Issue update_fundamentals  wiki (member only) 

Update MathML Fundamentals  
The current chapter is still largely from MathML2, and needs much work in later drafts. The outline of the contents is settled but, to a certain extent, the tone to be set depends on details still being settled by the Working Group. 

Resolution  None recorded 
MathML is an application of [XML], or Extensible Markup Language, and as such its syntax is governed by the rules of XML syntax. XML syntax is a notation for rooted labeled planar trees. Planarity means that the children of a node may be viewed as given a natural order. The grammar of MathML is in part specified by a DTD, or Document Type Definition or alternatively by an XML Schema. In other words, the details of using tags, attributes, entity references and the like 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 their possible relationships 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, found that the basic form of a DTD was not sufficiently flexible. Therefore, a W3C Working Group was created to develop a specification for XML Schemas [XMLSchemas], which are specification documents that will eventually supersede DTDs. Thus, there is a schema for MathML.
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.
Many MathML elements require a specific number of children or attach additional meanings to child elements 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.
A few elements have other requirements on the number or type of arguments. These additional requirements are described together with the individual elements.
An XML 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.
Attribute names are generally shown in a monospaced
font within descriptive text in this
specification, just as the monospaced
font is used
for examples.
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. When MathML applications process attribute values, whitespace is ignored except to separate letter and digit sequences into individual words or numbers. Attribute values may contain any MathML characters as specified in Chapter 6 Characters, Entities and Fonts permitted by the syntax restrictions for an attribute. Character data can be included directly in attribute values, or by using entity references as described in Section 6.2 Unicode Character Data.
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 extensions to the lists of predefined values. Similar sorts of considerations apply with XML 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 the CSS subsection below 
csscolorname  explained in the 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. See Chapter 6 Characters, Entities and Fonts for a full discussion of MathML characters. No syntax rule in MathML includes a string as only part of an attribute value; a string can only be the entire value.
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
CSS) immediately before unit identifiers, between the '' signs and digits
of negative numbers, or between #
and "rrggbb"
or "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 CSS. 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 size of a pixel in 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 D 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 font size
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 font size 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 CSS, except that if a font provides
specific values for em
and ex
which differ from
the values defined by CSS (the font size and "x"height
respectively), those values should be used.
Several MathML attributes, listed below, correspond closely to text rendering properties defined originally in [CSS1]. In MathML 1.01, the names and values of these attributes were aligned with the CSS Recommendation where possible. This was done so that renderers in CSS environments could query the environment for the corresponding property when determining the default values for the attributes.
Allowing style properties to be set both via MathML attributes and CSS style sheets has drawbacks. At a minimum, duplication is confusing, and at worst, it leads to the meaning of equations being inadvertently changed by documentwide CSS changes. For these reasons, these attributes have been deprecated. In their place, MathML 2.0 introduces four new mathematical style attributes. These attributes use logical values to better capture the abstract categories of letterlike symbols used in math, and afford a much cleaner separation between MathML and CSS. See Section 3.2.2 Mathematics style attributes common to token elements for more details.
For reference, a table showing the correspondence of the deprecated MathML 1.01 style attributes with their CSS counterparts is given below:
MathML attribute  CSS property  syntax symbol  MathML elements  refer to 

fontsize  fontsize    presentation tokens; mstyle 
Section 3.2.2 Mathematics style attributes common to token elements 
fontweight  fontweight    presentation tokens; mstyle 
Section 3.2.2 Mathematics style attributes common to token elements 
fontstyle  fontstyle    presentation tokens; mstyle 
Section 3.2.2 Mathematics style attributes common to token elements 
fontfamily  fontfamily  cssfontfamily  presentation tokens; mstyle 
Section 3.2.2 Mathematics style attributes common to token elements 
color  color  csscolorname  presentation tokens; mstyle 
Section 3.3.4 Style Change (mstyle) 
background  background  csscolorname  mstyle 
Section 3.3.4 Style Change (mstyle) 
See also Section 2.1.4 Attributes Shared by all MathML Elements below for a discussion
of the class
, style
and xml:id
attributes
for use with style sheets.
