Overview: Mathematical Markup Language (MathML) Version 2.0
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2 MathML Fundamentals
2.1 MathML Overview
2.1.1 Taxonomy of MathML Elements
2.1.2 Presentation Markup
2.1.3 Content Markup
2.1.4 Mixing Presentation and Content
2.2 Some MathML Examples
2.2.1 Presentation Examples
2.2.2 Content Examples
2.2.3 Mixed Markup Examples
2.3 MathML Syntax and Grammar
2.3.1 MathML Syntax and Grammar
2.3.2 An XML Syntax Primer
2.3.3 Children versus Arguments
2.3.4 MathML Attribute Values
2.3.5 Attributes Shared by all MathML Elements
2.3.6 Collapsing Whitespace in Input
This chapter introduces the basic ideas of MathML. The first section describes the overall design of MathML. The second section presents a number of motivating examples, to give the reader something concrete to refer to while reading subsequent chapters of the MathML Specification. The final section describes basic features of the MathML syntax and grammar, which apply to all MathML markup. In particular, Section 2.3 [MathML Syntax and Grammar] should be read before Chapter 3 [Presentation Markup], Chapter 4 [Content Markup] and Chapter 5 [Combining Presentation and Content Markup].
A fundamental challenge in defining a markup language for mathematics on the Web is reconciling the need to encode both the presentation of a mathematical notation and the content of the mathematical idea or object which it represents.
The relationship between a mathematical notation and a mathematical idea is subtle and deep. On a formal level, the results of mathematical logic raise unsettling questions about the correspondence between systems of symbolic logic and the phenomena they model. At a more intuitive level, anyone who uses mathematical notation knows the difference that a good choice of notation can make; the symbolic structure of the notation suggests the logical structure. For example, the Leibniz notation for derivatives `suggests' the chain rule of calculus through the symbolic cancellation of fractions: .
Mathematicians and teachers intuitively understand this very well; part of their expertise lies in choosing notation that emphasizes key aspects of a problem while hiding or diminishing extraneous aspects. It is commonplace in mathematics and science to write one thing when strictly technically something else is meant, because long experience shows this actually communicates the idea better at some higher level than rigorous detail.
In many other settings, though, mathematical notation is used to encode the full, precise meaning of a mathematical object. Mathematical notation is capable of prodigious rigor, and when used carefully, it can be virtually free of ambiguity. Moreover, it is precisely this lack of ambiguity which makes it possible to describe mathematical objects so that they can be used by software applications such as computer algebra systems and voice renderers. In situations where such interapplication communication is of paramount importance, the nuances of visual presentation generally play a minimal role.
MathML allows authors to encode both the notation which represents a mathematical object and the mathematical structure of the object itself. Moreover, authors can mix both kinds of encoding in order to specify both the presentation and content of a mathematical idea. The remainder of this section gives a basic overview of how MathML can be used in each of these ways.
All MathML elements fall into one of three categories: presentation elements, content elements and interface elements. Each of these categories is described in detail in Chapter 3 [Presentation Markup], Chapter 4 [Content Markup] and Chapter 7 [The MathML Interface], respectively.
Presentation elements describe mathematical notation's visually oriented
twodimensional structure. Typical examples are the mrow
element, which is usually employed to indicate a
horizontal row of pieces of expressions, and the msup
element, which is used to mark up a base
expression and a superscript to it. As a general rule, each presentation
element corresponds to a single kind of `???' such as digits,
letters, or other symbol characters.
Although this particular example involves mathematical notation, and hence presentation markup, the same observation about decomposition applies equally well to abstract mathematical objects, and hence to content markup. For example, in the context of content markup our superscript example would typically be denoted by an exponentiation operation that would require two operands: a `base' and an `exponent'. This is no coincidence, since as a general rule, mathematical notation's layout closely follows the logical structure of the underlying mathematical objects.
The recursive nature of mathematical objects and notation is strongly reflected in MathML markup. In use, most presentation or content elements contain some number of other MathML elements corresponding to the constituent pieces out of which the original object is recursively built. The original schema is commonly called the parent schema, and the constituent pieces are called child schemata. More generally, MathML expressions can be regarded as trees, where each node corresponds to a MathML element, the branches under a `parent' node correspond to its `children', and the leaves in the tree correspond to atomic notation or content units such as numbers, characters, etc.
Most leaf nodes in a MathML expression tree are either canonically
empty elements with no bodies, or token elements.
Canonically empty elements represent symbols directly in MathML, for
example, the content element <plus/>
does
this. MathML token elements are the only MathML elements permitted to
contain MathML character data. The MathML character data may consist of
Unicode characters and MathML <mchar/>
elements.
These <mchar/>
elements, such as <mchar name="alpha" />
and <mchar name="rightarrow" />
, typically
denote Unicode characters not in ASCII code and the name
attribute carries the information as to which
special symbol is being represented. The <mchar/>
construction supersedes the use of special MathML entities such as α
to encode special symbols specified in MathML 1
for compatibility with general XML mechanisms. A third kind of leaf node
permitted in MathML is the annotation
element,
which is used to hold data which is not in MathML format.
