<p>The <a href="#defDatatypeMap" class="termref">datatype map</a> which also contains 
  the set of all pairs of the form &lt;<code>http://www.w3.org/2001/XMLSchema#</code><i>sss</i>, 
  <em>sss</em>&gt;, where <em>sss</em> is a built-in datatype named <em>sss</em> 
  in <a href="http://www.w3.org/TR/xmlschema-2/">XML Schema Part 2: Datatypes</a> 
  [<cite><a
    href="#ref-xmls">XML-SCHEMA2</a></cite>] <span >and listed in the following 
  table</span>, is referred to here as XSD. <a name="XSDtable" id="XSDtable"></a></p>
<table  border="1">
  <tr> 
    <td class="ruletable"><strong>XSD datatypes</strong></td>
  </tr>
  <tr> 
    <td class="ruletable"><a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#string"><code>xsd:string</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#boolean"><code>xsd:boolean</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#decimal"><code>xsd:decimal</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#float"><code>xsd:float</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#double"><code>xsd:double</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#dateTime"><code>xsd:dateTime</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#time"><code>xsd:time</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#date"><code>xsd:date</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#gYearMonth"><code>xsd:gYearMonth</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#gYear"><code>xsd:gYear</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#gMonthDay"><code>xsd:gMonthDay</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#gDay"><code>xsd:gDay</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#gMonth"><code>xsd:gMonth</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#hexBinary"><code>xsd:hexBinary</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#base64Binary"><code>xsd:base64Binary</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#anyURI"><code>xsd:anyURI</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#normalizedString"><code>xsd:normalizedString</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#token"><code>xsd:token</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#language"><code>xsd:language</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#NMTOKEN"><code>xsd:NMTOKEN</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#Name"><code>xsd:Name</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#NCName"><code>xsd:NCName</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#integer"><code>xsd:integer</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#nonPositiveInteger"><code>xsd:nonPositiveInteger</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#negativeInteger"><code>xsd:negativeInteger</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#long"><code>xsd:long</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#int"><code>xsd:int</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#short"><code>xsd:short</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#byte"><code>xsd:byte</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#nonNegativeInteger"><code>xsd:nonNegativeInteger</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#unsignedLong"><code>xsd:unsignedLong</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#unsignedInt"><code>xsd:unsignedInt</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#unsignedShort"><code>xsd:unsignedShort</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#unsignedByte"><code>xsd:unsignedByte</code></a>, 
      <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#positiveInteger"><code>xsd:positiveInteger</code></a> 
    </td>
  </tr>
</table>
<p >The other built-in XML Schema datatypes are unsuitable for various reasons, 
  and <strong title="SHOULD NOT in RFC 2119 context" class="RFC2119">SHOULD NOT</strong> 
  be used: <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#duration"><code>xsd:duration</code></a> 
  does not have a well-defined value space (this may be corrected in later revisions 
  of XML Schema datatypes, in which case the revised datatype would be suitable 
  for use in RDF datatyping); <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#QName"><code>xsd:QName</code></a> 
  and <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#ENTITY"><code>xsd:ENTITY</code></a> 
  require an enclosing XML document context; <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#ID"><code>xsd:ID</code></a> 
  and <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#IDREF"><code>xsd:IDREF</code></a> 
  are for cross references within an XML document; <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#NOTATION"><code>xsd:NOTATION</code></a> 
  is not intended for direct use; <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#IDREFS"><code>xsd:IDREFS</code></a>, 
  <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#ENTITIES"><code>xsd:ENTITIES</code></a> 
  and <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#NMTOKENS"><code>xsd:NMTOKENS</code></a> 
  are sequence-valued datatypes which do not fit the RDF <a href="#defDatatype" class="termref">datatype</a> 
  model.</p>
<p><a name="defDinterp"></a>If D is a <a href="#defDatatypeMap" class="termref">datatype 
  map</a>, a <em>D-interpretation</em> of a vocabulary V is any <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a> 
  I of V union {aaa: &lt; aaa, x &gt; in D for some x } which satisfies the following extra conditions for every pair &lt; aaa, 
  x &gt; in D:</p>	
  <div class="title">General semantic conditions for datatypes.</div>
  <table   border="1" class="semantictable" summary="datatype semantic conditions">
    <tr> 
      <td>if &lt;aaa,x&gt; is in D then I(aaa) = x&nbsp;</td>
    </tr>
    <tr> 
      <td>if &lt;aaa,x&gt; is in D then ICEXT(x) is the value space of x and is 
        a subset of LV</td>
    </tr>
    <tr> 
      <td >if &lt;aaa,x&gt; is in D then for any typed literal &quot;sss&quot;^^ddd 
        in V with I(ddd) = x , <em><br />
        </em>&nbsp;&nbsp; if sss is in the lexical space of x then IL(&quot;sss&quot;^^ddd) 
        = L2V(x)(sss), otherwise IL(&quot;sss&quot;^^ddd) is not in LV</td>
    </tr>
    <tr> 
      <td >if &lt;aaa,x&gt; is in D then I(aaa) is in ICEXT(I(<code>rdfs:Datatype</code>))</td>
    </tr>
  </table>
    <p>The first condition ensures that I interprets the URI reference according to
      the <a href="#defDatatypeMap" class="termref">datatype map</a> provided. Note that this does not prevent other URI references
      from also denoting  the <a href="#defDatatype" class="termref">datatype</a>. </p>
    
<p>The second condition ensures that the datatype URI reference, when used as 
  a class name, refers to the value space of the <a href="#defDatatype" class="termref">datatype</a>, 
  and that all elements of a value space must be literal values.</p>
    
<p>The third condition ensures that typed literals in the vocabulary respect
  the datatype lexical-to-value mapping. For example, if I is an XSD-interpretation
  then I("15"^^<code>xsd:decimal</code>)
    must be the number fifteen. <a name="illformedliteral" id="illformedliteral"></a>The
     condition also requires that an <em>ill-typed</em> literal, where the literal
      string is not in the lexical space of the <a href="#defDatatype" class="termref">datatype</a>,
       not denote any literal value. Intuitively, such a name does not denote
      any value,  but in order to avoid the semantic complexities which arise
      from empty names,  the semantics requires such a typed literal to denote
      an 'arbitrary' non-literal  value. Thus for example, if I is an XSD-interpretation,
      then all that can be  concluded about I("arthur"^^<code>xsd:decimal</code>)
      is that it is not in LV,  i.e. not in ICEXT(I(<code>rdfs:Literal</code>)). <span >An
      ill-typed literal  does not in itself constitute an inconsistency, but
      a graph which entails that  an ill-typed literal has <code>rdf:type rdfs:Literal</code>, <span >or
       that an ill-typed XML literal has <code>rdf:type rdf:XMLLiteral</code>, </span>would
        be inconsistent.</span></p>
<p>Note that this third condition applies only to <a href="#defDatatype" class="termref">datatype</a>s 
  in the range of D. Typed literals whose type is not in the <a href="#defDatatypeMap" class="termref">datatype 
  map</a> of the interpretation are treated as before, i.e. as denoting some unknown 
  thing. The condition does not require that the URI reference in the typed literal 
  be the same as the associated URI reference of the <a href="#defDatatype" class="termref">datatype</a>; 
  this allows semantic extensions which can express identity conditions on URI 
  references to draw appropriate conclusions.</p>
    
<p>The fourth condition ensures that the class <code>rdfs:Datatype</code> contains 
  the <a href="#defDatatype" class="termref">datatype</a>s used in any satisfying 
  <a href="#defDinterp" class="termref">D-interpretation</a>. Notice that this 
  is a necessary, but not a sufficient, condition; it allows the class I(<code>rdfs:Datatype</code>) 
  to contain other <a href="#defDatatype" class="termref">datatype</a>s. </p>
      
    <p><a name="pfps8L1" id="pfps8L1"></a>The semantic conditions for <a href="#rdfinterpdef" class="termref">rdf-interpretation</a>s
      impose the correct interpretation on literals typed by <code>'rdf:XMLLiteral'</code>.
      However, a D-interpretation recognizes the <a href="#defDatatype" class="termref">datatype</a> to exist as an entity,
      rather than simply being a semantic condition imposed on the RDF typed
      literal syntax. Semantic extensions which can express identity conditions
      on resources could therefore draw stronger conclusions from <a href="#defDinterp" class="termref">D-interpretation</a>s
    than from <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s. </p>
<p ><a name="defdatatypeclash" id="defdatatypeclash"></a>If the <a href="#defDatatype" class="termref">datatype</a>s 
  in the datatype map D impose disjointness conditions on their value spaces, 
  it is possible for an RDF graph to have no <a href="#defDinterp" class="termref">D-interpretation</a> 
  which satisfies it. For example, XML Schema defines the value spaces of <code>xsd:string</code> 
  and <code>xsd:decimal</code> to be disjoint, so it is impossible to construct 
  a XSD-interpretation <a href="#glossSatisfy" class="termref">satisfying</a> 
  the graph</p>
    <p><code>&lt;ex:a&gt; &lt;ex:b&gt; "25"^^xsd:decimal .<br />
&lt;ex:b&gt; rdfs:range xsd:string .</code></p>
<p>This situation could be characterized by saying that the graph is XSD-inconsistent, 
  or more generally as a <em>datatype clash</em>. Note that it is possible to 
  construct a <a
    href="#glossSatisfy" class="termref">satisfying</a> <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a> 
  for this graph, but it would violate the XSD conditions, since the class extensions 
  of I(<code>xsd:decimal</code>) and I(<code>xsd:string</code>) would have a nonempty 
  intersection.</p>
<p>Datatype clashes can arise in several other ways. For example, any assertion 
  that something is in both of two disjoint dataype classes:</p>
<p><code>_:x rdf:type xsd:string .<br />
  _:x rdf:type xsd:decimal .</code></p>
<p>or that a property with an 'impossible' range has a value:</p>
<p><code>&lt;ex:p&gt; rdfs:range xsd:string .<br />
  &lt;ex:p&gt; rdfs:range xsd:decimal .<br />
  _:x &lt;ex:p&gt; _:y .</code></p>
<p>would constitute a datatype clash. A datatype clash may also arise from the 
  use of a particular lexical form, for example:</p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; &quot;2.5&quot;^^xsd:decimal .<br />
  &lt;ex:p&gt; rdfs:range xsd:integer .</code></p>
<p>or by the use of an ill-typed lexical form:</p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; &quot;abc&quot;^^xsd:integer .<br />
  &lt;ex:p&gt; rdfs:range xsd:integer .</code></p>
<p>Datatype clashes are the only <a
    href="#glossInconsistent" class="termref">inconsistencies</a> recognized by 
  this <a
    href="#glossModeltheory" class="termref">model theory</a>; note however that 
  datatype clashes involving XML literals can arise in RDFS.</p>
    <p><span >If D is a subset of D', then restricting interpretations
        of a graph to D'-interpretations amounts to a semantic extension compared
        to the same restriction with respect to D. In effect, the extension of
        the <a href="#defDatatypeMap" class="termref">datatype map</a> makes implicit assertions about typed literals, by requiring
        them
        to denote entities in the value space of a <a href="#defDatatype" class="termref">datatype</a>. The extra semantic
        constraints associated with the larger <a href="#defDatatypeMap" class="termref">datatype map</a> will force interpretations
        to make more triples true, but they may also reveal datatype clashes
    and violations, so that a D-consistent graph could be D'-inconsistent. </span></p>
    <p >Say that an RDF graph <em>recognizes</em> a datatype
    URI reference aaa when the graph rdfs-entails a <em>datatyping triple</em>    of the form</p>
    <p >aaa<code> rdf:type rdfs:Datatype .</code></p>
<p > The semantic conditions for <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a>s 
  require the built-in datatype URI reference <code>'rdf:XMLLiteral'</code> to 
  be recognized.</p>
    
<p >If every recognized URI reference in a graph is the name of a known <a href="#defDatatype" class="termref">datatype</a>, 
  then there is a natural <a href="#defDatatypeMap" class="termref">datatype map</a> 
  DG which pairs each recognized URI reference to that known datatype (and '<code>rdf:XMLLiteral</code>' 
  to <code>rdf:XMLLiteral</code>). Any <a href="#rdfsinterpdef" class="termref">rdfs-interpretation</a> 
  I of that graph then has a corresponding 'natural' DG-interpretation which is 
  like I except that I(aaa) is the appropriate <a href="#defDatatype" class="termref">datatype</a> 
  and the class extension of <code>rdfs:Datatype</code> is modified appropriately. 
  Applications <strong class="RFC2119" title="MAY in RFC2119 sense">MAY</strong> 
  require that RDF graphs be interpreted by <a href="#defDinterp" class="termref">D-interpretation</a>s 
  where D contains a natural datatype map of the graph. This amounts to treating 
  datatyping triples as 'declarations' of <a href="#defDatatype" class="termref">datatype</a>s 
  by the graph, and making the fourth semantic condition into an 'iff' condition. 
  Note however that a datatyping triple does not in itself provide the information 
  necessary to check that a graph satisfies the other datatype semantic conditions, 
  and it does not formally rule out other interpretations, so that adopting this 
  requirement as a formal entailment principle would violate the <a href="#GeneralMonotonicityLemma" class="termref">general 
  monotonicity lemma</a> described in <a href="#MonSemExt" class="termref">section 
  6</a>, below.</p>
<h3 ><a name="D_entailment" id="D_entailment"></a>5.2 D-entailment</h3>
<p >S <em>D-entails</em> E when every <a href="#defDinterp" class="termref">D-interpretation</a> 
  which satisfies every member of S also satisfies E. This follows the wording 
  of the definition of <a href="#defentail" class="termref">simple entailment</a> 
  in <a href="#entail" class="termref"> Section 2</a>, but refers only to <a href="#defDinterp" class="termref">D-interpretation</a>s 
  instead of all simple interpretations.<span > D-entailment is an example of 
  <a href="#vocabulary_entail" class="termref">vocabulary entailment</a>.</span> 
</p>
<p >As noted above, it is 
  possible that a graph which is consistent in one vocabulary becomes inconsistent 
  in a semantic extension defined on a larger vocabulary, and <a href="#defDinterp" class="termref">D-interpretation</a>s 
  allow for inconsistencies in an RDF graph. The definition of <a href="#vocabulary_entail" class="termref">vocabulary entailment</a>
  means that an inconsistent graph will entail <em>any</em> graph in the stronger 
  vocabulary entailment. For example, a D-inconsistent graph <a href="#D_entailment" class="termref"> D-entail</a>s any RDF 
  graph. However, it will usually not be appropriate to consider such 'trivial' 
  entailments as useful consequences, since they may not be <a href="#glossValid"
    class="termref">valid</a> entailments in a smaller vocabulary. </p>
<h2><a name="MonSemExt" id="MonSemExt"></a>6. Monotonicity of semantic extensions 
</h2>
<p>Given a set of RDF graphs, there are various ways in which one can 'add' information 
  to it. Any of the graphs may have some triples added to it; the set of graphs 
  may be extended by extra graphs; or the vocabulary of the graph may be interpreted 
  relative to a stronger notion of <a href="#vocabulary_entail" class="termref">vocabulary entailment</a>, i.e. with a larger set 
  of semantic conditions understood to be imposed on the interpretations. All 
  of these can be thought of as an addition of information, and may make more 
  entailments hold than held before the change. All of these additions are <a href="#glossMonotonic" class="termref"><em>monotonic</em></a>, 
  in the sense that entailments which hold before the addition of information, 
  also hold after it. We can sum up this in a single lemma:</p>
<p ><strong><a name="GeneralMonotonicityLemma" id="GeneralMonotonicityLemma"></a>General monotonicity lemma</strong>. Suppose
      that S, S' are sets of RDF graphs with every member of S  a subset
      of some member of S'. Suppose that Y indicates a semantic extension
      of&nbsp; X, S X-entails E, and   S and
      E satisfy any syntactic restrictions of Y. Then  S' Y-entails E.</p>
    
