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To be effective, MathML must work well with a wide variety of renderers, processors, translators and editors. This chapter addresses 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 HTML.
There are three kinds of interface issues that arise in embedding MathML in HTML. First, MathML must be semantically integrated into HTML. For example, there must be a mechanism for browsers to recognize MathML markup as embedded content, and not as an HTML syntax error. More generally, the semantic embedding of MathML in HTML is a special case of embedding XML in HTML, which involves issues such as name space management, and document validation.
Second, MathML rendering must be integrated into browser software. Until MathML is rendered natively by browsers, rendering will typically be done by embedded elements. However, to properly render mathematical notation in context in a Web document, improved coordination between browsers and embedded elements will be necessary. For example, embedded elements will need to be able to detect the ambient rendering environment, such as baseline, font family and color scheme, and respond appropriately to reader input such as font size changes. Support for printing is also essential.
Third, tools for generating MathML must be developed, including editors, translators, and export capabilities in computer algebra systems, and other scientific software.. Since MathML is designed to be powerful and flexible to accommodate a wide range of applications, while at the same time remaining structured and explicit for easy processing, MathML expressions tend to be lengthy, and prone to error when entered by hand. Therefore, special emphasis must be given to insuring that MathML can be easily generated by user-friendly conversion and authoring tools.
The W3C Math working group is committed to working with software vendors to develop a wide range of equation editors and translation tools, and plans to continue to do so in the future. In particular, the working group monitors the public www-math@w3.org mailing list, and will attempt to provide support to software developers with questions about the MathML specification. The working group also intends to try to stimulate the formation of MathML developer and user groups. For current information about MathML tools, applications and user support activities, consult the W3C Math home page.
MathML specifies a single top-level math element, which encapsulates each instance of MathML markup within an HTML page. As such, the math element provides an attachment point for information which affects a MathML expression as a whole. For example, the math element is the logical place to attach style sheet or macro information in the future, when these facilities become available for MathML.
Ideally, the math element should also serve as the interface for embedding MathML in HTML. To function in this capacity, the math element would have to simultaneously signal the semantic inclusion of MathML (XML) content in HTML, and provide the necessary machinery for rendering its content in a browser either by invoking an embedded element, or by specifying parameters for a native renderer in the browser. Both semantic inclusion and rendering present a number of issues that extend beyond the boundaries of W3C Math. To a large extent, the issues which arise for embedding MathML in HTML are the same as those for the more general problem of embedding XML in HTML. Resolving these issues will require the efforts of a number of World Wide Web Consortium Activities, including the HTML, XML, CSS and DOM activities.
In order to produce a complete and self-contained description of MathML, this document only specifies the attributes and usage of the math element as a top-level element for MathML, and not as an interface element. The W3C Math working group intends to continue working closely with other World Wide Web Consortium activities to insure that emerging standards for embedding XML in HTML accommodate seamless integration of MathML in HTML. Section 7.1.2 lists requirements which an interface element for MathML would have to meet in order to fully integrate MathML into HTML. However, it is important to note that the MathML specification is independent of the ultimate embedding mechanism.
As stated above, MathML specifies a single top-level math element. 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 which return subexpressions of other MathML expressions, for example as the result of a cut-and-paste operation, should always wrap them in <math> tags. The presence of enclosing <math> tags should be a reasonable heuristic test for MathML content. 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 a mrow element.
The attributes of the math element are:
The top-level math element described in the preceding section is concerned with encapsulating MathML content and defining attributes which affect the entire enclosed expression. It is, in a sense, "inward looking." However, to render MathML properly in a browser, and to integrate it properly into an HTML document, an "outward looking" interface element is also required. This interface element must be aware of its surrounding environment, and provide a mechanism for passing information between the browser, and the MathML renderer.
As noted above, the MathML interface element and the MathML top-level element should ideally be one and the same. The math element should not only serve to encapsulate MathML content, it should signal the semantic embedding of MathML content to an HTML processor, and admit additional attributes for controlling how the MathML renderer should interact with the browser.
