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|
<?xml version="1.0" encoding="utf-8"?><!DOCTYPE html
SYSTEM "mathml.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>Mixing Markup Languages for Mathematical Expressions</title><style type="text/css">
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
table.syntax th {
text-align: left;
}
table.attributes td { padding-left:0.5em; padding-right:0.5em; }
table.attributes td.attname { white-space:nowrap; vertical-align:top;}
table.attributes td.attdesc { background-color:#F0F0FF; padding-left:2em; padding-right:2em}
th.uname {font-size: 50%; text-align:left;}
code { font-family: monospace; }
div.constraint,
div.issue,
div.note,
div.notice { margin-left: 2em; }
li p { margin-top: 0.3em;
margin-bottom: 0.3em; }
div.exampleInner pre { margin-left: 1em;
margin-top: 0em; margin-bottom: 0em}
div.exampleOuter {border: 4px double gray;
margin: 0em; padding: 0em}
div.exampleInner { background-color: #d5dee3;
border-top-width: 4px;
border-top-style: double;
border-top-color: #d3d3d3;
border-bottom-width: 4px;
border-bottom-style: double;
border-bottom-color: #d3d3d3;
padding: 4px; margin: 0em }
div.exampleWrapper { margin: 4px }
div.exampleHeader { font-weight: bold;
margin: 4px}
a.mainindex {font-weight: bold;}
li.sitem {list-style-type: none;}
.error { color: red }
div.mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example h5 {
margin-top: -0.3em;
margin-bottom: -0.5em;}
var.meta {background-color:green}
var.transmeta {background-color:red}
pre.mathml {padding: 0.5em;
background-color: #FFFFDD;}
pre.mathml-fragment {padding: 0.5em;
background-color: #FFFFDD;}
pre.strict-mathml {padding: 0.5em;
background-color: #FFFFDD;}
.minitoc { border-style: solid;
border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
border-width: 1px ;
color: #5D0091;
background: #F9F5DE;
border-color: red;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
.IDL-definition { border-style: solid;
border-width: 1px ;
color: #001000;
background: #E0FFE0;
border-color: #206020;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.baseline {vertical-align: baseline}
#eqnoc1 {width: 10%}
#eqnoc2 {width: 80%; text-align: center; }
#eqnoc3 {width: 10%; text-align: right; }
div.div1 {margin-bottom: 1em;}
.h3style {
text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2;
font-size: 125%;
}
h4 { text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2; }
h5 { text-align: left;
font-family: sans-serif;
font-weight: bold;
color: #0050B2; }
th {background: #E0FFE0;}
p, blockquote, h4 { font-family: sans-serif; }
dt, dd, dl, ul, li { font-family: sans-serif; }
pre, code { font-family: monospace }
a.termref {
text-decoration: none;
color: black;
}
.mathml-render {
font-family: serif;
font-size: 130%;
border: solid 4px green;
padding-left: 1em;
padding-right: 1em;
}
</style><link rel="stylesheet" type="text/css"
href="../../../StyleSheets/TR/W3C-REC.css" />
</head>
<body>
<h1><a id="mixing"></a>5 Mixing Markup Languages for Mathematical Expressions
</h1>
<!-- TOP NAVIGATION BAR -->
<div class="minitoc">
Overview: <a href="Overview.xml.html">Mathematical Markup Language (MathML) Version 3.0</a><br class="html-compat" />
Previous: 4 <a href="chapter4.xml.html">Content Markup</a><br class="html-compat" />
Next: 6 <a href="chapter6.xml.html">Interactions with the Host Environment</a><br class="html-compat" /><br class="html-compat" />5 <a href="chapter5.xml.html">Mixing Markup Languages for Mathematical Expressions</a><br class="html-compat" />Â Â Â Â 5.1 <a href="chapter5.xml.html#mixing.semantic.annotations">Annotation Framework</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.1.1 <a href="chapter5.xml.html#mixing.annotation.elements">Annotation elements</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.1.2 <a href="chapter5.xml.html#mixing.annotation.keys">Annotation keys</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.1.3 <a href="chapter5.xml.html#mixing.alternate.representations">Alternate representations</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.1.4 <a href="chapter5.xml.html#mixing.content.equiv">Content equivalents</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.1.5 <a href="chapter5.xml.html#mixing.annotation.references">Annotation references</a><br class="html-compat" />Â Â Â Â 5.2 <a href="chapter5.xml.html#mixing.semantic.elements">Elements for Semantic Annotations</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.2.1 <a href="chapter5.xml.html#mixing.elements.semantics">The <code>semantics</code> element</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.1.1 <a href="chapter5.xml.html#id.5.2.1.1">Description</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.1.2 <a href="chapter5.xml.html#id.5.2.1.2">Attributes</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.2.2 <a href="chapter5.xml.html#mixing.elements.annotation">The <code>annotation</code> element</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.2.1 <a href="chapter5.xml.html#id.5.2.2.1">Description</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.2.2 <a href="chapter5.xml.html#id.5.2.2.2">Attributes</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.2.3 <a href="chapter5.xml.html#mixing.elements.annotation.xml">The <code>annotation-xml</code> element</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.3.1 <a href="chapter5.xml.html#id.5.2.3.1">Description</a><br class="html-compat" />Â Â Â Â Â Â Â Â Â Â Â Â 5.2.3.2 <a href="chapter5.xml.html#id.5.2.3.2">Attributes</a><br class="html-compat" />Â Â Â Â 5.3 <a href="chapter5.xml.html#mixing.markup">Combining Presentation and Content Markup</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.3.1 <a href="chapter5.xml.html#mixing.pmincm">Presentation Markup in Content Markup</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.3.2 <a href="chapter5.xml.html#mixing.cminpm">Content Markup in Presentation Markup</a><br class="html-compat" />Â Â Â Â 5.4 <a href="chapter5.xml.html#mixing.parallel">Parallel Markup</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.4.1 <a href="chapter5.xml.html#mixing.top.level">Top-level Parallel Markup</a><br class="html-compat" />Â Â Â Â Â Â Â Â 5.4.2 <a href="chapter5.xml.html#mixing.cr">Parallel Markup via Cross-References</a><br class="html-compat" /></div>
<div class="div1">
<p>MathML markup can be combined with other markup languages, and
these mixing constructions are realized by the semantic annotation
elements. The semantic annotation elements provide an important tool
for making associations between alternate representations of an
expression, and for associating semantic properties and other
attributions with a MathML expression. These elements allow
presentation markup and content markup to be combined in several
different ways. One method, known as <em>mixed markup</em>, is to
intersperse content and presentation elements in what is essentially a
single tree. Another method, known as <em>parallel markup</em>,
is to provide both explicit presentation markup and content markup in
a pair of markup expressions, combined by a single <code>semantics</code>
element.