<p >In particular, if D' is a <a href="#defDatatypeMap" class="termref">datatype map</a>, D a subset of D' and if S <a href="#D_entailment" class="termref"> D-entail</a>s 
  E then S also D'-entails E. </p>
<h2><span ><a name="rules" id="rules"></a></span>7. Entailment 
  rules (Informative)</h2>
<p>The following tables list some inference patterns which capture some of the 
  various forms of vocabulary entailment, and could be used as a guide for the 
  design of software to check RDF graphs for RDF and RDFS entailment. Implementations 
  may be based on applying the rules forwards, or treating the conclusion as a 
  pattern to be recognized in the consequent of a proposed entailment and searching 
  backwards for an appropriate match with other rule conclusions or the proposed 
  antecedent. Other strategies are also possible.</p>
<p>The rules are all stated with the form <em>add a triple to a graph when it 
  contains triples conforming to a pattern</em>, and they are all <a name="validrule" id="validrule"></a><a href="#glossValid" class="termref"><em>valid</em></a> 
  in the following sense: a graph entails (in the appropriate sense listed) any 
  larger graph that is obtained by applying the rules to the original graph. Notice 
  that applying such a rule to a graph amounts to forming a simple union with 
  the conclusion, rather than a <a href="#defmerge" class="termref">merge</a>, 
  and hence preserves any blank nodes already in the graph. </p>
<p>These rules all use the following conventions: aaa, bbb, etc., stand for any 
  URI reference, i.e. any possible predicate of a triple; uuu, vvv, etc. for any 
  URI reference or blank node identifier, i.e any possible subject of a triple; 
  xxx, yyy etc. for any URI reference, blank node identifier or literal, i.e. 
  any possible subject or object of a triple; lll for any literal; and _:nnn, 
  etc., for blank node identifiers.</p>
<h3 ><a name="simpleRules" id="simpleRules"></a>7.1 Simple Entailment Rules</h3>
<p >The <a href="#interplemma" class="termref">interpolation lemma</a> in <a href="#entail" class="termref">Section
     2</a> can be characterized by the following entailment rules which generate
     generalizations,  i.e. graphs which have the original graph as an instance.
     Being a subgraph requires  no explicit formulation as an inference rule
     on triples.</p>
  <div class="title">simple entailment rules.</div>
<table   border="1" summary="inference rules for existence">
  <tr class="ruletable"> 
    <td ><strong>Rule name</strong></td>
    <td ><strong>If E contains</strong></td>
    <td ><strong>then add</strong></td>
  </tr>
  <tr> 
    <td class="ruletable"><a name="rulese1" id="rulese1"></a>se1</td>
    <td class="ruletable">uuu aaa xxx<code> .</code></td>
    <td class="ruletable"><p>uuu aaa<code> _:</code>nnn<code> .</code></p>
      where <code>_:</code>nnn identifies a blank node <a href="#defallocated" class="termref">allocated</a> to xxx by rule 
      se1 or se2.</td>
  </tr>
  <tr> 
    <td class="ruletable" ><a name="rulese2" id="rulese2"></a>se2</td>
    <td class="ruletable" >uuu aaa xxx<code> .</code></td>
    <td class="ruletable" ><p><code>_:</code>nnn aaa xxx<code> .</code></p>
      where <code>_:</code>nnn identifies a blank node <a href="#defallocated" class="termref">allocated</a> to uuu by rule 
      se1 or se2.</td>
  </tr>
</table>
<p ><a name="defallocated" id="defallocated"></a>The terminology 'allocated 
  to' means that the blank node must have been created by an earlier application 
  of the specified rules on the same URI reference, blank node or literal, or 
  if there is no such blank node then it must be a 'new' node which does not occur 
  in the graph. This rather complicated condition ensures that the resulting graph, 
  obtained by adding the new blank-node triples, has the original graph as a proper 
  instance and that any such graph will have a subgraph which is the same as one 
  which can be generated by these rules. For example, the graph</p>
<p ><code>&lt;ex:a&gt; &lt;ex:p&gt; &lt;ex:b&gt; .<br />
  &lt;ex:c&gt; &lt;ex:q&gt; &lt;ex:a&gt; .</code></p>
<p >could be expanded as follows</p>
<p ><code>_:x &lt;ex:p&gt; &lt;ex:b&gt; . </code>by se1 using a new <code>_:x</code> 
  which is <a href="#defallocated" class="termref">allocated</a> to <code>ex:a<br />
  &lt;ex:c&gt; &lt;ex:q&gt; _:x . </code>by se2 using the same <code>_:x </code>allocated 
  to <code>ex:a<br />
  _:x &lt;ex:p&gt; _:y . </code> by se2 using a new <code>_:y</code> which is <a href="#defallocated" class="termref">allocated</a> to <code>ex:b</code></p>
<p >but it would not be correct to infer</p>
<p >** <code>_:x &lt;ex:q&gt; &lt;ex:a&gt; .</code> ** by se2 (** since<code> 
  _:x</code> is not <a href="#defallocated" class="termref">allocated</a> to <code>ex:c</code> )</p>
<p >These rules allow blank nodes to proliferate, producing highly non-<a href="#deflean" class="termref">lean</a> 
  graphs; they sanction entailments such as</p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; &lt;ex:b&gt; .<br />
  </code><code>&lt;ex:a&gt; &lt;ex:p&gt; _:x .</code> by xse1 with _:x <a href="#defallocated" class="termref">allocated</a> to ex:b<br />
  <code>&lt;ex:a&gt; &lt;ex:p&gt; _:y . </code>by xse1 with _:y <a href="#defallocated" class="termref">allocated</a> to _:x<br />
  <code>_:z &lt;ex:p&gt; _:y . </code>by xse2 with _:z <a href="#defallocated" class="termref">allocated</a> to ex:a <br />
  <code>_:u &lt;ex:p&gt; _:y .</code> by xse2 with _:u <a href="#defallocated" class="termref">allocated</a> to _:z<br />
  <code>_:u &lt;ex:p&gt; _:v .</code> by xse1 with _:v <a href="#defallocated" class="termref">allocated</a> to _:y </p>
<p>It is easy to see that S is an instance of E if and only if one can derive 
  E from S by applying these rules in a suitable sequence; the rule applications 
  required can be discovered by examination of the instance mapping. So, by the 
  interpolation lemma, S simply entails E iff one can derive (a graph containing) 
  E from S by the application of these rules. However, it is also clear that applying 
  these rules naively would not result in an efficient search process, since the 
  rules will not terminate and can produce arbitrarily many redundant derivations 
  of equivalent triples. </p>
<p>The general problem of determining simple entailment between arbitrary RDF 
  graphs is decideable but NP-complete. This can be shown by encoding the problem 
  of detecting a subgraph of an arbitrary directed graph as an RDF entailment, 
  using only blank nodes to represent the graph (observation due to Jeremy Carroll.)</p>
<p>Subsequent rule sets for detecting RDF and RDFS entailment use a special case 
  of rule se1 which applies only to literals:</p>
<div class="title"> literal generalization rule.</div>
<table   border="1" summary="inference rule connecting literals to blank nodes">
  <tr class="ruletable"> 
    <td ><strong>Rule Name</strong></td>
    <td ><strong>If E contains</strong></td>
    <td ><strong>then add</strong></td>
  </tr>
  <tr> 
    <td class="ruletable"><a name="ruleslg" id="ruleslg"></a>lg</td>
    <td class="ruletable"><p>uuu aaa lll<code> .</code> </p>
      
