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<div1 id="mixing" role="chapter5"><head>Combining Presentation and Content Markup</head>
<!-- $Id: mixing.xml,v 1.34 2001/01/29 10:55:15 davidc Exp $ -->
<!--
Principal authors
Stephen Watt: overal design; organization and original text of documentation.
Robert Miner: additional text.
-->
<p>Presentation markup and content markup can be combined in two
ways. The first manner is to intersperse content and presentation elements
in what is essentially a single tree. This is called <emph>mixed</emph>
markup. The second manner is to provide <emph>both</emph> an explicit
presentation and an explicit content in a pair of trees. This is called
<emph>parallel</emph> markup. This chapter describes both mixed and
parallel markup, and how they may used in conjunction with style sheets and
other tools.</p>
<div2 id="mixing_why"><head>Why Two Different Kinds of Markup?</head>
<p>Chapters 3 and 4 describe two kinds of markup for encoding mathematical
material in documents.</p>
<p><emph>Presentation markup</emph> captures <emph>notational
structure</emph>. It encodes the notational structure of an expression in a
sufficiently abstract way to facilitate rendering to various media. Thus,
the same presentation markup can be rendered with relative ease on screen
in either wide and narrow windows, in ASCII or graphics, in print, or it
can be enunciated in a sensible way when spoken. It does this by providing
information such as structured grouping of expression parts, classification
of symbols, etc.</p>
<p>Presentation markup does <emph>not</emph> directly concern itself with
the mathematical structure or meaning of an expression. In many situations,
notational structure and mathematical structure are closely related, so a
sophisticated processing application may be able to heuristically infer
mathematical meaning from notational structure, provided sufficient context
is known. However, in practice, the inference of mathematical meaning from
mathematical notation must often be left to the reader.</p>
<p> Employing presentation tags alone may limit the ability to re-use a
MathML object in another context, especially evaluation by external
applications.</p>
<p><emph>Content markup</emph> captures <emph>mathematical
structure</emph>. It encodes mathematical structure in a sufficiently
regular way in order to facilitate the assignment of mathematical meaning
to an expression by application programs. Though the details of mapping
from mathematical expression structure to mathematical meaning can be
extremely complex, in practice, there is wide agreement about the
conventional meaning of many basic mathematical constructions.
Consequently, much of the meaning of a content expression is easily
accessible to a processing application, independently of where or how it is
displayed to the reader. In many cases, content markup could be cut from a
Web browser and pasted into a mathematical software tool
with confidence that sensible
values will be computed.</p>
<p>Since content markup is <emph>not</emph> directly concerned with how an
expression is displayed, a renderer must infer how an expression should be
presented to a reader. While a sufficiently sophisticated renderer and
style sheet mechanism could in principle allow a user to read mathematical
documents using personalized notational preferences, in practice, rendering
content expressions with notational nuances may still require
intervention of some sort.</p>
<p> Employing content tags alone may limit the ability of the author to
precisely control how an expression is rendered.</p>
<p>Both content and presentation tags are
necessary in order to provide the full expressive capability one would
expect in a mathematical markup language. Often the same mathematical
notation is used to represent several completely different concepts. For
example, the notation <mi>x</mi><sup><mi>i</mi></sup> may be intended (in
polynomial algebra) as the <mi>i</mi>-th power of the variable <mi>x</mi>,
or as the <mi>i</mi>-th component of a vector <mi>x</mi> (in tensor calculus).
In other cases, the same mathematical concept may be displayed in one of various
notations. For instance, the factorial of a number might be expressed with
an exclamation mark, a Gamma function, or a Pochhammer symbol.</p>
<p>Thus the same notation may represent several mathematical ideas, and,
conversely, the same mathematical idea often has several notations. In
order to provide authors with the ability to precisely control notation
while at the same time encoding meanings in a machine-readable way,
both content and presentation markup are needed.</p>
<p>In general, if it is important to control exactly how an expression is
rendered, presentation markup will generally be more satisfactory. If it is
important that the meaning of an expression can be interpreted dependably
and automatically, then content markup will generally be more
satisfactory.</p>
</div2>
<div2 id="mixing_markup"><head>Mixed Markup</head>
<p>MathML offers authors elements for both content and presentation
markup. Whether to use one or the other, or a combination of both, depends
on what aspects of rendering and interpretation an author wishes to
control, and what kinds of re-use he or she wishes to facilitate.</p>
<div3 id="mixing_reasons"><head>Reasons to Mix Markup</head>
<p>In many common situations, an author or authoring tool may choose to
generate either presentation or content markup exclusively. For example, a
program for translating legacy documents would most likely generate pure
presentation markup. Similarly, an educational software package might very
well generate only content markup for evaluation in a computer algebra
system. However, in many other situations, there are advantages to mixing
both presentation and content markup within a single expression.</p>
<p>If an author is primarily presentation-oriented, interspersing some
content markup will often produce more accessible, more re-usable
results. For example, an author writing about linear algebra might write:
<eg role="mathml"><![CDATA[
<mrow>
<apply>
<power/>
<ci>x</ci>
<cn>2</cn>
</apply>
<mo>+</mo>
<msup>
<mi>v</mi>
<mn>2</mn>
</msup>
</mrow>
]]></eg>
where <mi>v</mi> is a vector and the superscript denotes a vector
component, and <mi>x</mi> is a real variable. On account of the linear
algebra context, a visually impaired reader may have directed his or her
voice synthesis software to render superscripts as vector components. By
explicitly encoding the power, the content markup yields a much better
voice rendering than would likely happen by default.</p>
<p>If an author is primarily content-oriented, there are two reasons to
intersperse presentation markup. First, using presentation markup provides
a way of modifying or refining how a content expression is rendered. For
example, one might write:
<eg role="mathml"><![CDATA[
<apply>
<in/>
<ci><mi mathvariant="bold">v</mi></ci>
<ci>S</ci>
</apply>
]]></eg>
In this case, the use of embedded presentation markup allows the
author to specify that <mi>v</mi> should be rendered in
boldface.