</p>
<div class="div2">
<h2><a id="mixing.semantic.annotations"></a>5.1 Annotation Framework
</h2>
<p>
An important concern of MathML is to represent associations between
presentation and content markup forms for an expression. Representing
associations between MathML expressions and data of other kinds is
also important in many contexts. For this reason, MathML provides a
general framework for annotation. A MathML expression may be
decorated with a sequence of pairs made up of a symbol that indicates
the kind of annotation, known as the <em>annotation key</em>, and
associated data, known as the <em>annotation value</em>.
</p>
<div class="div3">
<h3><a id="mixing.annotation.elements"></a>5.1.1 Annotation elements
</h3>
<p>
The <code>semantics</code>, <code>annotation</code>, and
<code>annotation-xml</code> elements are used together to represent
annotations in MathML. The <code>semantics</code> element provides the
container for a expression and its annotations. The
<code>annotation</code> element is the container for text
annotations, and the <code>annotation-xml</code> element is used for XML
annotations. The <code>semantics</code> element contains the expression
being annotated as its first child, followed by a sequence of zero or
more <code>annotation</code> and/or <code>annotation-xml</code> elements.
</p>
<table>
<tr>
<td><pre class="mathml"><semantics>
<mrow>
<mrow>
<mi>sin</mi>
<mo>&#x2061;<span style="color:#999900"><!--FUNCTION APPLICATION--></span></mo>
<mfenced><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
<annotation-xml encoding="application/openmath+xml">
<OMA xmlns="http://www.openmath.org/OpenMath">
<OMS cd="arith1" name="plus"/>
<OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA>
<OMI>5</OMI>
</OMA>
</annotation-xml>
</semantics></pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics>
<mrow>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mfenced><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
<annotation-xml encoding="application/openmath+xml">
<OMA xmlns="http://www.openmath.org/OpenMath">
<OMS cd="arith1" name="plus"/>
<OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA>
<OMI>5</OMI>
</OMA>
</annotation-xml>
</semantics></math>
</td>
</tr>
</table>
<p>The <code>semantics</code> element is considered to be both a presentation
element and a content element, and may be used in either context.
All MathML processors should process the <code>semantics</code> element,
even if they only process one of these two subsets of MathML.
</p>
</div>
<div class="div3">
<h3><a id="mixing.annotation.keys"></a>5.1.2 Annotation keys
</h3>
<p>An <em>annotation key</em> specifies the relationship between an
expression and an annotation. Many kinds of relationships are possible.
Examples include alternate representations, specification or clarification
of semantics, type information, rendering hints, and private data
intended for specific processors. The annotation key is the primary
means by which a processor determines whether or not to process an
annotation.
</p>
<p>The logical relationship between an expression and an annotation
can have a significant impact on the proper processing of the
expression. For example, a particular annotation form, called <em>semantic
attributions</em>, cannot be ignored without altering the meaning
of the annotated expression, at
least in some processing contexts. On the other hand, alternate
representations do not alter the meaning of an expression, but may
alter the presentation of the expression as they are frequently used
to provide rendering hints. Still other annotations carry private data or
metadata that are useful in a specific context, but do not alter either
the semantics or the presentation of the expression.
</p>
<p>In MathML 3, annotation keys are defined as symbols in <a href="chapter4.xml.html#contm.cds">Content Dictionaries</a>, and are specified using
of the <code>cd</code> and <code>name</code> attributes on the
<code>annotation</code> and <code>annotation-xml</code> elements. For backward
compatibility with MathML 2, an annotation key may also be referenced
using the <code>definitionURL</code> attribute as an alternative to the
<code>cd</code> and <code>name</code> attributes. Further details on
referencing symbols in Content Dictionaries are discussed in <a href="chapter4.xml.html#contm.csymbol">Section 4.2.3 Content Symbols <code><csymbol></code></a>. The symbol definition in a Content Dictionary
for an annotation key may have a <code>role</code> property. Two
particular roles are relevant for annotations: a role of
"attribution" identifies a generic annotation that can
be ignored without altering the meaning of the annotated term, and a
role of "semantic-attribution" indicates that the annotation
is a semantic annotation, that is, the annotation cannot be ignored
without potentially altering the meaning of the expression.