    </td>
    <td class="ruletable"><p>uuu aaa<code> _:</code>nnn<code> .</code></p>
      where <code>_:</code>nnn identifies a blank node <a href="#defallocated" class="termref">allocated</a> to the literal 
      lll by this rule.</td>
  </tr>
</table>
<p>Note that this rule is just sufficient to reproduce any subgraph of E consisting
   of triples all containing a given literal as an isomorphic subgraph in which
   the literal is replaced by a unique blank node. The significance of this lies
   in the fact the blank nodes can stand in a subject position in an RDF triple,
   allowing conclusions to be drawn by the other rules which express properties
  of the literal value denoted by that literal.</p>
<p>For RDFS entailment one additional rule is required, which inverts rule
     lg: </p>
	 <div class="title"> literal instantiation rule.</div>
<table   border="1" summary="inference rule connecting blank node to literal">
  <tr class="ruletable"> 
    <td ><strong>Rule Name</strong></td>
    <td ><strong>If E contains</strong></td>
    <td ><strong>then add</strong></td>
  </tr>
  <tr> 
    <td class="ruletable"><a name="rulesgl" id="rulesgl"></a>gl</td>
    <td class="ruletable"><p>uuu aaa _:nnn <code> .</code></p>
      <p>where <code>_:</code>nnn identifies a blank node <a href="#defallocated" class="termref">allocated</a> to
        the literal lll by <a href="#ruleslg" class="termref">rule lg</a>.</p></td>
    <td class="ruletable"><p>uuu aaa lll<code> .</code></p>      </td>
  </tr>
</table>
<p>Clearly rule gl will simply produce a redundancy, except in cases where the
  allocated blank node has been introduced into the object position of a new
  triple by some other inference rule. The effect of these rules
  is to ensure that any triple containing a literal, and the similar triple containing
  the allocated blank node,  are always derivable from one another, so that a
  literal can be identified with its allocated blank node for purposes of applying
  these rules. </p>
<h3 ><a name="RDFRules" id="RDFRules"></a>7.2 RDF Entailment Rules</h3>
<div class="title">RDF entailment rules</div> 
<table  border="1" summary="RDF entailment rules">
  <tbody>
    <tr class="ruletable"> 
      <td ><strong>Rule Name</strong> </td>
      <td ><strong>if E contains</strong></td>
      <td ><strong>then add</strong></td>
    </tr>
    <tr class="ruletable"> 
      <td ><a name="rulerdf1" id="rulerdf1"></a>rdf1</td>
      <td>uuu aaa yyy <code>.</code></td>
      <td>aaa <code>rdf:type rdf:Property .</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdf2" id="rulerdf2"></a>rdf2</td>
      <td><p>uuu aaa lll <code>.</code></p>
        where lll is a well-typed XML literal .</td>
      <td><p><code> _:</code>nnn <code>rdf:type rdf:XMLLiteral .</code></p>
        <p>where _:nnn identifies a blank node <a href="#defallocated" class="termref">allocated 
          to</a> lll by <a href="#ruleslg" class="termref">rule lg</a>.</p></td>
    </tr>
  </tbody>
</table>
<p>These rules are <em><a href="#glossComplete" class="termref"> complete</a></em> 
  in the following sense. </p>
<p><strong>RDF entailment lemma</strong>. S <a href="#rdf_entail" class="termref">rdf-entails</a> 
  E if and only if there is a graph which can be derived from S plus the <a href="#RDF_axiomatic_triples" class="termref">RDF 
  axiomatic triples</a> by the application of <a href="#ruleslg" class="termref">rule 
  lg</a> and the <a href="#RDFRules" class="termref">RDF entailment rules</a> 
  and which <a href="#defentail" class="termref">simply entails</a> E.<span > 
  </span>(<a href="#RDFEntailmentLemmaPrf" class="termref">Proof</a> in Appendix 
  A)</p>
<p>Note that this does not require the use of <a href="#rulesgl" class="termref">rule
  gl</a>. </p>
<h3><a name="RDFSRules" id="RDFSRules"></a>7.3 RDFS Entailment Rules</h3>
<div class="title">RDFS entailment rules.</div> 
<table  border="1" summary="RDFS entailment rules">
  <tbody>
    <tr class="ruletable"> 
      <th >Rule Name</th>
      <th   >If E contains:</th>
      <th >then add:</th>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs1" id="rulerdfs1"></a>rdfs1</td>
      <td> <p>uuu aaa lll<code>.</code></p>
        <p>where lll is a plain literal (with or without a language tag).</p></td>
      <td><p><code> _:</code>nnn <code>rdf:type rdfs:Literal .</code></p>
        <p>where <code>_:</code>nnn identifies a blank node <a href="#defallocated" class="termref">allocated 
          to</a> lll by rule <a href="#ruleslg" class="termref">rule lg</a>.</p></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs2" id="rulerdfs2"></a>rdfs2</td>
      <td> <p>aaa <code>rdfs:domain</code> xxx <code>.</code><br />
          uuu aaa yyy <code>.</code> </p></td>
      <td>uuu <code>rdf:type</code> xxx <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs3" id="rulerdfs3"></a>rdfs3</td>
      <td> <p>aaa <code>rdfs:range</code> xxx <code>.</code><br />
          uuu aaa vvv <code>.</code> </p></td>
      <td>vvv <code>rdf:type</code> xxx <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs4" id="rulerdfs4"></a>rdfs4a</td>
      <td>uuu aaa xxx <code>.</code></td>
      <td>uuu <code>rdf:type rdfs:Resource .</code></td>
    </tr>
    <tr class="ruletable"> 
      <td>rdfs4b</td>
      <td>uuu aaa vvv<code>.</code></td>
      <td>vvv <code>rdf:type rdfs:Resource .</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs5" id="rulerdfs5"></a>rdfs5</td>
      <td> <p>uuu <code>rdfs:subPropertyOf</code> vvv <code>.</code><br />
          vvv <code>rdfs:subPropertyOf</code> xxx <code>.</code></p></td>
      <td>uuu <code>rdfs:subPropertyOf</code> xxx <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs6" id="rulerdfs6"></a>rdfs6</td>
      <td>uuu <code>rdf:type rdf:Property .</code></td>
      <td>uuu <code>rdfs:subPropertyOf</code> uuu <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs7" id="rulerdfs7"></a>rdfs7</td>
      <td> <p>aaa <code>rdfs:subPropertyOf</code> bbb <code>.</code><br />
          uuu aaa yyy <code>.</code> </p></td>
      <td>uuu bbb yyy <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs8" id="rulerdfs8"></a>rdfs8</td>
      <td> <p>uuu <code>rdf:type rdfs:Class .</code></p></td>
      <td>uuu <code>rdfs:subClassOf rdfs:Resource .</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs9" id="rulerdfs9"></a>rdfs9</td>
      <td> <p>uuu <code>rdfs:subClassOf</code> xxx <code>.</code><br />
          vvv <code>rdf:type</code> uuu <code>.</code></p></td>
      <td>vvv <code>rdf:type</code> xxx <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs10" id="rulerdfs10"></a>rdfs10</td>
      <td>uuu <code>rdf:type rdfs:Class .</code></td>
      <td>uuu <code>rdfs:subClassOf</code> uuu <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs11" id="rulerdfs11"></a>rdfs11</td>
      <td> <p>uuu <code>rdfs:subClassOf</code> vvv <code>.</code><br />
          vvv <code>rdfs:subClassOf</code> xxx <code>.</code></p></td>
      <td>uuu <code>rdfs:subClassOf</code> xxx <code>.</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs12" id="rulerdfs12"></a>rdfs12</td>
      <td>uuu <code>rdf:type rdfs:ContainerMembershipProperty .</code></td>
      <td>uuu <code>rdfs:subPropertyOf rdfs:member .</code></td>
    </tr>
    <tr class="ruletable"> 
      <td><a name="rulerdfs13" id="rulerdfs13"></a>rdfs13</td>
      <td>uuu <code>rdf:type rdfs:Datatype .</code></td>
      <td>uuu <code>rdfs:subClassOf rdfs:Literal .</code></td>
    </tr>
  </tbody>
</table>
<p>Stating <a href="#glossComplete" class="termref">completeness</a> for these 
  rules requires more care, since it is possible for a graph to be rdfs-inconsistent 
  by virtue of containing unsatisfiable assertions about ill-typed XML literals, 
  for example:</p>
<p><code>&lt;ex:a&gt; rdfs:subClassOf rdfs:Literal .</code><br />
  <code> &lt;ex:b&gt; rdfs:range &lt;ex:a&gt; .</code><br />
  <code> &lt;ex:c&gt; rdfs:subPropertyOf &lt;ex:b&gt;.</code><br />
  <code> &lt;ex:d&gt; &lt;ex:c&gt; &quot;&lt;&quot;^^rdf:XMLLiteral .</code></p>
<p>where the denotation of the ill-typed XML literal is required not to be a literal 
  value but can be inferred to be on the basis of the other assertions. Since 
  such an rdfs-inconsistent graph <a href="#rdfs_entailment" class="termref">rdfs-entails</a> 
  all RDF graphs, even when they are syntactically unrelated to the antecedent, 
  we have to exclude this case. </p>
<p>There is a characteristic signal of inconsistency which can be recognized by 
  the entailment rules, by the derivation of a triple of the form</p>
<p>xxx <code>rdf:type rdfs:Literal .</code></p>
<p>where xxx is <a href="#defallocated" class="termref">allocated</a> to an ill-typed 
  XML literal by <a href="#ruleslg" class="termref">rule lg</a>. Call such a triple 
  an <em>XML clash</em>. The derivation of such a clash from the above example 
  is straightforward:</p>
<p><code>&lt;ex:d&gt; &lt;ex:c&gt; _:1 .</code> &nbsp;&nbsp;(by <a href="#ruleslg" class="termref">rule 
  lg</a>, with <code>_:1</code> <a href="#defallocated" class="termref">allocated</a> 
  to <code>&quot;&lt;&quot;^^rdf:XMLLiteral</code>)<br />
  <code>&lt;ex:d&gt; &lt;ex:b&gt; _:1 .</code> &nbsp; &nbsp;(by rule <a href="#rulerdfs7" class="termref">rdfs7</a>)<br />
  <code>_:1 rdf:type &lt;ex:a&gt;.</code> &nbsp;&nbsp;(by rule <a href="#rulerdfs3" class="termref">rdfs3</a>)<br />
  <code>_:1 rdf:type rdfs:Literal .</code> &nbsp;&nbsp;(by rule <a href="#rulerdfs9" class="termref">rdfs9</a>)</p>
<p>These rules are then<em><a href="#glossComplete" class="termref"> complete</a></em> 
  in the following sense: </p>
<p><strong>RDFS entailment lemma</strong>. S <a href="#rdfs_entailment" class="termref">rdfs-entails</a> 
  E if and only if there is a graph which can be derived from S plus the <a href="#RDF_axiomatic_triples" class="termref">RDF</a> 
  and <a href="#RDFS_axiomatic_triples" class="termref">RDFS axiomatic triples</a> 
  by the application of <a href="#ruleslg" class="termref">rule lg</a>, <a href="#rulesgl" class="termref">rule gl</a> and
  the 
  <a href="#RDFRules" class="termref">RDF</a> and <a href="#RDFSRules" class="termref">RDFS
   entailment rules</a> and which either <a href="#defentail" class="termref">simply
    entails</a> E or contains an XML clash. (<a href="#RDFSEntailmentLemmaPrf" class="termref">Proof</a> 
  in Appendix A)</p>
<p>The RDFS rules are somewhat redundant. All but one of the <a href="#RDF_axiomatic_triples" class="termref">RDF 
  axiomatic triples</a> can be derived from the rules <a href="#rulerdfs2" class="termref">rdfs2</a> 
  and <a href="#rulerdfs3" class="termref">rdfs3</a> and the <a href="#RDFS_axiomatic_triples" class="termref">RDFS 
  axiomatic triples</a>, for example.</p>
<p>The outputs of these rules will often trigger others. These rules will propagate 
  all <code>rdf:type</code> assertions in the graph up the subproperty and subclass 
  hierarchies, re-asserting them for all super-properties and superclasses. <a href="#rulerdf1" class="termref">rdf1</a> 
  will generate type assertions for all the property names used in the graph, 
  and <a href="#rulerdfs3" class="termref">rdfs3</a> together with the last <a href="#RDFS_axiomatic_triples" class="termref">RDFS 
  axiomatic triple</a> will add all type assertions for all the class names used. 
  Any subproperty or subclass assertion will generate appropriate type assertions 
  for its subject and object via <a href="#rulerdfs2" class="termref">rdfs2</a> 
  and <a href="#rulerdfs3" class="termref">rdfs3</a> and the domain and range 
  assertions in the RDFS axiomatic triple set. The rules will generate all assertions 
  of the form</p>
<p>uuu <code>rdf:type rdfs:Resource .</code></p>
<p>for every uuu in V, and of the form</p>
<p>uuu <code>rdfs:subClassOf rdfs:Resource .</code></p>
<p>for every class name uuu; and several more 'universal' facts, such as</p>
<p><code>rdf:Property rdf:type rdfs:Class .</code></p>
<h4><a name="RDFSExtRules" id="RDFSExtRules"></a>7.3.1 Extensional Entailment 
  Rules </h4>
<p>The stronger extensional semantic conditions described in <a href="#ExtensionalDomRang" class="termref"> 
  Section 4.1</a> sanction further entailments which are not covered by the RDFS 
  rules. The following table lists some entailment patterns which are valid in 
  this stronger semantics. This is not a <a href="#glossComplete" class="termref"> 
  complete</a> set of rules for the extensional semantic conditions. Note that 
  none of these rules are rdfs-valid; they apply only to semantic extensions which 
  apply the strengthened extensional semantic conditions described in <a href="#ExtensionalDomRang" class="termref"> 
  Section 4.1</a>. These rules have other consequences, eg that <code>rdfs:Resource</code> 
  is a domain and range of every property.</p>
<p>Rules ext5-ext9 follow a common pattern; they reflect the fact that the strengthened 
  extensional conditions require domains (and ranges for transitive properties) 
  of the properties in the rdfV and rdfsV vocabularies to be as large as possible, 
  so any attempt to restrict them will be subverted by the semantic conditions. 
  Similar rules apply to superproperties of <code>rdfs:range</code> and <span ><code>rdfs:domain</code></span>. 
  None of these cases are likely to arise in practice. </p>
<div class="title">Some additional inferences which would be valid under the extensional 
versions of the RDFS semantic conditions.</div> 
<table border="1">
  <tr class="ruletable"> 
    <td >ext1</td>
    <td > <p>uuu <code>rdfs:domain</code> vvv <code>.<br />
        </code> vvv <code>rdfs:subClassOf</code> zzz <code>.</code></p></td>
    <td >uuu <code>rdfs:domain</code> zzz <code>.</code></td>
  </tr>
  <tr class="ruletable"> 
    <td>ext2</td>
    <td> <p>uuu <code>rdfs:range</code> vvv <code>.<br />
        </code> vvv <code>rdfs:subClassOf</code> zzz <code>.</code></p></td>
    <td>uuu <code>rdfs:range</code> zzz <code>.</code></td>
  </tr>
  <tr class="ruletable"> 
    <td>ext3</td>
    <td>uuu <code>rdfs:domain</code> vvv <code>.<br />
      </code> www <code>rdfs:subPropertyOf</code> uuu <code>.</code></td>
    <td>www <code>rdfs:domain </code>vvv <code>.</code></td>
  </tr>
  <tr class="ruletable"> 
    <td>ext4</td>
    <td>uuu <code>rdfs:range</code> vvv <code>.<br />
      </code> www <code>rdfs:subPropertyOf</code> uuu <code>.</code></td>
    <td>www <code>rdfs:range</code> vvv <code>.</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >ext5</td>
    <td><span ><code>rdf:type</code><code> rdfs:subPropertyOf</code> www<code> 
      .</code><br />
      www <code>rdfs:domain</code> vvv<code> .</code> <br />
      </span> </td>
    <td ><code>rdfs:Resource </code><code>rdfs:subClassOf</code> vvv<code> .</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >ext6</td>
    <td><span ><code>rdfs:</code><code>subClassOf</code> <code>rdfs:subPropertyOf</code> 
      www<code> .</code><br />
      www <code>rdfs:domain</code> vvv<code> .</code> <br />
      </span> </td>
    <td ><code>rdfs:Class</code> <code>rdfs:subClassOf</code> vvv<code> .</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >ext7</td>
    <td><span ><code>rdfs:subPropertyOf</code> <code>rdfs:subPropertyOf</code> 
      www<code> .</code><br />
      www <code>rdfs:domain</code> vvv<code> .</code> <br />
      </span> </td>
    <td ><code>rdf:Property</code> <code>rdfs:subClassOf</code> vvv<code> .</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >ext8</td>
    <td><span ><code>rdfs:</code><code>subClassOf</code><code> rdfs:subPropertyOf</code> 
      www <code>.</code><br />
      www <code>rdfs:range</code> vvv<code> .</code> <br />
      </span> </td>
    <td ><code>rdfs:Class</code> <code>rdfs:subClassOf</code> vvv<code> .</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >ext9</td>
    <td><span ><code>rdfs:subPropertyOf</code><code> rdfs:subPropertyOf</code> 
      www <code>.</code><br />
      www <code>rdfs:range</code> vvv<code> .</code> <br />
      </span> </td>
    <td ><code>rdf:Property</code> <code>rdfs:subClassOf</code> vvv<code> .</code></td>
  </tr>
</table>
<p>&nbsp;</p>
<h3><a name="DtypeRules" id="DtypeRules"></a>7.4 Datatype 
  Entailment Rules</h3>
<p>In order to capture <a href="#D_entailment" class="termref"> datatype entailment</a> 
  in terms of assertions and entailment rules, the rules need to refer to information 
  supplied by the <a href="#defDatatype" class="termref">datatype</a>s themselves; 
  and to state the rules it is necessary to assume syntactic conditions which 
  can only be checked by consulting the <a href="#defDatatype" class="termref">datatype</a> 
  sources.</p>
<p>For each kind of information which is available about a <a href="#defDatatype" class="termref">datatype</a>, 
  inference rules for information of that kind can be stated, which can be thought 
  of as extending the table of RDFS entailment rules. These should be understood 
  as applying to <a href="#defDatatype" class="termref">datatype</a>s other than 
  the built-in datatype, the rules for which are part of the RDFS entailment rules. 
  The rules stated below assume that information is available about the <a href="#defDatatype" class="termref">datatype</a> 
  denoted by a recognized URI reference, and they use that URI reference to refer 
  to the <a href="#defDatatype" class="termref">datatype</a>.</p>
<p>The basic information specifies, for each literal string, whether or not it 
  is a legal lexical form for the datatype, i.e. one which maps to some value 
  under the lexical-to-value mapping of the <a href="#defDatatype" class="termref">datatype</a>. 
  This corresponds to the following rule, for each string sss that is a legal 
  lexical form for the <a href="#defDatatype" class="termref">datatype</a> denoted 
  by ddd:</p>
<table  border="1" summary="datatype rule">
  <tr class="ruletable"> 
    <td > <a name="rulerdfD1" id="rulerdfD1"></a>rdfD1</td>
    <td > <p>ddd <code>rdf:type rdfs:Datatype .</code><br />
        uuu aaa "sss"^^ddd <code>.</code></p></td>
    <td > <p> _:nnn <code>rdf:type</code> ddd <code>.</code></p>
      <p >where _:nnn identifies a blank node <a href="#defallocated" class="termref">allocated 
        to</a> &quot;sss&quot;^^ddd by rule <a href="#ruleslg" class="termref">rule 
        lg</a>.</p></td>
  </tr>
</table>
<p>Suppose it is known that two lexical forms sss and ttt map to the same value 
  under the <a href="#defDatatype" class="termref">datatype</a> denoted by ddd; then the following rule applies:</p>
<table  border="1" summary="datatype rule2">
  <tr class="ruletable"> 
    <td >rdfD2</td>
    <td > <p>ddd <code>rdf:type rdfs:Datatype .</code><br />
        uuu aaa "sss"^^ddd <code>.</code><br />
      </p></td>
    <td >uuu aaa "ttt"^^ddd <code>.</code></td>
  </tr>
</table>
<p>Suppose it is known that the lexical form sss of the <a href="#defDatatype" class="termref">datatype</a> denoted by ddd 
  and the lexical form ttt of the <a href="#defDatatype" class="termref">datatype</a> denoted by eee map to the same value. 
  Then the following rule applies:</p>
<table  border="1" summary="datatype rule3">
  <tr class="ruletable"> 
    <td >rdfD3</td>
    <td > <p>ddd <code>rdf:type rdfs:Datatype .</code><br />
        eee <code>rdf:type rdfs:Datatype .</code><br />
        uuu aaa "sss"^^ddd <code>.</code><br />
      </p></td>
    <td >uuu aaa "ttt"^^eee <code>.</code></td>
  </tr>
</table>
<p>In addition, if it is known that the value space of the <a href="#defDatatype" class="termref">datatype</a> 
  denoted by ddd is a subset of that of the <a href="#defDatatype" class="termref">datatype</a> 
  denoted by eee, then it would be appropriate to assert that </p>
<p>ddd <code>rdfs:subClassOf</code> eee <code>.</code></p>
<p>but this needs to be asserted explicitly; it does not follow from the subset 
  relationship alone.</p>
<p>Assuming that the information encoded in these rules is correct, applying these 
  and the earlier rules will produce a graph which is <a href="#D_entailment" class="termref"> 
  D-entail</a>ed by the original. </p>
<p>The rules rdfD2 and 3 are essentially substitutions by virtue of equations 
  between lexical forms. Such equations may be capable of generating infinitely 
  many conclusions, e.g. it is possible to add any number of leading zeros to 
  any lexical form for <code>xsd:integer</code> without it ceasing to be a correct 
  lexical form for <code>xsd:integer</code>. To avoid such <a href="#glossValid" class="termref">correct</a> 
  but unhelpful inferences, it is sufficient to restrict rdfD2 to cases which 
  replace a lexical form with the canonical form for the <a href="#defDatatype" class="termref">datatype</a> in question, 
  when such a canonical form is defined. In order not to omit some valid entailments, 
  however, such canonicalization rules should be applied to the conclusions as 
  well as the antecedents of any proposed entailments, and the corresponding rules 
  of type rdfD3 would need to reflect knowledge of identities between canonical 
  forms of the distinct <a href="#defDatatype" class="termref">datatype</a>.</p>
<p>In particular cases other information might be available, which could be expressed 
  using a particular RDFS vocabulary. Semantic extensions may also define further 
  such datatype-specific meanings.</p>
<p>These rules allow one to conclude that any well-formed typed literal of a recognized 
  datatype will denote something in the class <code>rdfs:Literal</code>.</p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; "sss"^^&lt;ex:dtype&gt; .<br />
  &lt;ex:dtype&gt; rdf:type rdfs:Datatype .</code></p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; _:nnn . </code>(by <a href="#ruleslg" class="termref">rule lg</a>)<code><br />
  _:nnn rdf:type &lt;ex:dtype&gt; .</code> (by rule <a href="#rulerdfD1" class="termref">rdfD1</a>)<code><br />
  &lt;ex:dtype&gt; rdfs:subClassOf rdfs:Literal .</code> (by rule <a href="#rulerdfs11" class="termref">rdfs11</a>)<code><br />
  _:nnn rdf:type rdfs:Literal .</code> (by rule <a href="#rulerdfs9" class="termref">rdfs9</a>)</p>
<p>The rule rdfD1 is sufficient to expose some cases of a <a href="#defdatatypeclash" class="termref">datatype 
  clash</a>, by a chain of reasoning of the following form:</p>
<p><code>&lt;ex:p&gt; rdfs:range &lt;ex:dtype&gt; .<br />
  &lt;ex:a&gt; &lt;ex:p&gt; "sss"^^&lt;ex:otherdtype&gt; .</code></p>
<p><code>&lt;ex:a&gt; &lt;ex:p&gt; _:nnn .<br />
  _:nnn rdf:type &lt;ex:otherdtype&gt; .</code> (by rule <a href="#rulerdfD1" class="termref">rdfD1</a>)<br />
  <code>_:nnn rdf:type &lt;ex:dtype&gt; .</code> (by rule <a href="#rulerdfs3" class="termref">rdfs3</a>)</p>
<p>These rules may not provide a complete set of inference principles for D-entailment, 
  since there may be valid D-entailments for particular datatypes which depend 
  on idiosyncratic properties of the particular datatypes, such as the size of 
  the value space (e.g. <code>xsd:boolean</code> has only two elements, so anything 
  established for those two values must be true for all literals with this datatype.)<a name="xsdstringlitnote" id="xsdstringlitnote"></a><span > 
  In particular, the value space and lexical-to-value mapping of the XSD datatype 
  <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#string"><code>xsd:string</code></a> 
  sanctions the identification of typed literals with plain literals without language 
  tags for all character strings which are in the lexical space of the datatype, 
  since both of them denote the Unicode character string which is displayed in 
  the literal; so the following inference rule is valid in all XSD-interpretations. 
  Here, 'sss' indicates any RDF string in the lexical space of 
  <a href="http://www.w3.org/TR/2001/REC-xmlschema-2-20010502/#string"><code>xsd:string</code></a>.</span></p>
<table  border="1" summary="xsd strings are same as plain literals">
  <tr class="ruletable"> 
    <td   >xsd 1a</td>
    <td  >uuu aaa &quot;sss&quot;<code>.</code></td>
    <td  >uuu aaa &quot;sss&quot;^^<code>xsd:string 
      .</code></td>
  </tr>
  <tr class="ruletable"> 
    <td >xsd 1b</td>
    <td >uuu aaa &quot;sss&quot;^^<code>xsd:string .</code></td>
    <td >uuu aaa &quot;sss&quot;<code>.</code></td>
  </tr>
</table>
<p >Again, as with the rules rdfD2 and rdfD3, applications may 
  use a systematic replacement of one of these equivalent forms for the other 
  rather than apply these rules directly. </p>
<h2><a name="prf" id="prf">Appendix A: Proofs of Lemmas (Informative)</a></h2>

