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introduces a function called multiplicity to replace card[Ω](μ) #110

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introduces a function called multiplicity to replace card[Ω](μ) #110

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98ef75a
introduces a function called multiplicity as a replacement for the no…
hartig Jul 3, 2023
6a440b1
revert back the change to BGP matching; this part may be adapted later
hartig Jul 3, 2023
a98dac6
uses the ReSpec <var> feature in the definition of the Multiplicity f…
hartig Sep 28, 2023
f1ae8c4
changes the presentation of the 'multiplicity' function, as discussed…
hartig Oct 3, 2023
7e6e6f9
uses the new 'multiplicity' function instead of card[..](..), and cha…
hartig Oct 5, 2023
5de0dbf
Use of <var> in the section about BGPs
hartig Oct 14, 2023
376e0c6
Use of <var> in the section about BGPs
hartig Oct 16, 2023
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104 changes: 64 additions & 40 deletions spec/index.html
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Expand Up @@ -3916,8 +3916,8 @@ <h4>DISTINCT</h4>
<h4>REDUCED</h4>
<p>While the <code>DISTINCT</code> modifier ensures that duplicate solutions are
eliminated from the solution set, <code>REDUCED</code> simply permits them to be
eliminated. The cardinality of any set of variable bindings in a <code>REDUCED</code>
solution set is at least one and not more than the cardinality of the solution set with
eliminated. The multiplicity of any solution in a <code>REDUCED</code>
solution set is at least one and not more than the multiplicity of the solution within the solution set with
no <code>DISTINCT</code> or <code>REDUCED</code> modifier. For example, using the data
above, the query</p>
<div class="queryGroup">
Expand Down Expand Up @@ -8880,7 +8880,7 @@ <h4>Converting Solution Modifiers</h4>
<li>Limit</li>
</ul>
<p>Step: ToList</p>
<p>ToList turns a multiset into a sequence with the same elements and cardinality. There is
<p>ToList turns a multiset into a sequence with the same elements and multiplicities. There is
no implied ordering to the sequence; duplicates need not be adjacent.</p>
<blockquote>
<p>Let M := ToList(Pattern)</p>
Expand Down Expand Up @@ -8942,15 +8942,15 @@ <h3>Basic Graph Patterns</h3>
<a href="https://en.wikipedia.org/w/index.php?title=Multiset">multiset</a>,
also known as a <i>bag</i>. A multiset is an
unordered collection of elements in which each element may appear more than once. It is
described by a set of elements and a cardinality function giving the number of occurrences of
each element from the set in the multiset.</p>
<p>Write μ for solution mappings.</p>
<p>Write μ<sub>0</sub> for the mapping such that dom(μ<sub>0</sub>) is the empty set.</p>
<p>Write Ω<sub>0</sub> for the multiset consisting of exactly the empty mapping
μ<sub>0,</sub> with cardinality 1. This is the join identity.</p>
<p>Write μ(x) for the solution mapping variable x to RDF term t : { (x, t) }</p>
<p>Write Ω(x) for the multiset consisting of exactly μ(?x-&gt;t), that is, { { (x, t) } }
with cardinality 1.</p>
described by a set of elements and a function giving the multiplicity of each of these
elements (i.e., the number of times the element is contained in the multiset).</p>
<p>Write <var>μ</var> for solution mappings.</p>
<p>Write <var>μ<sub>0</sub></var> for the mapping such that <var>dom</var>(<var>μ<sub>0</sub></var>) is the empty set.</p>
<p>Write <var>Ω<sub>0</sub></var> for the multiset consisting of exactly the empty mapping
<var>μ<sub>0</sub></var>, with multiplicity 1. This is the join identity.</p>
<p>Write <var>μ</var>(<var>x</var>) for the solution mapping variable <var>x</var> to RDF term <var>t</var> : { (<var>x</var>, <var>t</var>) }.</p>
<p>Write <var>Ω</var>(<var>x</var>) for the multiset consisting of exactly <var>μ</var>(<var>?x</var>-&gt;<var>t</var>), that is, <code>{ { (x, t) } }</code>
with multiplicity 1.</p>
<div class="defn">
<p><b>Definition: <span id="defn_algCompatibleMapping">Compatible Mappings</span></b></p>
<p>Two solution mappings μ<sub>1</sub> and μ<sub>2</sub> are compatible if, for every
Expand All @@ -8962,8 +8962,15 @@ <h3>Basic Graph Patterns</h3>
<p>If μ<sub>1</sub> and μ<sub>2</sub> are compatible then μ<sub>1</sub> ∪ μ<sub>2</sub> is
also a mapping. Write merge(μ<sub>1</sub>, μ<sub>2</sub>) for μ<sub>1</sub> ∪
μ<sub>2</sub></p>
<p>Write card[Ω](μ) for the cardinality of solution mapping μ in a multiset of mappings
Ω.</p>
<div class="defn">
<p><b>Definition: <span id="defn_Multiplicity">Multiplicity</span></b></p>
<p>Given a multiset&nbsp;<var>Ω</var> of solution mappings and a solution
mapping&nbsp;<var>μ</var>, we write <var>multiplicity</var>(<var>μ</var>&nbsp;|&nbsp;<var>Ω</var>)
to denote the number of times <var>μ</var> appears in <var>Ω</var>.</p>
<p>Similarly, given a solution sequence&nbsp;<var>Ψ</var> and a solution
mapping&nbsp;<var>μ</var>, we write <var>multiplicity</var>(<var>μ</var>&nbsp;|&nbsp;<var>Ψ</var>)
to denote the number of times <var>μ</var> appears in <var>Ψ</var>.</p>
</div>
<section id="BGPsparql">
<h4>SPARQL Basic Graph Pattern Matching</h4>
<p>A basic graph pattern is matched against the active graph for that part of the query.