</p>
<p>MathML 3 provides two predefined annotation keys for the most common
kinds of annotations: <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a> and <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#contentequiv">contentequiv</a> defined in the <a href="http://www.openmath.org/cd/mathmlkeys.xhtml">mathmlkeys</a> content
dictionary. The <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a> annotation key specifies that the
annotation value provides an alternate representation for an
expression in some other markup language, and the <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#contentequiv">contentequiv</a> annotation key specifies that
the annotation value provides a semantically equivalent alternative
for the annotated expression. Further details about the use of
these keys is given in the sections below.
</p>
<p>The default annotation key is <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a> when no annotation key is
explicitly specified on an <code>annotation</code> or
<code>annotation-xml</code> element.
</p>
<p>Typically, annotation keys specify only the logical nature of the
relationship between an expression and an annotation. The data format
for an annotation is indicated with the <code>encoding</code>
attribute. In MathML 2, the <code>encoding</code> attribute was the
primary information that a processor could use to determine whether or
not it could understand an annotation. For backward compatibility,
processors are encouraged to examine both the annotation key and
<code>encoding</code> attribute. In
particular, MathML 2 specified the predefined encoding values
<code>MathML</code>, <code>MathML-Content</code>, and
<code>MathML-Presentation</code>. The <code>MathML</code> encoding
value is used to indicate an <code>annotation-xml</code> element contains
a MathML expression. The use of the other values is more specific, as
discussed in following sections.
</p>
<p>While the predefined <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a> and <a href="http://www.openmath.org/cd/mathmlkeys.xhtml#contentequiv">contentequiv</a> keys cover many common use cases, user
communities are encouraged to define and standardize additional
content dictionaries as necessary. Annotation keys in user-defined,
public Content Dictionaries are preferred over private encoding
attribute value conventions, since content dictionaries are more
expressive, more open and more maintainable than private encoding
values. However, for backward compatibility with MathML 2, the
encoding attribute may also be used.
</p>
</div>
<div class="div3">
<h3><a id="mixing.alternate.representations"></a>5.1.3 Alternate representations
</h3>
<p>Alternate representation annotations are most often used to
provide renderings for an expression, or to provide an equivalent
representation in another markup language. In general, alternate
representation annotations do not alter the meaning of the annotated
expression, but may alter its presentation.
</p>
<p>A particularly important case is the use of a presentation MathML
expression to indicate a preferred rendering for a content MathML
expression. This case may be represented by labeling the annotation
with the <code>application/mathml-presentation+xml</code> value for
the <code>encoding</code> attribute. For backward compatibility with
MathML 2.0, this case can also be represented with the equivalent
<code>MathML-Presentation</code> value for the <code>encoding</code>
attribute. Note that when a presentation MathML annotation is
present in a <code>semantics</code> element, it may be used as the
default rendering of the <code>semantics</code> element, instead of
the default rendering of the first child.
</p>
<p>In the example below, the <code>semantics</code> element binds together
various alternate representations for a content MathML expression.
The presentation MathML annotation may be used as the
default rendering, while the other annotations give representations
in other markup languages. Since no attribution keys are explicitly
specified, the default annotation key
<a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a> applies
to each of the annotations.
</p>
<table>
<tr>
<td><pre class="mathml">
<semantics>
<apply>
<plus/>
<apply><sin/><ci>x</ci></apply>
<cn>5</cn>
</apply>
<annotation-xml encoding="MathML-Presentation">
<mrow>
<mrow>
<mi>sin</mi>
<mo>&#x2061;<span style="color:#999900"><!--FUNCTION APPLICATION--></span></mo>
<mfenced open="(" close=")"><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
</annotation-xml>
<annotation encoding="application/x-maple">
sin(x) + 5
</annotation>
<annotation encoding="application/vnd.wolfram.mathematica">
Sin[x] + 5
</annotation>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
<annotation-xml encoding="application/openmath+xml">
<OMA xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA>
<OMI>5</OMI>
</OMA>
</OMA>
</annotation-xml>
</semantics></pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<apply>
<plus/>
<apply><sin/><ci>x</ci></apply>
<cn>5</cn>
</apply>
<annotation-xml encoding="MathML-Presentation">
<mrow>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mfenced open="(" close=")"><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
</annotation-xml>
<annotation encoding="application/x-maple">
sin(x) + 5
</annotation>
<annotation encoding="application/vnd.wolfram.mathematica">
Sin[x] + 5
</annotation>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
<annotation-xml encoding="application/openmath+xml">
<OMA xmlns="http://www.openmath.org/OpenMath">
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA>
<OMI>5</OMI>
</OMA>
</OMA>
</annotation-xml>
</semantics></math>
</td>
</tr>
</table>
</div>
<div class="div3">
<h3><a id="mixing.content.equiv"></a>5.1.4 Content equivalents
</h3>
<p>Content equivalent annotations provide additional computational
information about an expression. Annotations with the
<a href="http://www.openmath.org/cd/mathmlkeys.xhtml#contentequiv">contentequiv</a> key cannot be ignored
without potentially changing the behavior of an expression.