    
<div class="title">&quot;One of the merits of a proof is that it instills a certain 
  doubt as to the result proved.&quot; -Bertrand Russell</div>
<p><strong><a name="emptygraphlemmaprf" id="emptygraphlemmaprf"></a>Empty Graph 
  Lemma.</strong> The empty set of triples is entailed by any graph, and does 
  not entail any graph except itself.</p>
<blockquote> 
  <p><strong>Proof.</strong> Let N be the empty set of triples. The semantic conditions 
    on graphs require that N is true in I, for any I; so the first part follows 
    from the definition of entailment. Suppose G is any nonempty graph and s p 
    o <code>.</code> is a triple in G, then an interpretation I with IEXT(I(p)) 
    = { } does not satisfy G but does satisfy N; so N does not entail G. <strong>QED</strong>.</p>
</blockquote>
<p>This means that most of the subsequent results are trivial for empty graphs, 
  which is what one would expect. </p>
<p><a name="subglemprf" id="subglemprf"><strong>Subgraph Lemma.</strong> A graph 
  entails all its subgraphs.</a></p>
<blockquote> 
  <p><strong>Proof.</strong> Obvious, from definitions of <a
      href="#defsubg" class="termref">subgraph</a> and <a
      href="#defentail" class="termref">entailment</a>. If the graph is true in 
    I then for some A, all its triples are true in I+A, so every subset of triples 
    is true in I. <strong>QED</strong></p>
</blockquote>
<p><a name="mergelemprf" id="mergelemprf"><strong>Merging lemma.</strong></a> 
  The <a href="#defmerge" class="termref">merge</a> of a set S of RDF graphs is 
  entailed by S, and entails every member of S.</p>
<blockquote> 
  <p><strong>Proof.</strong> Obvious, from definitions of <a
      href="#defentail" class="termref">entailment</a> and merge. All members 
    of S are true if and only if all triples in the merge of S are true. <strong>QED</strong>.</p>
</blockquote>
<p>This means that, as noted in the text, a set of graphs can be treated as a 
  single graph when discussing satisfaction and entailment. This convention will 
  be adopted in the rest of the appendix, where a reference to an interpretation 
  of a set of graphs, a set of graphs entailing a graph, and so on, should be 
  understood in each case to refer to the merge of the set of graphs, and references 
  to 'graph' in the following can be taken to refer to graphs or to sets of graphs.</p>
<p><a name="instlemprf" id="instlemprf"><strong>Instance Lemma.</strong> A graph 
  is entailed by all its instances.</a></p>
<blockquote> 
  <p><strong>Proof.</strong> Suppose I <a
      href="#defsatis" class="termref">satisfies</a> E' and E' is an <a
      href="#definst" class="termref">instance</a> of E. Then for some mapping 
    A on the blank nodes of E', I+A satisfies every triple in E'. For each blank 
    node b in E, define B(b)=I+A(c), where c is the blank node or <a href="#defname" class="termref">name</a> 
    that is substituted for b in E', or c=b if nothing was substituted for it. 
    Then I+B(E)=I+A(E')=true, so I satisfies E. But I was arbitrary; so E' <a href="#defentail" class="termref">entails</a> 
    E. <strong>QED</strong>.</p>
</blockquote>
<p><a id="defskolem" name="defskolem">Skolemization</a> is a syntactic transformation 
  routinely used in automatic inference systems in which existential variables 
  are replaced by 'new' functions - function names not used elsewhere - applied 
  to any enclosing universal variables. In RDF, Skolemization amounts to replacing 
  every blank node in a graph by a 'new' name, i.e. a URI reference which is guaranteed 
  to not occur anywhere else. In effect, it gives 'arbitrary' names to the anonymous 
  entities whose existence was asserted by the use of blank nodes: the arbitrariness 
  of the names ensures that nothing can be inferred that would not follow from 
  the bare assertion of existence represented by the blank node. (Using a literal 
  would not do. Literals are never 'new' in the required sense.) </p>
<p>To be precise, a <em>Skolemization</em> of E (with respect to V) is a ground
   instance of E with respect to V with a 1:1 instance mapping that maps each
  blank node in G to a URI reference that does not appear in G (so the Skolem
  vocabulary V must be disjoint from the vocabulary of E)</p>
<p>While not itself strictly a <a href="#glossValid"
    class="termref">valid</a> operation, Skolemization adds no new content to 
  an expression, in the sense that a Skolemized expression has the same entailments 
  as the original expression provided they do not contain names from the Skolem 
  vocabulary:</p>
<p><a name="skolemlemprf" id="skolemlemprf"><strong>Skolemization Lemma.</strong> 
  Suppose sk(E) is a skolemization of E with respect to V. Then sk(E) entails 
  E; and if sk(E) entails F and the vocabulary of F is disjoint from V, then E 
  entails F .</a></p>
<blockquote> 
  <p><strong>Proof.</strong> sk(E) entails E by the instance lemma.</p>
  <p>Now, suppose that sk(E) entails F where F shares no vocabulary with V; and 
    suppose I is some interpretation satisfying E. Then for some mapping A from 
    the blank nodes of E, I+A satisfies E. Define an interpretation I' of the 
    vocabulary of sk(E) by: IR'=IR, IEXT'=IEXT, I'(x)=I(x) for x in the vocabulary 
    of E, and I'(x)=[I+A](y) for x in V, where y is the blank node in E that is 
    replaced by x in sk(E). Clearly I' satisfies sk(E), so I' satisfies F. But 
    I'(F)=[I+A](F) since the vocabulary of F is disjoint from that of V; so I 
    satisfies F. But I was arbitrary; so E entails F. </p>
  <p><strong>QED</strong>.</p>
</blockquote>
<p> Intuitively, this lemma shows that asserting a Skolemization expresses a similar 
  content to asserting the original graph, in many respects. However, a graph 
  should not be thought of as being equivalent to its Skolemization, since these 
  'arbitrary' names would have the same status as any other URI references once 
  published. Also, Skolemization would not be an appropriate operation when applied 
  to anything other than the antecendent of an entailment. A Skolemization of 
  a query would represent a completely different query. Nevertheless, for many 
  purposes when proving results about entailment, we need only consider ground 
  graphs: for provided E does not contain any Skolem vocabulary, S entails E iff 
  S' entails E.</p>
<p>The proof of the subsequent lemmas uses a way of constructing an interpretation 
  of a graph by using the lexical items in the graph itself. (This was <a
    href="http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Mathematicians/Herbrand.html"> 
  Herbrand</a>'s idea; we here modify it slightly to handle literals appropriately.) 
  Given a nonempty graph G, <a name="defherbinterp" id="defherbinterp"></a>the 
  <em>(simple) Herbrand interpretation</em> of G, written <a name="defHerb" id="defHerb"></a>Herb(G), 
  is the interpretation defined as follows.</p>
<table  border="1">
  <tr>
    <td><p>LV<sub>Herb(G)</sub> = the set of all plain literals in G; </p>
      <p>IR<sub>Herb(G)</sub> = the set of all <a href="#defname" class="termref">name</a>s 
        and blank nodes which occur in a subject or object position in a triple 
        in G; </p>
      <p>IP<sub>Herb(G)</sub> = the set of URI references which occur in the property 
        position of a triple in G;</p>
      <p>IEXT<sub>Herb(G)</sub> = {&lt;s,o&gt;: G contains a triple s p o <code>.</code> 
        } </p>
      <p>IS<sub>Herb(G)</sub> and IL<sub>Herb(G)</sub> are both identity mappings 
        on the appropriate parts of the vocabulary of G.</p>    </td>
  </tr>
</table>
<p>Clearly Herb(G)+B, where B is the identity map on blank nodes in G, satisfies 
  every triple in G, by construction, so Herb(G) satisfies G.</p>
<p><a href="#defherbinterp" class="termref">Herbrand interpretation</a>s treat 
  URI references and typed literals (and blank nodes) in the same way as plain literals, i.e. as denoting their own syntactic forms. Of course this may not 
  be what was intended by the writer of the RDF, but the construction shows that 
  any graph <i>can</i> be interpreted in this way. This therefore establishes 
  that any RDF graph has a <a href="#glossSatisfy"
    class="termref">satisfying</a> simple interpretation, i.e. there cannot be 
  a simple inconsistency in RDF. </p>
<p>Notice that the universe of the Herbrand interpretation of G contains the blank 
  nodes of G; they are 'standing for' the entities that they assert the existence 
  of, in effect. Since blank nodes must be interpreted to denote themselves in 
  order to satisfy the graph, the Herbrand interpretation of a Skolemization of 
  a graph is isomorphic with the Herbrand interpretation of the graph together 
  with the blank node mapping: Herb(sk(G)) = Herb(G)+B (using a familiar abuse 
  of notation where a blank node in a Herbrand interpretation is treated as a 
  Skolem name.)</p>
<p><a name="interplemmaprf"
    id="interplemmaprf"><strong>Interpolation Lemma.</strong> S entails E if and 
  only if a subgraph of S is an instance of E</a>.</p>
<blockquote> 
  <p><strong>Proof.</strong> 'if' follows from the subgraph and instance lemmas. 
  </p>
  <p>'only if' uses the Herbrand construction. Suppose S simply entails E. <a href="#defHerb" class="termref">Herb(</a>S) 
    satisfies S, so <a href="#defHerb" class="termref">Herb(</a>S) satisfies E, 
    i.e. for some mapping A from the blank nodes of E to IR<sub>Herb(S)</sub>, 
    [<a href="#defHerb" class="termref">Herb</a>(S)+A] satisfies every triple 
    <br />
    s p o <code>.</code> <br />
    in E, so S must contain the triple <br />
    [<a href="#defHerb" class="termref">Herb(</a>E)+A](s) p [<a href="#defHerb" class="termref">Herb(</a>E)+A](o) 
    <code>.</code> <br />
    which is the instance of the previous triple under the instance mapping A. 
    So the set of all such triples is a subgraph of S which is an instance of 
    E. </p>
  <p><strong> QED</strong></p>
</blockquote>
<p>The following are direct consequences of the interpolation lemma:</p>
<p><a name="Anonlem1prf" id="Anonlem1prf"><strong>Anonymity lemma.</strong></a> 
  Suppose E is a <a href="#deflean" class="termref">lean</a> graph and E' is a 
  proper instance of E. Then E does not entail E'.</p>
<blockquote> 
  <p><strong>Proof.</strong> Suppose E entails E', then a subgraph of E is an 
    instance of E' and therefore a proper instance of E; so E is not <a
      href="#deflean" class="termref">lean</a> , contrary to hypothesis. So E 
    does not entail E'.<strong><br />
    QED</strong></p>
</blockquote>
<p><strong><a name="compactlemmaprf" id="compactlemmaprf"></a>Compactness Lemma</strong>. 
  If S entails E and E is a finite graph, then some finite subset S' of S entails 
  E.</p>
<blockquote> 
  <p><strong>Proof.</strong> By the interpolation lemma, a subgraph S' of S is 
    an instance of E; so S' is finite, and S' entails E. <strong><br />
    QED</strong></p>
</blockquote>
<p>Although compactness is trivial for simple entailment, it becomes progressively 
  less trivial in more elaborate semantic extensions. </p>
<p><strong><a name="monotonicitylemmaprf" id="monotonicitylemmaprf"></a>Monotonicity 
  Lemma</strong>. Suppose S is a subgraph of S' and S entails E. Then S' entails 
  E. (Special case of <a href="#GeneralMonotonicityLemmaprf" class="termref">general 
  monotonicity lemma</a>) <strong>QED</strong></p>
<p><strong><a name="GeneralMonotonicityLemmaprf" id="GeneralMonotonicityLemmaprf"></a>General 
  monotonicity lemma</strong>. Suppose that S, S' are sets of RDF graphs with 
  every member of S a subset of some member of S'. Suppose that Y indicates a 
  semantic extension of&nbsp; X, S X-entails E, and S and E satisfy any syntactic 
  restrictions of Y. Then S' Y-entails E.</p>
<blockquote> 
  <p><strong>Proof. </strong>This follows simply by tracing the definitions. Suppose 
    that I is a Y-interpretation of S'; then since Y is a semantic extension of 
    X, I is an X-interpretation; and by the subgraph and merge lemmas, I satisfies 
    S; so I satisfies E.<br />
    <strong>QED</strong></p>
</blockquote>
<p>Both of the following proofs follow a common pattern which generalizes that 
  used in the proof of the interpolation lemma, by using a modification of the 
  Herbrand construction applied to a 'closure' obtained by applying rules to exhaustion. 
  The proofs operate by showing that the resulting interpretation is both appropriate 
  to the vocabulary and also acts similarly to the <a href="#defHerb" class="termref">Herbrand 
  interpretation</a>. Much of the complexity of the proofs arises from the need 
  to adapt the Herbrand construction in order to take proper account of literal 
  values. Herbrand interpretations ignore literal typing and treat all typed literals 
  as non-literal values; this is irrelevant when considering simple entailment, 
  which treats typed literals simply as denoting names; but more care will be 
  needed when considering rdf- and rdfs-interpretations.</p>
<p>Both proofs use a single basic idea which requires rather an awkward notation 
  but is basically straightforward to understand. The simple Herbrand interpretation 
  treats all vocabulary items as denoting themselves, and builds the interpretation 
  out of these syntactic items. The semantic conditions on rdf- and rdfs-interpretations 
  do not permit this in all cases: XML literals in particular are required to 
  denote other kinds of entity. We therefore distinguish between the 'real' semantic 
  values and their syntactic 'surrogates' in these cases, by defining a mapping 
  <em>sur</em> from the universe of the intepretation to the vocabulary of the 
  graph (plus blank nodes) which is as close as possible to being an identity 
  mapping, but which when applied to these special literal values, identifies 
  the particular blank node which acts as a witness for that value in the graph. 
  In the case of RDFS, the surrogate mapping is extended to all literal values, 
  since the blank node allocated to the literal can occur in a subject position, 
  and hence record information about the literal value which must be applied back 
  to that value in the interpretation. </p>
<p><strong><a name="RDFEntailmentLemmaPrf" id="RDFEntailmentLemmaPrf"></a></strong><strong>RDF 
  entailment lemma</strong>. S <a href="#rdf_entail" class="termref">rdf-entails</a> 
  E if and only if there is a graph which can be derived from S plus the <a href="#RDF_axiomatic_triples" class="termref">RDF 
  axiomatic triples</a> by the application of <a href="#ruleslg" class="termref">rule 
  lg</a> and the <a href="#RDFRules" class="termref">RDF entailment rules</a> 
  and which simply entails E.<span > </span></p>
<blockquote> 
  <p><strong>Proof. </strong> To show 'if' one has only to check that the rules 
    are rdf-valid, which is left as an exercise for the reader; and if S or E 
    is empty then the result follows trivially; so suppose they are both non-empty.</p>
  <p>To establish 'only if', the proof proceeds by constructing an <em>rdf Herbrand 
    interpretation</em> RH of S which serves the same role for rdf-interpretations 
    that the simple Herbrand interpretation does for simple interpretations. The 
    construction follows the Herbrand construction as far as possible, but interprets 
    well-formed XML literals so as to satisfy the RDF semantic conditions, guided 
    by the triples in the <em>RDF closure</em>, C, defined to be the graph resulting 
    from the following process:</p>
  <p>add to S all the <span ><a href="#RDF_axiomatic_triples" class="termref">RDF 
    axiomatic triples</a></span>; <br />
    apply <span ><a href="#simpleRules" class="termref"></a> </span> <a href="#ruleslg" class="termref">rule 
    lg</a> to any triple containing a well-typed XML literal until the graph is 
    unchanged by the rule;<br />
    apply rule <a href="#rulerdf2" class="termref">rdf2</a> until the graph is 
    unchanged;<br />
    apply rule <a href="#rulerdf1" class="termref">rdf1</a> until the graph is 
    unchanged.</p>
  <p>Note that C contains precisely one new blank node _:nnn <a href="#defallocated" class="termref">allocated</a> 
    to each literal in S by the rule <a href="#ruleslg" class="termref">rule lg</a>, 
    and that the subgraph of triples in S containing any well-typed XML literal 
    is reproduced exactly in C with this blank node replacing the literal and 
    with the extra triple<br />
    _:nnn <code>rdf:type</code> <code>rdf:XMLLiteral .</code><br />
    introduced by rule <a href="#rulerdf2" class="termref">rdf2</a>. Note also 
    that the proof only requires that <a href="#ruleslg" class="termref">rule 
    lg</a> is used on well-typed XML literals, so that it actually establishes 
    a slightly tighter result.</p>
  <p>Blank nodes introduced by <a href="#ruleslg" class="termref">rule lg</a> 
    stand as <em>surrogates</em> for well-formed XML literals in the subject position 
    of a triple. (In the proof of the next lemma, this will be extend to all literals.) 
    In order to construct an RDF interpretation, XML literals and their surrogates 
    must be replaced by the appropriate literal values in the domain of the interpretation, 
    but the proof requires that each XML literal value be uniquely associated 
    with a lexical item that denotes it. This requires some delicacy in the following 
    construction. </p>
  <p>If lll is a well-formed XML literal, let <em>xml</em>(lll) be the XML value 
    of lll; and for each XML value x of any well-formed XML literal in C, let 
    <em>sur</em>(x) be the blank node allocated to that XML literal by <a href="#ruleslg" class="termref">rule 
    lg</a>; and extend <em>sur</em> to be the identity mapping on URI references, 
    blank nodes and all other literals in C. </p>
  <p>RH is then defined by:</p>
  <table  border="1">
    <tr>
        <td><p>LV<sub>RH</sub> = all plain literals in C plus {<em>xml</em>(x): 
          x a well-typed XML literal in S}</p>
        <p>IR<sub>RH</sub> = LV<sub>RH</sub> plus the set of URI references, blank 
          nodes and other typed literals occurring in C<sub></sub> </p>
          