CSS or analogous style sheets can specify changes to rendering
properties of selected MathML elements. Since rendering properties
can also be changed by attributes on an element, or be changed automatically
by the renderer, it is necessary to specify the order in which changes
requested by various sources should 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 CSS, 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)
Automatic changes to properties or attributes based on the type of the
parent element, and this element's position in the parent, as for the
changes to 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
From a style sheet from the reader: styles which are not declared "important"
Explicit attribute settings on this MathML element
From a style sheet from the author: styles which are not declared "important"
From a style sheet from the author: styles which are declared "important"
From a style sheet from the reader: styles which are declared "important"
(last changes; highest priority)
Note that the order of the changes derived from CSS style sheets is specified by CSS itself (this is the order specified by CSS2). 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 CSS 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 attributes, 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 2.3.2 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 2.1 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 use with style sheet mechanisms such as
[XSLT] and [CSS2]
all MathML elements accept class
, style
, and xml: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 style sheet mechanisms have more restrictive syntaxes for
them. That is, any value for them is valid in MathML.
In order to facilitate compatibility with linking mechanisms, all
MathML elements accept the xlink:href
attribute.
All MathML elements also accept the xref
attribute for use in parallel markup (Section 5.5 Parallel Markup). The xml:id
is also used
in this context.
Every MathML element, because of a legacy from MathML 1.0, also
accepts the deprecated attribute
other
(Section 2.3.3 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 discouraged when there are already other ways
within MathML of passing specific information.
See also Section 3.2.2 Mathematics style 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 6.6 NonMarking Characters. For example, compare
<mtext> Theorem 1: </mtext>
with
<mtext>  <!NOBREAK SPACE>Theorem  <!NOBREAK SPACE>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", 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.
Issue update_interface  wiki (member only) 

Update Interface  
The current section needs continuing and updating further in later drafts. 

Resolution  None recorded 
To be effective, MathML must work well with a wide variety of renderers, processors, translators and editors. This section raises some of the interface issues involved in generating and rendering MathML. Since MathML exists primarily to encode mathematics in Web documents, perhaps the most important interface issues are related to embedding MathML in [HTML4] and [XHTML], and in a newer HTML5 when it appears.
There are three kinds of interface issues that arise in embedding MathML in other XML documents. First, MathML must be semantically integrated. MathML markup must be recognized as valid embedded XML content, and not as an error. This could be seen as primarily a question of managing namespaces in XML [Namespaces]. However, the implementation of XML namespaces and their management has not been well supported by recent commercial software. So there have grown up other ways of dealing with 'foreign content' in an XML document which is viewed as of a particular type. The Compound Document Formats Working Group (CDF WG) of the W3C has been grappling with the questions of putting together XML vocabularies and has been defining ways to do so for particular combinations of vocabularies. Their initial success has been with specifying profiles for combining XHTML and SVG, with special attention paid to the needs of mobile phone technology. The W3C Math WG is working with the CDF WG to define profiles for full scientific documents involving XHTML for text, MathML for equations and SVG for diagrams and images.
Second, in the case of HTML/XHTML, MathML rendering must be integrated with browser software. Some browsers already implement MathML rendering natively, and one can expect more browsers will do so in the future. At the same time, other browsers have developed infrastructure to facilitate the rendering of MathML and other embedded XML content by thirdparty software or other builtin technology. Examples of this builtin technology are the sophisticated CSS rendering engines now available, and the powerful implementations of ECMAscript (or JavaScript) that are becoming common. Using these browserspecific mechanisms generally requires additional interface markup of some sort to activate them. In the case of CSS, there is a special restricted form of MathML3 tailored for use with presentday CSS, up to CSS2.1, which is specified in "A MathML for CSS profile" [MathMLforCSS]. This does not offer the full expressiveness afforded by MathML3 but provides a portable simpler form that can be rendered acceptably on the screen by modern CSS engines.
Third, other tools for generating and processing MathML must be able to intercommunicate. A number of MathML tools have been or are being developed, including editors, translators, computer algebra systems, and other scientific software. However, since MathML expressions tend to be lengthy, and prone to error when entered by hand, special emphasis must be given to ensuring that MathML can be easily generated by userfriendly conversion and authoring tools, and that these tools work together in a dependable, platform and vendor independent way.
To be continued ...
Information is increasingly generated, processed and rendered by software tools. The exponential growth of the Web is fueling the development of advanced systems for automatically searching, categorizing, and interconnecting information. Thus, although MathML can be written by hand and read by humans, the future of MathML is largely tied to the ability to process it with software tools.