</p>
<p>An important case arises when a content MathML annotation is used
to disambiguate the meaning of a presentation MathML expression.
This case may be represented by labeling the annotation with the
<code>application/mathml-content+xml</code> value for the
<code>encoding</code> attribute. In
MathML 2, this type of annotation was represented with the equivalent
<code>MathML-Content</code> value for the <code>encoding</code> attribute,
so processors are urged to support this usage for backward compatibility.
A content MathML annotation, whether in MathML 2 or 3, may be used for
other purposes as well, such as for other kinds of semantic assertions.
Consequently, in MathML 3, the
<a href="http://www.openmath.org/cd/mathmlkeys.xhtml#contentequiv">contentequiv</a> annotation key should
be used to make an explicit assertion that the annotation provides a
definitive content markup equivalent for an expression.
</p>
<p>In the example below, an ambiguous presentation MathML expression
is annotated with a <code>MathML-Content</code> annotation clarifying
its precise meaning.
</p>
<table>
<tr>
<td><pre class="mathml">
<semantics>
<mrow>
<mrow>
<mi>a</mi>
<mfenced open="(" close=")">
<mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow>
</mfenced>
</mrow>
</mrow>
<annotation-xml cd="mathmlkeys" name="contentequiv"
encoding="MathML-Content">
<apply>
<ci>a</ci>
<apply><plus/><ci>x</ci><cn>5</cn></apply>
</apply>
</annotation-xml>
</semantics></pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mrow>
<mi>a</mi>
<mfenced open="(" close=")">
<mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow>
</mfenced>
</mrow>
</mrow>
<annotation-xml cd="mathmlkeys" name="contentequiv"
encoding="MathML-Content">
<apply>
<ci>a</ci>
<apply><plus/><ci>x</ci><cn>5</cn></apply>
</apply>
</annotation-xml>
</semantics></math>
</td>
</tr>
</table>
</div>
<div class="div3">
<h3><a id="mixing.annotation.references"></a>5.1.5 Annotation references
</h3>
<p>In the usual case, each annotation element includes either character data
content (in the case of <code>annotation</code>) or XML markup data (in the case
of <code>annotation-xml</code>) that represents the <em>annotation value</em>.
There is no restriction on the type of annotation that may appear within a
<code>semantics</code> element. For example, an annotation could provide a
T<sub>E</sub>X encoding, a linear input form for a computer algebra system,
a rendered image, or detailed mathematical type information.
</p>
<p>In some cases the alternative children of a <code>semantics</code> element
are not an essential part of the behavior of the annotated expression, but
may be useful to specialized processors. To enable the availability of
several annotation formats in a more efficient manner, a <code>semantics</code>
element may contain empty <code>annotation</code> and <code>annotation-xml</code>
elements that provide <code>encoding</code> and <code>src</code> attributes
to specify an external location for the annotation value associated with
the annotation. This type of annotation is known as an <em>annotation
reference</em>.
</p><pre class="mathml-fragment">
<semantics>
<mfrac><mi>a</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfrac>
<annotation encoding="image/png" src="333/formula56.png"/>
<annotation encoding="application/x-maple" src="333/formula56.ms"/>
</semantics></pre><p>Processing agents that anticipate that consumers of exported markup may
not be able to retrieve the external entity referenced by such annotations
should request the content of the external entity at the indicated location
and replace the annotation with its expanded form.
</p>
<p>An annotation reference follows the same rules as for other annotations
to determine the annotation key that specifies the relationship between
the annotated object and the annotation value.
</p>
</div>
</div>
<div class="div2">
<h2><a id="mixing.semantic.elements"></a>5.2 Elements for Semantic Annotations
</h2>
<p>This section explains the semantic mapping elements <code>semantics</code>,
<code>annotation</code>, and <code>annotation-xml</code>. These elements associate
alternate representations for a presentation or content expression, or
associate semantic or other attributions that may modify the meaning of
the annotated expression.
</p>
<div class="div3">
<h3><a id="mixing.elements.semantics"></a>5.2.1 The <code>semantics</code> element
</h3>
<div class="div4">
<h4><a id="id.5.2.1.1"></a>5.2.1.1 Description
</h4>
<p>The <code>semantics</code> element is the container element that
associates annotations with a MathML expression. The
<code>semantics</code> element has as its first child the expression to be
annotated. Any MathML expression may appear as the first child of the
<code>semantics</code> element. Subsequent <code>annotation</code> and
<code>annotation-xml</code> children enclose the annotations.
An annotation represented in XML is enclosed in an
<code>annotation-xml</code> element. An annotation represented
in character data is enclosed in an <code>annotation</code> element.
</p>
<p>As noted above, the <code>semantics</code> element is considered to be
both a presentation element and a content element, since it can act
as either, depending on its content. Consequently, all MathML
processors should process the <code>semantics</code> element, even if they
process only presentation markup or only content markup.
</p>
<p>The default rendering of a <code>semantics</code> element is the default
rendering of its first child. A renderer may use the information contained
in the annotations to customize its rendering of the annotated element.
