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snapshot of addressing #228 in Sec.18.5 'Property Path Patterns'
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‎spec/index.html‎

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@@ -9687,45 +9687,49 @@ <h4>Treatment of Blank Nodes</h4>
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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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property path express and an object endpoint. The
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<a href="#sparqlTranslatePathExpressions">translation of property path expressions</a> converts some
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forms to other SPARQL expressions, such as converting property paths of length one to triple
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patterns, which in turn are combined into basic graph patterns. This leaves property path
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operators ZeroOrOnePath, ZeroOrMorePath, OneOrMorePath and NegatedPropertySets and also path
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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, 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>,
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<a href="#defn_ppeZeroOrMorePath" class="ppeOp">ZeroOrMorePath</a>,
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<a href="#defn_ppeOneOrMorePath" class="ppeOp">OneOrMorePath</a>,
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and <a href="#defn_ppeNPS" class="ppeOp">NegatedPropertySets</a>,
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as well as algebraic property path
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expressions contained within these operators.</p>
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<p>All remaining property path expressions are present in the algebra in the form
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<code>Path(X, path, Y)</code> for endpoints X and Y. For example: syntax<code>(:p/:q)*</code>
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is a ZeroOrMorePath expression involving a sequence property path becoming the algebra
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expession <code>ZeroOrMorePath(seq(link(:p), link(:q)))</code>.</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|.
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For example, the <a href="#defn_PropertyPathExpr">property path expression</a> <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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<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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<div class="defn">
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<div id="pp-eval-notation">
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<b>Notation</b>
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</div>
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<p>Write</p>
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<pre>eval(Path(X, PP, Y))</pre>
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<p>for the evaluation of the property path patterns. This produces a multiset of solution
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mappings μ, each solution mapping having a binding for variables used (each of X and Y can
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<p>For every <a href="#defn_AlgebraicQueryExpression">algebraic query expression</a> of the form
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<a href="#defn_absPath" class="absOp">Path</a>(|X|, |ppe|, |Y|) we write</p>
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<pre class="nohighlight">ppeval(<var>X</var>, <var>ppe</var>, <var>Y</var>)</pre>
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<p>to denote the evaluation of the property path pattern. This produces a multiset of solution
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mappings, each solution mapping having a binding for variables used (each of |X| and |Y| can
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be a variable). Some operators only produce a set of solution mappings.</p>
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<p>Write</p>
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<pre>
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Var(x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>n</sub>) = { x<sub>i</sub> | i in 1...n and x<sub>i</sub> is a variable
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}
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</pre>
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<p>for the variables in <code>x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>n</sub></code>.</p>
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<p>Write</p>
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<table style="border-collapse: collapse; border-color: #000000; border-spacing: 10px ; border-width: 1px">
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<tbody>
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<tr>
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<td><code>x:term</code></td>
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<td>when <code>x</code> is an RDF term</td>
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<td><code>|x|:term</code></td>
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<td>when <code>|x|</code> is an RDF term</td>
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</tr>
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<tr>
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<td><code>x:var</code></td>
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<td>when <code>x</code> is a variable</td>
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</tr>
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<tr>
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<td><code>x:path</code></td>
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<td>when <code>x</code> is a path expression</td>
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<td><code>|x|:var</code></td>
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<td>when <code>|x|</code> is a variable</td>
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</tr>
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</tbody>
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</table>
@@ -9735,46 +9739,44 @@ <h3>Property Path Patterns</h3>
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in each definition for clarity.</p>
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<div class="defn">
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<p><b>Definition: <span id="defn_evalPP_predicate">Evaluation of Predicate Property Path</span></b></p>
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<p>Let Path(X, link(iri), Y) be an predicate inverse property path pattern, using some IRI
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iri.</p>
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<pre>
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eval(Path(X, link(iri), Y)) = <a href="#BasicGraphPattern">evaluation of basic graph pattern</a> {X iri Y}
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<p>Let |iri| be an IRI.</p>
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<pre class="nohighlight">
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ppeval(<var>X</var>, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>) = <a href="#BasicGraphPattern">evaluation of basic graph pattern</a> {<var>X</var> <var>iri</var> <var>Y</var>}