Expand All @@ -8986,8 +8993,9 @@ <h4>SPARQL Basic Graph Pattern Matching</h4>
<p>μ is a <b>solution</b> for BGP from G when there is a pattern instance mapping P such
that P(BGP) is a subgraph of G and μ is the restriction of P to the query variables in
BGP.</p>
<p>card[Ω](μ) = card[Ω](number of distinct RDF instance mappings, σ, such that P = μ(σ)
is a pattern instance mapping and P(BGP) is a subgraph of G).</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Ω ) =
number of distinct RDF instance mappings, σ, such that P = μ(σ)
is a pattern instance mapping and P(BGP) is a subgraph of G.</p>
</div>
<p>If a basic graph pattern is the empty set, then the solution is Ω<sub>0</sub>.</p>
</section>
Expand Down Expand Up @@ -9255,7 +9263,8 @@ <h3>SPARQL Algebra</h3>
<p>Let Ω be a multiset of solution mappings and expr be an expression. We define:</p>
<p>Filter(expr, Ω) = { μ | μ in Ω and expr(μ) is an expression that has an
effective boolean value of true }</p>
<p>card[Filter(expr, Ω)](μ) = card[Ω](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Filter(expr, Ω) )
= <a href="#defn_Multiplicity">multiplicity</a>( μ | Ω )</p>
<blockquote>
Note that evaluating an <code>exists(pattern)</code> expression uses the dataset and
active graph, D(G). See the <a href="#defn_evalFilter">evaluation of filter</a>.
Expand All @@ -9267,24 +9276,25 @@ <h3>SPARQL Algebra</h3>
<p>Join(Ω<sub>1</sub>, Ω<sub>2</sub>) = { merge(μ<sub>1</sub>, μ<sub>2</sub>) |
μ<sub>1</sub> in Ω<sub>1</sub> and μ<sub>2</sub> in Ω<sub>2</sub>, and μ<sub>1</sub> and
μ<sub>2</sub> are compatible }</p>
<p>card[Join(Ω<sub>1</sub>, Ω<sub>2</sub>)](μ) =<br>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Join(Ω<sub>1</sub>, Ω<sub>2</sub>) ) =<br>
&nbsp;&nbsp;&nbsp; for each merge(μ<sub>1</sub>, μ<sub>2</sub>), μ<sub>1</sub> in
Ω<sub>1</sub> and μ<sub>2</sub> in Ω<sub>2</sub> such that μ = merge(μ<sub>1</sub>,
μ<sub>2</sub>),<br>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; sum over (μ<sub>1</sub>, μ<sub>2</sub>),
card[Ω<sub>1</sub>](μ<sub>1</sub>)*card[Ω<sub>2</sub>](μ<sub>2</sub>)</p>
<a href="#defn_Multiplicity">multiplicity</a>( μ<sub>1</sub> | Ω<sub>1</sub> ) * <a href="#defn_Multiplicity">multiplicity</a>( μ<sub>2</sub> | Ω<sub>2</sub> )</p>
</div>
<p>It is possible that a solution mapping μ in a Join can arise in different solution
mappings, μ<sub>1</sub> and μ<sub>2</sub> in the multisets being joined. The cardinality
of&nbsp; μ is the sum of the cardinalities from all possibilities.</p>
mappings, μ<sub>1</sub> and μ<sub>2</sub> in the multisets being joined. The multiplicity
of&nbsp; μ is the sum of the multiplicities from all possibilities.</p>
<div class="defn">
<p><b>Definition: <span id="defn_algDiff">Diff</span></b></p>
<p>Let Ω<sub>1</sub> and Ω<sub>2</sub> be multisets of solution mappings and expr be an
expression. We define:</p>
<p>Diff(Ω<sub>1</sub>, Ω<sub>2</sub>, expr) = { μ | μ in Ω<sub>1</sub> such that ∀ μ′ in
Ω<sub>2</sub>, either μ and μ′ are not compatible or μ and μ' are compatible and
expr(merge(μ, μ')) does not have an effective boolean value of true }</p>
<p>card[Diff(Ω<sub>1</sub>, Ω<sub>2</sub>, expr)](μ) = card[Ω<sub>1</sub>](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Diff(Ω<sub>1</sub>, Ω<sub>2</sub>, expr) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω<sub>1</sub> )</p>
</div>
<p>The evaluation of expr(merge(μ, μ')) does not have an effective boolean