There are many different kinds of MathML processors: editors for authoring MathML expressions, translators for converting to and from other encodings, validators for checking MathML expressions, computation engines that evaluate, manipulate or compare MathML expressions, and rendering engines that produce visual, aural or tactile representations of mathematical notation. What it means to support MathML varies widely between applications. For example, the issues that arise with a validating parser are very different from those for a equation editor.
In this section, guidelines are given for describing different types of MathML support, and for quantifying the extent of MathML support in a given application. Developers, users and reviewers are encouraged to use these guidelines in characterizing products. The intention behind these guidelines is to facilitate reuse and interoperability between MathML applications by accurately characterizing their capabilities in quantifiable terms.
The W3C Math Working Group maintains MathML 2.0 Conformance Guidelines. Consult this document for future updates on conformance activities and resources.
A valid MathML expression is an XML construct determined by the MathML DTD together with the additional requirements given in this specification.
Define a "MathML processor" to mean any application that can accept, produce, or "roundtrip" a valid MathML expression. An example of an application that might roundtrip a MathML expression might be an editor that writes a new file even though no modifications are made.
Three forms of MathML conformance are specified:
A MathMLinputconformant processor must accept all valid MathML expressions, and faithfully translate all MathML expressions into applicationspecific form allowing native application operations to be performed.
A MathMLoutputconformant processor must generate valid MathML, faithfully representing all applicationspecific data.
A MathMLroundtripconformant processor must preserve MathML equivalence. Two MathML expressions are "equivalent" if and only if both expressions have the same interpretation (as stated by the MathML DTD and specification) under any circumstances, by any MathML processor. Equivalence on an elementbyelement basis is discussed elsewhere in this document.
Beyond the above definitions, the MathML specification makes no demands of individual processors. In order to guide developers, the MathML specification includes advisory material; for example, there are many suggested rendering rules throughout Chapter 3 Presentation Markup. However, in general, developers are given wide latitude in interpreting what kind of MathML implementation is meaningful for their own particular application.
To clarify the difference between conformance and interpretation of what is meaningful, consider some examples:
In order to be MathMLinputconformant, a validating parser needs only to accept expressions, and return "true" for expressions that are valid MathML. In particular, it need not render or interpret the MathML expressions at all.
A MathML computeralgebra interface based on content markup might choose to ignore all presentation markup. Provided the interface accepts all valid MathML expressions including those containing presentation markup, it would be technically correct to characterize the application as MathMLinputconformant.
An equation editor might have an internal data representation that makes it easy to export some equations as MathML but not others. If the editor exports the simple equations as valid MathML, and merely displays an error message to the effect that conversion failed for the others, it is still technically MathMLoutputconformant.
As the previous examples show, to be useful, the concept of MathML conformance frequently involves a judgment about what parts of the language are meaningfully implemented, as opposed to parts that are merely processed in a technically correct way with respect to the definitions of conformance. This requires some mechanism for giving a quantitative statement about which parts of MathML are meaningfully implemented by a given application. To this end, the W3C Math Working Group has provided a test suite.
The test suite consists of a large number of MathML expressions categorized by markup category and dominant MathML element being tested. The existence of this test suite makes it possible, for example, to characterize quantitatively the hypothetical computer algebra interface mentioned above by saying that it is a MathMLinputconformant processor which meaningfully implements MathML content markup, including all of the expressions in the content markup section of the test suite.
Developers who choose not to implement parts of the MathML specification in a meaningful way are encouraged to itemize the parts they leave out by referring to specific categories in the test suite.
For MathMLoutputconformant processors, there is also a MathML validator accessible over the Web. Developers of MathMLoutputconformant processors are encouraged to verify their output using this validator.
Customers of MathML applications who wish to verify claims as to which parts of the MathML specification are implemented by an application are encouraged to use the test suites as a part of their decision processes.
MathML 2.0 contains a number of MathML 1.x features which are now deprecated. The following points define what it means for a feature to be deprecated, and clarify the relation between deprecated features and MathML 2.0 conformance.
In order to be MathMLoutputconformant, authoring tools may not generate MathML markup containing deprecated features.
In order to be MathMLinputconformant, rendering/reading tools must support deprecated features if they are to be in conformance with MathML 1.x. They do not have to support deprecated features to be considered in conformance with MathML 2.0. However, all tools are encouraged to support the old forms as much as possible.
In order to be MathMLroundtripconformant, a processor need only preserve MathML equivalence on expressions containing no deprecated features.