In the same way, it is sometimes the case that a completely different
notation is desired for a content expression. For example, here
we express a fact about factorials, <mi>n</mi> = <mi>n</mi>!/(<mi>n</mi>-1)!,
using the ascending factorial notation:
<eg role="mathml"><![CDATA[
<apply>
<equivalent/>
<ci>n</ci>
<apply>
<divide/>
<semantics>
<apply>
<factorial/>
<ci>n</ci>
</apply>
<annotation-xml encoding="MathML-Presentation">
<msup>
<mn>1</mn>
<mover accent="true">
<mi>n</mi>
<mo>‾</mo>
</mover>
</msup>
</annotation-xml>
</semantics>
<semantics>
<apply>
<factorial/>
<apply><minus/><ci>n</ci><cn>1</cn></apply>
</apply>
<annotation-xml encoding="MathML-Presentation">
<msup>
<mn>1</mn>
<mover accent="true">
<mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow>
<mo>‾</mo>
</mover>
</msup>
</annotation-xml>
</semantics>
</apply>
</apply>
]]></eg>
This content expression would render using the given notation as:
<graphic role="inline" source="image/factid.gif"
alt="\frac{1^{\bar{n}}}{1^{\frac{}{n-1}}}"/>
</p>
<p>A second reason to use presentation within content markup is that
there is a continually growing list of areas of discourse that do not have
pre-defined content elements for encoding their objects and operators. As a
consequence, any system of content markup inevitably requires an extension
mechanism that combines notation with semantics in some way. MathML
content markup specifies several ways of attaching an external semantic
definitions to content objects. It is necessary, however, to use MathML
presentation markup to specify how such user-defined semantic extensions
should be rendered.</p>
<p>For example, the <quote>rank</quote> operator from linear algebra is not
included as a pre-defined MathML content element. Thus, to express
the statement rank(<mi>u</mi><sup>T</sup><mi>v</mi>)=1
we use a <kw role="element">semantics</kw> element to bind a semantic
definition to the symbol <mi>rank</mi>.
<eg role="mathml"><![CDATA[
<apply>
<eq/>
<apply>
<semantics>
<mi>rank</mi>
<annotation-xml encoding="OpenMath">
<OMS name="rank" cd="linalg3" xmlns="http://www.openmath.org/OpenMath"/>
</annotation-xml>
</semantics>
<apply>
<times/>
<apply> <transpose/> <ci>u</ci> </apply>
<ci>v</ci>
</apply>
</apply>
<cn>1</cn>
</apply>
]]></eg>
Here, the semantics of rank have been
given using a symbol from an OpenMath content dictionary (CD).</p>
</div3>
<div3><head>Combinations that are prohibited</head>
<p>The main consideration when presentation markup and content markup are
mixed together in a single expression is that the result should still make
sense. When both kinds of markup are contained in a presentation
expression, this means it should be possible to render the resulting mixed
expressions simply and sensibly. Conversely, when mixed markup appears in a
content expression, it should be possible to simply and sensibly assign a
semantic interpretation to the expression as whole. These requirements
place a few natural constraints on how presentation and content markup can
be mixed in a single expression, in order to avoid ambiguous or otherwise
problematic expressions.</p>
<p>Two examples illustrate the kinds of problems that must
be avoided in mixed markup. Consider:
<eg role="mathml-error"><![CDATA[
<mrow>
<bvar> x </bvar> <mo> + </mo> <bvar> y </bvar>
</mrow>
]]></eg>
In this example, the content element <kw role="element">bvar</kw> has been
indiscriminately embedded in a presentation expression.