value of true if it evaluates to false or if it raises an error.</p>
Expand All @@ -9295,26 +9305,29 @@ <h3>SPARQL Algebra</h3>
expression. We define:</p>
<p>LeftJoin(Ω<sub>1</sub>, Ω<sub>2</sub>, expr) = Filter(expr, Join(Ω<sub>1</sub>,
Ω<sub>2</sub>)) ∪ Diff(Ω<sub>1</sub>, Ω<sub>2</sub>, expr)</p>
<p>card[LeftJoin(Ω<sub>1</sub>, Ω<sub>2</sub>, expr)](μ) = card[Filter(expr,
Join(Ω<sub>1</sub>, Ω<sub>2</sub>))](μ) + card[Diff(Ω<sub>1</sub>, Ω<sub>2</sub>,
expr)](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | LeftJoin(Ω<sub>1</sub>, Ω<sub>2</sub>, expr) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Filter(expr,Join(Ω<sub>1</sub>, Ω<sub>2</sub>)) ) +
<a href="#defn_Multiplicity">multiplicity</a>( μ | Diff(Ω<sub>1</sub>, Ω<sub>2</sub>,
expr) )</p>
</div>

<div class="defn">
<p><b>Definition: <span id="defn_algUnion">Union</span></b></p>
<p>Let Ω<sub>1</sub> and Ω<sub>2</sub> be multisets of solution mappings. We define:</p>
<p>Union(Ω<sub>1</sub>, Ω<sub>2</sub>) = { μ | μ in Ω<sub>1</sub> or μ in Ω<sub>2</sub>
}</p>
<p>card[Union(Ω<sub>1</sub>, Ω<sub>2</sub>)](μ) = card[Ω<sub>1</sub>](μ) +
card[Ω<sub>2</sub>](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Union(Ω<sub>1</sub>, Ω<sub>2</sub>) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω<sub>1</sub> ) +
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω<sub>2</sub> )</p>
</div>
<div class="defn">
<p><b>Definition: <span id="defn_algMinus">Minus</span></b></p>
<p>Let Ω<sub>1</sub> and Ω<sub>2</sub> be multisets of solution mappings. We define:</p>
<p>Minus(Ω<sub>1</sub>, Ω<sub>2</sub>) = { μ | μ in Ω<sub>1</sub> . ∀ μ' in
Ω<sub>2</sub>, either μ and μ' are not compatible or dom(μ) and dom(μ') are disjoint
}</p>
<p>card[Minus(Ω<sub>1</sub>, Ω<sub>2</sub>)](μ) = card[Ω<sub>1</sub>](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Minus(Ω<sub>1</sub>, Ω<sub>2</sub>) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω<sub>1</sub> )</p>
</div>
<p>The additional restriction on dom(μ) and dom(μ') is added because otherwise if there is
a solution mapping in Ω<sub>2</sub> that has no variables in common with the solution
Expand All @@ -9332,45 +9345,55 @@ <h3>SPARQL Algebra</h3>
<p>Extend(Ω, var, expr) = { Extend(μ, var, expr) | μ in Ω }</p>
</div>
<p>Write [ x | C ] for a sequence of elements where C is a condition on x.</p>
<p>Write card[L](x) to be the cardinality of x in L.</p>
<div class="defn">
<p><b>Definition: <span id="defn_algToList">ToList</span></b></p>
<p>Let Ω be a multiset of solution mappings. We define:</p>
<p>ToList(Ω) = a sequence of mappings μ in Ω in any order, with card[Ω](μ) occurrences of
<p>ToList(Ω) = a sequence of mappings μ in Ω in any order, with
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω ) occurrences of
μ</p>
<p>card[ToList(Ω)](μ) = card[Ω](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | ToList(Ω) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ω )</p>
</div>
<div class="defn">
<p><b>Definition: <span id="defn_algOrdered">OrderBy</span></b></p>
<p>Let Ψ be a sequence of solution mappings. We define:</p>
<div id="defn_algOrderBy">
OrderBy
</div>(Ψ, condition) = [ μ | μ in Ψ and the sequence satisfies the ordering condition]
<p>card[OrderBy(Ψ, condition)](μ) = card[Ψ](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | OrderBy(Ψ, condition) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ψ )</p>
</div>
<div class="defn">
<p><b>Definition: <span id="defn_algProjection">Project</span></b></p>