Since a general mechanism for embedding XML in HTML is anticipated in the near future which may not be compatible with using the top-level math element for the interface element as well, the remainder of this section describes attributes and functionality that a MathML interface element should ultimately provide. In the near term, implementors attempting to provide interim solutions for rendering MathML in browsers should try to give authors some way of passing the following interface attributes to the renderer:
Attributes which apply to the MathML interface element necessarily take effect when the document is first loaded, and therefore suffer the limitation that they cannot change in response to reader interaction. The height and width attributes are good examples; if the reader changes the current font size, the height and width of the embedded math fragments also need to change. Therefore, in order to properly render MathML, an embedded element must be able to communicate with the browser, and react to reader input.
At present, browser support for embedded elements is too limited to provide acceptable rendering for MathML. The W3C Math working group is working closely with the Document Object Model working group in an effort to provide better communication between embedded MathML renderers and browsers. Some of the most needed improvements are:
Until MathML is natively supported by browsers, we anticipate that MathML rendering will be carried out via embedded objects such as plug-ins, applets, or helper applications. In the near term, the W3C Math working group advocates the use of MIME types to bind embedded MathML to renderers. Mechanisms for assigning MIME types already exist in HTML, and mechanisms for registering and automatically invoking embedded elements such as plug-ins based on MIME type already exist in Web browsers.
The type attribute, described in the previous section as a requirement for the MathML interface element, is intended to associate a MIME type with its content. The HTML element META is proposed as a means of specifying document-wide default MIME types for an element.
We propose a simple MIME type naming convention which is flexible enough to accommodate several common situations:
We propose that generic MathML be assigned the MIME type
text/mathml, and for browser registry, we suggest the
standard file extension .mml be used. To invoke specific
renderers, we suggest assigning a MIME type of the following format:
text/mathml-renderer
Example:
A user downloads and installs renderer A, and
registers it with the browser for the
text/mathml MIME type to process generic MathML.
However renderer A also accepts TeX as an input syntax, and therefore
during the installation process, it requests to be registered for
application/x-tex as well. Later, the user discovers
renderer B provides additional features, such as
cut and paste capability. Therefore, the user downloads,
installs and registers renderer B for the
text/mathml-rendererB MIME type.
An author then creates a document that contains the the following line in the document header:
<META Content-math-Type="text/mathml">
Later, the document contains the following expressions:
When our hypothetical reader views this document, renderer A is invoked to process the first expression, while renderer B is invoked for the second. Later, when our hypothetical reader later views a document with MIME type<math> <msup><mi>x</mi><mn>2</mn></msup> </math> <math type="text/mathml-rendererB"> <mi>α<mi><mo>=</mo><mn>0.4</mn> </math>
application/x-tex, renderer A is again
invoke, this time in TeX processing mode.
Although rendering MathML expressions typically occurs in place in a Web browser, other MathML processing functions take place more naturally in other applications. Particularly common tasks include opening a MathML expression in an equation editor or computer algebra system.
At present, there is no standard way of specifying that embedded content should be rendered with one application, edited in another, and evaluated by a third. As work progresses on coordination between browsers and embedded elements and the Document Object Model (DOM), providing this kind of functionality should be a priority. Both authors and readers should be able to indicate a preference about what MathML application to use in a given context. For example, one might imagine that some mouse gesture over a MathML expression would cause a browser to present the reader with a pop-up menu, showing the various kinds of MathML processing available on the system, and the MathML processors recommended by the author.
Since MathML will probably be widely generated by authoring tools, it is particularly important that opening a MathML expression in an editor should be easy to do and to implement. In many cases, it will be desirable for an authoring tool to record some information about its internal state along with a MathML expression, so that an author can pick up editing where he or she left off. The MathML specification does not explicitly contain provisions for recording authoring tool information. In some circumstances, it may be possible to include authoring tool information which applies to an entire document as meta data; interested readers are encouraged to consult the W3C Metadata Activity for current information about metadata and resource definition. For encoding authoring tool state information that applies to a particular MathML instance, readers are referred to the possible use of the semantics element for this purpose.
In order to be fully integrated into HTML, it should be possible not only to embed MathML in HTML, but also to embed HTML in MathML. However, the problem of supporting HTML in MathML presents many difficulties. Moreover, the problems are not specific to MathML; they are problems for XML applications in HTML generally. Therefore, at present, the MathML specification does not permit any HTML elements within a MathML expression, although this may be subject to change in a future revision of MathML, when.mechanisms for embedding XML in HTML have been further developed.