</p>
<table>
<tr>
<td><pre class="mathml"><semantics>
<mrow>
<mrow>
<mi>sin</mi>
<mo>&#x2061;<span style="color:#999900"><!--FUNCTION APPLICATION--></span></mo>
<mfenced><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
<annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content">
<apply>
<plus/>
<apply><sin/><ci>x</ci></apply>
<cn>5</cn>
</apply>
</annotation-xml>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
</semantics></pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics>
<mrow>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mfenced><mi>x</mi></mfenced>
</mrow>
<mo>+</mo>
<mn>5</mn>
</mrow>
<annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content">
<apply>
<plus/>
<apply><sin/><ci>x</ci></apply>
<cn>5</cn>
</apply>
</annotation-xml>
<annotation encoding="application/x-tex">
\sin x + 5
</annotation>
</semantics></math>
</td>
</tr>
</table>
</div>
<div class="div4">
<h4><a id="id.5.2.1.2"></a>5.2.1.2 Attributes
</h4>
<table border="1" class="attributes">
<thead>
<tr>
<th>Name</th>
<th>values</th>
<th>default</th>
</tr>
</thead>
<tbody>
<tr>
<td rowspan="2" class="attname">definitionURL</td>
<td><a href="chapter2.xml.html#type.uri"><em>URI</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The location of an external source for semantic information</td>
</tr>
<tr>
<td rowspan="2" class="attname">encoding</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The encoding of the external semantic information</td>
</tr>
</tbody>
</table>
<p>The <code>semantics</code> element takes the <code>definitionURL</code> and
<code>encoding</code> attributes, which reference an external source for some
or all of the semantic information for the annotated element, as modified
by the annotation. The use of these attributes on the <code>semantics</code>
element is deprecated in MathML3.
</p>
</div>
</div>
<div class="div3">
<h3><a id="mixing.elements.annotation"></a>5.2.2 The <code>annotation</code> element
</h3>
<div class="div4">
<h4><a id="id.5.2.2.1"></a>5.2.2.1 Description
</h4>
<p>The <code>annotation</code> element is the container element for a semantic
annotation whose representation is parsed character data in a non-XML
format. The <code>annotation</code> element should contain the character
data for the annotation, and should not contain XML markup elements.
If the annotation contains one of the XML reserved characters
<code>&</code>, <code><</code>, <code>></code>,
<code>'</code>, or <code>"</code>, then these characters must
be encoded using an XML entity reference or an XML <code>CDATA</code>
section.
</p>
</div>
<div class="div4">
<h4><a id="id.5.2.2.2"></a>5.2.2.2 Attributes
</h4>
<table border="1" class="attributes">
<thead>
<tr>
<th>Name</th>
<th>values</th>
<th>default</th>
</tr>
</thead>
<tbody>
<tr>
<td rowspan="2" class="attname">definitionURL</td>
<td><a href="chapter2.xml.html#type.uri"><em>URI</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The location of the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">encoding</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The encoding of the semantic information in the annotation</td>
</tr>
<tr>
<td rowspan="2" class="attname">cd</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td>mathmlkeys</td>
</tr>
<tr>
<td colspan="2" class="attdesc">The content dictionary that contains the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">name</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td>alternate-representation</td>
</tr>
<tr>
<td colspan="2" class="attdesc">The name of the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">src</td>
<td><a href="chapter2.xml.html#type.uri"><em>URI</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The location of an external source for semantic information</td>
</tr>
</tbody>
</table>
<p>Taken together, the <code>cd</code> and <code>name</code> attributes
specify the annotation key symbol, which identifies the relationship
between the annotated element and the annotation, as described in
<a href="chapter5.xml.html#mixing.annotation.elements">Section 5.1.1 Annotation elements</a>. The <code>definitionURL</code>
attribute provides an alternate way to reference the annotation key
symbol as a single attribute. If none of these attributes are present,
the annotation key symbol is the symbol
<a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a>
from the <a href="http://www.openmath.org/cd/mathmlkeys.xhtml">mathmlkeys</a> content dictionary.
</p>
<p>The <code>encoding</code> attribute describes the content type of the
annotation. The value of the <code>encoding</code> attribute may contain
a media type that identifies the data format for the encoding data. For
data formats that do not have an associated media type, implementors may
choose a self-describing character string to identify their content type.
</p>
<p>The <code>src</code> attribute provides a mechanism to attach external
entities as annotations on MathML expressions.
</p><pre class="mathml-fragment">
<annotation encoding="image/png" src="333/formula56.png"/></pre><p>The <code>annotation</code> element is a semantic mapping element that may
only be used as a child of the <code>semantics</code> element. While there is
no default rendering for the <code>annotation</code> element, a renderer may
use the information contained in an annotation to customize its rendering
of the annotated element.
</p>
</div>
</div>
<div class="div3">
<h3><a id="mixing.elements.annotation.xml"></a>5.2.3 The <code>annotation-xml</code> element
</h3>
<div class="div4">
<h4><a id="id.5.2.3.1"></a>5.2.3.1 Description
</h4>
<p>The <code>annotation-xml</code> element is the container element for a
semantic annotation whose representation is structured markup in an XML
format. The <code>annotation-xml</code> element should contain the markup
elements, attributes, and character data for the annotation.