        <p>IP<sub>RH</sub> = {x: C contains the triple: x <code>rdf:type rdf:Property 
          .</code>}</p>
          
        <p>If x is in IP<sub>RH</sub> then IEXT<sub>RH</sub>(x) = {&lt;s,o&gt;: 
          C contains the triple <em>sur</em>(s) x <em>sur</em>(o) <code>.</code> 
          } </p>
          <p> IS<sub>RH</sub> is the identity map on URIrefs in S</p>
          <p>If x is a well-formed XML literal in S then IL<sub>RH</sub>(x) = 
            <em>xml</em>(x), otherwise IL<sub>RH</sub>(x) = x</p>
  </td>
    </tr>
  </table>
   
    
  <p>Define a mapping B on blank nodes in C as follows: B(x)=<em>xml</em>(lll) 
    if x is allocated to a well-formed XML literal lll, otherwise B(x)=x, then 
    clearly [RH+B] satisfies C and therefore S, so RH satisfies S. </p>
  <p>Since C contains all the required RDF axiomatic triples, RH satisfies them. 
  </p>
  <p>It is easy to see that J satisfies the first two RDF semantic conditions, 
    by construction; for the triples introduced by rule <a href="#rulerdf2" class="termref">rdf2</a> 
    require that IEXT<sub>RH</sub>(rdf:type) contains &lt;<em>xml</em>(lll),<code>rdf:XMLLiteral</code>&gt; 
    for every well-typed XML literal lll. </p>
  <p>The <a href="#rdfsemcond3" class="termref">third RDF semantic condition</a> 
    is the only negative semantic condition which cannot be satisfied simply by 
    construction, but it is satisfied trivially. Ill-typed XML literals denote 
    themselves in RH, and so are excluded from LV<sub>RH </sub>by construction. 
    The pair &lt;lll, <code>rdf:XMLLiteral</code>&gt; cannot occur in IEXT<sub>RH</sub>(<code>rdf:type</code>) 
    because a literal cannot occur in subject position; so the condition is satisfied, 
    so RH is an rdf-interpretation.</p>
  <p>Since S rdf-entails E, RH satisfies E: so for some mapping A from the blank 
    nodes of E to IR<sub>RH</sub>, [RH+A] satisfies every triple <br />
    s p o <code>.</code><br />
    in E, i.e. IEXT<sub>RH</sub>(p) contains &lt;[RH+A](s),[RH+A](o)&gt;, i.e.C 
    contains a triple <br />
    <em>sur</em>([RH+A](s)) p <em>sur</em>([RH+A](o))<code>.<br />
    </code> but this is an instance of the first triple under the instantiation 
    mapping x -&gt; <em>sur</em>(A(x)); so a subgraph of C is an instance of E; 
    so C simply entails E.</p>
  <p><strong>QED</strong></p>
</blockquote>
<p>This lemma also shows that any graph has a satisfying rdf-interpretation, and 
  the proof illustrates how to construct it from a Herbrand interpretation of 
  the closure, by interpreting well-formed XML literals appropriately and allowing 
  the possible existence of properties which have no extensions. Note that if 
  E is finite then the derived subgraph of C is also finite.</p>
<p>The proof of the RDFS entailment lemma is similar in structure and uses closely 
  similar definitions, but is of course longer and requires a more elaborate construction 
  to ensure that the class extensions of <code>rdfs:Literal</code> and <code>rdfs:Resource</code> 
  contain all literal values. </p>
<p><a name="RDFSEntailmentLemmaPrf" id="RDFSEntailmentLemmaPrf"></a><strong>RDFS 
  entailment lemma</strong>. S <a href="#rdfs_entailment" class="termref">rdfs-entails</a> 
  E if and only if there is a graph which can be derived from S plus the <a href="#RDF_axiomatic_triples" class="termref">RDF</a> 
  and <a href="#RDFS_axiomatic_triples" class="termref">RDFS axiomatic triples</a> 
  by the application of <a href="#ruleslg" class="termref">rule lg</a>, <a class="termref" href="#rulesgl">rule 
  gl</a> and the 
  <a href="#RDFRules" class="termref">RDF</a> and <a href="#RDFSRules" class="termref">RDFS 
  entailment rules</a> and which either simply entails E or is an XML clash. </p>
<blockquote> 
  <p><strong>Proof. </strong> Again, to show 'if' it is sufficient to show that 
    the <span ><a href="#RDFSRules" class="termref">RDFS entailment 
    rules</a></span> are rdfs-valid, which is again left as an exercise; and again, 
    the empty cases are trivial. </p>
  <p>The proof of 'only if' is similar to that used in the previous lemma, and 
    similar constructions and terminology will be used, except that the RDFS closure, 
    D, is defined to be the graph resulting from the following process:</p>
  <p>add to S all the <span ><a href="#RDF_axiomatic_triples" class="termref">RDF</a></span> 
    and <a href="#RDFS_axiomatic_triples" class="termref">RDFS axiomatic triples</a>; 
    <br />
    apply <span ><a href="#simpleRules" class="termref"></a> </span> <a href="#ruleslg" class="termref">rule 
    lg</a> to any triple containing a literal until the graph is unchanged by 
    the rule;<br />
    apply rules <a href="#rulerdf2" class="termref">rdf2</a> and <a href="#rulerdfs1" class="termref">rdfs1</a> 
    until the graph is unchanged;<br />
    apply rule <a href="#rulerdf1" class="termref">rdf1</a>, <a href="#rulesgl" class="termref">rule gl</a> and the
    remaining 
    <a href="#RDFSRules" class="termref">RDFS entailment rules</a> until the
    graph  is unchanged.</p>
  <p>Unlike the previous
    lemma, this proof requires that <a href="#ruleslg" class="termref">rule
       lg</a> is applied to all literals, even ill-typed XML literals, and it
       requires the inverse <a href="#rulesgl" class="termref">rule gl</a>.  <a href="#rulesgl" class="termref">Rule
       gl</a> needs to be used only after an application of rules 
	   <a href="#rulerdfs6" class="termref">rdfs6</a> or <a href="#rulerdfs10" class="termref">rdfs10</a>,
       since those are the only rules which can move a blank node from subject
       to object position. Note
       that D contains precisely one new blank node _:nnn <a href="#defallocated" class="termref">allocated</a> 
    to each literal in S by the rule <a href="#ruleslg" class="termref">rule
    lg</a>, 
    and that the subgraph of triples in S containing any literal is reproduced
     exactly in D with this blank node replacing the literal and with the extra
     triple<br />
    _:nnn <code>rdf:type</code> <code>rdfs:Literal .<br />
    </code>introduced by rule <a href="#rulerdfs1" class="termref">rdfs1</a>, 
    and possibly also <br />
    _:nnn <code>rdf:type</code> <code>rdf:XMLLiteral .</code><br />
    introduced by rule <a href="#rulerdf2" class="termref">rdf2</a> when appropriate.
     This means that after this point in the construction, literals can effectively
     be ignored, since any of the subsequent rules which applies to triples containing
     a literal will also apply equally well to the similar triples where the
    literal  is replaced by its allocated blank node. The rest of the proof uses
    this by  requiring literal values in the interpretation to satisfy just the
    semantic  conditions imposed on the blank nodes allocated to the corresponding
    literal,  and ignoring triples in the graph which contain literals. The use
    of <a href="#rulesgl" class="termref">rule gl</a> ensures that D contains
    any triple containing a literal if and only if it contains the similar triple
    with the literal replaced by its allocated blank node. </p>
  <p>As in the previous proof, if lll is a well-formed XML literal, let <em>xml</em>(lll) 
    be the XML value of lll; the surrogate mapping <em>sur</em> is then extended 
    as follows. First, the domain of <em>sur</em> is the set containing just the 
    URI references, literals and blank nodes occurring in D and all XML values 
    of well-formed XML literals in D. (This is the universe of the rdfs-Herbrand 
    interpretation, defined below.) Now, if lll is a well-formed XML literal in 
    D, let <em>sur</em>(<em>xml</em>(lll)) be the blank node allocated to lll 
    by <a href="#ruleslg" class="termref">rule lg</a>; for any other literal lll 
    in D, let <em>sur</em>(lll) be the blank node allocated to lll by <a href="#ruleslg" class="termref">rule 
    lg</a>, and for all URI references and blank nodes in D, let <em>sur</em>(x) 
    = x . Note that the range of <em>sur</em> contains only URI references 
    and blank nodes which occur in D. </p>
  <p>The <em>rdfs-Herbrand interpretation</em> SH of S is then constructed similarly 
    to the previous lemma. </p>
  <table  border="1">
    <tr>
        <td ><p>LV<sub>SH</sub> = {x: D contains the triple: <em>sur</em>(x) <code>rdf:type 
          rdfs:Literal .</code>}</p>
        <p>IR<sub>SH</sub> = LV<sub>SH</sub> plus the set of URI references, blank 
          nodes and literals other than well-formed XML literals occurring in D </p>
        <p>IP<sub>SH</sub> = {x: D contains the triple: <em>sur</em>(x)<code></code><code> 
          rdf:type rdf:Property .</code>}</p>
          