The most important presentation token elements are mi
, mn
and mo
for representing identifiers, numbers and operators
respectively. Typically a renderer will employ slightly different
typesetting styles for each of these kinds of character data: numbers are
usually in upright font, identifiers in italics, and operators have extra
space around them. In content markup, there are only three tokens, ci
, cn
and csymbol
, for identifiers, numbers and new symbols
introduced in the document itself, respectively. In content markup,
separate elements are provided for commonly used functions and
operators. The fn
element is provided for
userdefined extensions to the base set.
In terms of markup, most MathML elements are denoted by
start tag and an end tag, which enclose the
markup for their contents. In the case of tokens, the content is character
data, and in most other cases, the content is the markup for child
elements. A third category of elements, called canonically empty elements,
don't require any contents, and denoted by a single tag of the form <name/>
. An example of this kind of markup is <plus/>
in content markup.
Returning to the example of (a + b)^{2}, we can now see how the principles discussed above play out in practice. One form of presentation markup for this example is:
<msup> <mfenced> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfenced> <mn>2</mn> </msup>
This example demonstrates a number of presentation elements. The first
element, one that is used a great deal is mrow
.
This element is used to denote a row of horizontally aligned material. The
material contained between the <mrow>
and </mrow>
tags is considered to be an argument to the mrow
element. Thus the whole expression here is
contained in an mrow
element. As previously noted,
almost all mathematical expressions decompose into subexpressions. These
subexpressions can can, in turn, also be contained in an mrow
element. For example, a+b is also contained in an
mrow
.
The mfenced
element is used to provide fences
(braces, brackets, and parentheses) around formula material. It defaults
to using parentheses.
Note the use of the mi
element for displaying
the variables a and b and the mo
element for
marking the + operator.
The msup
element is for expressions involving
superscripts and takes two arguments, in order, the base expression (here,
(a+b)) and the exponent expression (here, 2).
The content markup for the same example is:
<apply> <power/> <apply> <plus/> <ci>a</ci> <ci>b</ci> </apply> <cn>2</cn> </apply>
Here, the apply
content element means apply an
operation to an expression. In this example, the power
element (for exponentiation), which requires no
body, and the similar plus
element (for addition)
are both applied. Observe that both operators take two
arguments, the order being particularly significant in the case of the
power operator.
Note the use of the ci
element to denote the
variables a and b, and the cn
element for denoting
the number 2.
MathML presentation markup consists of about 30 elements which accept
over 50 attributes. Most of the elements correspond to layout
schemata, which contain other presentation elements. Each layout
schema corresponds to a twodimensional notational device, such as a
superscript or subscript, fraction or table. In addition, there are the
presentation token elements mi
, mn
and mo
introduced above, as
well as several other less commonly used token elements. The remaining few
presentation elements are empty elements, and are used mostly in connection
with alignment.
The layout schemata fall into several classes. One group of elements is
concerned with scripts, and contains elements such as msub
, munder
, and mmultiscripts
. Another group focuses on more general
layout and includes mrow
, mstyle
, and mfrac
. A third
group deals with tables. The maction
element is in
a category by itself, and allows coding of various kinds of actions on
notation, such as occur in an expression which toggles between two pieces
of notation.
An important feature of many layout schemata is that the order of child
schemata is significant. For example, the first child of an mfrac
element is the numerator and the second child is
the denominator. Since the order of child schemata is not enforced at the
XML level by the MathML DTD, the information added by ordering is only
available to a MathML processor, as opposed to a generic XML processor.
When we want to emphasize that a MathML element such as mfrac
requires children in a specific order, we will
refer to them as arguments, and think of the mfrac
element as a notational
`constructor'.
Content markup consists of about 100 elements accepting roughly a dozen
attributes. The majority of these elements are empty elements corresponding
to a wide variety of operators, relations and named functions. Examples of
this sort include partialdiff
, leq
and tan
. Others such as
matrix
and set
are used to
encode various mathematical data types, and a third, important category of
content elements such as apply
are used to apply
operations to expressions and also to make new mathematical objects from
others.
The apply
element is perhaps the single most
important content element. It is used to apply a function or operation to a
collection of arguments. The positions of the child schemata are again
significant, with the first child denoting the function to be applied, and
the remaining children denoting the arguments of the function in
order. Note that the apply
construct always uses
prefix notation, like the programming language LISP. In particular, even
binary operations like subtraction are marked up by applying a prefix
subtraction operator to two arguments. For example, a 
b would be marked up as
<apply> <minus/> <ci>a</ci> <ci>b</ci> </apply>
A number of functions and operations require one or more quantifiers to
be welldefined. For example, in addition to an integrand, a definite
integral must specify the limits of integration and the bound variable. For
this reason, there are several qualifier schemata such as bvar
and lowlimit
. They are
used with operators such as diff
and int
.
The declare
construct is especially important
for content markup that might be evaluated by a computer algebra system.
The declare
element provides a basic assignment
mechanism, where a variable can be declared to be of a certain type, with a
certain value.
Different kinds of markup will be found most appropriate for different kinds of tasks. Documents written before the worldwide web became important were most often intended only for visual communication of information, so that legacy data is probably best translated into pure presentation markup, since semantic information about what the author meant can only be guessed at heuristically. By contrast, some mathematical applications and pedagogicallyoriented authoring tools will likely choose to be entirely contentbased. The majority of applications fall somewhere in between these extremes. For these applications, the most appropriate markup is a mixture of both presentation and content markup.