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</pre>
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</div>
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<p>If both X and Y are variables, this is the same as:</p>
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<pre>
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eval(Path(X:var, link(iri), Y:var)) = { (X, xn) (Y, yn) | xn and yn are RDF terms and triple (xn iri yn) is in the active graph }</pre>
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<p>If X is a variable and Y an RDF term:</p>
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<pre>
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eval(Path(X:var, link(iri), Y:term)) = { (X, xn) | xn is an RDF term and triple (xn iri Y) is in the active graph
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}
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</pre>
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<p>If X is an RDF term and Y is a variable:</p>
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<pre>
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eval(Path(X:term, link(iri), Y:var)) = { (Y, yn) | yn is an RDF term and triple (X iri yn) is in the active graph
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}
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</pre>
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<p>If both X and Y are RDF terms:</p>
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<pre>
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eval(Path(X:term, link(iri), Y:term))
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= { μ<sub>0</sub> } if triple (X iri Y) is in the active graph
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<p>If both |X| and |Y| are variables, this is the same as:</p>
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<pre class="nohighlight">
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ppeval(<var>X</var>:var, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>:var) = { (<var>X</var>, <var>xn</var>) (<var>Y</var>, <var>yn</var>) | <var>xn</var> and <var>yn</var> are RDF terms and triple (<var>xn</var>, <var>iri</var>, <var>yn</var>) is in the active graph }</pre>
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<p>If |X| is a variable and |Y| an RDF term:</p>
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<pre class="nohighlight">
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ppeval(<var>X</var>:var, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>:term) = { (<var>X</var>, <var>xn</var>) | <var>xn</var> is an RDF term and triple (<var>xn</var>, <var>iri</var>, <var>Y</var>) is in the active graph }</pre>
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<p>If |X| is an RDF term and |Y| is a variable:</p>
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<pre class="nohighlight">
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ppeval(<var>X</var>:term, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>:var) = { (<var>Y</var>, <var>yn</var>) | <var>yn</var> is an RDF term and triple (<var>X</var>, <var>iri</var>, <var>yn</var>) is in the active graph }</pre>
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<p>If both |X| and |Y| are RDF terms:</p>
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<pre class="nohighlight">
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ppeval(<var>X</var>:term, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>:term)
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= { μ<sub>0</sub> } if triple (<var>X</var>, <var>iri</var>, <var>Y</var>) is in the active graph
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= { { } } = Ω<sub>0</sub>
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eval(Path(X:term, link(iri), Y:term)) = { } if triple (X iri Y) is not in the active graph
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ppeval(<var>X</var>:term, <a href="#defn_ppeLink" class="ppeOp">link</a>(<var>iri</var>), <var>Y</var>:term) = { } if triple (<var>X</var>, <var>iri</var>, <var>Y</var>) is not in the active graph
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</pre>
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<p>Informally, evaluating a Predicate Property Path is the same as executing a subquery
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<code>SELECT * { <i>X P Y</i> }</code> at that point in the query evaluation.</p>
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<code>SELECT * { |X| |iri| |Y|</i> }</code> at that point in the query evaluation.</p>
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<div class="defn">
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<p><b>Definition: <span id="defn_evalPP_inverse">Evaluation of Inverse Property Path</span></b></p>
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<p>Let P be a property path expression, then:</p>
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<pre>eval(Path(X, inv(P), Y)) = eval(Path(Y, P, X))</pre>
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<p>Let |ppe| be an <a href="#defn_AlgebraicPropertyPathExpression">algebraic property path expression</a>, then:</p>
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<pre class="nohighlight">ppeval(<var>X</var>, <a href="#defn_ppeInv" class="ppeOp">inv</a>(<var>ppe</var>), <var>Y</var>) = ppeval(<var>Y</var>, <var>ppe</var>, <var>X</var>)</pre>
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</div>
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<div class="defn">
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<p><b>Definition: <span id="defn_evalPP_sequence">Evaluation of Sequence Property Path</span></b></p>
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<p>Let P and Q be property path expressions. Let V be a fresh variable.</p>
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<pre>A = Join( eval(Path(X, P, V)), eval(Path(V, Q, Y)) )</pre>
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<pre>eval(Path(X, seq(P,Q), Y)) = Project(A, Var(X,Y))</pre>
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<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>. Let |V| be a fresh variable.</p>
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<pre class="nohighlight"><var>A</var> = <a href="#defn_algJoin" class="algFct">Join</a>( ppeval(<var>X</var>, <var>ppe<sub>1</sub></var>, <var>V</var>), ppeval(<var>V</var>, <var>ppe<sub>2</sub></var>, <var>Y</var>) )</pre>
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<pre class="nohighlight">ppeval(<var>X</var>, <a href="#defn_ppeSeq" class="ppeOp">seq</a>(<var>ppe<sub>1</sub></var>,<var>ppe<sub>2</sub></var>), <var>Y</var>) = <a href="#defn_algProject" class="algFct">Project</a>(<var>A</var>, <var>PV</var>)</pre>
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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>
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</div>
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<div class="ednote">Olaf, continue here!</div>
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<p>Informally, this is the same as:</p>
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<pre>SELECT * { X P _:a . _:a Q Y }</pre>
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<p>using the fact that a blank node <code>_:a</code> acts like a variable (under simple

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