        <p>If x is in IP<sub>RH</sub> then IEXT<sub>RH</sub>(x) = {&lt;s,o&gt;: 
          D contains the triple <em>sur</em>(s) x <em>sur</em>(o) <code>.</code> 
          } </p>
        <p> IS<sub>SH</sub> is the identity map on URI references in S</p>
          
        <p>If x is a well-formed XML literal in S then IL<sub>SH</sub>(x) = <em>xml</em>(x), 
          otherwise IL<sub>SH</sub>(x) = x</p>
  </td>
    </tr>
  </table>
   
    
  <p>Define B(x) as follows: if x is a blank node allocated to a well-formed XML 
    literal lll in D then B(x) = <em>xml</em>(lll); if it is allocated to any 
    other literal lll in D then B(x)=lll; and otherwise B(x)=x; then clearly [SH+B] 
    satisfies D and therefore S, so SH satisfies S. </p>
  <p>As in the previous lemma, SH satisfies all the required RDF and RDFS axiomatic 
    triples and the first two RDF semantic conditions by construction.</p>
  <p>SH satisfies the third RDF semantic condition just in case D does not contain 
    an XML clash. Note that the presence of surrogates for ill-typed XML literals 
    invalidates the argument used in the previous lemma to the effect that this 
    condition is trivally satisfied. So assume that D does not contain an XML 
    clash. </p>
  <p>As noted in the text, we can regard the first RDFS semantic condition as 
    defining ICEXT and IC: we will do so without further comment and describe 
    all the conditions in terms of IEXT. To show that SH satisfies 
    the remaining RDFS semantic conditions we argue case by case, using the minimality 
    of the Herbrand interpretation and the completeness of the closure. </p>
  <p>All of these conditions can be mirrored by a corresponding sequence of rule 
    applications. The general form of the argument can be illustrated with the 
    case of the <a href="#rdfssemcond2" class="termref">second RDFS semantic condition</a>. 
    Suppose &lt;x,y&gt; is in IEXT<sub>SH</sub>(<code>rdfs:domain</code>) and 
    &lt;u,v&gt; is in IEXT<sub>SH</sub>(x); then D must contain the triples</p>
  <p><em>sur</em>(x<code>) rdfs:domain </code><em>sur</em>(y<code>) .<br />
    </code><em>sur</em>(u)<em> </em>x <em>sur</em>(v)<code>.</code></p>
  <p>so x must be a URIref, so <em>sur</em>(x)=x; and then by rule <a href="#rulerdfs2" class="termref">rdfs2</a>, 
    it must also contain the triple</p>
  <p><em>sur</em>(u)<code> rdf:type</code> <em>sur</em>(y)<code>.</code></p>
  <p>so IEXT<sub>SH</sub>(<code>rdf:type</code>) contains &lt;u,v&gt; ; so the 
    condition is satisfied.</p>
  <p>The other cases proceed similarly, by translating the semantic condition 
    into a derivation using the rules and axiomatic triples. The argument forms 
    are summarized in the following table. Some of the semantic conditions split 
    into several subconditions, and some also have special subcases.</p>
  <table border="1">
    <tr> 
      <td  ><strong>RDFS semantic condition</strong></td>
      <td colspan="2" ><strong>Derivation</strong></td>
    </tr>
    <tr> 
      <td rowspan="4">if x in IR then<br /> &lt;x,rdfs:Resource&gt; in IEXT(rdf:type)</td>
      <td >URIref or bnode in subject:<br />
        x a b<br />
        x rdf:type rdfs:Resource</td>
      <td ><br /> <br />
        <a href="#rulerdfs4" class="termref">rdfs4a</a></td>
    </tr>
    <tr> 
      <td > Literal:<br />
        _:x rdf:type rdfs:Literal<br />
        rdfs:type rdfs:range rdfs:Class<br />
        rdfs:Literal rdf:type rdfs:Class<br />
        rdfs:Literal rdfs:subClassOf rdfs:Resource<br />
        _:x rdf:type rdfs:Resource</td>
      <td><br />
        see below<br />
        axiomatic<br />
        <a href="#rulerdfs3" class="termref">rdfs3</a><br />
        <a href="#rulerdfs8" class="termref">rdfs8</a><br />
        <a href="#rulerdfs9" class="termref">rdfs9</a></td>
    </tr>
    <tr> 
      <td>URIref or bnode in object:<br />
        a b x<br />
        x rdf:type rdfs:Resource</td>
      <td><br /> <br />
        <a href="#rulerdfs4" class="termref">rdfs4b</a></td>
    </tr>
    <tr> 
      <td><p>URIref in predicate:<br />
          a x b<br />
          x rdf:type rdf:Property<br />
          rdf:type rdfs:domain rdfs:Resource<br />
          x rdf:type rdfs:Resource</p></td>
      <td><br /> <br />
        <a href="#rulerdf1" class="termref">rdf1</a><br />
        axiomatic<br />
        <a href="#rulerdfs2" class="termref">rdfs2</a></td>
    </tr>
    <tr> 
      <td rowspan="2">x in LV iff<br /> &lt;x,rdfs:Literal&gt; in IEXT(rdf:type)</td>
      <td>well-typed XML literal lll:<br />
        a b lll<br />
        _:x rdf:type rdf:XMLLiteral<br />
        rdf:XMLLiteral rdfs:subClassOf rdfs:Literal<br />
        _:x rdf:type rdfs:Literal</td>
      <td><br /> <br />
        <a href="#ruleslg" class="termref">lg</a>, <a href="#rulerdf2" class="termref">rdf2</a>, 
        <em>sur</em>(<em>xml</em>(lll))=_:x <br />
        axiomatic<br />
        <a href="#rulerdfs9" class="termref">rdfs9</a></td>
    </tr>
    <tr> 
      <td><p>other literal lll :<br />
          a b lll<br />
          _:x rdf:type rdfs:Literal</p></td>
      <td><br /> <br />
        <a href="#ruleslg" class="termref">lg</a>, <a href="#rulerdfs1" class="termref">rdfs1</a>, 
        <em>sur</em>(lll)=_:x </td>
    </tr>
    <tr> 
      <td>if <br /> &lt;x,y&gt; in IEXT(rdfs:domain) and &lt;u,v&gt; in IEXT(x) 
        <br />
        then <br /> &lt;u,y&gt; in IEXT(rdf:type)</td>
      <td>x rdfs:domain y .<br />
        u x v .<br />
        u rdf:type y . </td>
      <td><br /> <br />
        <a href="#rulerdfs2" class="termref">rdfs2</a></td>
    </tr>
    <tr> 
      <td>if <br /> &lt;x,y&gt; in IEXT(rdfs:range) and &lt;u,v&gt; in IEXT(x) 
        <br />
        then <br /> &lt;v,y&gt; in IEXT(rdf:type)</td>
      <td>x rdfs:range y .<br />
        u x v .<br />
        v rdf:type y . </td>
      <td><br /> <br />
        <a href="#rulerdfs3" class="termref">rdfs3</a></td>
    </tr>
    <tr> 
      <td><p>if<br />
          &lt;x,rdf:Property&gt; in IEXT(rdf:type)<br />
          then<br />
          &lt;x,x&gt; in IEXT(rdfs:subPropertyOf)</p></td>
      <td>x rdf:type rdf:Property<br />
        x rdfs:subPropertyOf x</td>
      <td><br />
        <a href="#rulerdfs6" class="termref">rdfs6</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdf:Property&gt; in IEXT(rdf:type)<br /> &lt;y,rdf:Property&gt; 
        in IEXT(rdf:type)<br /> &lt;z,rdf:Property&gt; in IEXT(rdf:type)<br /> 
        &lt;x,y&gt; in IEXT(rdfs:subPropertyOf)<br /> &lt;y,z&gt; in IEXT(rdfs:subPropertyOf)<br />
        then<br /> &lt;x,z&gt; in IEXT(rdfs:subPropertyOf)</td>
      <td>x rdfs:subPropertyOf y<br />
        y rdfs:subPropertyOf z<br />
        x subPropertyOf z</td>
      <td><br /> <br />
        <a href="#rulerdfs5" class="termref">rdfs5</a></td>
    </tr>
    <tr> 
      <td><p>if<br />
          &lt;x,y&gt; in IEXT(rdfs:subPropertyOf)<br />
          &lt;u,v&gt; in IEXT(x)<br />
          then<br />
          &lt;x,rdf:Property&gt; in IEXT(rdf:type)<br />
          &lt;y,rdf:Property&gt; in IEXT(rdf:type)<br />
          &lt;u,v&gt; in IEXT(y)</p></td>
      <td>x rdfs:subPropertyOf y<br />
        u x v<br />
        rdfs:subPropertyOf rdfs:domain rdf:Property<br />
        x type rdf:Property<br />
        rdfs:subPropertyOf rdfs:domain rdf:Property<br />
        y rdf:type rdf:Property<br />
        u y v</td>
      <td><br /> <br />
        axiomatic triple<br />
        <a href="#rulerdfs2" class="termref">rdfs2</a><br />
        axiomatic triple<br />
       <a href="#rulerdfs3" class="termref">rdfs3</a><br />
        <a href="#rulerdfs7" class="termref">rdfs7</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdfs:Class&gt; in IEXT(rdf:type)<br />
        then<br /> &lt;x,rdfs:Resource&gt; in IEXT(rdfs:subClassOf)</td>
      <td>x rdf:type rdfs:Class<br />
        x rdfs:subClassOf rdfs:Resource</td>
      <td><br />
        <a href="#rulerdfs8" class="termref">rdfs8</a></td>
    </tr>
    <tr> 
      <td><p>if<br />
          &lt;x,y&gt; in IEXT(rdfs:subClassOf)<br />
          &lt;u,x&gt; in IEXT(rdf:type)<br />
          then<br />
          &lt;x,rdfs:Class&gt; in IEXT(rdf:type)<br />
          &lt;y,rdfs:Class&gt; in IEXT(rdf:type)<br />
          &lt;u,y&gt; in IEXT(rdf:type) </p></td>
      <td><p>x rdfs:subClassOf y<br />
          u rdf:type x<br />
          rdfs:subClassOf rdfs:domain rdfs:Class<br />
          x rdf:type rdfs:Class<br />
          rdfs:subClassOf rdfs:range rdfs:Class<br />
          y rdf:type rdfs:Class<br />
          u rdf:type y</p></td>
      <td><br /> <br />
        axiomatic triple<br />
        <a href="#rulerdfs2" class="termref">rdfs2</a><br />
        axiomatic triple<br />
        <a href="#rulerdfs3" class="termref">rdfs3</a><br />
        <a href="#rulerdfs9" class="termref">rdfs9</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdfs:Class&gt; in IEXT(rdf:type)<br />
        then<br /> &lt;x,x&gt; in IEXT(rdfs:subClassOf)</td>
      <td>x rdf:type rdfs:Class<br />
        x rdfs:subClassOf x</td>
      <td><br />
        <a href="#rulerdfs10" class="termref">rdfs10</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdfs:Class&gt; in IEXT(rdf:type)<br /> &lt;y,rdfs:Class&gt; 
        in IEXT(rdf:type)<br /> &lt;z,rdfs:Class&gt; in IEXT(rdf:type)<br /> &lt;x,y&gt; 
        in IEXT(rdfs:subClassOf)<br /> &lt;y,z&gt; in IEXT(rdfs:subClassOf)<br />
        then<br /> &lt;x,z&gt; in IEXT(rdfs:subClassOf)</td>
      <td>x rdfs:subClassOf y<br />
        y rdfs:subClassOf z<br />
        x rdfs:subClassOf z</td>
      <td><br /> <br />
        <a href="#rulerdfs11" class="termref">rdfs11</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdfs:ContainerMembershipProperty&gt; in IEXT(rdf:type)<br />
        then<br /> &lt;x,rdfs:member&gt; in IEXT(rdfs:subPropertyOf)</td>
      <td>x rdf:type rdfs:ContainerMembershipProperty<br />
        x rdfs:subPropertyOf rdfs:member</td>
      <td><br />
        <a href="#rulerdfs12" class="termref">rdfs12</a></td>
    </tr>
    <tr> 
      <td>if<br /> &lt;x,rdfs:Datatype&gt; in IEXT(rdf:type)<br />
        then<br /> &lt;x,rdfs:Literal&gt; in IEXT(rdfs:subClassOf)</td>
      <td>x rdf:type rdfs:Datatype<br />
        x rdfs:subClassOf rdfs:Literal</td>
      <td><br />
        <a href="#rulerdfs13" class="termref">rdfs13</a></td>
    </tr>
  </table>
  <p>so SH is an rdfs-interpretation. </p>
  <p>Since S rdfs-entails E, SH satisfies E: so for some mapping A from the blank
     nodes of E to IR<sub>SH</sub>, [SH+A] satisfies every triple <br />
    s p o <code>.</code><br />
    in E, i.e. IEXT<sub>SH</sub>(p) contains &lt;[SH+A](s),[SH+A](o)&gt;, i.e.D 
    contains a triple </p>
  <p><em>sur</em>([SH+A](s)) p <em>sur</em>([SH+A](o))<code>.</code></p>
  <p>which is an instance of the first triple under the instantiation mapping
    x -&gt; <em>sur</em>(A(x)), unless o is a literal. If o is a
    literal, then <em>sur</em>([SH+A](o) is the blank node
    allocated to o, and so D also contains the triple </p>
  <p><em>sur</em>([SH+A](s)) p o <code>.</code></p>
  <p>which is an instance of the first triple under the instantiation mapping
    x -&gt; <em>sur</em>(A(x)). So a subgraph of D is an instance of E under
    the  instantiation mapping
    x -&gt; <em>sur</em>(A(x)); so D simply entails E.</p>
  <p>So either D contains an XML clash or D simply entails E, so D satisfies the 
    conditions of the lemma.</p>
  <p><strong>QED</strong></p>
</blockquote>
<p>Note that if E is finite, or if D contains an XML clash, then a finite subgraph 
  of D also satisfies the conditions of the lemma.</p>
<h2><a name="gloss" id="gloss"></a>Appendix B: Glossary of Terms (Informative)</h2>

