MathML

12 Aug 2005

MathML

These slides are in XHTML, and follow the approach described on a separate page for presentation. That page describes the minimum environment you should have to display these slides with the proper MathML content.

Table of Content:

Why MathML?
Why MathML? (cont.)
What is MathML?
Who developed MathML?
A Flavour of MathML: Quadratic formula in MathML
A Bit of History
MathML structure
Same Formula in Different Markups
Why two markups?
Presentation Elements
Main Token Presentation Elements
Controlling the Rendering
Controlling the Rendering (cont.)
General Layout
General Layout (cont.)
Script and Limits
Tables and Matrices
Tables and Matrices (cont.)
Enlivening Expressions
Content Markup
Content Markup Details
Content Markup
Fundamental Content Elements
Fundamental Content Elements (cont.)
Fundamental Content Elements
Fundamental Content Elements
Content Operators and Functions
Content Operators and Functions
Mixing Content and Presentation Markups
Semantics of Content Elements
Adding semantics to an expression
Using MathML with Other Markups
Embedding MathML
Using “object”
Full XML Based Usage: Namespaces
Full XML Based Usage: Common Approach
Further Information

Why MathML?

Current solutions to display math on the Web:

Use HTML

eg: a_i (a<sub>i</sub>))

HTML is not rich enough
typesetting mathematical formulae is hard!

Use embedded images

eg:

requires specialized editors
difficult to position/size the image properly
not searchable
accessibility problems (what should a voice browser do?)

Why MathML? (cont.)

Mathematics are ubiquitous in technical documentation
Having mathematics on the web is essential for digital documents!
Large amount of legacy document should be converted (mostly in T_EX)
Hence MathML…

What is MathML?

Mathematical formulae described in XML
- it is searchable
- can be embedded in other, XML-based markups
- voice browsers can be adapted to "read" formulae
- is human editable (sort of… specialized editors are better)
Reminiscent to T_EX (ie, converter tools are feasible)

Who developed MathML?

Elsevier, INRIA, Univ. of Waterloo (Maple), IBM, American Mathematical Society, the OpenMath consortium, Wolfram Research (Mathematica),…
Software are now available:
- IBM Techexplorer, or MathPlayer plugins
- Firefox displays MathML natively
- DesignSite with MSN
- Mathematica and Maple export and render MathML
- LaTeX2HTML MathML and Omega packages convert T_EX to MathML
- … (see the MathML Activity home page also for further links)

A Flavour of MathML: Quadratic formula in MathML

<mrow>

  <mi>x</mi> <mo>=</mo>

  <mfrac>

    <mrow>

      <mrow><mo>-</mo><mi>b</mi></mrow>

      <mo>±</mo>

      <msqrt>

        <mrow>

          <msup><mi>b</mi><mn>2</mn></msup>

          <mo>-</mo>

          <mrow>

            <mi>c</mi>

          </mrow>

        </mrow>

      </msqrt>

    </mrow>

    <mrow>

      <mi>a</mi>

    </mrow>

  </mfrac>

</mrow>

A Bit of History

MathML 1.0 was published in 1998 with an update in 1999
- MathML 1.0 was the first XML based W3C Recommendation!
MathML 2.0 was published in 2002, with an update in 2003
- “modernized” MathML 1.0 for the evolution of XML
  - styling, namespaces, usage of Unicode, etc.
- MathML 2.0, 2nd. edition is the authoritative version!

MathML structure

XML Elements in MathML are:

Presentation markup elements:: Describing the notation directly (see previous example): rows, fractions, operators, etc. It captures the notational structure.
Content markup elements:: Explicit encoding of the mathematical content. It captures the mathematical structure.
The (only) interface element:: math element, which binds a formula to the surrounding XML (eg., XHTML)

Note: earlier versions relied on XML Entities for special signs (integrals, greek letters, etc), but Unicode has taken over…

Same Formula in Different Markups

Presentation Markup	Content Markup
<msup> <mfenced> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfenced> <mn>2</mn> </msup>	<apply> <power/> <apply> <plus/> <ci>a</ci> <ci>b</ci> </apply> <cn>2</cn> </apply>

Why two markups?

Legacy data transformed into presentation markup
Authoring tools may prefer presentation markup
High quality renderers may prefer to use content markup:
- difference between an inline vs standalone formula
- differences in local traditions, disciplines:
  - 3+4i resp. 3+4j!
Pedagogic tools would use content markup
Content markup represents abstract information
Content markup is more accessible
The two markups may be mixed!

Presentation Elements

Presentation elements fall in two categories:

Token elements: Individual symbols, names, numbers, alignment control, ...
Layout elements: Rows, fractions, square roots, superscripts, tables, ...