Since <kw role="element">bvar</kw> requires an enclosing context for its
meaning, this expression is unclear.</p>
<p>Similarly, consider:
<eg role="mathml-error"><![CDATA[
<apply>
<ci> x </ci> <mo> + </mo> <ci> y </ci>
</apply>
]]></eg>
Here, the <kw role="element">mo</kw> element is problematic. Should a
renderer infer that the usual arithmetic operator is intended, and act as
if the prefix content element <kw role="element">plus</kw> had been used?
Or should this be literally interpreted as the operator <mi>x</mi>
applied to two arguments,
<kw role="starttag">mo</kw>+<kw role="endtag">mo</kw> and
<kw role="starttag">mi</kw>y<kw role="endtag">mi</kw> ?
Even if we were to decide that
<kw role="starttag">mo</kw>+<kw role="endtag">mo</kw> was the operator, then what
should its meaning be?
These questions do not have particularly compelling answers, so this kind
of mixing of content and presentation markup is also prohibited.</p>
</div3>
<div3 id="mixing_pmincm"><head>Presentation Markup Contained in Content Markup</head>
<p>The use of presentation markup within content markup is limited to
situations that do not effect the ability of content markup to
unambiguously encode mathematical meaning. Specifically,
presentation markup may only appear in content markup in three ways:
<olist>
<item><p>within <kw role="element">ci</kw> and
<kw role="element">cn</kw> token elements</p>
</item>
<item><p>within the <kw role="element">csymbol</kw> element</p>
</item>
<item><p>within the <kw role="element">semantics</kw> element</p>
</item>
</olist>
Any other presentation markup occurring within a content markup is a
MathML error. More detailed discussion of these three cases follows:
<glist>
<gitem>
<label>Presentation markup within token elements.</label>
<def><p>The token elements <kw role="element">ci</kw> and <kw
role="element">cn</kw> are permitted to contain any sequence of
MathML characters (defined in <specref ref="chars"/>),
presentation elements, and <kw role="element">sep</kw> empty elements.
Contiguous blocks of MathML characters in <kw role="element">ci</kw> and
<kw role="element">cn</kw> elements are rendered as if they were wrapped in
<kw role="element">mi</kw> and <kw role="element">mn</kw>
elements respectively. If a token element contains both MathML characters
and presentation elements, contiguous blocks of MathML characters (if any)
are treated as
if wrapped in <kw role="element">mi</kw> or <kw role="element">mn</kw>
elements as appropriate, and the resulting collection of presentation
elements are rendered as if wrapped in an <kw role="element">mrow</kw>
element.</p></def>
</gitem>
<gitem>
<label>Presentation markup within the <kw role="element">csymbol</kw>
element.</label>
<def><p>The <kw role="element">csymbol</kw> element may contain either
MathML characters interspersed with presentation markup, or content elements
of the container type. It is a MathML error for a <kw
role="element">csymbol</kw> element to contain both presentation and
content elements. When the <kw role="element">csymbol</kw> element contains
both raw data and presentation markup, the same rendering rules that apply
to content elements of the token type should be used.</p></def>
</gitem>
<gitem>
<label>Presentation markup within the <kw role="element">semantics</kw> element.</label>
<def><p>One of the main purposes of the
<kw role="element">semantics</kw> element is to provide a mechanism for
incorporating arbitrary MathML expressions into content markup in a
semantically meaningful way. In particular, any valid presentation
expression can be embedded in a content expression by placing it as the
first child of a <kw
role="element">semantics</kw> element. The
meaning of this wrapped expression should be indicated by one or more
annotation elements also contained in the <kw role="element">semantics</kw>
element. Suggested rendering for a <kw role="element">semantics</kw>
element is discussed in <specref ref="contm_synsem"/>.</p></def>
</gitem>
</glist>
</p>
</div3>
<div3 id="mixing_cminpm"><head>Content Markup Contained in
Presentation Markup</head>
<p>The guiding principle for embedding content markup within presentation
expressions is that the resulting expression should still have an
unambiguous rendering. In general, this means that embedded content
expressions must be semantically meaningful, since rendering of content
markup depends on its meaning.</p>
<p>Certain content elements
derive part of their semantic meaning from the surrounding context, such as
whether a <kw role="element">bvar</kw> element is qualifying an integral,
logical quantifier or lambda expression. Another example
would be whether a <kw role="element">degree</kw> element occurs in
a <kw role="element">root</kw> or <kw role="element">partialdiff</kw> element.
Thus, in a presentation context, elements such as these do not have a clearly
defined meaning, and hence there is no obvious choice for a rendering.
Consequently, they are not allowed.</p>
<p>Using the terminology of <specref ref="contm_cats"/>,
we see that operator, relation, container, constant and symbol
elements make sense on their own, while elements of the
qualifier and condition type do not.
(Note that <kw role="element">interval</kw> may be used either as
a general container, or as a qualifier.)</p>
<p>Outside these categories, certain elements deserve specific comment:
the elements <kw role="element">declare</kw>, <kw role="element">sep</kw>,
<kw role="element">annotation</kw> and <kw role="element">annotation-xml</kw>
can only appear in very specific contexts and consequently are not permitted
as direct sub-expressions of any presentation element.