    <p><strong><a name="glossAntecedent"
    id="glossAntecedent"></a>Antecedent</strong> (n.) In an <a
    href="#glossInference" class="termref">inference</a>, the
    expression(s) from which the <a href="#glossConsequent"
    class="termref">conclusion</a> is derived. In an <a
    href="#glossEntail" class="termref">entailment</a> relation, the
    entailer. Also <em>assumption</em>.</p>

    <p><strong><a name="glossAssertion"
    id="glossAssertion"></a>Assertion</strong> (n.) (i) Any expression
    which is claimed to be true. (ii) The act of claiming something to
    be true.</p>

    <p><strong><a name="glossClass" id="glossClass"></a>Class</strong>
    (n.) A general concept, category or classification. Something<a
    href="#glossResource" class="termref"></a> used primarily to
    classify or categorize other things. Formally, in RDF, a <a
    href="#glossResource" class="termref">resource</a> of type
    <code>rdfs:Class</code> with an associated set of <a
    href="#glossResource" class="termref">resources</a> all of which
    have the class as a value of the <code>rdf:type</code> property.
    Classes are often called 'predicates' in the formal logical
    literature.</p>

    <p>(RDF distinguishes <em>class</em> from <em>set</em>, although the two are often
    identified. Distinguishing classes from sets allows RDF more
    freedom in constructing class hierarchies, as <a
    href="#technote">explained earlier</a>.)</p>
<p><strong><a name="glossComplete"
    id="glossComplete"></a>Complete</strong> (adj., of an inference system). (1) 
  Able to detect all <a
    href="#glossEntail" class="termref">entailment</a>s between any two expressions. 
  (2) Able to draw all <a href="#glossValid"
    class="termref">valid</a> inferences. See <a href="#glossInference"
    class="termref"><em>Inference</em></a>. Also used with a qualifier: able to 
  detect entailments or draw all <a href="#glossValid"
    class="termref">valid</a> inferences in a certain limited form or kind (e.g. 
  between expressions in a certain normal form, or meeting certain syntactic conditions.)</p>
<p>(These definitions are not exactly equivalent, since the first requires that 
  the system has access to the consequent of the entailment, and may be unable 
  to draw 'trivial' inferences such as (p and p) from p. Typically, efficient 
  mechanical inference systems may be complete in the first sense but not necessarily 
  in the second.) </p>

    <p><strong><a name="glossConsequent"
    id="glossConsequent"></a>Consequent</strong> (n.) In an inference,
    the expression constructed from the <a href="#glossAntecedent"
    class="termref">antecedent</a>. In an entailment relation, the
    entailee. Also <em>conclusion</em>.</p>
<p><strong><a name="glossConsistent" 
	id="glossConsistent"></a>Consistent</strong> (adj., of an expression) Having 
  a satisfying <a href="#glossInterpretation"
    class="termref">interpretation</a>; not internally contradictory. (Also used 
  of an inference system as synonym for <em>Correct</em>.) </p>
<p><strong><a name="glossCorrect"
    id="glossCorrect"></a>Correct</strong> (adj., of an inference system). Unable 
  to draw any invalid inferences, or unable to make false claims of entailment. See <em>Inference</em>.</p>
<p><strong><a name="glossDecideable" id="glossDecideable"></a>Decideable</strong> 
  (adj., of an inference system). Able to determine for any pair of expressions, 
  in a finite time with finite resources, whether or not the first entails the 
  second. (Also: adj., of a logic:) Having a decideable inference system which 
  is complete and correct for the semantics of the logic.</p>
<p>(Not all logics can have inference systems which are both complete and decideable, 
  and decideable inference systems may have arbitrarily high computational complexity. 
  The relationships between logical syntax, semantics and complexity of an inference 
  system continue to be the subject of considerable research.)</p>

    <p><strong><a name="glossEntail"
    id="glossEntail"></a>Entail</strong> (v.),
    <strong>entailment</strong> (n.). A semantic relationship between
    expressions which holds whenever the truth of the first guarantees
    the truth of the second. Equivalently, whenever it is logically
    impossible for the first expression to be true and the second one
    false. Equivalently, when any <a href="#glossInterpretation"
    class="termref">interpretation</a> which <a href="#glossSatisfy"
    class="termref">satisfies</a> the first also satisfies the second.
    (Also used between a set of expressions and an expression.)</p>

    <p><strong><a name="glossEquivalent"
    id="glossEquivalent"></a>Equivalent</strong> (prep., with
    <em>to</em>) True under exactly the same conditions; making
    identical claims about the world, when asserted. <a
    href="#glossEntail" class="termref">Entails</a> and is entailed
    by.</p>

    <p><strong><a name="glossExtensional"
    id="glossExtensional"></a>Extensional</strong> (adj., of a logic) A
    set-based theory or logic of classes, in which classes are
    considered to be sets, properties considered to be sets of
    &lt;object, value&gt; pairs, and so on. A theory which admits no
    distinction between entities with the same extension. See <a
    href="#glossIntensional"
    class="termref"><em>Intensional</em></a>.</p>

    <p><strong><a name="glossFormal"
    id="glossFormal"></a>Formal</strong> (adj.) Couched in language
    sufficiently precise as to enable results to be established using
    conventional mathematical techniques.</p>

    <p><strong><a name="glossIff" id="glossIff"></a>Iff</strong>
    (conj.) Conventional abbreviation for 'if and only if'. Used to
    express necessary and sufficient conditions.</p>
<p><a name="glossInconsistent"
    id="glossInconsistent"></a><strong>Inconsistent</strong> (adj.) False under 
  all interpretations; impossible to <a
    href="#glossSatisfy" class="termref">satisfy</a>. <strong>Inconsistency</strong> 
  (n.), any inconsistent expression or graph.</p>
<p>(<a
    href="#glossEntail" class="termref">Entailment</a> and inconsistency are closely 
  related, since A entails B just when (A and not-B) is inconsistent, c.f. the 
  second definition for <a
    href="#glossEntail" class="termref">entailment</a>. This is the basis of many 
  mechanical inference systems. </p>
<p>Although the definitions of <a href="#glossConsistent" class="termref">consistency</a> 
  and inconsistency are exact duals, they are computationally dissimilar. It is 
  often harder to detect consistency in all cases than to detect inconsistency 
  in all cases<a
    href="#glossEntail" class="termref"></a>.)</p>
<p><strong><a name="glossIndexical"
    id="glossIndexical"></a>Indexical</strong> (adj., of an expression) having 
  a meaning which implicitly refers to the context of use. Examples from English 
  include words like 'here', 'now', 'this'.</p>
<p><strong><a name="glossInference"
    id="glossInference"></a>Infer</strong><strong>ence</strong> (n.) An act or 
  process of constructing new expressions from existing expressions, or the result 
  of such an act or process. Inferences corresponding to <a href="#glossEntail"
    class="termref">entailments</a> are described as <em>correct</em> or <em>valid</em>. 
  <strong>Inference rule</strong>, formal description of a type of inference; 
  <strong>inference system</strong>, organized system of inference rules; also, 
  software which generates inferences or checks inferences for validity.</p>

    <p><strong><a name="glossIntensional"
    id="glossIntensional"></a>Intensional</strong> (adj., of a logic)
    Not <a href="#glossExtensional" class="termref">extensional</a>. A
    logic which allows distinct entities with the same extension.</p>

    
<p>(The merits and demerits of intensionality have been extensively debated in 
  the philosophical logic literature. Extensional semantic theories are simpler, 
  and conventional semantics for formal logics usually assume an extensional view, 
  but conceptual analysis of ordinary language often suggests that intensional 
  thinking is more natural. Examples often cited are that an extensional logic 
  is obliged to treat all 'empty' extensions as identical, so must identify 'round 
  square' with 'santa clause', and is unable to distinguish concepts that 'accidentally' 
  have the same instances, such as human beings and bipedal hominids without body 
  hair. The semantics described in this document is basically intensional.)</p>

    <p><strong><a name="glossInterpretation"
    id="glossInterpretation"></a>Interpretation</strong>
    (<strong>of</strong>) (n.) A minimal formal description of those
    aspects of a <a href="#glossWorld" class="termref">world</a> which
    is just sufficient to establish the truth or falsity of any
    expression of a logic.</p>

    <p>(Some logic texts distinguish between a <em>interpretation
    structure</em>, which is a 'possible world' considered as something
    independent of any particular vocabulary, and an <em>interpretation
    mapping</em> from a vocabulary into the structure. The RDF
    semantics takes the simpler route of merging these into a single
    concept.)</p>

    <p><strong><a name="glossLogic" id="glossLogic"></a>Logic</strong>
    (n.) A formal language which expresses <a href="#glossProposition"
    class="termref">propositions</a>.</p>

    <p><a name="glossMetaphysical"
    id="glossMetaphysical"></a><strong>Metaphysical</strong> (adj.).
    Concerned with the true nature of things in some absolute or
    fundamental sense.</p>

    <p><a name="glossModeltheory"
    id="glossModeltheory"></a><strong>Model Theory</strong> (n.) A
    formal semantic theory which relates expressions to
    interpretations.</p>

    <p>(The name 'model theory' arises from the usage, traditional in
    logical semantics, in which a satisfying interpretation is called a
    "model". This usage is often found confusing, however, as it is
    almost exactly the inverse of the meaning implied by terms like
    "computational modelling", so has been avoided in this
    document.)</p>

    <p><strong><a name="glossMonotonic"
    id="glossMonotonic"></a>Monotonic</strong> (adj., of a logic or
    inference system) Satisfying the condition that if S entails E then
    (S + T) entails E, i.e. adding information to some antecedents
    cannot invalidate a <a href="#glossValid"
    class="termref">valid</a> entailment.</p>

    <p>(All logics based on a conventional <a href="#glossModeltheory"
    class="termref">model theory</a> and a standard notion of
    entailment are monotonic. Monotonic logics have the property that
    entailments remain <a href="#glossValid"
    class="termref">valid</a> outside of the context in which they were
    generated. This is why RDF is designed to be monotonic.)</p>

    <p><strong><a name="glossNonmonotonic"
    id="glossNonmonotonic"></a>Nonmonotonic</strong> (adj.,of a logic
    or inference system) Not <a href="#glossMonotonic"
    class="termref">monotonic</a>. Non-monotonic formalisms have been
    proposed and used in AI and various applications. Examples of
    nonmonotonic inferences include <em>default reasoning</em>, where
    one assumes a 'normal' general truth unless it is contradicted by
    more particular information (birds normally fly, but penguins
    don't fly); <em>negation-by-failure</em>, commonly assumed in logic
    programming systems, where one concludes, from a failure to prove a
    proposition, that the proposition is false; and <em>implicit
    closed-world assumptions</em>, often assumed in database
    applications, where one concludes from a lack of information about
    an entity in some corpus that the information is false (e.g. that
    if someone is not listed in an employee database, that he or she is not
    an employee.)</p>

