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Added the semantics of triple terms, according to the "alternative semantics"
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@franconi
franconi authored Dec 2, 2024
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Expand Up @@ -465,6 +465,7 @@ <h6 class="defcap">Definition of a Simple Interpretation</h6>
<li>A mapping <em>IEXT</em> from <em>IP</em> into the powerset of <em>IR</em> x <em>IR</em>, i.e., the set of sets of pairs <em>&lt;x, y&gt;</em> with <em>x</em> and <em>y</em> in <em>IR</em>.</li>
<li>A mapping <em>IS</em> from <em>IRIs</em> into (<em>IR</em> union <em>IP</em>).</li>
<li>A partial mapping <em>IL</em> from literals into <em>IR</em>.</li>
<li><span style="background-color: Cyan;color: Black">6. An injective mapping RE from IR x IP x IR into IR, called the denotation of triple terms</span> </li>
</ol>
</div>
</section>
Expand Down Expand Up @@ -506,16 +507,19 @@ <h6 class="defcap">Definition of a Simple Interpretation</h6>
from its set-theoretic extension.
A similar technique is used in the ISO/IEC Common Logic standard [[?ISO24707]].
</p>
<p>The referent of a ground RDF graph in a simple interpretation <em>I</em> is then given by the following rules,
<p><span style="background-color: Cyan;color: Black"><s>The referent of a ground RDF graph in a simple interpretation <em>I</em> is then given by the following rules,
where the interpretation is also treated as a function from expressions (<a>names</a>, <a>triples</a> and <a>graphs</a>) to elements of the universe and truth values:
</p>
</s></span></p>
<p><span style="background-color: Cyan;color: Black">The referent of a ground RDF graph in a simple interpretation I is given by an <i>interpretation</i> function I from expressions (<a>names</a>, <a>triples</a> and <a>graphs</a>) to elements of the universe and truth values, defined as follows:</span></p>
<section class="notoc">
<h6 class="defcap">Semantic Conditions for Ground Graphs</h6>
<div class="semantictable">
<p>If <em>E</em> is a literal, then <em>I(E) = IL(E)</em></p>
<span class="space"></span>
<p>If <em>E</em> is an IRI, then <em>I(E) = IS(E)</em></p>
<span class="space"></span>
<p><span style="background-color: Cyan;color: Black">if <em>E</em> is a Triple Term then <em>I(E) = RE(I(E.s), I(E.p), I(E.o))</em><br> where we write the subject, predicate, object of <em>E</em> as <em>E.s, E.p, E.o</em>, respectively.</span></p>
<span class="space"></span>
<p>If <em>E</em> is a ground triple `s p o.`, <em>I(p)</em>
is in <em>IP</em> and the pair &lt;<em>I(s)</em>, <em>I(o)</em>&gt;
is in <em>IEXT(I(p))</em>, then <em>I(E) = true</em>;
Expand Down Expand Up @@ -957,6 +961,9 @@ <h6 class="defcap"><dfn>RDF vocabulary</dfn></h6>
<li><code>rdf:subject</code></li>
<li><code>rdf:predicate</code></li>
<li><code>rdf:object</code></li>
<li><code><span style="background-color: Cyan;color: Black">rdf:TripleTerm</span></code></li>
<li><code><span style="background-color: Cyan;color: Black">rdf:ReificationProperty</span></code></li>
<li><code><span style="background-color: Cyan;color: Black">rdf:InverseReificationProperty</span></code></li>
<li><code>rdf:first</code></li>
<li><code>rdf:rest</code></li>
<li><code>rdf:value</code></li>
Expand Down Expand Up @@ -985,6 +992,36 @@ <h6 class="defcap">RDF Semantic Conditions</h6>
<em>IEXT(I(<code>rdf:type</code>))</em> if and only if <em>x</em> is in the
<a>value space</a> of <em>I(aaa)</em>.
</p>
<span class="space"></span>

<p id="rdfsemcond1"><span style="background-color: Cyan;color: Black">
if <em>[I+A](E) = TRUE</em> and <em>E.s</em> is a triple term, then
<em><[I+A](E.s), I(<code>rdf:TripleTerm</code>)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em>.
</span></p>
<span class="space"></span>

<p id="rdfsemcond1"><span style="background-color: Cyan;color: Black">
if <em>[I+A](E) = TRUE</em> and <em>E.o</em> is a triple term, then
<em><[I+A](E.o), I(<code>rdf:TripleTerm</code>)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em>.
</span></p>
<span class="space"></span>

<p id="rdfsemcond1"><span style="background-color: Cyan;color: Black">
if <em>[I+A](E) = TRUE</em> and
<em><[I+A](E.o), [I+A](rdf:TripleTerm)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em> and
<em>E.p ≠ <code>rdf:type</code></em>, then
<em><[I+A](E.p), I(<code>rdf:ReificationProperty</code>)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em>.
</span></p>
<span class="space"></span>