Finally, the element <kw role="element">semantics</kw>
carries with it sufficient information to be permitted in presentation.</p>
<p>The complete list of content elements that <emph>cannot</emph> appear as a child
in a presentation element is:
<kw role="element">annotation</kw>,
<kw role="element">annotation-xml</kw>,
<kw role="element">sep</kw>,
<kw role="element">declare</kw>,
<kw role="element">bvar</kw>,
<kw role="element">condition</kw>,
<kw role="element">degree</kw>,
<kw role="element">logbase</kw>,
<kw role="element">lowlimit</kw>,
<kw role="element">uplimit</kw>.
</p>
<p>Note that within presentation markup, content expressions may only
appear in locations where it is valid for any MathML expression to
appear. In particular, content expressions may not appear within
presentation token elements. In this regard mixing presentation and content
are asymmetrical.</p>
<p>Note that embedding content markup in presentation will often require
applications to render operators outside of an <kw role="element">apply</kw>
context. E.g., it may be necessary to render
<kw role="element">abs</kw>,
<kw role="element">plus</kw>,
<kw role="element">root</kw> or
<kw role="element">sin</kw>
outside of an application. Content/presentation mixing does not
introduce any new requirements, however, since unapplied operators
are already permitted in content expressions, for example:
<eg role="mathml"><![CDATA[
<apply>
<compose/>
<sin/>
<apply>
<inverse/>
<root/>
</apply>
</apply>
]]></eg></p>
</div3>
</div2>
<div2 id="mixing_parallel"><head>Parallel Markup</head>
<p>Some applications are able to make use of <emph>both</emph> presentation
and content information. For these applications it is desirable to provide
both forms of markup for the same mathematical expression. This is called
<emph>parallel markup</emph>.</p>
<p>Parallel markup is achieved with the <kw
role="element">semantics</kw> element. Parallel markup for an expression
can be used on its own, or can be incorporated as part of a larger content
or presentation tree.
</p>
<div3><head>Top-level Parallel Markup</head>
<p>
In many cases what is desired is to provide
presentation markup and content markup for a mathematical
expression as a whole.
To achieve this, a single <kw role="element">semantics</kw> element is used
pairing two markup trees, with the first branch being the MathML presentation
markup, and the second branch being the MathML content markup.</p>
<p>The following example encodes the boolean arithmetic expression
(<mi>a</mi>+<mi>b</mi>)(<mi>c</mi>+<mi>d</mi>) in this way.
<eg role="mathml"><![CDATA[
<semantics>
<mrow>
<mrow><mo>(</mo><mi>a</mi> <mo>+</mo> <mi>b</mi><mo>)</mo></mrow>
<mo>⁢</mo>
<mrow><mo>(</mo><mi>c</mi> <mo>+</mo> <mi>d</mi><mo>)</mo></mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply><and/>
<apply><xor/><ci>a</ci> <ci>b</ci></apply>
<apply><xor/><ci>c</ci> <ci>d</ci></apply>
</apply>
</annotation-xml>
</semantics>
]]></eg>
This example is non-trivial in the sense that the content markup
could not be easily derived from the presentation markup alone.
</p>
</div3>
<div3><head>Fine-grained Parallel Markup</head>
<p>Top-level pairing of independent presentation and content markup is
sufficient for many, but not all, situations. Applications that allow
treatment of <emph>sub-expressions</emph> of mathematical objects require
the ability to associate presentation, content or information with the
<emph>parts</emph> of an object with mathematical markup. Top-level
pairing with a <kw role="element">semantics</kw> element is insufficient in
this type of situation; identification of a sub-expression in one branch of
<kw role="element">semantics</kw> element gives no indication of the
corresponding parts in other branches.</p>
<p>The ability to identify corresponding sub-expressions is required in
applications such as mathematical expression editors. In this situation,
selecting a sub-expression on a visual display can identify a particular
portion of a presentation markup tree. The application then needs to
determine the corresponding annotations of the sub-expressions; in
particular, the application requires the sub-expressions of the
<kw role="element">annotation-xml</kw> tree in MathML content notation.</p>
<p>It is, in principle, possible to provide annotations for
each presentation node by incorporating
<kw role="element">semantics</kw> elements recursively.
