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addresses #228 in Sec.18.5 'Property Path Patterns', and addresses #218 
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‎spec/index.html‎

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@@ -9686,14 +9686,18 @@ <h4>Treatment of Blank Nodes</h4>
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<section id="PropertyPathPatterns">
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<h3>Property Path Patterns</h3>
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<p>This section defines the evaluation of <a href="#defn_PropertyPathPattern">property path
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patterns</a>. A property path pattern is a subject endpoint (an RDF term or a variable), a
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<a href="#defn_PropertyPathExpr">property path expression</a>, and an object endpoint. The
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<a href="#sparqlTranslatePathExpressions">translation of property path expressions</a> converts every
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<a href="#defn_PropertyPathExpr">property path expression</a> into an
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<a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
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Thereafter, <a href="#sparqlTranslatePathPatterns">translation of property path pattern</a> converts
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some of these algebraic property path expressions
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to other SPARQL graph pattern, such as converting property paths of length one to triple
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patterns</a>. A property path pattern consists of a subject endpoint (an RDF term or a variable), a
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<a href="#defn_PropertyPathExpr">property path expression</a>, and an object endpoint.</p>
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<p>The <a href="#sparqlTranslatePathExpressions">translation of property path expressions</a>
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converts every <a href="#defn_PropertyPathExpr">property path expression</a>
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into an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
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For example, the property path expression <code>(:p/:q)*</code>
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is a ZeroOrMorePath expression involving a sequence property path,
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and is translated into the algebraic property path expression
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<a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>( <a href="#defn_ppeSeq" class="ppeOp">seq</a>(<a href="#defn_ppeLink" class="ppeOp">link</a>(:p), <a href="#defn_ppeLink" class="ppeOp">link</a>(:q)) ).</p>
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<p>Thereafter, the <a href="#sparqlTranslatePathPatterns">translation of property path patterns</a>
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converts some of these algebraic property path expressions
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to other SPARQL graph patterns, such as converting property paths of length one to triple
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patterns, which in turn are combined into basic graph patterns.
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This leaves algebraic property path expressions with the operators
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<a href="#defn_ppeZeroOrOnePath" class="ppeOp">ZeroOrOnePath</a>,
@@ -9704,12 +9708,7 @@ <h3>Property Path Patterns</h3>
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expressions contained within these operators.</p>
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<p>The property path patterns with these remaining algebraic property path expressions
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are present in the <a href="#algebraicSyntax">algebraic syntax</a> in the form
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<a href="#defn_absPath" class="absOp">Path</a>(|X|, |ppe|, |X|) for endpoints |X| and |Y|.
9708-
For example, the <a href="#defn_PropertyPathExpr">property path expression</a> <code>(:p/:q)*</code>
9709-
is a ZeroOrMorePath expression involving a sequence property path,
9710-
and is translated into the algebraic property path expression
9711-