The rules for mixing presentation and content markup derive from the general principle that mixed content should only be allowed in places where it makes sense. For content markup embedded in presentation markup this basically means that any content fragments should be semantically meaningful, and should not require additional arguments or quantifiers to be fully specified. For presentation markup embedded in content markup, this usually means that presentation markup must be contained in a content token element, so that it will be treated as an indivisible notational unit used as a variable or function name.
Another option is to use a semantics
element.
The semantics
element is used to bind MathML
expressions to various kinds of annotations. One common use for the semantics
element is to bind a piece of content markup
to some presentation markup as a semantic annotation. In this way, an
author can specify a nonstandard notation to be used when displaying a
particular content expression. Another use of the semantics
element is to bind some other kind of
semantic specification, such as an OpenMath expression, to a MathML
expression. In this way, the semantics
element can
be used to extend the scope of MathML content markup.
Notation: x^{2} + 4x + 4 = 0.
Markup:
<mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <mrow> <mn>4</mn> <mo>⁢</mo> <mi>x</mi> </mrow> <mo>+</mo> <mn>4</mn> </mrow> <mo>=</mo> <mn>0</mn> </mrow>
The mfrac and msqrt elements are used for generating fractions and square roots, respectively.
Note the use of nested mrow
elements to denote
terms, for example, the lefthand side of the equation functioning as an
operand of `='. Marking terms greatly facilitates spacing for
visual rendering, voice rendering, and line breaking. The InvisibleTimes
MathML character entity is used here to
indicate to a renderer that there are special spacing rules between the 4
and the x, and that the 4 and the x should not be broken onto separate
lines. In fact, this use of an entity is now explicitly deprecated in
favor of the use of <mchar name="InvisibleTimes" />
but was introduced
in MathML 1.0. The new version, which was mentioned above, will be used in
the examples below, and is explcitly discussed in Section 4.4.1 [Token Elements].
Notation: .
Markup:
<mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo></mo> <mi>b</mi> </mrow> <mo><mchar name="PlusMinus"/></mo> <msqrt> <mrow> <msup> <mi>b</mi> <mn>2</mn> </msup> <mo></mo> <mrow> <mn>4</mn> <mo><mchar name="InvisibleTimes"/></mo> <mi>a</mi> <mo><mchar name="InvisibleTimes"/></mo> <mi>c</mi> </mrow> </mrow> </msqrt> </mrow> <mrow> <mn>2</mn> <mo><mchar name="InvisibleTimes"/></mo> <mi>a</mi> </mrow> </mfrac> </mrow>
Notice that the `plus or minus' sign is given by a special
element used for specific symbol names <mchar name="PlusMinus" />
.
Then the same construction is used with <mchar name="InvisibleTimes" />
, instead of the old form in the previous
example. MathML provides a very comprehensive list of character names for
mathematical symbols. In addition to the mathematical symbols needed for
screen and print rendering, MathML provides symbols to facilitate audio
rendering. For audio rendering, it is important to be able to
automatically determine whether
<mrow> <mi>z</mi> <mfenced> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> </mfenced> </mrow>
should be read as `z times the quantity x plus
y' or `z of x plus y'. The markup
elements <mchar name="InvisibleTimes" />
and <mchar name="ApplyFunction" />
provide a way for authors to directly
encode the distinction for audio renderers. For instance, in the first
case <mchar name="InvisibleTimes" />
should be inserted after the line
containing the z. MathML also introduces entities like <mchar name="dd" />
which represents a `differential d' which
renders with slightly different spacing in print, and can be rendered as
`d' or `with respect to' in speech. Unless
content tags, or some other mechanism, are used to eliminate the ambiguity,
authors should always use these entities, in order to make their documents
more accessible.
Notation: .
Markup:
<mrow> <mi>A</mi> <mo>=</mo> <mfenced open="[" close="]"> <mtable> <mtr> <mtd><mi>x</mi></mtd> <mtd><mi>y</mi></mtd> </mtr> <mtr> <mtd><mi>z</mi></mtd> <mtd><mi>w</mi></mtd> </mtr> </mtable> </mfenced> </mrow>
Most elements have a number of attributes that control the details of their
screen and print rendering. For example, there are several attributes for
the mfenced
element that controls what delimiters
should be used at the beginning and the end of the grouped expression
above. The attributes for operator elements given using <mo>
are set to default values determined by a
dictionary. For the suggested MathML operator dictionary, see Appendix D [Operator Dictionary].
Notation: x^{2} + 4x + 4 = 0.
Markup:
<apply> <eq/> <apply> <plus/> <apply> <power/> <ci>x</ci> <cn>2</cn> </apply> <apply> <times/> <cn>4</cn> <ci>x</ci> </apply> <cn>4</cn> </apply> <cn>0</cn> </apply>
Note that the apply
element is used for
relations, operators and functions.
Notation: .
Markup:
<apply> <eq/> <ci>x</ci> <apply> <divide/> <apply> <fn><mo><mchar name='PlusMinus'/></mo></fn> <apply> <minus/> <ci>b</ci> </apply> <apply> <root/> <apply> <minus/> <apply> <power/> <ci>b</ci> <cn>2</cn> </apply> <apply> <times/> <cn>4</cn> <ci>a</ci> <ci>c</ci> </apply> </apply> <cn>2</cn> </apply> </apply> <apply> <times/> <cn>2</cn> <ci>a</ci> </apply> </apply> </apply>
MathML content markup does not directly contain an element for the
`plus or minus' operation. Therefore, we use the fn
element to declare that we want the presentation
markup for this operator to act as a content operator. This is a simple
example of how presentation and content markup can be mixed to extend
content markup.