<p>Let Ψ be a sequence of solution mappings and PV a set of variables.</p>
<p>For mapping μ, write Proj(μ, PV) to be the restriction of μ to variables in PV.</p>
<p>Project(Ψ, PV) = [ Proj(μ, PV) | μ in Ψ ]</p>
<p>card[Project(Ψ, PV)](μ) = sum(card[Ψ](ν) | ν in Ψ such that ν = Proj(μ, PV))</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Project(Ψ, PV) ) =
sum( <a href="#defn_Multiplicity">multiplicity</a>( ν | Ψ ) | ν in Ψ such that ν = Proj(μ, PV))</p>
<p>The order of Project(Ψ, PV) must preserve any ordering given by OrderBy.</p>
</div>
<div class="defn">
<p><b>Definition: <span id="defn_algDistinct">Distinct</span></b></p>
<p>Let Ψ be a sequence of solution mappings. We define:</p>
<p>Distinct(Ψ) = [ μ | μ in Ψ ]</p>
<p>card[Distinct(Ψ)](μ) = 1</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Distinct(Ψ) ) = 1
for every μ ∈ Distinct(Ψ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Distinct(Ψ) ) = 0
for every μ ∉ Distinct(Ψ)</p>
<p>The order of Distinct(Ψ) must preserve any ordering given by OrderBy.</p>
</div>
<div class="defn">
<p><b>Definition: <span id="defn_algReduced">Reduced</span></b></p>
<p>Let Ψ be a sequence of solution mappings. We define:</p>
<p>Reduced(Ψ) = [ μ | μ in Ψ ]</p>
<p>card[Reduced(Ψ)](μ) is between 1 and card[Ψ](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Reduced(Ψ) ) is
between 1 and <a href="#defn_Multiplicity">multiplicity</a>( μ | Ψ )
for every μ ∈ Reduced(Ψ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | Reduced(Ψ) ) = 0
for every μ ∉ Reduced(Ψ)</p>
<p>The order of Reduced(Ψ) must preserve any ordering given by OrderBy.</p>
</div>
<p>The Reduced solution sequence modifier does not guarantee a defined cardinality.</p>
<p>The Reduced solution sequence modifier does not guarantee a defined multiplicity.</p>
<div class="defn">
<p><b>Definition: <span id="defn_algSlice">Slice</span></b></p>
<p>Let Ψ be a sequence of solution mappings. We define:</p>
Expand All @@ -9382,15 +9405,16 @@ <h3>SPARQL Algebra</h3>
<p><b>Definition: <span id="defn_algToMultiSet">ToMultiSet</span></b></p>
<p>Let Ψ be a solution sequence. We define:</p>
<p>ToMultiSet(Ψ) = { μ | μ in Ψ }</p>
<p>card[ToMultiSet(Ψ)](μ) = card[Ψ](μ)</p>
<p><a href="#defn_Multiplicity">multiplicity</a>( μ | ToMultiSet(Ψ) ) =
<a href="#defn_Multiplicity">multiplicity</a>( μ | Ψ )</p>
</div>
<p>ListEval is a function which is used to evaluate a list of expressions against a
solution and return a list of the resulting values.</p>
<div class="defn">
<div id="defn_algToMultiset">
<b>Definition: ToMultiset</b>
</div>
<p>ToMultiset turns a sequence into a multiset with the same elements and cardinality as
<p>ToMultiset turns a sequence into a multiset with the same elements and multiplicities as
the sequence. The order of the sequence has no effect on the resulting multiset, and
duplicates are preserved.</p>
</div>
Expand Down Expand Up @@ -9740,8 +9764,8 @@ <h3>Evaluation Semantics</h3>
the result is R
</pre>
</div>
<p>The evaluation of graph uses the SPARQL algebra union operator. The cardinality of a
solution mapping is the sum of the cardinalities of that solution mapping in each join
<p>The evaluation of graph uses the SPARQL algebra union operator. The multiplicity of a
solution mapping is the sum of the multiplicities of that solution mapping in each join
operation.</p>
<div class="defn">
<div id="defn_evalGroup">
Expand Down
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