<a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>( <a href="#defn_ppeSeq" class="ppeOp">seq</a>(<a href="#defn_ppeLink" class="ppeOp">link</a>(:p), <a href="#defn_ppeLink" class="ppeOp">link</a>(:q)) ).</p>
9712-
<div class="ednote">The example at the end of the previous paragraph fits much better into the first paragraph of the section and, thus, should be moved there.</div>
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<a href="#defn_absPath" class="absOp">Path</a>(|X|, |ppe|, |X|) for endpoints |X| and |Y|.</p>
97139712
<div class="defn">
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<div id="pp-eval-notation">
97159714
<b>Notation</b>
@@ -9776,139 +9775,150 @@ <h3>Property Path Patterns</h3>
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<p>where |PV| = { |v| ∈ {|X|,|Y|} | |v| is a variable}.</p>
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<div class="issue" data-number="266"><a href="#defn_algJoin" class="algFct">Join</a> produces a multiset of solution mappings but <a href="#defn_algProject" class="algFct">Project</a> expects a sequence as its first argument. Moreover, <a href="#defn_algProject" class="algFct">Project</a> produces a sequence but ppeval(..) should be a multiset.</div>
97789777
</div>
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<div class="ednote">Olaf, continue here!</div>
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<p>Informally, this is the same as:</p>
9781-
<pre>SELECT * { X P _:a . _:a Q Y }</pre>
9782-
<p>using the fact that a blank node <code>_:a</code> acts like a variable (under simple
9779+
<pre class="nohighlight">SELECT * { <var>X</var> <var>P<sub>1</sub></var> _:a . _:a <var>P<sub>2</sub></var> <var>Y</var> }</pre>
9780+
<p>where <var>P<sub>1</sub></var> is a
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<a href="#defn_AlgebraicPropertyPathExpression">property path expression</a>
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that can be <a href="#sparqlTranslatePathExpressions">translated</a> into the
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<a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a> <var>ppe<sub>1</sub></var>
9784+
and <var>P<sub>2</sub></var> can be translated into <var>ppe<sub>2</sub></var>.
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This observation is based on the fact that a blank node <code>_:a</code> acts like a variable (under simple
97839786
entailment) except it does not appear in the results from <code>SELECT *</code>.</p>
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<div class="defn">
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<p><b>Definition: <span id="defn_evalPP_alternative">Evaluation of Alternative Property Path</span></b></p>
9786-
<p>Let P and Q be property path expressions.</p>
9787-
<pre>
9788-
eval(Path(X, alt(P,Q), Y)) =
9789-
Union(eval(Path(X, P, Y)), eval(Path(X, Q, Y)))
9789+
<p>Let <var>ppe<sub>1</sub></var> and <var>ppe<sub>2</sub></var> be <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expressions</a>.
9790+
<pre class="nohighlight">
9791+
ppeval(<var>X</var>, <a href="#defn_ppeAlt" class="ppeOp">alt</a>(<var>ppe<sub>1</sub></var>,<var>ppe<sub>2</sub></var>), <var>Y</var>) =
9792+
<a href="#defn_algUnion" class="algFct">Union</a>( ppeval(<var>X</var>, <var>ppe<sub>1</sub></var>, <var>Y</var>), ppeval(<var>X</var>, <var>ppe<sub>2</sub></var>, <var>Y</var>) )
97909793
</pre>
97919794
</div>
97929795
<p>Informally, this is the same as:</p>
9793-
<pre>SELECT * { { X P Y } UNION { X Q Y } }</pre>
9796+
<pre class="nohighlight">SELECT * { { <var>X</var> <var>P<sub>1</sub></var> <var>Y</var> } UNION { <var>X</var> <var>P<sub>2</sub></var> <var>Y</var> } }</pre>
9797+
<p>where <var>P<sub>1</sub></var> is a
9798+
<a href="#defn_AlgebraicPropertyPathExpression">property path expression</a>
9799+
that can be <a href="#sparqlTranslatePathExpressions">translated</a> into the
9800+
<a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a> <var>ppe<sub>1</sub></var>
9801+
and <var>P<sub>2</sub></var> can be translated into <var>ppe<sub>2</sub></var>.</p>
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<div class="defn">
97959803
<p><b>Definition: <span id="defn_nodeSet">Node set of a graph</span></b></p>
9796-
<p>The node set of a graph G, nodes(G), is:</p>
9797-
<p>nodes(G) = { n | n is an RDF term that is used as a subject or object of a triple of
9798-
G}</p>
9804+
<p>The node set of a graph |G|, <a href="#defn_nodeSet">nodes</a>(|G|), is:</p>
9805+
<p><a href="#defn_nodeSet">nodes</a>(|G|) = { |n| | |n| is an RDF term that is used as a subject or object of a triple of