MathML 2.0 defines three extension mechanisms: The mglyph
element provides a way of displaying glyphs for nonUnicode
characters, and glyph variants for existing Unicode characters; the
maction
element uses attributes from other namespaces to obtain
implementationspecific parameters; and content markup makes use of
the definitionURL
attribute to point to external
definitions of mathematical semantics.
These extension mechanisms are important because they provide a way
of encoding concepts that are beyond the scope of MathML 2.0, which
allows MathML to be used for exploring new ideas not yet susceptible
to standardization. However, as new ideas take hold, they may become
part of future standards. For example, an emerging character that
must be represented by an mglyph
element today may be
assigned a Unicode codepoint in the future. At that time,
representing the character directly by its Unicode codepoint would be
preferable.
Because the possibility of future obsolescence is inherent in the
use of extension mechanisms to facilitate the discussion of new ideas,
MathML 2.0 makes no conformance requirement concerning the use of
extension mechanisms, even when alternative standard markup is
available. For example, using an mglyph
element to represent
an 'x' is permitted. However, authors and implementors are
strongly encouraged to use standard markup whenever possible.
Similarly, maintainers of documents employing MathML 2.0 extension
mechanisms are encouraged to monitor relevant standards activity
(e.g. Unicode, OpenMath, etc) and update documents as more
standardized markup becomes available.
If a MathMLinputconformant application receives input containing one or
more elements with an illegal number or type of attributes or child
schemata, it should nonetheless attempt to render all the input in an
intelligible way, i.e. to render normally those parts of the input that
were valid, and to render error messages (rendered as if enclosed in
an merror
element) in place of invalid expressions.
MathMLoutputconformant applications such as editors and translators may
choose to generate merror
expressions to signal
errors in their input. This is usually preferable to generating valid, but
possibly erroneous, MathML.
The MathML attributes described in the MathML specification are necessary for presentation and content markup. Ideally, the MathML attributes should be an openended list so that users can add specific attributes for specific renderers. However, this cannot be done within the confines of a single XML DTD. Although it can be done using extensions of the standard DTD, some authors will wish to use nonstandard attributes to take advantage of rendererspecific capabilities while remaining strictly in conformance with the standard DTD.
To allow this, the MathML 1.0 specification [MathML1]
allowed the attribute other
on all elements, for use as a hook to pass
on rendererspecific information. In particular, it was intended as a hook for
passing information to audio renderers, computer algebra systems, and for pattern
matching in future macro/extension mechanisms. The motivation for this approach to
the problem was historical, looking to PostScript, for example, where comments are
widely used to pass information that is not part of PostScript.
In the meantime, however, the development of a general XML namespace
mechanism has made the use of the other
attribute obsolete. In MathML 2.0, the other
attribute is deprecated
in favor of the use of namespace
prefixes to identify nonMathML attributes.
For example, in MathML 1.0, it was recommended that if additional information
was used in a rendererspecific implementation for the maction
element
(Section 3.6.1 Bind Action to SubExpression
(maction)),
that information should be passed in using the other
attribute:
<maction actiontype="highlight" other="color='#ff0000'"> expression </maction>
In MathML 2.0, a color
attribute from another
namespace would be used:
<body xmlns:my="http://www.example.com/MathML/extensions"> ... <maction actiontype="highlight" my:color="#ff0000"> expression </maction> ... </body>
Note that the intent of allowing nonstandard attributes is not to encourage software developers to use this as a loophole for circumventing the core conventions for MathML markup. Authors and applications should use nonstandard attributes judiciously.
If MathML is to remain useful in the future, it is to be expected that MathML will need to be extended and revised in various ways. Some of these extensions can be easily foreseen; for example, as work on behavioral extensions to CSS proceeds, MathML will likely need to be extended as well.
Similarly, there are several kinds of functionality that are fairly obvious candidates for future MathML extensions. These include macros, style sheets, and perhaps a general facility for "labeled diagrams". However, there will no doubt be other desirable extensions to MathML that will only emerge as MathML is widely used. For these extensions, the W3C Math Working Group relies on the extensible architecture of XML, and the common sense of the larger Web community.
The development of stylesheet mechanisms for XML is part of the ongoing XML activity of the World Wide Web Consortium. Both XSL and CSS are working to incorporate greater support for mathematics.
In particular, XSL Transformations [XSLT] are likely to have a large impact on the future development of MathML. Macros have traditionally contributed greatly the usability and effectiveness of mathematics encodings. Further work developing applications of XSLT tailored specifically to MathML is clearly called for.