Notation: .
Markup:
<apply> <eq/> <ci>A</ci> <matrix> <matrixrow> <ci>x</ci> <ci>y</ci> </matrixrow> <matrixrow> <ci>z</ci> <ci>w</ci> </matrixrow> </matrix> </apply>
Note that, by default, the rendering of the content element matrix
includes enclosing parentheses, so we need not
directly encode them. This is quite different from the presentation element
mtable
which may or may not refer to a matrix, and
hence requires explicit encoding of parentheses if they are desired.
Notation: .
Markup:
<semantics> <mrow> <msubsup> <mo><mchar name='int'/></mo> <mn>0</mn> <mi>t</mi> </msubsup> <mfrac> <mrow> <mo><mchar name='dd'/></mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <annotationxml encoding="MathMLContent"> <apply> <int/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><ci>t</ci></uplimit> <apply> <divide/> <cn>1</cn> <ci>x</ci> </apply> </apply> </annotationxml> </semantics>
In this example, we use the semantics
element to
provide a MathML content expression to serve as a `semantic
annotation' for a presentation expression. The semantics
element has as its first child the expression
being annotated, and the subsequent children are the annotations. There is
no restriction on the kind of annotation that can be attached using the semantics
element. For example, one might give a T_{E}X
encoding, or computer algebra input in an annotation. The type of
annotation is specified by the encoding
attribute
and the annotation
and annotationxml
elements.
Another common use of the semantics
element arises when one wants to use a content coding, and
provide a suggestion for its presentation. In such a case,
applied to the formula above we would have the markup:
<semantics> <apply> <int/> <bvar><ci>x</ci></bvar> <lowlimit><cn>0</cn></lowlimit> <uplimit><ci>t</ci></uplimit> <apply> <divide/> <cn>1</cn> <ci>x</ci> </apply> </apply> <annotationxml encoding="MathMLPresentation"> <mrow> <msubsup> <mo><mchar name='int'/></mo> <mn>0</mn> <mi>t</mi> </msubsup> <mfrac> <mrow> <mo><mchar name='dd'/></mo> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> </annotationxml> </semantics>
This kind of annotation is useful when something other than the default rendering of the content encoding is desired. For example, by default, some renderers might layout the integrand something like `(1/x) dx'. Specifying that the integrand should by preference render as `dx/x' instead can be accomplished with the use of a MathML Presentation annotation as shown. Be aware, however, that renderers are not required to take into account information contained in annotations, and what use is made of them, if any, will depend on the renderer.
MathML is an application of XML, or Extensible Markup Language [Bray1998], and as such its syntax is governed by the rules of XML syntax, and its grammar is in part specified by a DTD, or Document Type Definition. In other words, the details of using tags, attributes, entity references and so on are defined in the XML language specification, and the details about MathML element and attribute names, which elements can be nested inside each other, and so on are specified in the MathML DTD. This is in Appendix A [Parsing MathML]
The W3C in seeking to increase the flexibility of the use of XML for the Web, and to encourage modularization of applications built with XML, has found that the basic form of a DTD is not ideally suited. Therefore a W3C Working Group was created to develop a specification for XML Schemas [XMLSchemas], which are specification documents that will eventually supersede DTDs. MathML 2.0 is consciously designed so that math may take advantage of the latest in the evolving Web technology. Thus there is to be a Schema for MathML. For further information on a MathML Schema see Appendix A [Parsing MathML] and the MathML Home Page ().
However, MathML also specifies some syntax and grammar rules in addition to the general rules it inherits as an XML application. These rules allow MathML to encode a great deal more information than would ordinarily be possible with pure XML, without introducing many more elements, and using a substantially more complex DTD or schema. A grammar for content markup expressions is given in Appendix B [Content Markup Validation Grammar]. Of course, one drawback to using MathML specific rules is that they are invisible to generic XML processors and validators.
There are basically two kinds of additional MathML grammar and syntax rules. One kind involves placing additional criteria on attribute values. For example, it is not possible in pure XML to require that an attribute value be a positive integer. The second kind of rule specifies more detailed restrictions on the child elements (for example on ordering) than are given in the DTD or even a schema. For example, it is not possible in XML to specify that the first child be interpreted one way, and the second in another.
The following sections discuss features both of XML syntax and grammar in general, and of MathML in particular. Throughout the remainder of the MathML specification, we will usually take care to distinguish between usage required by XML syntax and the MathML DTD (and schema) and usage required by MathML specific rules. However, we will frequently allude to `MathML errors' without identifying which part of the specification is being violated.
Since MathML is an application of XML, the MathML Specification uses the
terminology of XML to describe it. Briefly, XML data is composed of Unicode
characters (which include ordinary ASCII characters), `entity
references' (informally called `entities') such as <
which usually represent `extended
characters', and `elements' such as <mi
fontstyle="normal"> x </mi>
.