<eg role="mathml"><![CDATA[
<semantics>
<mrow>
<semantics>
<mrow><mo>(</mo><mi>a</mi> <mo>+</mo> <mi>b</mi><mo>)</mo></mrow>
<annotation-xml encoding="MathML-Content">
<apply><plus/><ci>a</ci> <ci>b</ci></apply>
</annotation-xml>
</semantics>
<mo>⁢</mo>
<semantics>
<mrow><mo>(</mo><mi>c</mi> <mo>+</mo> <mi>d</mi><mo>)</mo></mrow>
<annotation-xml encoding="MathML-Content">
<apply><plus/><ci>c</ci> <ci>d</ci></apply>
</annotation-xml>
</semantics>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply><times/>
<apply><plus/><ci>a</ci> <ci>b</ci></apply>
<apply><plus/><ci>c</ci> <ci>d</ci></apply>
</apply>
</annotation-xml>
</semantics>
]]></eg>
To be complete this example would be much more verbose, wrapping each of
the individual leaves <kw role="element">mi</kw>, <kw
role="element">mo</kw> and <kw role="element">mn</kw> in a further seven
<kw role="element">semantics</kw> elements.</p>
<p>This approach is very general and works for all kinds of annotations
(including non-MathML annotations and multiple annotations). It leads,
however, to O(<mi>n</mi><sup>2</sup>) increase in size of the document.
This is therefore not a suitable approach for fine-grained parallel markup
of large objects.</p>
</div3>
<div3><head>Parallel Markup via Cross-References:
<kw role="attrib">id</kw> and <kw role="attrib">xref</kw></head>
<p>To better accommodate applications that must deal with sub-expressions
of large objects, MathML uses cross-references between the branches of a
<kw role="element">semantics</kw> element to identify corresponding
sub-structures.</p>
<p>Cross-referencing is achieved using <kw role="attrib">id</kw> and <kw
role="attrib">xref</kw> attributes within the branches of a containing <kw
role="element">semantics</kw> element. These attributes may optionally be
placed on MathML elements of any type.</p>
<p>The following example shows this cross-referencing for the
boolean arithmetic expression
(<mi>a</mi>+<mi>b</mi>)(<mi>c</mi>+<mi>d</mi>).
<eg role="mathml"><![CDATA[
<semantics>
<mrow id="E">
<mrow id="E.1">
<mo id="E.1.1">(</mo>
<mi id="E.1.2">a</mi>
<mo id="E.1.3">+</mo>
<mi id="E.1.4">b</mi>
<mo id="E.1.5">)</mo>
</mrow>
<mo id="E.2">⁢</mo>
<mrow id="E.3">
<mo id="E.3.1">(</mo>
<mi id="E.3.2">c</mi>
<mo id="E.3.3">+</mo>
<mi id="E.3.4">d</mi>
<mo id="E.3.5">)</mo>
</mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply xref="E">
<and xref="E.2"/>
<apply xref="E.1">
<xor xref="E.1.3"/><ci xref="E.1.2">a</ci><ci xref="E.1.4">b</ci>
</apply>
<apply xref="E.3">
<xor xref="E.3.3"/><ci xref="E.3.2">c</ci><ci xref="E.3.4">d</ci>
</apply>
</apply>
</annotation-xml>
</semantics>
]]></eg></p>
<p>An <kw role="attrib">id</kw> attribute and a corresponding <kw
role="attrib">xref</kw> appearing within the same <kw
role="element">semantics</kw> element create a correspondence between
sub-expressions.</p>
<p>In creating these correspondences by cross-reference, <emph>all</emph>
of the <kw role="attrib">id</kw> attributes referenced by any <kw
role="attrib">xref</kw> must be in the <emph>same</emph> branch of an
enclosing <kw role="element">semantics</kw> element. This constraint
guarantees that these correspondences do not create unintentional cycles.
(Note that this restriction does <emph>not</emph> exclude the use of <kw
role="attrib">id</kw> attributes within the other branches of the enclosing
<kw role="element">semantics</kw> element. It does, however, exclude
references to these other <kw role="attrib">id</kw> attributes
originating in the same <kw role="element">semantics</kw> element.)</p>
<p>There is no restriction on which branch of the <kw
role="element">semantics</kw> element may contain the destination <kw
role="attrib">id</kw> attributes. It is up to the application to determine
which branch to use.</p>
<p>In general, there will not be a one-to-one correspondence between nodes
in parallel branches. For example, a presentation tree may contain
elements, such as parentheses, that have no correspondents in the content
tree. It is therefore often useful to put the <kw role="attrib">id</kw>
attributes on the branch with the finest-grained node structure. Then all
of the other branches will have <kw role="attrib">xref</kw> attributes
to some subset of the <kw role="attrib">id</kw> attributes.</p>
<p>In absence of other criteria, the first branch of the <kw
role="element">semantics</kw> element is a sensible choice to contain the
<kw role="attrib">id</kw> attributes. Applications that add or remove
annotations will then not have to re-assign attributes to the <kw
role="element">semantics</kw> trees.</p>
<p>In general, the use of <kw role="attrib">id</kw> and <kw
role="attrib">xref</kw> attributes allows a full correspondence between
sub-expressions to be given in text that is at most a constant factor
larger than the original. The direction of the references should not be
taken to imply that sub-expression selection is intended to be permitted
only on one child of the <kw role="element">semantics</kw> element. It is
equally feasible to select a subtree in any branch and to recover the
corresponding subtrees of the other branches.</p>
</div3>
<div3><head>Annotation Cross-References using XLink:
<kw role="attrib">id</kw> and <kw role="attrib">href</kw></head>
<p>It is possible to give cross-references between a MathML expression and
a non-MathML XML annotation using the XLink protocol <bibref ref="XLink"/>.