Some of the possible uses of macro capabilities for MathML include:
One common use of macros is for abbreviation. Authors needing to repeat some complicated but constant notation can define a macro. This greatly facilitates hand authoring. Macros that allow for substitution of parameters facilitate such usage even further.
By defining macros for semantic objects, for example a binomial coefficient, or a Bessel function, one can in effect extend the content markup for MathML. Such a macro could include an explicit semantic binding, or such a binding could be easily added by an external application. Narrowly defined disciplines should be able to easily introduce standardized content markup by using standard macro packages. For example, the OpenMath project could release macro packages for attaching OpenMath content markup.
Another basic way in which macros are often used is to provide a way of controlling style and rendering behavior by replacing highlevel macro definitions. This is especially important for controlling the rendering behavior of MathML content tags in a context sensitive way. Such a macro capability is also necessary to provide a way of attaching renderings to userdefined XML extensions to the MathML core.
Readercontrolled style sheets are important in providing
accessibility to MathML. For example, a reader listening to a voice
renderer might, by default, hear a bit of MathML presentation markup read as
"D sub x sup 2 of f". Knowing the context to be multivariable
calculus, the reader may wish to use a style sheet or macro package that
instructs the renderer to render this <msubsup>
element as "second derivative with respect to x of f".
The set of elements and attributes specified in the MathML specification are necessary for rendering common mathematical expressions. It is recognized that not all mathematical notation is covered by this set of elements, that new notations are continually invented, and that subcommunities within mathematics often have specialized notations; and furthermore that the explicit extension of a standard is a necessarily slow and conservative process. This implies that the MathML standard could never explicitly cover all the presentational forms used by every subcommunity of authors and readers of mathematics, much less encode all mathematical content.
In order to facilitate the use of MathML by the widest possible audience, and to enable its smooth evolution to encompass more notational forms and more mathematical content (perhaps eventually covered by explicit extensions to the standard), the set of tags and attributes is openended, in the sense described in this section.
MathML is described by an XML DTD, which necessarily limits the elements and attributes to those occurring in the DTD. Renderers desiring to accept nonstandard elements or attributes, and authors desiring to include these in documents, should accept or produce documents that conform to an appropriately extended XML DTD that has the standard MathML DTD as a subset.
MathML renderers are allowed, but not required, to accept
nonstandard elements and attributes, and to render them in any way. If a
renderer does not accept some or all nonstandard tags, it is encouraged
either to handle them as errors as described above for elements with the
wrong number of arguments, or to render their arguments as if they were
arguments to an mrow
, in either case rendering all
standard parts of the input in the normal way.
While MathML can be used in isolation as a language for exchanging mathematical expressions between MathMLaware applications, the primary anticipated use of MathML is to encode mathematical expression within larger documents. MathML is ideal for embedding math expressions in other applications of XML.
In particular, the focus here is on the mechanics of embedding MathML in [XHTML]. XHTML is a W3C Recommendation formulating a family of current and future XMLbased document types and modules that reproduce, subset, and extend HTML. While [HTML4] is the dominant language of the Web at the time of this writing, one may anticipate a shift from HTML to XHTML. Indeed, XHTML can already be made to render properly in most HTML user agents.
Since MathML and XHTML share a common XML framework, namespaces provide a standard mechanism for embedding MathML in XHTML. While some popular user agents also support inclusion of MathML directly in HTML as "XML data islands," this is a transitional strategy. Consult user agent documentation for specific information on its support for embedding XML in HTML.
Embedding MathML in XMLbased documents in general, and XHTML in particular, is a matter of managing namespaces. See the W3C Recommendation "Namespaces in XML" [Namespaces] for full details.
An XML namespace is a collection of names identified by a URI. The URI for the MathML namespace is:
http://www.w3.org/1998/Math/MathML
Using namespaces, embedding a MathML expression in a larger XML document is merely a matter of identifying the MathML markup as residing in the MathML namespace. This can be accomplished by either explicitly identifying each MathML element name by attaching a namespace prefix, or by declaring a default namespace on an enclosing element.
To declare a namespace, one uses an xmlns
attribute, or an attribute
with an xmlns
prefix. When the xmlns
attribute
is used alone, it sets
the default namespace for the element on which it
appears, and for any children elements.