An element quite often encloses other XML data called its
`content', or `body', between a `start
tag' (sometimes called a `begin tag') and an `end
tag', much as in HTML. There are also `empty elements'
such as <plus/>
, whose start tag ends with
/>
to indicate that the element has no content or end
tag. The start tag can contain named parameters called
`attributes', such as fontstyle="normal"
in the
example above. For further details on XML, consult the XML specification
[Bray1998].
As XML is casesensitive, MathML element and attribute names are casesensitive. For reasons of legibility, the MathML defines them almost all in lowercase.
In formal discussions of XML markup a distinction is maintained between
an element, such as an mrow
element, and the tags
<mrow>
and </mrow>
marking
it. What is between the <mrow>
start tag and the </mrow>
end tag is the content or body of the mrow
element. An `empty element' such as
none
is defined to have no body and so has a single
tag of the form <none/>
. Usually, the distinction
between elements and tags will not be so finely drawn in this
specification. For instance, we will sometimes refer to the <mrow>
and <none/>
elements,
really meaning the elements whose tags these are, in order that references
to elements are visually distinguishable from references to attributes.
However, the words `element' and `tag' themselves
will be used strictly in accordance with XML terminology.
Many MathML elements require a specific number of child elements or attach additional meanings to children in certain positions. As noted above, these kinds of requirements are MathML specific, and cannot be given entirely using XML syntax and grammar. When the children of a given MathML element are subject to these kinds of additional conditions, we will often refer to them as arguments instead of merely as children in order to emphasize their MathML specific usage. Note that especially in Chapter 3 [Presentation Markup] the term `argument' is usually used in this technical sense, unless otherwise noted, and therefore refers to a child element.
In the detailed discussions of element syntax given with each element throughout the MathML specification, the number of required arguments and their order is implicitly indicated by giving names for the arguments at various positions. This information is also given for presentation elements in the table of argument requirements in Section 3.1.3 [Required Arguments], and for content elements in Appendix B [Content Markup Validation Grammar].
A few elements have other requirements on the number or type of arguments. These additional requirements are described together with the individual elements.
According to the XML language specification, attributes given to elements must have one of the forms
attributename = "value"
or
attributename = 'value'
where whitespace around the '=' is optional.
Attribute names are generally shown in a monospaced
font within descriptive text in this
specification, just as the monospaced
font is used
for examples.
An attribute's value, which in general in MathML can be a string of
arbitrary characters, must be surrounded by a pair of either double quotes
("
) or single quotes ('
). The kind of quotes
not used to surround the value may be included within it.
MathML uses a more complicated syntax for attribute values than the generic XML syntax required by the MathML DTD. These additional rules are intended for use by MathML applications, and it is a MathML error to violate them, though they cannot be enforced by XML processing. The MathML syntax of each attribute value is specified in the table of attributes provided with the description of each element, using a notation described below. In MathML applications these attribute values should be further processed as follows, unless otherwise specified: whitespace is to be ignored except to separate letter and digit sequences into individual words or numbers; and the numerical Unicode references (listed in Chapter 6 [Characters, Entities and Fonts]) which can be used within token elements to represent characters can be used to represent those characters in attribute values (whenever those characters would be permitted by that attribute value's syntax). Note that the use of entity references for most synbols is now deprecated in MathML 2. Thus the use of a numerical Unicode character reference is better, if the input encoding does not allow you to use it directly.
In particular, the characters "
, '
,
&
and <
can be included in MathML
attribute values (when permitted by the attribute value syntax) using the
entity references "
, '
,
&
and <
, respectively.
The MathML DTD provided in Appendix A [Parsing MathML] declares most
attribute value types as CDATA
strings. This permits increased
interoperability with existing SGML and XML software and allows extension
to the lists of predefined values. Similar sorts of considerations apply
with schemas.
To describe the MathMLspecific syntax of permissible attribute values, the following conventions and notations are used for most attributes in the present document.
Notation  What it matches 
number  decimal integer or rational number (a string of digits with one decimal point), optionally starting with '' 
unsignednumber  decimal integer or real number, no sign 
integer  decimal integer, optionally starting with '' 
positiveinteger  decimal integer, unsigned, not 0 
string  arbitrary string (always the entire attribute value) 
character  single nonwhitespace character, or MathML entity reference; whitespace separation is optional 
#rrggbb  RGB color value; the three pairs of hexadecimal digits in the example #5599dd define proportions of red, green and blue on a scale of x00 through xFF, which gives a strong sky blue. 
hunit  unit of horizontal length (allowable units are listed below) 
vunit  unit of vertical length (allowable units are listed below) 
cssfontfamily  explained in CSS subsection, below 
csscolorname  explained in CSS subsection, below 
other italicized words  explained in the text for each attribute 
form +  one or more instances of 'form' 
form *  zero or more instances of 'form' 
f1 f2 ... fn  one instance of each form, in sequence, perhaps separated by whitespace 
f1  f2  ...  fn  any one of the specified forms 
[ form ]  an optional instance of 'form' 
( form )  same as form 
word in plain text  that word, literally present in the attribute value (unless it is obviously part of an explanatory phrase) 
quoted symbol  that symbol, literally present in attribute value (e.g. "+" or '+') 
The order of precedence of the syntax notation operators is, from highest to lowest precedence:
A string can contain arbitrary characters which are
specifiable within XML CDATA
attribute values; it must use entity
references for certain characters, as described earlier. It can contain
XMLformat entity or character references for any of the characters listed
in Chapter 6 [Characters, Entities and Fonts]. No syntax rule in MathML includes a
string as only part of an attribute value, only as the entire
value.