Main Token Presentation Elements

identifier, number	<mi>a</mi> <mn>2</mn>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
operator, separator, or accent	<mi>a</mi><mo>→</mo><mi>b</mi>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
text	<mtext>Theorem 1:</mtext>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Controlling the Rendering

Attributes to control the precise rendering, eg,

<mo> ( </mo>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
<mo maxsize="1"> ( </mo>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Controlling the Rendering (cont.)

Other examples:
- font size, color, …
- leftspace, rightspace, …
- operator form (infix, prefix, postfix)
  - operator dictionary for the characterization of the operators
  - invisible operators (eg, "invisible times"), important for, eg, voice browsers!

General Layout

subexpressions	<mrow><mi>a</mi><mi>b</mi></mrow>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
fraction	<mfrac><mi>a</mi><mi>b</mi></mfrac>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
square root	<msqrt><mi>a</mi></msqrt>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

General Layout (cont.)

radical	<mroot><mi>a</mi><mi>b</mi></mroot>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
surround content	<mfenced open="[" separators=";"> <mi>a</mi><mi>b</mi> </mfenced>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Script and Limits

subscript	<msub><mi>x</mi><mi>i</mi></msub>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
superscript	<msub><mi>x</mi><mi>i</mi></msub>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
tensor indices	<mmultiscripts><mi>R</mi> <mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi> <mprescripts/><mi>a</mi><none/><mi>b</mi> ... </mmultiscripts>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Tables and Matrices

table	<mrow><mo> [ </mo> <mtable> <mtr> <mtd><mn>1</mn></mtd> <mtd><mn>0</mn></mtd> <mtd><mn>0</mn></mtd> </mtr> ... </mtable> <mo> ] </mo></mrow>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
row
entry

Tables and Matrices (cont.)

align

<mtable

  groupalign="decimalpoint left ...">

   <mtr><mtd><mrow>

        <maligngroup/>

        <mn> 8.44 </mn>

        <maligngroup/>

        <mi> x </mi>

...

   </mrow></mtd></mtr>

   <mtr><mtd><mrow>

        <maligngroup/>

        <mn> 3.1 </mn>

        <maligngroup/>

        <mi> x </mi>

...

   </mrow></mtd></mtr>

</mtable>

Enlivening Expressions

<maction actiontype="toggle" selection="2">

    <mrow>...</mrow>

    <mrow>...</mrow>

</maction>

Other possible action types (depending on the rendering engine)

tooltip
statusline
highlight background, foreground
(popup) menu

Content Markup

Content markup reflects the structure of formulae:

schematic view of the structure of an integr120al expression

Content Markup Details

Content markup reflects the structure of formulae:

  <apply><int/>

    <bvar><ci>t</ci></bvar>

    <lowlimit><cn>0</cn></lowlimit>

    <uplimit><cn>1</cn></uplimit>

    <apply><power/>

      <ci>t</ci>

      <cn>2</cn>

    </apply>

  </apply>

MathML does not define the exact mapping from content markup to rendering (only a "Default Rendering" is defined).

Content Markup

MathML defines a small number of commonplace mathematical constructs, covering
- arithmetic, algebra, linear algebra, logic and relations
- calculus, sequences and series, set theory
- trigonometry
- statistics
Adequate for an A-level (US) or Baccalaureate (Europe) mathematics

Fundamental Content Elements

Number	<cn type="complex-cartesian"> 3<sep/>4 </cn>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
Variable	<apply> <ci>F</ci> <ci>x</ci> </apply>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
Apply construct	<apply> <ci>F</ci> <ci>x</ci> </apply>
Numeric interval	<interval closure="open-closed"> <cn>3</cn><cn>4</cn> </interval>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Fundamental Content Elements (cont.)

Functions and operators	<apply><laplacian/> <ci>f</ci> </apply>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->
Inverse function	<apply> <apply><inverse/><sin/></apply> <ci>x</ci> </apply>	<!--This is ugly, but necessary to get slidy and firefox's mathml coexist...-->

Fundamental Content Elements

Condition

<apply>

  <max/>

   <condition>

    <apply><and/>

      <apply><in/>

        <ci>x</ci>

        <ci type="set">B</ci>

      </apply>

      <apply><notin/>

        <ci>x</ci>

        <ci type="set">C</ci>

      </apply>

    </apply>

   </condition>

  <ci>x</ci>

</apply>

Fundamental Content Elements

Matrices

<matrix>

  <matrixrow>

     <cn> 1 </cn> <cn> 2 </cn>

  </matrixrow>

  <matrixrow>

     <cn> 3 </cn> <cn> 4 </cn>

  </matrixrow>

</matrix>

Piecewise

<piecewise>

  <piece>

    <apply><minus/>

      <ci> x </ci>

    </apply>

    <apply><lt/>

      <ci> x </ci>

      <cn> 0 </cn>

    </apply>

  </piece>

  <piece>...</piece>

</piecewise>

Content Operators and Functions

Arithmetic, Algebra, Logic: quotient, exp, max, min, root, and, xor, forall, exists, floor ...
Relations: eq, neq, equivalent, ...
Calculus: ln, int, partialdiff, diff, limit, laplacian ...
Set theory: list, union, intersect, cartesian product ...