</p>
</div>
<div class="div4">
<h4><a id="id.5.2.3.2"></a>5.2.3.2 Attributes
</h4>
<table border="1" class="attributes">
<thead>
<tr>
<th>Name</th>
<th>values</th>
<th>default</th>
</tr>
</thead>
<tbody>
<tr>
<td rowspan="2" class="attname">definitionURL</td>
<td><a href="chapter2.xml.html#type.uri"><em>URI</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The location of the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">encoding</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The encoding of the semantic information in the annotation</td>
</tr>
<tr>
<td rowspan="2" class="attname">cd</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td>mathmlkeys</td>
</tr>
<tr>
<td colspan="2" class="attdesc">The content dictionary that contains the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">name</td>
<td><a href="chapter2.xml.html#type.string"><em>string</em></a></td>
<td>alternate-representation</td>
</tr>
<tr>
<td colspan="2" class="attdesc">The name of the annotation key symbol</td>
</tr>
<tr>
<td rowspan="2" class="attname">src</td>
<td><a href="chapter2.xml.html#type.uri"><em>URI</em></a></td>
<td><em>none</em></td>
</tr>
<tr>
<td colspan="2" class="attdesc">The location of an external source for semantic information</td>
</tr>
</tbody>
</table>
<p>Taken together, the <code>cd</code> and <code>name</code> attributes
specify the annotation key symbol, which identifies the relationship
between the annotated element and the annotation, as described in
<a href="chapter5.xml.html#mixing.annotation.elements">Section 5.1.1 Annotation elements</a>. The <code>definitionURL</code>
attribute provides an alternate way to reference the annotation key
symbol as a single attribute. If none of these attributes are present,
the annotation key symbol is the symbol
<a href="http://www.openmath.org/cd/mathmlkeys.xhtml#alternate-representation">alternate-representation</a>
from the <a href="http://www.openmath.org/cd/mathmlkeys.xhtml">mathmlkeys</a> content dictionary.
</p>
<p>The <code>encoding</code> attribute describes the content type of the
annotation. The value of the <code>encoding</code> attribute may contain
a media type that identifies the data format for the encoding data. For
data formats that do not have an associated media type, implementors may
choose a self-describing character string to identify their content type.
In particular, as described above and in <a href="chapter6.xml.html#encoding-names">Section 6.2.3 Names of MathML Encodings</a>, MathML specifies
<code>MathML</code>, <code>MathML-Presentation</code>, and
<code>MathML-Content</code> as predefined values for the
<code>encoding</code> attribute. Finally, The <code>src</code>
attribute provides a mechanism to attach external XML entities as
annotations on MathML expressions.
</p><pre class="mathml-fragment"><annotation-xml cd="mathmlkeys" name="contentequiv" encoding="MathML-Content">
<apply>
<plus/>
<apply><sin/><ci>x</ci></apply>
<cn>5</cn>
</apply>
</annotation-xml>
<annotation-xml encoding="application/openmath+xml">
<OMA xmlns="http://www.openmath.org/OpenMath">
<OMS cd="arith1" name="plus"/>
<OMA><OMS cd="transc1" name="sin"/><OMV name="x"/></OMA>
<OMI>5</OMI>
</OMA>
</annotation-xml></pre><p>When the annotation value is represented in an XML dialect other than MathML, the
namespace for the XML markup for the annotation should be identified by means of namespace
attributes and/or namespace prefixes on the annotation value. For instance:
</p><pre class="mathml-fragment">
<annotation-xml encoding="application/xhtml+xml">
<html xmlns="http://www.w3.org/1999/xhtml">
<head><title>E</title></head>
<body>
<p>The base of the natural logarithms, approximately 2.71828.</p>
</body>
</html>
</annotation-xml></pre><p>The <code>annotation-xml</code> element is a semantic mapping element that may only be used
as a child of the <code>semantics</code> element. While there is no default rendering for the
<code>annotation-xml</code> element, a renderer may use the information contained in an
annotation to customize its rendering of the annotated element.
</p>
</div>
</div>
</div>
<div class="div2">
<h2><a id="mixing.markup"></a>5.3 Combining Presentation and Content Markup
</h2>
<p>
Presentation markup encodes the <em>notational structure</em> of an expression.
Content markup encodes the <em>functional structure</em> of an expression. In certain
cases, a particular application of MathML may require a combination of both presentation
and content markup. This section describes specific constraints that govern the use of
presentation markup within content markup, and vice versa.
</p>
<div class="div3">
<h3><a id="mixing.pmincm"></a>5.3.1 Presentation Markup in Content Markup
</h3>
<p>
Presentation markup may be embedded within content markup so long as the
resulting expression retains an unambiguous function application structure.
Specifically, presentation markup may only appear in content markup
in three ways:
</p>
<ol type="1">
<li>
<p>within <code>ci</code> and <code>cn</code> token elements
</p>
</li>
<li>
<p>within the <code>csymbol</code> element
</p>
</li>
<li>
<p>within the <code>semantics</code> element
</p>
</li>
</ol>
<p>
Any other presentation markup occurring within content markup is a
MathML error. More detailed discussion of these three cases follows:
</p>
<dl>
<dt class="label">Presentation markup within token elements.</dt>
<dd>The token elements <code>ci</code> and <code>cn</code> are permitted to
contain any sequence of MathML characters (defined in <a href="chapter7.xml.html">Chapter 7 Characters, Entities and Fonts</a>)
and/or presentation elements. Contiguous blocks of MathML characters in
<code>ci</code> or <code>cn</code> elements are treated as if wrapped in
<code>mi</code> or <code>mn</code> elements, as appropriate, and the resulting
collection of presentation elements is rendered as if wrapped in an
implicit <code>mrow</code> element.