In most cases, HTML elements either do not apply in mathematical contexts (headings, paragraphs, lists, etc), or MathML already provides equivalent or better functionality specifically tailored to mathematical content (tables, style changes, etc). However, there are two notable exceptions.
MathML has no element which corresponds to the HTML anchor element A. In HTML, anchors are used both to make links, and to provide locations to link to. MathML, as an XML application, defines links by the use of the XML-LINK attribute. However, MathML at present does not provide a way for other documents to make links into a MathML expression. One reason for this omission is that linking into embedded XML content is better addressed as part of a general mechanism for embedding XML in HTML. Moreover, until browsers either natively implement MathML rendering, or substantially better coordination between embedded elements and browsers becomes possible, there is no reasonable way of implementing links into MathML expressions.
MathML linking elements are generic XML linking elements as described in the Extensible Markup Language (XML): Part 2. Linking working draft. The reader is cautioned, however, that this working draft is less mature than the XML syntax working draft, and is therefore more subject to future revision. Since the MathML linking mechanism is defined in terms of the XML linking specification, the same proviso holds for it as well.
A MathML element is designated as a link by the presence of the XML-LINK attribute. The possible values for the this attribute are "simple", "extended", "locator", "group" and "document". Although all of these values are valid, MathML renderers need only implement "simple" XML links to be MathML compliant. How links are indicated to the reader is left to the individual MathML processing application.
Elements which specify the value of the XML-LINK attribute as "simple" must also specify a value for the HREF attribute. These two attributes fully specify a "simple" XML link. Thus, a typical MathML link might look like:
<mrow XML-LINK="simple" HREF="http://www.w3.org"> ... </mrow>
MathML designates that almost all elements can be used as an XML linking element. The only elements which cannot serve as linking elements are those such as the <sep/> element which exist primarily to disambiguate other MathML constructs and in general do not correspond to any part of a typical visual rendering. The full list of exceptional elements which cannot be used as linking elements is given below in table 7.1.5.1.
| <prescripts/> | <none/> | <sep/> |
| <power/> | <malignmark/> | <maligngroup/> |
Table 7.1.5.1 MathML Elements Which Cannot Be Linking Elements
The IMG element has no MathML equivalent. The decision to omit a general image inclusion mechanism in MathML was based on several factors. First, a simple mechanism for including images in MathML along the lines of the IMG element would not be more closely tied to mathematical content or notation than the HTML IMG element itself. Therefore, such an element would likely be superseded by the IMG element if it becomes possible to mix XML and HTML generally.
Another reason for not providing an image facility is that MathML takes great pains to make the notational structure and mathematical content it encodes easily available to processors while information contained in images is only available to a human reader looking at a visual representation. Thus, for example, in the MathML paradigm, it would be preferable to introduce new glyphs by the creation of special symbol fonts, rather than simply including them as images.
Finally, apart from the introduction of new glyphs, many of the situations where one might be inclined to use an image amount to some sort of labeled diagram. For example, knot diagrams, Venn diagrams, Dynkin diagrams, Feynman diagrams and complicated commutative diagrams all fall into this category. As such, their content would be better encoded via some combination of structured graphics and MathML markup. Because of the generality of the "labeled diagram" construction, the definition of a markup language to encode such constructions extends beyond the scope of the W3C Math activity. However, it may be possible to provide such functionality in a future extension of MathML.
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 also tied to the ability to process it with software tools.
Many different kinds of MathML editors, translators, processors and renderers will be implemented. In addition to supporting the MathML core language, it is reasonable to assume that some of these renderers will provide additional specialized capabilities. Consequently, it is important to specify what one can and cannot expect from a generic MathML compliant application, and in what ways MathML can be extended, or used to pass additional information directly to specific application that can take advantage of it.
It is important to clearly specify what it means to be a MathML compliant processor. Specifying MathML compliance serves two purposes. First, authors can be assured that their documents will be generally accessible if they refrain from using proprietary extensions. Second, software developers can be assured of the criteria for interoperability.
A well-formed MathML expression is a XML construct determined by the MathML DTD together with the additional requirements given in the specifications of the MathML document.