Example:
<math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow>...</mrow> </math>
When the xmlns
attribute is used as a
prefix, it declares a
prefix which can then be used to explicitly associate other elements
and attributes with a particular namespace.
Example:
<body xmlns:m="http://www.w3.org/1998/Math/MathML"> ... <m:math><m:mrow>...</m:mrow></m:math> ... </body>
These two methods of namespace declaration can be used together.
For example, by using both an explicit documentwide namespace prefix,
and default namespace declarations on individual mathematical
elements, it is possible to localize namespace related markup to the
toplevel math
element.
Example:
<body xmlns:m="http://www.w3.org/1998/Math/MathML"> ... <m:math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow>...<mrow> </m:math> ... </body>
The use of namespace prefixes creates an issue for DTD validation of documents embedding MathML. DTD validation requires knowing the literal (possibly prefixed) element names used in the document. However, the Namespaces in XML Recommendation [Namespaces] allows the prefix to be changed at arbitrary points in the document, since namespace prefixes may be declared on any element.
The 'historical' method of bridging this gap was to write a DTD with a fixed prefix, or in the case of XHTML and MathML, with no prefix, and mandate that the specified form must be used throughout the document. However, this is somewhat restricting for a modular DTD that is intended for use in conjunction with another DTD, which is exactly the situation with MathML in XHTML. In essence, the MathML DTD would have to allocate a prefix for itself and hope no other module uses the same prefix to avoid name clashes, thus losing one of the main benefits of XML namespaces.
One strategy for addressing this problem is to make every element name in the DTD be accessed by an entity reference. This means that by declaring a couple of entities to specify the prefix before the DTD is loaded, the prefix can be chosen by a document author, and compound DTDs that include several modules can, without changing the module DTDs, specify unique prefixes for each module to avoid clashes. The MathML DTD has been designed in this fashion. See Section A.3 Using the MathML DTD and [Modularization] for details.
An extra issue arises in the case where explicit prefixes are used
on the toplevel math
element, but a default
namespace is used for other MathML elements. In this case, one wants
the MathML module to be included into XHTML with the prefix set to
empty. However, the 'driver' DTD file that sets up the inclusion of
the MathML module would then need to define a new element called
m:math. This would allow the toplevel math
element to use an explicit prefix, for attaching rendering behaviors
in current browsers, while the contents would not need an explicit
prefix, for ease of interoperability between authoring tools, etc.
While the use of namespaces to embed MathML in other XML applications is completely described by the relevant W3C Recommendations, a certain degree of pragmatism is still called for at present. Support for XML, namespaces and rendering behaviors in popular user agents is not always fully in alignment with W3C Recommendations. In some cases, the software predates the relevant standards, and in other cases, the relevant standards are not yet complete.
During the transitional period, in which some software may not be fully namespaceaware, a few conventional practices will ease compatibility problems:
When using namespace prefixes with MathML markup, use m: as a conventional prefix for the MathML namespace. Using an explicit prefix is probably safer for compatibility in current user agents.
When using namespace prefixes, pick one and use it consistently within a document.
Explicitly declare the MathML namespace on all
math
elements.
Examples.
<body> ... <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow>...<m:mrow> </m:math> ... </body>
Or
<body> ... <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow>...<mrow> </math> ... </body>
Note that these suggestions alone may not be sufficient for creating functional Web pages containing MathML markup. It will generally be the case that some additional documentwide markup will be required. Additional work may also be required to make all MathML instances in a document compatible with documentwide declarations. This is particularly true when documents are created by cutting and pasting MathML expressions, since current tools will probably not be able to query global namespace information.
Consult the W3C Math Working Group home page for compatibility and implementation suggestions for current browsers and other MathMLaware tools.
math
Element
MathML specifies a single toplevel or root math
element,
which encapsulates each instance of
MathML markup within a document. All other MathML content must be
contained in a math
element; equivalently,
every valid, complete MathML expression must be contained in
<math>
tags. The math
element must always be the outermost element in a MathML expression;
it is an error for one math
element to contain
another.
Applications that return subexpressions of other MathML
expressions, for example, as the result of a cutandpaste operation,
should always wrap them in <math>
tags. Ideally, the presence of enclosing <math>
tags should be a very good heuristic test for MathML
material. Similarly, applications which insert MathML expressions in
other MathML expressions must take care to remove the
<math>
tags from the inner expressions.