</dd>
<dt class="label">Presentation markup within the <code>csymbol</code> element.
</dt>
<dd>The <code>csymbol</code> element may contain either MathML characters
interspersed with presentation markup, or content markup. It is a MathML
error for a <code>csymbol</code> element to contain both presentation and
content elements. When the <code>csymbol</code> element contains
character data and presentation markup, the same rendering rules that apply
to the token elements <code>ci</code> and <code>cn</code> should be used.
</dd>
<dt class="label">Presentation markup within the <code>semantics</code> element.
</dt>
<dd>One of the main purposes of the <code>semantics</code> 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 <code>semantics</code> element. The meaning of this
wrapped expression should be indicated by one or more annotation elements
also contained in the <code>semantics</code> element.
</dd>
</dl>
</div>
<div class="div3">
<h3><a id="mixing.cminpm"></a>5.3.2 Content Markup in Presentation Markup
</h3>
<p>
Content markup may be embedded within presentation markup so long as the
resulting expression has an unambiguous rendering. That is, it must be
possible, in principle, to produce a presentation markup fragment for
each content markup fragment that appears in the combined expression.
The replacement of each content markup fragment by its corresponding
presentation markup should produce a well-formed presentation markup
expression. A presentation engine should then be able to process this
presentation expression without reference to the content markup bits
included in the original expression.
</p>
<p>
In general, this constraint means that each embedded content expression
must be well-formed, as a content expression, and must be able to stand
alone outside the context of any containing content markup element. As
a result, the following content elements may not appear as an immediate
child of a presentation element:
<code>annotation</code>, <code>annotation-xml</code>,
<code>bvar</code>, <code>condition</code>, <code>degree</code>,
<code>logbase</code>, <code>lowlimit</code>, <code>uplimit</code>.
</p>
<p>
In addition, within presentation markup, content markup may not appear
within presentation token elements.
</p>
</div>
</div>
<div class="div2">
<h2><a id="mixing.parallel"></a>5.4 Parallel Markup
</h2>
<p>
Some applications are able to use <em>both</em> presentation
and content information. <em>Parallel markup</em> is a way to
combine two or more markup trees for the same mathematical expression.
Parallel markup is achieved with the <code>semantics</code> element.
Parallel markup for an expression may appear on its own, or as part
of a larger content or presentation tree.
</p>
<div class="div3">
<h3><a id="mixing.top.level"></a>5.4.1 Top-level Parallel Markup
</h3>
<p>
In many cases, the goal is to provide presentation markup and content
markup for a mathematical expression as a whole.
A single <code>semantics</code> element may be used to pair two markup trees,
where one child element provides the presentation markup, and the
other child element provides the content markup.
</p>
<p>
The following example encodes the Boolean arithmetic expression
(<var>a</var>+<var>b</var>)(<var>c</var>+<var>d</var>) in this way.
</p>
<table>
<tr>
<td><pre class="mathml">
<semantics>
<mrow>
<mrow><mo>(</mo><mi>a</mi> <mo>+</mo> <mi>b</mi><mo>)</mo></mrow>
<mo>&#x2062;<span style="color:#999900"><!--INVISIBLE TIMES--></span></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></pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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></math>
</td>
</tr>
</table>
<p>
Note that the above markup annotates the presentation markup as
the first child element, with the content markup as part of the
<code>annotation-xml</code> element. An equivalent form could be given
that annotates the content markup as the first child element, with
the presentation markup as part of the <code>annotation-xml</code> element.
</p>
</div>
<div class="div3">
<h3><a id="mixing.cr"></a>5.4.2 Parallel Markup via Cross-References
</h3>
<p>
To accommodate applications that must process sub-expressions of large
objects, MathML supports cross-references between the branches of a
<code>semantics</code> element to identify corresponding sub-structures.
These cross-references are established by the use of the <code>id</code>
and <code>xref</code> attributes within a <code>semantics</code> element.
This application of the <code>id</code> and <code>xref</code> attributes within
a <code>semantics</code> element should be viewed as best practice to enable
a recipient to select arbitrary sub-expressions in each alternative
branch of a <code>semantics</code> element. The <code>id</code> and
<code>xref</code> attributes may be placed on MathML elements of any type.
</p>
<p>
The following example demonstrates cross-references for the
Boolean arithmetic expression
(<var>a</var>+<var>b</var>)(<var>c</var>+<var>d</var>).
</p>
<table>
<tr>
<td><pre class="mathml">
<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">&#x2062;<span style="color:#999900"><!--INVISIBLE TIMES--></span></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>
</pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
</td>
</tr>
</table>
<p>
An <code>id</code> attribute and associated <code>xref</code> attributes
that appear within the same <code>semantics</code> element establish the
cross-references between corresponding sub-expressions.
</p>
<p>
For parallel markup, all of the <code>id</code> attributes referenced by any <code>xref</code>
attribute should be in the <em>same</em> branch of an enclosing
<code>semantics</code> element. This constraint guarantees that the
cross-references do not create unintentional cycles. This restriction
does <em>not</em> exclude the use of <code>id</code> attributes within
other branches of the enclosing <code>semantics</code> element. It does,
however, exclude references to these other <code>id</code> attributes
originating from the same <code>semantics</code> element.