A character is a single nonwhitespace Unicode character,
or a character entity reference, or an expression using the mchar
and giving a name character. Examples of this
last form are given by the hundreds in Chapter 6 [Characters, Entities and Fonts].
As a simple example, the permissible values of boolean attributes are
specified as true  false
, meaning that the entire attribute
value should be either true
or false
.
Adjacent keywords and numbers must be separated by whitespace in the
actual attribute values, except for unit identifiers (denoted by hunit
or vunit
syntax symbols)
following numbers. Whitespace is not otherwise required, but is permitted
between any of the tokens listed above, except (for compatibility with
CSS1) immediately before unit identifiers, between the '' signs and digits
of negative numbers, or between #
or rrggbb
and rgb
Numerical attribute values for dimensions that should depend upon the
current font can be given in fontrelated units, or in named absolute units
(described in a separate subsection below). Horizontal dimensions are
conventionally given in em
's, and vertical dimensions in
ex
's, by immediately following a number by one of the unit
identifiers em
or ex
. For
example, the horizontal spacing around an operator such as `+'
is conventionally given in em
s, though other units
can be used. Using fontrelated units is usually preferable to using
absolute units, since it allows renderings to grow or shrink in proportion
to the current font size.
For most numerical attributes, only those in a subset of the expressible values are sensible; values outside this subset are not errors, unless otherwise specified, but rather are rounded up or down (at the discretion of the renderer) to the closest value within the allowed subset. The set of allowed values may depend on the renderer, and is not specified by MathML.
If a numerical value within an attribute value syntax description is
declared to allow a minus sign (''), e.g. number
or
integer
, it is not a syntax error when one is
provided in cases where a negative value is not sensible. Instead, the
value should be handled by the processing application as described in the
preceding paragraph. An explicit plus sign ('+') is not allowed as part of
a numerical value except when it is specifically listed in the syntax (as a
quoted '+' or "+"), and its presence can change the meaning of the
attribute value (as documented with each attribute which permits it).
The symbols hunit
, vunit
, cssfontfamily
,
and csscolorname
are explained in the
following subsections.
Some attributes accept horizontal or vertical lengths as numbers
followed by a `unit identifier' (often just called a
`unit'). The syntax symbols hunit
and
vunit
refer to a unit for horizontal or vertical
length, respectively. The possible units and the lengths they refer to are
shown in the table below; they are the same for horizontal and vertical
lengths, but the syntax symbols are distinguished in attribute syntaxes as
a reminder of the direction each is used in.
The unit identifiers and meanings are taken from CSS1. However, the syntax of numbers followed by unit identifiers in MathML is not identical to the syntax of length values with units in CSS style sheets, since numbers in CSS cannot end with decimal points, and are allowed to start with '+' signs.
The possible horizontal or vertical units in MathML are:
Unit identifier  Unit description 
em  em (fontrelative unit traditionally used for horizontal lengths) 
ex  ex (fontrelative unit traditionally used for vertical lengths) 
px  pixels, or pixel size of the current display 
in  inches (1 inch = 2.54 centimeters) 
cm  centimeters 
mm  millimeters 
pt  points (1 point = 1/72 inch) 
pc  picas (1 pica = 12 points) 
%  percentage of default value 
The typesetting units em
and ex
are defined in Appendix F [Glossary], and
discussed further under `Additional notes' below.
%
is a `relative unit'; when an attribute
value is given as n%
(for any numerical value n
), the value being specified is the default value for
the property being controlled multiplied by n
divided by 100. The default value (or the way in which it is obtained, when
it is not constant) is listed in the table of attributes for each element,
and its meaning is described in the subsequent documentation about that
attribute. (The mpadded
element has its own syntax
for %
and does not allow it as a unit identifier.)
For consistency with CSS, length units in MathML are rarely
optional. When they are, the unit symbol is enclosed in square brackets in
the attribute syntax, following the number to which it applies,
e.g. number [ hunit ]
. The meaning of specifying no unit is
given in the documentation for each attribute; in general it is that the
number given is a multiplier for the default value of the attribute. (In
such cases, specifying the number nnn
without a unit
is equivalent to specifying the number nnn
times 100
followed by %
. For example, <mo maxsize="2"> (
</mo>
is equivalent to <mo maxsize="200%"> (
</mo>
.)
As a special exception (also consistent with CSS), a numerical value equal to 0 need not be followed by a unit identifier even if the syntax specified here requires one. In such cases, the unit identifier (or lack of one) would not matter, since 0 times any unit is 0.
For most attributes, the typical unit which would be used to describe
them in typesetting is chosen as the one used in that attribute's default
value in this specification; when a specific default value is not given,
the typical unit is usually mentioned in the syntax table or in the
documentation for that attribute. The most common units are em
or ex
. However, any unit can be used, unless otherwise
specified for a specific attribute.
Note that some attributes, e.g. framespacing
on
a <mtable>
, can contain more than one numerical
value, each followed by its own unit.