We define a "MathML processor" to mean any application that can accept, produce, or "roundtrip" a well-formed MathML expression. An example of an application that might round-trip a MathML expression might be an editor that writes a new file even though no modifications are made.
We specify three forms of MathML compliance:
For example a MathML-input-compliant validating parser which implements the MathML specification returns a truth value. A MathML-input-compliant renderer faithfully translates a MathML expression into application-specific form allowing native application operations to be performed.
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 element-by-element basis is discussed elsewhere in this document.
We note that being roundtrip-compliant may be very difficult for processors that convert MathML input into an internal form that is structurally very different from the XML expression model. The first generation of processors may very well be input-compliant and output-compliant, but not roundtrip-compliant. Nevertheless, we expect roundtrip-compliant processors to be eventually produced with the wide-spread acceptance of MathML.
Beyond the above, the MathML core specification makes no demands of individual processors. However, in order to guide developers, the MathML specification includes advisory material; for example, there are suggested rendering rules included in section 3. The remainder of this section makes additional suggestions about a number of interface issues a MathML processor should address in some fashion.
If a MathML-input-compliant application receives input containing one or more elements with an illegal number or type of attributes or children schemata, it should nonetheless attempt to render all the input in an intelligible way, i.e. to render normally those parts of the input which were well-formed, and to render error messages (rendered as if enclosed in an <merror> element) in place of ill-formed expressions.
MathML-output-compliant applications such as editors and translators may choose to generate <merror> expressions to signal errors in their input. This is usually preferable to generating well-formed, but possibly erroneous, MathML.
The MathML attributes described in the MathML specification are necessary for display and content markup. Ideally, the MathML attributes should be an open-ended list so that users could add specific attributes for specific renderers. However, this can't 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 while remaining strictly in compliance with the standard DTD.
To allow this, this specification also allows the attribute other="..." for all elements, for use as a hook to pass on renderer-specific information. In particular, it can be used as a hook for passing information to audio renderers, computer algebra systems, and for pattern matching in any future macro/extension mechanism. This idea is used in other languages. For example, Postscript comments are widely used to pass information that is not part of Postscript.
At the same time, the intent of the other attribute is not to encourage software developers to use this as a loophole for circumventing the MathML core markup conventions. We trust both authors and applications will use the other attribute judiciously.
The value of the other attribute should be a string containing an attribute list in valid XML format (i.e., attr1="val1" attr2="val2"; ..., with appropriate escaping of the double quotes). Renderers which accept nonstandard attributes directly should also accept them when they occur within the string value of the other attribute. This is not required for attributes specifically documented by the MathML standard.
MathML is in its infancy; 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; as noted repeatedly in this chapter, the mechanisms for fully integrating MathML into HTML are not yet developed, and these mechanisms may have a significant impact on some aspects of MathML
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 "labeled diagram" facility. However, there will also no doubt be other desirable extensions to MathML which 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 definition of a style sheet mechanism for XML is part of the ongoing XML activity at the World Wide Web Consortium. Although it is too soon to say what this mechanism will ultimately be like, it is likely that it will accommodate the needs of MathML. It is also possible that such a style sheet mechanism will be sufficiently powerful to provide basic macro capability as well.
Macros, however, play a very important and useful role in encoding mathematical content and meaning. Moreover, it is difficult to devise a coherent, general macro system for MathML, because there are so many distinct applications for MathML macros. Therefore, the W3C Math working group plans to investigate the definition of a macro mechanism specifically tailored to MathML, in addition to participating in general ongoing XML style sheet and macro facility activities.
Some of the possible uses of MathML macros include:
The set of elements and attributes specified in the MathML specification are necessary for rendering common math expressions. It is recognized that not all mathematical notation is covered by this set of elements, that new notations are continually invented, and that sub-communities 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 sub-community 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 open-ended, in the sense described in this section.
MathML is described by an XML-compliant DTD, which necessarily limits the elements and attributes to those which occur in the DTD. Renderers desiring to accept nonstandard elements or attributes, and authors desiring to include these in documents, should accept or produce documents which conform to an appropriately extended XML-compliant DTD which has the standard MathML DTD as a subset.
MathML compliant 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 to either 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 normally.