As an example, the boolean expression of the previous section can
be annotated with OpenMath, and cross-linked as follows:
<eg role="mathml-extension"><![CDATA[
<semantics>
<mrow id="E">
<mrow id="E.1">
<mo id="E.1.1">(</mo>
<mi id="E.1.2">a</mi>
<mo id="E.1.3">+</mo>
<mi id="E.1.4">b</mi>
<mo id="E.1.5">)</mo>
</mrow>
<mo id="E.2">⁢</mo>
<mrow id="E.3">
<mo id="E.3.1">(</mo>
<mi id="E.3.2">c</mi>
<mo id="E.3.3">+</mo>
<mi id="E.3.4">d</mi>
<mo id="E.3.5">)</mo>
</mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply xref="E">
<and xref="E.2"/>
<apply xref="E.1">
<xor xref="E.1.3"/><ci xref="E.1.2">a</ci><ci xref="E.1.4">b</ci>
</apply>
<apply xref="E.3">
<xor xref="E.3.3"/><ci xref="E.3.2">c</ci><ci xref="E.3.4">d</ci>
</apply>
</apply>
</annotation-xml>
<annotation-xml encoding="OpenMath"
xmlns:om="http://www.openmath.org/OpenMath"
xmlns:xlink="http://www.w3.org/1999/xlink">
<om:OMA xlink:href="#xpointer(id('E'))" xlink:type="simple">
<om:OMS name="and" cd="logic1"
xlink:href="#xpointer(id('E'))" xlink:type="simple"/>
<om:OMA xlink:href="#xpointer(id('E.1'))" xlink:type="simple">
<om:OMS name="xor" cd="logic1"
xlink:href="#xpointer(id('E.1.3'))" xlink:type="simple"/>
<om:OMV name="a"
xlink:href="#xpointer(id('E.1.2'))" xlink:type="simple"/>
<om:OMV name="b"
xlink:href="#xpointer(id('E.1.4'))" xlink:type="simple"/>
</om:OMA>
<om:OMA xlink:href="#xpointer(id('E.3'))" xlink:type="simple">
<om:OMS name="xor" cd="logic1"
xlink:href="#xpointer(id('E.3.3'))" xlink:type="simple"/>
<om:OMV name="c"
xlink:href="#xpointer(id('E.3.2'))" xlink:type="simple"/>
<om:OMV name="d"
xlink:href="#xpointer(id('E.3.4'))" xlink:type="simple"/>
</om:OMA>
</om:OMA>
</annotation-xml>
</semantics>
]]></eg>
Here
<kw role="element">OMA</kw>, <kw role="element">OMS</kw> and
<kw role="element">OMV</kw> are elements defined in the OpenMath
standard for representing application, symbol and variable, respectively.
The references from the OpenMath annotation are given by the
<kw role="attrib">xlink:href</kw> attributes which in this case
use XPointer <bibref ref="XPointer"/> to refer to an
<kw role="attrib">id</kw>s within the current document.</p>
<p>Note that the application might or might not have a mechanism for
extending DTDs. It will be the case, therefore that some applications
will give well-formed, but not "valid," XML within
<kw role="element">annotation-xml</kw> elements.
Consequently, some MathML applications using
<kw role="element">annotation-xml</kw> will not be validated.
More flexibility is offered by the use of XML Schemas.</p>
</div3>
</div2>
<div2 id="mixing_tools"><head>Tools, Style Sheets and Macros for Combined Markup</head>
<p>The interaction of presentation and content markup can be greatly
enhanced through the use of various tools. While the set of tools and
standards for working with XML applications is rapidly evolving at the
present, we can already outline some specific techniques.</p>
<p>In general, the interaction of content and presentation is handled via
transformation rules on MathML trees. These transformation rules are
sometimes called <quote>macros</quote>. In principle, these rules can be
expressed using any one of a number of mechanisms, including DSSSL, Java
programs operating on a DOM, etc. We anticipate, however, that the
principal mechanism for these transformations in most applications shall be
XSLT.</p>
<p>In this section we discuss transformation rules for two specific purposes:
for notational style sheets, and to simplify parallel markup.</p>
<div3 id="mixing_notsheet"><head>Notational Style Sheets</head>
<p>Authors who make use of content markup may be required to deploy their
documents in locales with notational conventions different than the default
content rendering. It is therefore expected that transformation tools will
be used to determine notations for content elements in different
settings. Certain elements, e.g. <kw role="element">lambda</kw>,
<kw role="element">mean</kw> and <kw role="element">transpose</kw>, have
widely varying common notations and will often require notational
selection. Some examples of notational variations are given below.