It is conventional to use the fontrelative unit ex
mainly
for vertical lengths, and em
mainly for horizontal lengths,
but this is not required. These units are relative to the font and fontsize
which would be used for rendering the element in whose attribute value they
are specified, which means they should be interpreted after
attributes such as fontfamily
and fontsize
are processed, if those occur on the same
element, since changing the current font or fontsize can change the length
of one of these units.
The definition of the length of each unit (but not the MathML syntax for
length values) is as specified in CSS1, except that if a font provides
specific values for em
and ex
which differ from
the values defined by CSS1 (the font size and `x'height
respectively), those values should be used.
Several MathML attributes, listed below, correspond closely to text rendering properties defined by Cascading Style Sheets, Level 1 (CSS1).
The names and acceptable values of these attributes have been aligned with the CSS1 recommendation where possible. In general, the MathML syntax for each attribute is intended to be a subset of the CSS syntax for the corresponding property. Differences of detail, where they exist, are explained with the documentation about each attribute, in the sections of this specification listed in the table above.
The syntax of certain attributes is partially specified, in the tables
of attribute syntax in this specification, using one of the symbols cssfontfamily
or csscolorname
,
as shown in the following table. These symbols refer to syntaxes from
other W3C Recommendations, and are explained in the sections of this
specification referred to in the table.
MathML attribute  CSS property  syntax symbol  MathML elements  refer to 
fontsize  fontsize    presentation tokens; mstyle 

fontweight  fontweight    presentation tokens; mstyle 

fontstyle  fontstyle    presentation tokens; mstyle 

fontfamily  fontfamily  cssfontfamily  presentation tokens; mstyle 

color  color  csscolorname  presentation tokens; mstyle 

background  background  csscolorname  mstyle 
See also Section 2.3.5 [Attributes Shared by all MathML Elements] below for a discussion of the class
, style
and id
attributes for use with style sheets.
CSS or analogous style sheets can specify changes to rendering
properties of selected MathML elements; the selection of the elements can
happen in various ways). Either the properties listed above, or other
MathML rendering attributes or properties supported by a style sheet
mechanism, can be affected, in principle, for any element. Since rendering
properties can also be changed by attributes on an element or automatically
it is necessary to specify the order in which changes from various sources
occur. An example of automatic adjustment is what happens for fontsize
, as explained in the discussion on scriptlevel
in Section 3.3.4 [Style Change (mstyle
)]. In the
case of `absolute' changes, i.e. setting a new property value
independent of the old value (as opposed to `relative'
changes, such as increments or multiplications by a factor), the absolute
change performed last will be the only absolute change which is effective,
so the sources of changes which should have the highest priority must be
processed last.
In the case of CSS1, the order of processing of changes from various sources which affect one MathML element's rendering properties should be as follows:
(first changes; lowest priority)
fontsize
in relation to scriptlevel
mentioned above; such changes will usually
be implemented by the parent element itself before it passes a set of
rendering properties to this element(last changes; highest priority)
Note that the order of the changes derived from CSS style sheets is specified by CSS itself. The following rationale is related only to the issue of where in this preexisting order the changes caused by explicit MathML attribute settings should be inserted.
Rationale: MathML rendering attributes are analogous to HTML rendering
attributes such as align
, which the CSS1 section on
cascading order specifies should be processed with the same priority.
Furthermore, this choice of priority permits readers, by declaring certain
CSS styles as `important', to decide which of their style
preferences should override explicit attribute settings in MathML. Since
MathML expressions, whether composed of `presentation' or
`content' elements, are primarily intended to convey meaning,
with their `graphic design' (if any) intended mainly to aid in
that purpose but not to be essential in it, it is likely that readers will
often want their own style preferences to have priority; the main exception
will be when a rendering attribute is intended to alter the meaning
conveyed by an expression, which is generally discouraged in the
presentation attributes of MathML.
Default values for MathML attributes are in general given along with the detailed descriptions of specific elements in the text. Default values shown in plain text in the tables of attributes for an element are literal (unless they are obviously explanatory phrases), but when italicized are descriptions of how default values can be computed.
Default values described as inherited are taken from the
rendering environment, as described under mstyle
,
or in some cases (described individually) from the values of other
attributes of surrounding elements, or from certain parts of those
values. The value used will always be one which could have been specified
explicitly, had it been known; it will never depend on the content or
attributes of the same element, only on its environment. (What it means
when used may, however, depend on those attributes or the content.)
Default values described as automatic should be computed by a MathML renderer in a way which will produce a highquality rendering; how to do this is not usually specified by the MathML specification. The value computed will always be one which could have been specified explicitly, had it been known, but it will usually depend on the element content and possibly on the rendering environment.
Other italicized descriptions of default values which appear in the tables of attributes are explained for each attribute individually.
The single or double quotes which are required around attribute values in an XML start tag are not shown in the tables of attribute value syntax for each element, but are shown around example attribute values in the text.
