Improvements on the semantics

spec/semantics.html

@@ -306,8 +306,8 @@ <h3>Terms, Triples and Graphs</h3>
 -->
 
       <div class=note>
-        In this context, an RDF graph is a graph that does not contain graph terms nor lists, does not have literals as subjects or predicates, does not have variables as predicates, and for which $U=\emptyset$.
-        The blank nodes occurring in the graph would be labeled and stored in $E$.
+        In this context, an RDF graph is a graph that does not contain graph terms nor lists, does not have literals as subjects or predicates, does not have variables as predicates, for which $U=\emptyset$,
+        and $E$ is the set of blank nodes occurring in the graph.
       </div>
 
       <div class="issue">The translation of nested blank nodes as explained here and for example applied in <a href="#example3"> Example 3</a> follows
@@ -320,7 +320,7 @@ <h3>Terms, Triples and Graphs</h3>
         as<br/> 
         <span class="asex">$\qquad (\{\},\{x\},\{(\text{:socrates}, \text{:says}, &lt;\{\},\{\},\{(x,\text{rdf:type}, \text{:Mortal} )\}&gt;) \})$</span><br/>
          and <strong>not</strong> as<br/>
-         <span class="asex"> $(\{\},\{\},\{(\text{:socrates}, \text{:says}, &lt;\{\},\{x\},\{(x,\text{rdf:type}, \text{:Mortal} )\}&gt;) \})$</span>.<br/>
+         <span class="asex">$\qquad (\{\},\{\},\{(\text{:socrates}, \text{:says}, &lt;\{\},\{x\},\{(x,\text{rdf:type}, \text{:Mortal} )\}&gt;) \})$</span>.<br/>
          A solution for that problem is currently being discussed. The difference, however, does not influence the formal semantics of the <em>abstract</em> syntax, as quantification is always explicit there.
        </div>
             </section>
@@ -387,11 +387,11 @@ <h2>Variables</h2>
 
             <p class=note>Note that the definition makes a difference between <a>graphs</a> and <a>graph terms</a>.
               A graph term is a <a>direct constituent</a> of the graph ("containing graph") in which it occurs as a triple component. However, the direct constituents of the graph term's graph, and their direct constituents (and so on), are not direct constituents of the containing graph. Rather, they are <a>constituents</a> of the containing graph.
-              This difference is crucial for the definitions below: a graph is not the same as a graph term.</p>
+              This difference between graph and graph term is crucial for the definitions below.</p>
 <p>
 With that definition, we now introduce <dfn data-lt="free">free variables</dfn>: these are variables which occur directly or nested in the graph but which are not covered by a <a>quantification set</a>.
 </p>
-            The set of free variables ($FV$) of a term or a set of triples is defined as follows:
+            <p>The set of free variables ($FV$) of a <a>term</a> or a set of <a>triples</a> is defined as follows:
 
               <ul>
                 <li>$FV(x) = \emptyset$ if $x ∈ R ∪ L$,</li>
@@ -402,13 +402,12 @@ <h2>Variables</h2>
                 <li>$FV(x) = FV(F)\setminus(U ∪ E)$ if $x=&lt;U,E,F&gt;$ is a <a>graph term</a>.</li>
               </ul>
 
-              The set $FV$ for a <a>graph</a> $G=(U,E,F)$ and a variable $v\in V\cap C(G)$:
+              <p>Consider a <a>graph</a> $G=(U,E,F)$ and a variable $v\in V\cap C(G)$:
               <ul>
                 <li>We say that $v$  is <a>free</a> in $G$ iff $v\not\in U\cup E$ and there exists $x\in DC(G)$ such that $v\in FV(x)$;</li>
                 <li>Otherwise we call $v$ <dfn class="lint-ignore" data-lt="scoped variable">scoped</dfn> in $G$;</li>
-                <li>We denote the set of variables which are <a>free</a> in $G$ as $FV(G)$.</li>
               </ul>
-<p>
+              <p>We denote the set of variables which are free in $G$ as $FV(G)$.
 
 <p>Examples:</p>
 <ol>
@@ -421,8 +420,8 @@ <h2>Variables</h2>
 </ol>
 <p>This enables us to introduce the notion of closed graphs and ground terms.</p>
 <ul>
-   <li>A <a>graph</a> $G=(U,E,F)$ is <dfn data-lt="closed graph">closed</dfn> if all variables $v\in C(G)\cap V$ are scoped in $G$.</li>
-   <li>A term is <dfn data-lt="ground term">ground</dfn> if it is either
+   <li>A <a>graph</a> $G=(U,E,F)$ is <dfn data-lt="closed graph">closed</dfn> if all variables $v\in V \cap C(G)$ are scoped in $G$.</li>
+   <li>A <a>term</a> is <dfn data-lt="ground term">ground</dfn> if it is either
     <ul>
       <li>an IRI,</li>
       <li>a Literal,</li>
@@ -434,7 +433,7 @@ <h2>Variables</h2>
 
 <p>We denote the set of <a>ground terms</a> as $T_G$.
 
-<p>A <a>triple</a> is <a>ground</a> if all of its <a>direct constituents</a> are <a>ground</a>.</p>
+<p>A <a>triple</a> is <a>ground</a> if its subject, predicate and object are all <a>ground</a>.</p>
 
 <p>A graph $G=(U,E,F)$ is <a>ground</a> if $E=U=\emptyset$ and all <a>triples</a> in $F$ are <a>ground</a>.</p>
 
@@ -454,7 +453,7 @@ <h2>Variables</h2>
   the variable $x$ will be a free variable (as it was quantified in the containing N3 graph).
 </p>
 
-<p>An abstract <a>graph</a> $G=(U,E,F)$ can also contain quantified variables $v\in U\cup E$ which do not occur in any <a>triple</a> of $F$. 
+<p>An abstract <a>graph</a> $G=(U,E,F)$ can also contain quantified variables $v\in U\cup E$ which do not occur <a>free</a> in any <a>triple</a> of $F$. 
    These variables do not contribute to the interpretation of the graph. In order to simplify our considerations for the following sections, 
    we introduce the notion of normalised graphs:</p>
 
@@ -767,7 +766,7 @@ <h2>Graphs with variables</h2>
   The mapping $A_2$ assigns variables to terms, and will be used to <em>ground free variables within graph terms</em>. 
   Note that given an interpretation $I$ and two assignments $A$ for a set $V_1$, and $B$ for a set $V_2$ of variables, 
   the <a>combination</a> $A\bullet B$ is again an assignment for $V_1\cup V_2$.</p>
-  <!----    </section>
+  <!--    </section>
 
       <section>
         <h2></h2>