<p id="rdfsemcond1"><span style="background-color: Cyan;color: Black">
if <em>[I+A](E) = TRUE</em> and
<em><[I+A](E.s), [I+A](rdf:TripleTerm)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em>
and <em>E.p ≠ <code>rdf:type</code></em>, then
<em><[I+A](E.p), I(<code>rdf:InverseReificationProperty</code>)></em> is in <em>IEXT(I(<code>rdf:type</code>))</em>.
</span></p>


</div>
</section>
<p>and <a>satisfies</a> every triple in the following infinite set:</p>
Expand All @@ -1002,6 +1039,7 @@ <h6 class="defcap">RDF Axioms</h6>
rdf:nil rdf:type rdf:List .
rdf:_1 rdf:type rdf:Property .
rdf:_2 rdf:type rdf:Property .
<span style="background-color: Cyan;color: Black">rdf:reifies rdf:type rdf:ReificationProperty .</span>
...
</pre>
</div>
Expand Down Expand Up @@ -1110,6 +1148,29 @@ <h4>Patterns of RDF entailment (Informative)</h4>
_:x rdf:type xsd:boolean .
_:x rdf:type xsd:integer .
</pre>

<p><span style="background-color: Cyan;color: Black">
The semantics conditions for triple terms give the following RDF entailment pattern:
</span></p>

<table style="background-color: Cyan;color: Black">
<caption style="background-color: Cyan;color: Black">RDF entailment patterns with triple terms.</caption>
<tbody style="background-color: Cyan;color: Black">
<tr style="background-color: Cyan;color: Black">
<th > </th>
<th ><strong>if <em>S</em> contains</strong></th>
<th ><strong>then <em>S</em> RDF entails</strong></th>
</tr>
<tr>
<td class="othertable" style="background-color: Cyan;color: Black"><dfn>rdfREIF1</dfn></td>
<td class="othertable" style="background-color: Cyan;color: Black">sss aaa ttt <code>.</code><br/> for ttt a triple term</td>
<td class="othertable" style="background-color: Cyan;color: Black">aaa <code>rdf:type</code> <code>rdf:ReificationProperty .</code><br/>sss aaa _:nnn <code>.</code><br/>_:nnn <code>rdf:type</code><code> rdf:TripleTerm </code><code>.</code></td>
</tr>
</tbody>
</table>



</section>
</section>
</section>
Expand Down Expand Up @@ -1619,6 +1680,38 @@ <h2>Entailment rules (Informative)</h2>
<p>As noted earlier, detecting datatype entailment for larger sets of datatype IRIs
requires attention to idiosyncratic properties of the particular datatypes.
</p>

<table style="background-color: Cyan;color: Black">
<caption style="background-color: Cyan;color: Black">G-RDF entailment patterns with triple terms.</caption>
<tbody style="background-color: Cyan;color: Black">
<tr style="background-color: Cyan;color: Black">
<th > </th>
<th ><strong>if <em>S</em> contains</strong></th>
<th ><strong>then <em>S</em> RDF entails</strong></th>
</tr>
<tr >
<td class="othertable" style="background-color: Cyan;color: Black"><dfn>G-rdfREIF1</dfn></td>
<td class="othertable" style="background-color: Cyan;color: Black">sss aaa ttt <code>.</code><br/> ttt <code>rdf:type</code> <code>rdf:TripleTerm .</code><br/>for aaa ≠ <code>rdf:type</code></td>
<td class="othertable" style="background-color: Cyan;color: Black">aaa <code>rdf:type</code> <code>rdf:ReificationProperty .</code></td>
</tr>
<tr>
<td class="othertable" style="background-color: Cyan;color: Black"><dfn>G-rdfREIF2</dfn></td>
<td class="othertable" style="background-color: Cyan;color: Black">ttt aaa ooo <code>.</code><br/> ttt <code>rdf:type</code> <code>rdf:TripleTerm .</code><br/>for aaa ≠ <code>rdf:type</code></td>
<td class="othertable" style="background-color: Cyan;color: Black">aaa <code>rdf:type</code> <code>rdf:InverseReificationProperty .</code></td>
</tr>
<tr>
<tr>
<td class="othertable" style="background-color: Cyan;color: Black"><dfn>G-rdfTT1</dfn></td>
<td class="othertable" style="background-color: Cyan;color: Black">sss aaa ttt <code>.</code><br/> for ttt a triple term</td>
<td class="othertable" style="background-color: Cyan;color: Black">ttt <code>rdf:type</code> <code>rdf:TripleTerm .</code></td>
</tr>
<td class="othertable" style="background-color: Cyan;color: Black"><dfn>G-rdfTT2</dfn></td>
<td class="othertable" style="background-color: Cyan;color: Black">ttt aaa ooo <code>.</code><br/> for ttt a triple term</td>
<td class="othertable" style="background-color: Cyan;color: Black">ttt <code>rdf:type</code> <code>rdf:TripleTerm .</code></td>
</tr>
</tbody>
</table>

</section>
<section id="finite_interpretations" class="informative appendix">
<h2>Finite interpretations</h2>
Expand Down

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