9806+
|G|}</p>
97999807
</div>
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<div class="defn">
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<p><b>Definition: <span id="defn_evalPP_ZeroOrOnePath">Evaluation of ZeroOrOnePath</span></b></p>
9802-
<pre>
9803-
eval(Path(X:term, ZeroOrOnePath(P), Y:var)) =
9804-
{ (Y, yn) | yn = X or {(Y, yn)} in eval(Path(X,P,Y)) }
9810+
<p>Let |ppe| be an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
9811+
Let <var>G</var> be the <a href="#defn_ActiveGraph">active graph</a>.</p>
9812+
<pre class="nohighlight">
9813+
ppeval(<var>X</var>:term, <a href="#defn_ppeZeroOrOnePath" class="ppeOp">ZeroOrOnePath</a>(<var>ppe</var>), <var>Y</var>:var) =
9814+
{ (<var>Y</var>, <var>yn</var>) | <var>yn</var> = <var>X</var> or {(<var>Y</var>, <var>yn</var>)} in ppeval(<var>X</var>,<var>ppe</var>,<var>Y</var>) }
98059815
</pre>
9806-
<pre>
9807-
eval(Path(X:var, ZeroOrOnePath(P), Y:term)) =
9808-
{ (X, xn) | xn = Y or {(X, xn)} in eval(Path(X,P,Y)) }
9816+
<pre class="nohighlight">
9817+
ppeval(<var>X</var>:var, <a href="#defn_ppeZeroOrOnePath" class="ppeOp">ZeroOrOnePath</a>(<var>ppe</var>), <var>Y</var>:term) =
9818+
{ (<var>X</var>, <var>xn</var>) | <var>xn</var> = <var>Y</var> or {(<var>X</var>, <var>xn</var>)} in ppeval(<var>X</var>,<var>ppe</var>,<var>Y</var>) }
98099819
</pre>
9810-
<pre>
9811-
eval(Path(X:term, ZeroOrOnePath(P), Y:term)) =
9812-
{ {} } if X = Y or eval(Path(X,P,Y)) is not empty
9813-
{ } othewise</pre>
9814-
<pre>
9815-
eval(Path(X:var, ZeroOrOnePath(P), Y:var)) =
9816-
{ (X, xn) (Y, yn) | either (yn in nodes(G) and xn = yn) or {(X,xn), (Y,yn)} in eval(Path(X,P,Y)) }</pre>
9820+
<pre class="nohighlight">
9821+
ppeval(<var>X</var>:term, <a href="#defn_ppeZeroOrOnePath" class="ppeOp">ZeroOrOnePath</a>(<var>ppe</var>), <var>Y</var>:term) =
9822+
{ {} } if <var>X</var> = <var>Y</var> or ppeval(<var>X</var>,<var>ppe</var>,<var>X</var>) is not empty
9823+
{ } otherwise</pre>
9824+
<pre class="nohighlight">
9825+
ppeval(<var>X</var>:var, <a href="#defn_ppeZeroOrOnePath" class="ppeOp">ZeroOrOnePath</a>(<var>ppe</var>), <var>Y</var>:var) =
9826+
{ (<var>X</var>, <var>xn</var>) (<var>Y</var>, <var>yn</var>) | either (<var>yn</var> in <a href="#defn_nodeSet">nodes</a>(<var>G</var>) and <var>xn</var> = <var>yn</var>) or {(<var>X</var>,<var>xn</var>), (<var>Y</var>,<var>yn</var>)} in ppeval(<var>X</var>,<var>ppe</var>,<var>Y</var>) }</pre>
98179827
</div>
9818-
<p>We define an auxillary function, ALP, used in the definitions of ZeroOrMorePath and
9819-
OneOrMorePath. Note that the algorithm given here serves to specify the feature. An
9828+
<p>We define an auxiliary function, <a href="#defn_evalALP_1">ALP</a>, used in the definitions of <a href="#defn_evalZeroOrMorePath">ZeroOrMorePath</a> and
9829+
<a href="#defn_evalOneOrMorePath">OneOrMorePath</a>. Note that the algorithm given here serves to specify the feature. An
98209830
implementation is free to implement evaluation by any method that produces the same results
9821-
for the query overall. The ZeroOrMorePath and OneOrMorePath forms return matches based on
9831+
for the query overall. The <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a> and <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a> forms return matches based on
98229832
distinct nodes connected by the path.</p>
98239833
<p>The matching algorithm is based on following all paths, and detecting when a graph node
98249834
(subject or object), has been already visited on the path.</p>
98259835
<p>Informally, this algorithm attempts to extend the multiset of results by one application
9826-
of <code>path</code> at each step, noting which nodes it has visited for this particular path. If
9836+
of the given <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a> |ppe| at each step, noting which nodes it has visited for this particular path. If
98279837
a node has been visited for the path under consideration, it is not a candidate for another
98289838
step.</p>
98299839
<div class="defn">
98309840
<p><b>Definition: <span id="defn_evalALP_1">Function ALP</span></b></p>
9831-
<pre>
9832-
Let eval(x:term, path) be the evaluation of 'path', starting at RDF term x,
9841+
<pre class="nohighlight">
9842+
Let <var>ppe</var> be an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
9843+
Let <var>eval</var>(<var>x</var>:term, <var>ppe</var>) be the evaluation of <var>ppe</var>, starting at RDF term <var>x</var>,
98339844
and returning a multiset of RDF terms reached
9834-
by repeated matches of path.
9845+
by repeated matches of <var>ppe</var>.
98359846