</p>
<p>
There is no restriction on which branch of the <code>semantics</code> element
may contain the destination <code>id</code> 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 <code>id</code> attributes on the branch with
the finest-grained node structure. Then all of the other branches will have
<code>xref</code> attributes to some subset of the <code>id</code> attributes.
</p>
<p>
In absence of other criteria, the first branch of the <code>semantics</code>
element is a sensible choice to contain the <code>id</code> attributes.
Applications that add or remove annotations will then not have to re-assign
these attributes as the annotations change.
</p>
<p>
In general, the use of <code>id</code> and <code>xref</code> 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 <code>semantics</code> element.
It is equally feasible to select a subtree in any branch and
to recover the corresponding subtrees of the other branches.
</p>
<p>
Parallel markup with cross-references may be used in any XML-encoded branch
of the semantic annotations, as shown by the following example where
the Boolean expression of the previous section is
annotated with OpenMath markup that includes cross-references:
</p>
<table>
<tr>
<td><pre class="mathml">
<semantics>
<mrow id="EE">
<mrow id="EE.1">
<mo id="EE.1.1">(</mo>
<mi id="EE.1.2">a</mi>
<mo id="EE.1.3">+</mo>
<mi id="EE.1.4">b</mi>
<mo id="EE.1.5">)</mo>
</mrow>
<mo id="EE.2">&#x2062;<span style="color:#999900"><!--INVISIBLE TIMES--></span></mo>
<mrow id="EE.3">
<mo id="EE.3.1">(</mo>
<mi id="EE.3.2">c</mi>
<mo id="EE.3.3">+</mo>
<mi id="EE.3.4">d</mi>
<mo id="EE.3.5">)</mo>
</mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply xref="EE">
<and xref="EE.2"/>
<apply xref="EE.1">
<xor xref="EE.1.3"/><ci xref="EE.1.2">a</ci><ci xref="EE.1.4">b</ci>
</apply>
<apply xref="EE.3">
<xor xref="EE.3.3"/><ci xref="EE.3.2">c</ci><ci xref="EE.3.4">d</ci>
</apply>
</apply>
</annotation-xml>
<annotation-xml encoding="application/openmath+xml">
<om:OMA xmlns:om="http://www.openmath.org/OpenMath" href="EE">
<om:OMS name="and" cd="logic1" href="EE.2"/>
<om:OMA href="EE.1">
<om:OMS name="xor" cd="logic1" href="EE.1.3"/>
<om:OMV name="a" href="EE.1.2"/>
<om:OMV name="b" href="EE.1.4"/>
</om:OMA>
<om:OMA href="EE.3">
<om:OMS name="xor" cd="logic1" href="EE.3.3"/>
<om:OMV name="c" href="EE.3.2"/>
<om:OMV name="d" href="EE.3.4"/>
</om:OMA>
</om:OMA>
</annotation-xml>
</semantics>
</pre></td>
<td class="mathml-render" valign="middle">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow id="EE">
<mrow id="EE.1">
<mo id="EE.1.1">(</mo>
<mi id="EE.1.2">a</mi>
<mo id="EE.1.3">+</mo>
<mi id="EE.1.4">b</mi>
<mo id="EE.1.5">)</mo>
</mrow>
<mo id="EE.2">⁢</mo>
<mrow id="EE.3">
<mo id="EE.3.1">(</mo>
<mi id="EE.3.2">c</mi>
<mo id="EE.3.3">+</mo>
<mi id="EE.3.4">d</mi>
<mo id="EE.3.5">)</mo>
</mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<apply xref="EE">
<and xref="EE.2"/>
<apply xref="EE.1">
<xor xref="EE.1.3"/><ci xref="EE.1.2">a</ci><ci xref="EE.1.4">b</ci>
</apply>
<apply xref="EE.3">
<xor xref="EE.3.3"/><ci xref="EE.3.2">c</ci><ci xref="EE.3.4">d</ci>
</apply>
</apply>
</annotation-xml>
<annotation-xml encoding="application/openmath+xml">
<om:OMA xmlns:om="http://www.openmath.org/OpenMath" href="EE">
<om:OMS name="and" cd="logic1" href="EE.2"/>
<om:OMA href="EE.1">
<om:OMS name="xor" cd="logic1" href="EE.1.3"/>
<om:OMV name="a" href="EE.1.2"/>
<om:OMV name="b" href="EE.1.4"/>
</om:OMA>
<om:OMA href="EE.3">
<om:OMS name="xor" cd="logic1" href="EE.3.3"/>
<om:OMV name="c" href="EE.3.2"/>
<om:OMV name="d" href="EE.3.4"/>
</om:OMA>
</om:OMA>
</annotation-xml>
</semantics>
</math>
</td>
</tr>
</table>
<p>
Here <code>OMA</code>, <code>OMS</code>
and <code>OMV</code> are elements defined in the OpenMath
standard for representing application, symbol, and variable, respectively.
The references from the OpenMath annotation are given by the
<code>href</code> attributes.
</p>
</div>
</div>
</div>
<div class="minitoc">
Overview: <a href="Overview.xml.html">Mathematical Markup Language (MathML) Version 3.0</a><br class="html-compat" />
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