improves definitions of all algebra operators that involve expressions

[hartig]

This PR addresses #254

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Preview | Diff

[TallTed]

It seems likely that this section (and likely others I'm not seeing right now) will benefit from HTML markup that delivers to the spec reader something like the subparagraphs which have been produced within the HTML markup. Perfect world, the lowercase Roman numerals could be reproduced in an ordered list, but these would be OK as a unordered list (to which I would suggest it be changed if there's no importance to the i/ii/iii) following the either (which I would suggest be changed to one of the following).

This is a rough suggestion (I have no HTML preview, at present), and it may itself need substantial change, if not a full rewrite, before merging.

Suggested change

	<p>Let <var>Ω<sub>1</sub></var> and <var>Ω<sub>2</sub></var> be multisets of solution mappings,
	<var>expr</var> be an <a href="#expressions">expression</a>,
	|D| be a <a href="#sparqlDataset">dataset</a>,
	and |G| be the <a href="#defn_ActiveGraph">active graph</a>.
	We define:</p>
	<p><a href="#defn_algDiff" class="algFct">Diff</a>(<var>Ω<sub>1</sub></var>, <var>Ω<sub>2</sub></var>, <var>expr</var>, |D|, |G|) = { <var>μ</var> | <var>μ</var> in <var>Ω<sub>1</sub></var> such that ∀ <var>μ'</var> in
	<var>Ω<sub>2</sub></var>, either i)&nbsp;<var>μ</var> and <var>μ'</var> are not <a href="#defn_algCompatibleMapping">compatible</a>,
	or
	ii)&nbsp;<var>μ</var> and <var>μ'</var> are <a href="#defn_algCompatibleMapping">compatible</a> and
	|expr|(merge(<var>μ</var>, <var>μ'</var>), |D|, |G|) is an [=error=],
	or
	iii)&nbsp;<var>μ</var> and <var>μ'</var> are <a href="#defn_algCompatibleMapping">compatible</a> and
	|expr|(merge(<var>μ</var>, <var>μ'</var>), |D|, |G|) is an RDF term |t| for which
	<a href="#func-ebv">EBV</a>(|t|) is not `"true"^^xsd:boolean` }</p>
	<p><a href="#defn_Multiplicity">multiplicity</a>( <var>μ</var> | <a href="#defn_algDiff" class="algFct">Diff</a>(<var>Ω<sub>1</sub></var>, <var>Ω<sub>2</sub></var>, <var>expr</var>, |D|, |G|) ) =
	<a href="#defn_Multiplicity">multiplicity</a>( <var>μ</var> | <var>Ω<sub>1</sub></var> )</p>
	<p>
	where, for every solution mapping <var>μ</var>,
	|expr|(<var>μ</var>, |D|, |G|) is the result of
	<a href="#expression-evaluation">evaluating</a> expression |expr|
	with respect to <var>μ</var>, in the context of dataset |D| with active graph |G|.
	</p>
	<p>Let <var>Ω<sub>1</sub></var> and <var>Ω<sub>2</sub></var>
	be multisets of solution mappings,
	<var>expr</var> be an <a href="#expressions">expression</a>,
	|D| be a <a href="#sparqlDataset">dataset</a>,
	and |G| be the <a href="#defn_ActiveGraph">active graph</a>.
	We define the following:</p>
	<ul>
	<li><a href="#defn_algDiff"
	class="algFct">Diff</a>(<var>Ω<sub>1</sub></var>,
	<var>Ω<sub>2</sub></var>, <var>expr</var>, |D|, |G|) =
	{ <var>μ</var> | <var>μ</var> in <var>Ω<sub>1</sub></var>
	such that ∀ <var>μ'</var> in
	<var>Ω<sub>2</sub></var>, either of the following
	<ol type="i">
	<li><var>μ</var> and <var>μ'</var> are not
	<a href="#defn_algCompatibleMapping">compatible</a></li>
	<li><var>μ</var> and <var>μ'</var> are
	<a href="#defn_algCompatibleMapping">compatible</a> and
	|expr|(merge(<var>μ</var>, <var>μ'</var>), |D|, |G|)
	is an [=error=]</li>
	<li><var>μ</var> and <var>μ'</var> are
	<a href="#defn_algCompatibleMapping">compatible</a> </li>
	</ol>
	<p>and |expr|(merge(<var>μ</var>, <var>μ'</var>), |D|, |G|)
	is an RDF term |t| for which
	<a href="#func-ebv">EBV</a>(|t|)
	is not `"true"^^xsd:boolean` }</p></li>
	<li><a href="#defn_Multiplicity">multiplicity</a>( <var>μ</var> | <a href="#defn_algDiff" class="algFct">Diff</a>(<var>Ω<sub>1</sub></var>, <var>Ω<sub>2</sub></var>, <var>expr</var>, |D|, |G|) ) =
	<a href="#defn_Multiplicity">multiplicity</a>( <var>μ</var> | <var>Ω<sub>1</sub></var> )</li></ul>
	<p>
	where, for every solution mapping <var>μ</var>,
	|expr|(<var>μ</var>, |D|, |G|) is the result of
	<a href="#expression-evaluation">evaluating</a> expression |expr|
	with respect to <var>μ</var>, in the context of dataset |D| with active graph |G|.
	</p>

[hartig]

Good idea. I have implemented it in commit 0c316c4

[TallTed]

Great! Similar should be applied to the other blocks that are similarly prettified as HTML, without prettifying their rendering in the browser.

[hartig]

Great! Similar should be applied to the other blocks that are similarly prettified as HTML, without prettifying their rendering in the browser.

There are no other such blocks within the scope of this PR (whose purpose is to fix issues in the formalization of algebra operators that involve expressions).

[TallTed]

I did look before posting the above ... but apparently I was hallucinating at the time. Sorry!

[hartig]

No worries Ted.

So, you are okay now with this PR?

[TallTed]

Yes. Merge away. :-)

[hartig]

Yes. Merge away. :-)

Thanks Ted!

I will still wait for @afs' review.