9836-
ALP(x:term, path) =
9837-
Let V = empty set
9838-
ALP(x:term, path, V)
9839-
return is V
9847+
<a href="#defn_evalALP_1">ALP</a>(<var>x</var>:term, <var>ppe</var>) =
9848+
Let <var>V</var> = empty set
9849+
<a href="#defn_evalALP_1">ALP</a>(<var>x</var>:term, <var>ppe</var>, <var>V</var>)
9850+
return is <var>V</var>
98409851

9841-
# V is the set of nodes visited
9852+
# <var>V</var> is the set of nodes visited
98429853

9843-
ALP(x:term, path, V:set of RDF terms) =
9844-
if ( x in V ) return
9845-
add x to V
9846-
X = eval(x,path)
9847-
For n:term in X
9848-
ALP(n, path, V)
9854+
<a href="#defn_evalALP_1">ALP</a>(<var>x</var>:term, <var>ppe</var>, <var>V</var>:set of RDF terms) =
9855+
if ( <var>x</var> in <var>V</var> ) return
9856+
add <var>x</var> to <var>V</var>
9857+
<var>X</var> = <var>eval</var>(<var>x</var>, <var>ppe</var>)
9858+
For <var>n</var>:term in <var>X</var>
9859+
<a href="#defn_evalALP_1">ALP</a>(<var>n</var>, <var>ppe</var>, <var>V</var>)
98499860
End
98509861
</pre>
98519862
</div>
98529863
<div class="defn">
9853-
<p><b>Definition: Evaluation of</b></p>
9854-
<div id="defn_evalZeroOrMorePath">
9855-
<b>ZeroOrMorePath</b>
9856-
</div>
9857-
<pre>
9858-
eval(Path(X:term, ZeroOrMorePath(path), vy:var)) =
9859-
{ { (vy, n) } | n in ALP(X, path) }
9864+
<p><b>Definition: <span id="defn_evalZeroOrMorePath">Evaluation of ZeroOrMorePath</span></b></p>
9865+
<p>Let <var>ppe</var> be an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
9866+
Let <var>G</var> be the <a href="#defn_ActiveGraph">active graph</a>.</p>
9867+
<pre class="nohighlight">
9868+
ppeval(<var>X</var>:term, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>vy</var>:var) =
9869+
{ { (<var>vy</var>, <var>n</var>) } | <var>n</var> in <a href="#defn_evalALP_1">ALP</a>(<var>X</var>, <var>ppe</var>) }
98609870

9861-
eval(Path(vx:var, ZeroOrMorePath(path), vy:var)) =
9862-
{ { (vx, t), (vy, n) } | t in nodes(G), (vy, n) in eval(Path(t, ZeroOrMorePath(path), vy)) }
9871+
ppeval(<var>vx</var>:var, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>vy</var>:var) =
9872+
{ { (<var>vx</var>, <var>t</var>), (<var>vy</var>, <var>n</var>) } | <var>t</var> in <a href="#defn_nodeSet">nodes</a>(<var>G</var>), (<var>vy</var>, <var>n</var>) in ppeval(<var>t</var>, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>vy</var>) }
98639873

9864-
eval(Path(vx:var, ZeroOrMorePath(path), y:term)) =
9865-
eval(Path(y:term, ZeroOrMorePath(inv(path)), vx:var))
9874+
ppeval(<var>vx</var>:var, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>y</var>:term) =
9875+
ppeval(<var>y</var>:term, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<a href="#defn_ppeInv" class="ppeOp">inv</a>(<var>ppe</var>)), <var>vx</var>:var)
98669876

9867-
eval(Path(x:term, ZeroOrMorePath(path), y:term)) =
9868-
{ { } } if { (vy:var,y) } in eval(Path(x, ZeroOrMorePath(path) vy)
9877+
ppeval(<var>x</var>:term, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>y</var>:term) =
9878+
{ { } } if { (<var>vy</var>:var,<var>y</var>) } in ppeval(<var>x</var>, <a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>(<var>ppe</var>), <var>vy</var>)
98699879
{ } otherwise
98709880
</pre>
98719881
</div>
98729882
<div class="defn">
9873-
<p><b>Definition: Evaluation of</b></p>
9874-
<div id="defn_evalOneOrMorePath">
9875-
<b>OneOrMorePath</b>
9876-
</div>
9877-
<p>eval(Path(X, OneOrMorePath(path), Y))</p>
9878-
<pre>
9879-
# For OneOrMorePath, we take one step of the path then start
9883+
<p><b>Definition: <span id="defn_evalOneOrMorePath">Evaluation of OneOrMorePath</span></b></p>
9884+
<p>Let <var>ppe</var> be an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>.
9885+
Let <var>G</var> be the <a href="#defn_ActiveGraph">active graph</a>.</p>
9886+
<pre class="nohighlight">
9887+
# For <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>, we take one step of the path then start
98809888
# recording nodes for results.
98819889