The math
element can contain an arbitrary number
of children schemata. The children schemata render by default as if they
were contained in an mrow
element.
The attributes of the math
element are:
Provided for use with stylesheets.
Provided along with xml:id
for use in parallel markup (Section 5.5 Parallel Markup)
This attribute provides a way of pointing to
external macro definition files. Macros are not part of the MathML
specification, and much of the functionality provided by macros in MathML can be
accommodated by XSL transformations [XSLT]. However, the
macros
attribute is provided to make possible future
development of more streamlined, MathMLspecific macro mechanisms. The
value of this attribute is a sequence of URLs or URIs, separated by
whitespace
The mode
attribute specifies whether
the enclosed MathML expression should be rendered in a display style
or an inline style. Allowed values are
"display" and
"inline" (default).
This attribute is deprecated in
favor of the new display
attribute, or the
CSS2
'display' property with the analogous block and
inline values.
The display
attribute replaces the
deprecated mode
attribute.
It specifies whether
the enclosed MathML expression should be rendered in a display style
or an inline style. Allowed values are
"block" and
"inline" (default).
The dir
attribute specifies the overall directionality
of layout. Allowed values are "ltr"(default) or "rtl".
This attribute, in addition to the directionality of the text content
of token elements, is used for presentation of mathematics in RighttoLeft scripts.
See Section 3.1.5 Directionality for further discussion.
The attributes of the math
element affect
the entire enclosed expression. They are, in a sense, "inward
looking". However, to render MathML properly in a browser, and
to integrate it properly into an XHTML document, a second collection
of "outward looking" attributes are also useful.
While general mechanisms for attaching rendering behaviors to elements in XML documents are under development, wide variations in strategy and level of implementation remain between various existing user agents. Consequently, the remainder of this section describes attributes and functionality that are desirable for integrating thirdparty rendering modules with user agents:
In cases where size negotiation is not possible or fails (for example in the case of an expression that is too long to fit in the allowed width), this attribute is provided to suggest a processing method to the renderer. Allowed values are:
(Default) The expression will be broken across several lines. The line breaking algorithm is not specified, but it is recommended that line breaking should try to keep meaningful subexpressions together and indent lines in a manner that aids in understanding the expression.
The window provides a viewport into the larger complete display of the mathematical expression. Horizontal or vertical scrollbars are added to the window as necessary to allow the viewport to be moved to a different position.
The display is abbreviated by removing enough of it so that the remainder fits into the window. For example, a large polynomial might have the first and last terms displayed with "+ ... +" between them. Advanced renderers may provide a facility to zoom in on elided areas.
The display is abbreviated by simply truncating it at the right and bottom borders. It is recommended that some indication of truncation is made to the viewer.
The fonts used to display the mathematical expression are chosen so that the full expression fits in the window. Note that this only happens if the expression is too large. In the case of a window larger than necessary, the expression is shown at its normal size within the larger window.
This attribute provides a graceful fallback for browsers that do not support embedded elements. The value of the attribute is an URL.
This attribute provides a graceful fallback for browsers that do not support embedded elements or images. The value of the attribute is a text string.
This attribute provides a width for the altimg
(if any). The value of attribute is an hunit
. This value is useful for high resolution images which, if displayed at their full resolution, would be too large.
If neither altimgwidth
nor altimgheight
is given, then for those renderers that use an image, they should use the image's natural size. If only the width is given,
the renderer should scale the height so as to preserve the aspect ration of the image.
This attribute provides a total height for the altimg
(if any). The value of attribute is a vunit
. This value is useful for high resolution images which, if displayed at their full resolution, would be too large.
If neither altimgwidth
nor altimgheight
is given, then for those renderers that use an image, they should use the image's natural size. If only the width is given,
the renderer should scale the width so as to preserve the aspect ration of the image.
By default, the bottom of the image aligns to the current baseline.
The valign
attribute specifies the alignment
point within the image. The value of attribute is a vunit
.
A positive value of valign
shifts the bottom of the image below the current baseline, while
a negative value will raise it above the baseline.
Issue linebreakcontrol  wiki (member only) 

linebreak control  
Should there be a way to specify some sort of control over how line breaks are choosen (eg, before or after an infix operator, or if the infix operator is duplicated)? 

Resolution  None recorded 
Issue indentcontrol  wiki (member only) 

indent control  
Should there be a way to specify some sort of indenting style? 

Resolution  None recorded 