Note that, in general, there is no value which can be given explicitly
for a MathML attribute which will simulate the effect of not specifying the
attribute at all for attributes which are inherited or
automatic. Giving the words `inherited' or
`automatic' explicitly will not work, and is not generally
allowed. Furthermore, even for presentation attributes for which a
specific default value is documented here, the mstyle
element (Section 3.3.4 [Style Change (mstyle
)]) can be
used to change this for the elements it contains. Therefore, the MathML DTD
declares most presentation attribute default values as #IMPLIED
,
which prevents XML preprocessors from adding them with any specific default
value. This point of view is carried through to the MathML schema
In an XML DTD, allowed attribute values can be declared as general strings, or they can be constrained in various ways, either by enumerating the possible values, or by declaring them to be certain special data types. The choice of an XML attribute type affects the extent to which validity checks can be performed using a DTD.
The MathML DTD specifies formal XML attribute types for all MathML
attributqes, including enumerations of legitimate values in some cases. In
general, however, the MathML DTD is relatively permissive, frequently
declaring attribute values as strings; this is done to provide for
interoperability with SGML parsers while allowing multiple attributes on
one MathML element to accept the same values (such as true
and false
), and also to
allow extension to the lists of predefined values.
At the same time, even though an attribute value may be declared as a string in the DTD, only certain values are legitimate in MathML, as described above and in the rest of this specification. For example, many attributes expect numerical values. In the sections which follow, the allowed attribute values are described for each element. To determine when these constraints are actually enforced in the MathML DTD, consult Appendix A [Parsing MathML]. However, lack of enforcement of a requirement in the DTD does not imply that the requirement is not part of the MathML language itself, or that it will not be enforced by a particular MathML renderer. (See Section 7.2.2 [Handling of Errors] for a description of how MathML renderers should respond to MathML errors.)
Furthermore, the MathML DTD is provided for convenience; although it is intended to be fully compatible with the text of the specification, the text should be taken as definitive if there is a contradiction. (Any contradictions which may exist between various chapters of the text should be resolved by favoring Chapter 6 [Characters, Entities and Fonts] first, then Chapter 3 [Presentation Markup], Chapter 4 [Content Markup], then Section 2.3 [MathML Syntax and Grammar], and then other parts of the text.) For the MathML Schema the situation will be the same: the published Recommendation text takes precedence. Though this is what is intended to happen, there is a practical difficulty. If the system processing the MathML uses a validating parser, whether it be based on a DTD or on a Schema, the process will probably simply stop when it hits something held to be incorrect syntax, whether or not further MathML processing in full harmony with the specification would have processed the piece correctly.
In order to facilitate compatibility with Cascading Style Sheets,
Level 1 (CSS1), all MathML elements accept class
, style
, and id
attributes in addition to the attributes described
specifically for each element. MathML renderers not supporting CSS may
ignore these attributes. MathML specifies these attribute values as general
strings, even if stylesheet mechanisms have more restrictive syntaxes for
them. That is, any value for them is valid in MathML.
Renderers supporting CSS (or analogous style sheet mechanisms) may use these attributes to help determine which MathML elements should be subject to which style sheetinduced changes to various rendering properties. The properties that can be affected, and how these changes affect them, are discussed in Section 2.3.4.3 [CSScompatible attributes] above.
Every MathML element, because of a legacy from MathML 1.0, also accepts
the deprecated attribute other
(Section 7.2.3 [Attributes for unspecified data]) which was conceived for passing nonstandard
attributes without violating the MathML DTD. MathML renderers are only
required to process this attribute if they respond to any attributes which
are not standard in MathML. However, the use of other
is strongly deprecated when there are already
other ways within MathML of passing specific information.
See also Section 3.2.1 [Attributes common to token elements] for a list of MathML attributes which can be used on most presentation token elements.
MathML ignores whitespace occurring outside token elements. Nonwhitespace characters are not allowed there. Whitespace occurring within the content of token elements is `trimmed' from the ends, i.e. all whitespace at the beginning and end of the content is removed. Whitespace internal to content of MathML elements is `collapsed' canonically, i.e. each sequence of 1 or more whitespace characters is replaced with one space character (sometimes called a blank character).
In MathML, as in XML, `whitespace' means simple spaces,
tabs, newlines, or carriage returns, i.e. characters with hexadecimal
Unicode codes U+0020
, U+0009
, U+000a
, or
U+000d
, respectively.
For example, <mo> ( </mo>
is equivalent to
<mo>(</mo>
, and
<mtext> Theorem 1: </mtext>
is equivalent to
<mtext>Theorem 1:</mtext>
.
Authors wishing to encode whitespace characters at the start or end of
the content of a token, or in sequences other than a single space, without
having them ignored, must use
or other
`whitespace' nonmarking entities as described in Section 6.1.4 [NonMarking Characters]. For example, compare
<mtext> Theorem 1: </mtext>
with
<mtext> Theorem
 1: </mtext>
When the first example is rendered, there is no whitespace before `Theorem', one space between `Theorem' and `1:', and no whitespace after `1:'. In the second example, a single space is rendered before `Theorem', a new line is placed after `Theorem', two spaces are rendered before `1:', and there is no whitespace after the `1:'.
Note that the xml:space
attribute does not apply
in this situation since XML processors pass whitespace in tokens to a
MathML processor; it is the MathML processing rules which specify that
whitespace is trimmed and collapsed.
For whitespace occurring outside the content of the token elements mi
, mn
, mo
, ms
, mtext
, ci
, cn
and annotation
, an mspace
element should be used, as opposed to an mtext
element containing only `whitespace'
entities.
Overview: Mathematical Markup Language (MathML) Version 2.0
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