Content Operators and Functions

Sequences and series: sum, product, ...
Statistics: mean, variance, moment ...
Trigonometry: sin, arccos, ...
Linear algebra: vector, matrix, vector product, ...
Constants: integers, reals, pi, infinity, ...

Altogether cca. 100 elements!

Mixing Content and Presentation Markups

Content markup helps accessibility (eg, voice reader); but …
…it cannot account for all situations! For example:

  <apply>

    <mrow>

      <msub>

        <mi>F</mi>

        <mi>i</mi>

      </msub>

    </mrow>

    <ci>x</ci>

  </apply>

Semantics of Content Elements

Elements can use the definitonURL to override default semantics (eg, infix, postfix)
Recommendation includes a formalism for defining semantics
Other specifications through csymbol:

<!-- reference to OpenMath definition of Bessel function -->

<apply>

  <csymbol encoding="OpenMath"

    definitionURL="http://www.openmath.org/...">

    <msub><mi>J</mi><mn>0</mn></msub>

  </csymbol>

  <ci>y</ci>

</apply>

Adding semantics to an expression

<semantics>

  <apply>

  <divide/>

    <cn>123</cn>

    <cn>456</cn>

  </apply>

  <annotation encoding="Mathematica">

       N[123/456, 39]

  </annotation>

  <annotation encoding="Maple">

       evalf(123/456, 39);

  </annotation>

  <annotation-xml encoding="OpenMath">

    <OMA xmlns="http://www.openmath.org/OpenMath">

	  <OMS cd="arith1" name="divide"/>

	  <OMI>123</OMI>

	  <OMI>456</OMI>

    </OMA>

  </annotation-xml>

</semantics>

Using MathML with Other Markups

The math element:
- “root” element of MathML
- has attributes to control font, colour, background colour, overflow control,etc,
XSLT can also be used to “transform” markups on the fly:
- adapt to local conventions (“tan” vs “tg”, “[a,b]” vs “]a;b[”, etc)
- is a powerful technology that MathML “inherits” from other XML recommendations

Embedding MathML

Usage of embed:

<embed src="mmls/mixExample.mml" height="60" width="110" />

Non-standard tag
Not all browsers may support it
Might be the only way to use with some plugins… (sigh!)

Using “object”

Alternative should be object:

<object type="application/mathml+xml"

   data="mmls/mixExample.mml"

   height="60" width="110"></object>

Proper HTML4 and XHTML tag
The iframe can also be an alternative in some cases…

Full XML Based Usage: Namespaces

Using XHTML, the math content should be directly includable:

<body>

  <p>Bla bla bla</p>

  <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">

    <m:mrow>...<m:mrow>

  </m:math>

</body>

Works in Firefox (and derivatives), in Amaya, in Gaelon
- content markup may not be implemented
- content must be served as xml, not as text/html
IE 6.+ with MathPlayer or TechExplorer plugins
- file has to follow IE’s behavior rules…
There is a separate Working group at W3C cleaning up this approach (Compound Document Formats)

Full XML Based Usage: Common Approach

MathML activity has developed some XSLT scripts to unify IE and Mozilla/Netscape
- “Universal Math XSL stylesheet (UMSS)”
Same file can be displayed on both platforms
- the IE “behavior” requirements are hidden, for example
- if necessary content markup is transformed into presentation markup “on the fly”
Caveat (again): the file must be served as XML and not in HTML
This is how the file should look like:

    <?xml version="1.0" encoding="iso-8859-1" ?>

    <?xml-stylesheet type="text/xsl" href="mathml.xsl"?>

    <html xmlns="http://www.w3.org/1999/xhtml">

      <head>

…

See the separate, dedicated page for the details: http://www.w3.org/Math/XSL/

Further Information

These slides are at:: http://www.w3.org/Consortium/Offices/Presentations/MathML/
W3C MathML Activity Page:: http://www.w3.org/Math/
The latest release of the Recommendation: http://www.w3.org/TR/MathML2/
More information about W3C:: http://www.w3.org/Consortium/
Contact information:: http://www.w3.org/Consortium/Contact
Mail me:: ivan@w3.org