<ulist>
<item><p><graphic role="inline" source="image/f5001.gif" alt="\mathbf{V}"/>
versus <graphic role="inline" source="image/f5002.gif" alt="\vec{V}"/>
</p></item>
<item><p>tan <mi>x</mi> versus tg <mi>x</mi>
</p></item>
<item><p><graphic role="inline" source="image/f5003.gif" alt="{n \choose m}"/>
versus <graphic role="inline" source="image/f5004.gif" alt="{}_nC^m"/>
versus <graphic role="inline" source="image/f5005.gif" alt="C^n_m"/>
versus <graphic role="inline" source="image/f5006.gif" alt="C^m_n"/>
</p></item>
<item><p><graphic role="inline" source="image/f5007.gif"
alt="a_0 + \frac{ ~1 ~ |}{| ~a_1 ~} + \ldots + \frac{~1 ~ |}{| ~ a_k~}"/>
versus <graphic role="inline" source="image/f5008.gif" alt="[a_0, a_1, \ldots, a_k]"/>
</p></item>
</ulist>
Other elements, for example <kw role="element">plus</kw> and <kw
role="element">sin</kw>, are less likely to require these features.</p>
<p>Selection of notational style is sometimes necessary
for correct understanding of documents by locale. For instance,
the binomial coefficient
<graphic role="inline" source="image/f5005.gif" alt="C^n_m"/>
in French notation is equivalent to
<graphic role="inline" source="image/f5006.gif" alt="C^m_n"/>
in Russian notation.</p>
<p>A natural way for a MathML application to bind a particular notation to
the set of content tags is with an XSLT style sheet <bibref
ref="XSLT"/>. The examples of this section shall assume this is the
mechanism to express style choices. (Other choices are equally possible,
for example an application program may provide menus offering a number of
rendering choices for all content tags.)
</p>
<p>When writing XSLT style sheets for mathematical notation, some
transformation rules can be purely local, while others will require
multi-node context to determine the correct output notation. The following
example gives a local transformation rule that could be included in a
notational style sheet displaying open intervals as
]<mi>a</mi>,<mi>b</mi>[ rather than as (<mi>a</mi>,<mi>b</mi>).
<eg><![CDATA[
<xsl:template match="m:interval">
<m:mrow>
<xsl:choose>
<xsl:when test="@closure='closed'">
<m:mfenced open="[" close="]" separators=",">
<xsl:apply-templates/>
</m:mfenced>
</xsl:when>
<xsl:when test="@closure='open'">
<m:mfenced open="]" close="[" separators=",">
<xsl:apply-templates/>
</m:mfenced>
</xsl:when>
<xsl:when test="@closure='open-closed'">
<m:mfenced open="]" close="]" separators=",">
<xsl:apply-templates/>
</m:mfenced>
</xsl:when>
<xsl:when test="@closure='closed-open'">
<m:mfenced open="[" close="[" separators=",">
<xsl:apply-templates/>
</m:mfenced>
</xsl:when>
<xsl:otherwise>
<m:mfenced open="[" close="]" separators=",">
<xsl:apply-templates/>
</m:mfenced>
</xsl:otherwise>
</xsl:choose>
</m:mrow>
</xsl:template>
]]></eg>
Here <kw role="element">m</kw> is established as the MathML namespace.</p>
<p>An example of a rule requiring context information would be:
<eg><![CDATA[
<xsl:template match="m:apply[m:factorial]">
<m:mrow>
<xsl:choose>
<xsl:when test="not(*[2]=m:ci) and not(*[2]=m:cn)">
<m:mrow>
<m:mo>(</m:mo>
<xsl:apply-templates select="*[2]" />
<m:mo>)</m:mo>
</m:mrow>
</xsl:when>
<xsl:otherwise>
<xsl:apply-templates select="*[2]" />
</xsl:otherwise>
</xsl:choose>
<m:mo>!</m:mo>
</m:mrow>
</xsl:template>
]]></eg>
Other examples of context-dependent transformations would be, e.g.
for the <kw role="element">apply</kw> of a <kw role="element">plus</kw>
to render <mi>a</mi>-<mi>b</mi>+<mi>c</mi>, rather than
<mi>a</mi>+ -<mi>b</mi>+<mi>c</mi>, or for the
<kw role="element">apply</kw> of a <kw role="element">power</kw>
to render sin<sup>2</sup><mi>x</mi>, rather than
sin <mi>x</mi><sup>2</sup>.</p>
<p>Notational variation will occur both for built-in content elements
as well as extensions. Notational style for extensions can be handled
as described above, with rules matching the names of any extension
tags, and with the content handling (in a content-faithful style sheet)
proceeding as described in <specref ref="mixing_sheet"/>.</p>
</div3>
<div3><head>Content-Faithful Transformations</head>
<p>There may be a temptation to view notational style sheets as a
transformation from content markup to equivalent presentation markup.