9882-
eval(Path(x:term, OneOrMorePath(path), vy:var)) =
9883-
Let X = eval(x, path)
9884-
Let V = the empty multiset
9885-
For n in X
9886-
ALP(n, path, V)
9890+
ppeval(<var>x</var>:term, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>vy</var>:var) =
9891+
Let <var>X</var> = <var>eval</var>(<var>x</var>, <var>ppe</var>)
9892+
Let <var>V</var> = the empty multiset
9893+
For <var>n</var> in <var>X</var>
9894+
<a href="#defn_evalALP_1">ALP</a>(<var>n</var>, <var>ppe</var>, <var>V</var>)
98879895
End
9888-
result is V
9896+
result is <var>V</var>
9897+
<div class="issue" data-number="267"><var>V</var> is not a multiset of solution mappings but a set of RDF terms.</div>
98899898

9890-
eval(Path(vx:var, OneOrMorePath(path), vy:var)) =
9891-
{ { (vx, t), (vy, n) } | t in nodes(G), (vy, n) in eval(Path(t, OneOrMorePath(path), vy)) }
9899+
ppeval(<var>vx</var>:var, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>vy</var>:var) =
9900+
{ { (<var>vx</var>, <var>t</var>), (<var>vy</var>, <var>n</var>) } | <var>t</var> in <a href="#defn_nodeSet">nodes</a>(<var>G</var>), (<var>vy</var>, <var>n</var>) in ppeval(<var>t</var>, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>vy</var>) }
98929901

9893-
eval(Path(vx:var, OneOrMorePath(path), y:term)) =
9894-
eval(Path(y:term, OneOrMorePath(inv(path)), vx))
9902+
ppeval(<var>vx</var>:var, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>y</var>:term) =
9903+
ppeval(<var>y</var>:term, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<a href="#defn_ppeInv" class="ppeOp">inv</a>(<var>ppe</var>)), <var>vx</var>)
98959904

9896-
eval(Path(x:term, OneOrMorePath(path), y:term)) =
9897-
{ { } } if { (vy:var, y) } in eval(Path(x, OneOrMorePath(path), vy))
9905+
ppeval(<var>x</var>:term, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>y</var>:term) =
9906+
{ { } } if { (<var>vy</var>:var, <var>y</var>) } in ppeval(<var>x</var>, <a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>(<var>ppe</var>), <var>vy</var>)
98989907
{ } otherwise
98999908
</pre>
99009909
</div>
99019910
<div class="defn">
99029911
<p><b>Definition: <span id="eval_negatedPropertySet">Evaluation of NegatedPropertySet</span></b></p>
9903-
<pre class="code nohighlight">
9904-
Write μ' as the extension of a solution mapping:
9905-
μ'(μ,x) = μ(x) if x is a variable
9906-
μ'(μ,t) = t if t is a RDF term
9912+
<pre class="nohighlight">
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Write <var>μ'</var> as the extension of a solution mapping:
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<var>μ'</var>(<var>μ</var>, <var>x</var>) = <var>μ</var>(<var>x</var>) if <var>x</var> is a variable
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<var>μ'</var>(<var>μ</var>, <var>t</var>) = <var>t</var> if <var>t</var> is a RDF term
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</pre>
9908-
<pre class="code nohighlight">
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Let x and y be variables or RDF terms, and S a set of IRIs:
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<pre class="nohighlight">
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Let <var>x</var> and <var>y</var> be variables or RDF terms, <var>S</var> a set of IRIs,
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and <var>G</var> the <a href="#defn_ActiveGraph">active graph</a>.
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eval(Path(x, NPS(S), y)) = { μ | ∃ triple(μ'(μ,x), p, μ'(μ,y)) in G, such that the IRI of p ∉ S }
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ppeval(<var>x</var>, <a href="#defn_ppeNPS" class="ppeOp">NPS</a>(<var>S</var>), <var>y</var>) = { <var>μ</var> | ∃ triple(<var>μ'</var>(<var>μ</var>,<var>x</var>), <var>p</var>, <var>μ'</var>(<var>μ</var>,<var>y</var>)) in <var>G</var>, such that the IRI of <var>p</var> ∉ <var>S</var> }
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</pre>
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</div>
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</section>

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