    <p>(The relationship between monotonic and nonmonotonic inferences
    is often subtle. For example, if a closed-world assumption is made
    explicit, e.g. by asserting explicitly that the corpus is complete
    and providing explicit provenance information in the conclusion,
    then closed-world reasoning is monotonic; it is the implicitness
    that makes the reasoning nonmonotonic. Nonmonotonic conclusions can
    be said to be <a href="#glossValid"
    class="termref">valid</a> only in some kind of 'context', and are liable
    to be incorrect or misleading when used outside that context.
    Making the context explicit in the reasoning and visible in the
    conclusion is a way to map them into a monotonic framework.)</p>

    <p><strong><a name="glossOntological"
    id="glossOntological"></a>Ontological</strong> (adj.) (Philosophy)
    Concerned with what kinds of things really exist. (Applied)
    Concerned with the details of a formal description of some topic or
    domain.</p>

    <p><strong><a name="glossProposition"
    id="glossProposition"></a>Proposition</strong> (n.) Something that
    has a truth-value; a statement or expression that is true or
    false.</p>

    <p>(Philosophical analyses of language traditionally distinguish
    propositions from the expressions which are used to state them, but
    model theory does not require this distinction.)</p>

    <p><strong><a name="glossReify" id="glossReify"></a>Reify</strong>
    (v.), <strong>reification</strong> (n.) To categorize as an object;
    to describe as an entity. Often used to describe a convention
    whereby a syntactic expression is treated as a semantic object and
    itself described using another syntax. In RDF, a reified triple is
    a description of a triple-token using other RDF triples.</p>

    <p><strong><a name="glossResource"
    id="glossResource"></a>Resource</strong> (n.)(as used in RDF)(i) An
    entity; anything in the universe. (ii) As a class name: the class
    of everything; the most inclusive category possible.</p>

    <p><strong><a name="glossSatisfy"
    id="glossSatisfy"></a>Satisfy</strong> (v.t.),
    <strong>satisfaction</strong>,(n.) <strong>satisfying</strong>
    (adj., of an interpretation). To make true. The basic semantic
    relationship between an interpretation and an expression. X
    satisfies Y means that if <a href="#glossWorld" class="termref">the
    world</a> conforms to the conditions described by X, then Y must be
    true.</p>

    <p><strong><a name="glossSemantic"
    id="glossSemantic"></a>Semantic</strong> (adj.) ,
    <strong>semantics</strong> (n.). Concerned with the specification
    of meanings. Often contrasted with <em>syntactic</em> to emphasize
    the distinction between expressions and what they denote.</p>

    <p><a name="glossSkolemization" id="glossSkolemization"></a><a
    href="#skolemlemprf"><strong>Skolemization</strong></a> (n.) A
    syntactic transformation in which blank nodes are replaced by 'new'
    names.</p>

    <p>(Although not strictly <a href="#glossValid"
    class="termref">valid</a>, Skolemization retains the
    essential meaning of an expression and is often used in mechanical
    inference systems. The full logical form is more complex. It is
    named after the logician <a
    href="http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Skolem.html">
    A. T. Skolem</a>)</p>

    <p><a name="glossToken" id="glossToken"></a><strong>Token</strong>
    (n.) A particular physical inscription of a symbol or expression in
    a document. Usually contrasted with <em>type</em>, the abstract
    grammatical form of an expression.</p>

    <p><strong><a name="glossUniverse"
    id="glossUniverse"></a>Universe</strong> (n., also <em>Universe of
    discourse</em>) The universal classification, or the set of all
    things that an interpretation considers to exist. In RDF/S, this is
    identical to the set of resources.</p>

    <p><strong><a name="glossUse" id="glossUse"></a>Use</strong> (v.)
    contrasted with <em>mention</em>; to use a piece of syntax to
    denote or refer to something else. The normal way that language is
    used.</p>

    <p>("Whenever, in a sentence, we wish to say something about a
    certain thing, we have to use, in this sentence, not the thing
    itself but its name or designation." - <a
    href="http://www.philosophypages.com/dy/t.htm">Alfred
    Tarski</a>)</p>

    <p><strong><a name="glossValid" id="glossValid"></a>Valid</strong>
    (adj., of an inference or inference process) Corresponding to an <a
    href="#glossEntail" class="termref">entailment</a>, i.e. the
    conclusion of the inference is entailed by the antecedent of the
    inference. Also <em>correct</em>.</p>

    <p><a name="glossWellformed"
    id="glossWellformed"></a><strong>Well-formed</strong> (adj., of an
    expression). Syntactically legal.</p>

    <p><strong><a name="glossWorld" id="glossWorld"></a>World</strong>
    (n.) (with <em>the:</em>) (i) The actual world. (with
    <em>a:</em>) (ii) A way that the actual world might be arranged.
    (iii) An <a href="#glossInterpretation"
    class="termref">interpretation</a> (iv) A possible world.</p>

    
<p>(The metaphysical status of '<a
    href="http://plato.stanford.edu/entries/actualism/possible-worlds.html">possible 
  worlds</a>' is highly controversial. Fortunately, one does not need to commit 
  oneself to a belief in parallel universes in order to use the concept in its 
  second and third senses, which are sufficient for semantic purposes.)</p>

    
<h2><a name="ack" id="ack"></a>Appendix C: Acknowledgements</h2>
<p>This document reflects the joint effort of the members of the <a
    href="http://www.w3.org/2001/sw/RDFCore/">RDF Core Working Group</a>. Particular 
  contributions were made by Jeremy Carroll, Dan Connolly, Jan Grant, R. V. Guha, 
  Graham Klyne, Ora Lassilla, Brian McBride, Sergey Melnick, Jos deRoo and Patrick 
  Stickler. </p>

    <p>The basic idea of using an explicit extension mapping to allow
    self-application without violating the axiom of foundation was
    suggested by Christopher Menzel.</p>
<p>Peter Patel-Schneider and Herman ter Horst found several major problems in 
  earlier drafts, and suggested several important technical improvements.</p>
<p>Patrick Hayes' work on this document was supported in part by DARPA under contract 
  #2507-225-22. </p>
<h2><a name="refs" id="refs">References</a></h2>
<h2><a name="normative" id="normative">Normative</a></h2>

    
<dl>

          <dt><a id="ref-rdf-concepts"
          name="ref-rdf-concepts"></a>[RDF-CONCEPTS]</dt>
<dd><cite><a href="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, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/ . <a href="http://www.w3.org/TR/rdf-concepts/">Latest version</a> available at http://www.w3.org/TR/rdf-concepts/ .</dd>


          <dt><a id="ref-rdf-syntax"
          name="ref-rdf-syntax"></a>[RDF-SYNTAX]</dt>
<dd><cite><a href="http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/">RDF/XML Syntax Specification (Revised)</a></cite>, Dave Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/ . <a href="http://www.w3.org/TR/rdf-syntax-grammar/">Latest version</a> available at http://www.w3.org/TR/rdf-syntax-grammar/ .</dd>

          <dt><a id="ref-rdf-tests"
          name="ref-rdf-tests"></a>[RDF-TESTS]</dt>
<dd><cite><a href="http://www.w3.org/TR/2004/REC-rdf-testcases-20040210/">RDF Test Cases</a></cite>, Jan Grant and Dave Beckett, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-testcases-20040210/ . <a href="http://www.w3.org/TR/rdf-testcases/">Latest version</a> available at http://www.w3.org/TR/rdf-testcases/ .</dd>


  <dt><a name="ref-rdfms" id="ref-rdfms"></a>[RDFMS]</dt>
  <dd> <cite><a href="http://www.w3.org/TR/1999/REC-rdf-syntax-19990222/">Resource 
    Description Framework (RDF) Model and Syntax Specification</a></cite> , O. 
    Lassila and R. Swick, Editors. World Wide Web Consortium. 22 February 1999. 
    This version is http://www.w3.org/TR/1999/REC-rdf-syntax-19990222/. The <a href="http://www.w3.org/TR/REC-rdf-syntax/">latest 
    version of RDF M&amp;S</a> is available at http://www.w3.org/TR/REC-rdf-syntax/. 
  </dd>
  <dt><a id="ref-2119" name="ref-2119"></a>[RFC 2119]</dt>
  <dd> <cite><a href="http://www.ietf.org/rfc/rfc2119.txt">RFC 2119 - Key words 
    for use in RFCs to Indicate Requirement Levels</a></cite> , S. Bradner, IETF. 
    March 1997. This document is http://www.ietf.org/rfc/rfc2119.txt. </dd>
  <dt><a name="ref-2369" id="ref-2369"></a>[RFC 2396]</dt>
  <dd><cite><a href="http://www.isi.edu/in-notes/rfc2396.txt">RFC 2396 - Uniform 
    Resource Identifiers (URI): Generic Syntax</a></cite> Berners-Lee,T., Fielding 
    and Masinter, L., August 1998</dd>
  <dt><a id="ref-xmls" name="ref-xmls"></a>[XSD]</dt>
  <dd><cite><a href="http://www.w3.org/TR/xmlschema-2/">XML Schema Part 2: Datatypes</a></cite>, 
    Biron, P. V., Malhotra, A. (Editors) World Wide Web Consortium Recommendation, 
    2 May 2001</dd>
</dl>

    <h2><a name="nonnormative" id="nonnormative">Non-Normative</a></h2>
<dl>
<dt><a name="ref-owl" id="ref-owl"></a>[OWL]</dt>
<dd><cite><a href="http://www.w3.org/TR/2004/REC-owl-ref-20040210/">OWL Web Ontology Language Reference</a></cite>, Mike Dean and Guus Schreiber, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-ref-20040210/ . <a href="http://www.w3.org/TR/owl-ref/">Latest version</a> available at http://www.w3.org/TR/owl-ref/ .</dd>

  <dt><a id="ref-ConKla"
      name="ref-ConKla"></a>[Conen&amp;Klapsing]</dt>
  <dd><cite><a
      href="http://nestroy.wi-inf.uni-essen.de/rdf/logical_interpretation/index.html"> 
    A Logical Interpretation of RDF</a></cite>, Conen, W., Klapsing, R..Circulated 
    to <a
      href="http://lists.w3.org/Archives/Public/www-rdf-interest/2000Aug/0122.html"> 
    RDF Interest Group</a>, August 2000.</dd>
  <dt><a name="ref-daml" id="ref-daml">[DAML]</a></dt>
  <dd>Frank van Harmelen, Peter F. Patel-Schneider, Ian Horrocks (editors), <a
      href="http://www.daml.org/2001/03/reference.html"><em>Reference Description 
    of the DAML+OIL (March 2001) ontology markup language</em></a></dd>
  <dt><a id="ref-HayMen"
      name="ref-HayMen"></a>[Hayes&amp;Menzel]</dt>
  <dd><cite><a
      href="http://reliant.teknowledge.com/IJCAI01/HayesMenzel-SKIF-IJCAI2001.pdf"> 
    A Semantics for the Knowledge Interchange Format</a></cite>, Hayes, P., Menzel, 
    C., Proceedings of 2001 <a
      href="http://reliant.teknowledge.com/IJCAI01/">Workshop on the IEEE Standard 
    Upper Ontology</a>, August 2001.</dd>
  <dt><a name="ref-KIF" id="ref-KIF">[KIF]</a></dt>
  <dd>Michael R. Genesereth et. al., <a
      href="http://logic.stanford.edu/kif/dpans.html"><em>Knowledge Interchange 
    Format</em></a>, 1998 (draft American National Standard).</dd>
  <dt><a id="ref-MarSaa"
      name="ref-MarSaa"></a>[Marchiori&amp;Saarela]</dt>
  <dd><cite><a
      href="http://www.w3.org/TandS/QL/QL98/pp/metalog.html">Query + Metadata 
    + Logic = Metalog</a></cite>, Marchiori, M., Saarela, J. 1998.</dd>
  <dt><a id="ref-Lbase" name="ref-Lbase"></a>[LBASE]</dt>
  <dd><cite><a href="http://www.w3.org/TR/2003/NOTE-lbase-20031010/">Lbase: Semantics 
    for Languages of the Semantic Web</a></cite>, Guha, R. V., Hayes, P., W3C 
    Note, 10 October 2003.</dd>
  <dt><a id="ref-daml-axiomat"
      name="ref-daml-axiomat"></a>[McGuinness&amp;al]</dt>
  <dd><cite><a
      href="http://www.ksl.stanford.edu/people/dlm/papers/daml-oil-ieee-abstract.html"> 
    DAML+OIL:An Ontology Language for the Semantic Web</a></cite>, McGuinness, 
    D. L., Fikes, R., Hendler J. and Stein, L.A., IEEE Intelligent Systems, Vol. 
    17, No. 5, September/October 2002.</dd>
	

  <dt><a id="ref-rdf-primer" name="ref-rdf-primer">[RDF-PRIMER]</a>
  </dt>
<dd><cite><a href="http://www.w3.org/TR/2004/REC-rdf-primer-20040210/">RDF Primer</a></cite>, Frank Manola and Eric Miller, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-primer-20040210/ . <a href="http://www.w3.org/TR/rdf-primer/">Latest version</a> available at http://www.w3.org/TR/rdf-primer/ .</dd>



          <dt><a id="ref-rdf-vocabulary"
          name="ref-rdf-vocabulary"></a>[RDF-VOCABULARY]</dt>

<dd><cite><a href="http://www.w3.org/TR/2004/REC-rdf-schema-20040210/">RDF Vocabulary Description Language 1.0: RDF Schema</a></cite>, Dan Brickley and R. V. Guha, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-schema-20040210/ . <a href="http://www.w3.org/TR/rdf-schema/">Latest version</a> available at http://www.w3.org/TR/rdf-schema/ .</dd>


  </dl>
	<hr />
 
  
<h2><a id="changes" name="changes"></a><a name="change" id="change"></a>Appendix D: Change Log. (Informative)</h2>
<p><strong>Changes since the <a href="http://www.w3.org/TR/2003/PR-rdf-mt-20031215/">15 December 2003 Proposed Recommendation</a>.</strong></p>
<p>An error in the wording of the RDFS entailment lemma in appendix A was corrected. 
  Some typos in the glossary and main text were corrected.</p>
<p>After considering <a href="http://lists.w3.org/Archives/Public/www-rdf-comments/2003OctDec/0233.html">comments 
  by ter Horst</a>, the definition of D-interpretation has been modified so as 
  to apply to an extended vocabulary including the datatype names. </p>
<p>Older entries in the change log were removed.   They can be found in <a href="http://www.w3.org/TR/2003/PR-rdf-mt-20031215/#change">the previous version.</a></p>
  
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