This viewpoint is explicitly discouraged, since information will be
lost and content-oriented applications will not function properly.</p>
<p>We define a <quote>content-faithful</quote> transformation to be a
transformation that retains the original content in parallel markup
(<specref ref="mixing_parallel"/>).</p>
<p>Tools that support MathML should be <quote>content-faithful</quote>,
and not gratuitously convert content elements to presentation elements in
their processing. Notational style sheets should be content-faithful
whenever they may be used in interactive applications.</p>
<p>It is possible to write content-faithful style sheets in a number
of ways. Top-level parallel markup can be achieved by incorporating the
following rules in an XSLT style sheet:
<eg><![CDATA[
<xsl:template match="m:math">
<m:semantics>
<xsl:apply-templates/>
<m:annotation-xml m:encoding="MathML-Content">
<xsl:copy-of select="."/>
</m:annotation-xml>
</m:semantics>
</xsl:template>
]]></eg>
The notation would be generated by additional rules for producing presentation
from content, such as those in <specref ref="mixing_notsheet"/>.
Fine-grained parallel markup can be achieved with additional rules
treating <kw role="attrib">id</kw> attributes. A detailed example
is given in <bibref ref="RodWatt2000"/>. </p>
</div3>
<div3 id="mixing_sheet"><head>Style Sheets for Extensions</head>
<p>The presentation tags of MathML form a closed vocabulary of notational
structures, but are quite rich and can be used to express a rendering of
most mathematical notations. Complex notations can be composed from
the basic elements provided for presentation markup.
In this sense, the presentation ability of MathML is open-ended.
It is often useful, however, to give a
name to new notational schemas if they are going to be used often.
For example, we can shorten and clarify the ascending factorial example of
<specref ref="mixing_reasons"/>, with a rule which replaces
<eg role="mathml-extension"><![CDATA[
<mx:a-factorial>X</mx:a-factorial>
]]></eg>
with
<eg role="mathml"><![CDATA[
<semantics>
<apply> <factorial/> <mi>X</mi> </apply>
<annotation-xml encoding="MathML-Presentation">
<msup>
<mn>1</mn>
<mover accent="true">
<mi>X</mi>
<mo>‾</mo>
</mover>
</msup>
</annotation-xml>
</semantics>
]]></eg>
Then the example would be more clearly written as:
<eg role="mathml-extension"><![CDATA[
<apply>
<equivalent/>
<ci>n</ci>
<apply>
<divide/>
<mx:a-factorial><ci>n</ci></mx:a-factorial>
<mx:a-factorial>
<apply><minus/><ci>n</ci><cn>1</cn></apply>
</mx:a-factorial>
</apply>
</apply>
]]></eg></p>
<p>Likewise, the content tags form a fixed vocabulary of concepts covering
the types of mathematics seen in most common applications. It is not
reasonable to
expect users to compose existing MathML content tags to construct new
content concepts. (This approach is fraught with technical difficulties
even for professional mathematicians.) Instead, it is anticipated that
applications whose mathematical content concepts extend beyond what is
offered by MathML will use annotations and attributes within
<kw role="element">semantics</kw> and <kw role="element">csymbol</kw> elements,
and that these annotations will use
content description languages outside of MathML.</p>
<p>Often the naming of a notation and the identification of a new semantic
concept are related. This allows a single transformation rule to capture
both a presentation and a content markup for an expression. This is one of
the areas of MathML that benefits most strongly from the use of macro
processing.
<eg role="mathml-extension"><![CDATA[
<mx:rank/>
]]></eg>
and
<eg role="mathml-extension"><![CDATA[
<mx:tr>X</mx:tr>
]]></eg>
and respectively transform them to
<eg role="mathml"><![CDATA[
<semantics>
<ci><mo>rank</mo></ci>
<annotation-xml encoding="OpenMath">
<OMS name="rank" cd="linalg3" xmlns="http://www.openmath.org/OpenMath"/>
</annotation-xml>
</semantics>
]]></eg>
and
<eg role="mathml"><![CDATA[
<apply>
<transpose/>
<ci>X</ci>
</apply>
]]></eg>
The lengthy sample encoding of
rank(<mi>u</mi><sup>T</sup><mi>v</mi>)=1,
from <specref ref="mixing_reasons"/> could then be condensed to
<eg role="mathml-extension"><![CDATA[
<apply>
<eq/>
<apply>
<mx:rank/>
<apply> <times/> <mx:tr>u</mx:tr> <ci>v</ci> </apply>
</apply>
<cn>1</cn>
</apply>
]]></eg>
From this example we see how the combination of presentation and content
markup could become much simpler and effective to generate as standard
style sheet libraries become available.
</p>
</div3>
</div2>
</div1>