It may be noted that the dates of MathML development overlap considerably with the time frame for the OpenMath Esprit project. At the time of the original Esprit proposal, the W3C did not have an activity for producing a markup language for Mathematics. However the Math Working Group was chartered, and MathML 1.0 was produced shortly after the OpenMath consortium effectively began work at the end of 1997. The emergence of MathML has had a large impact on the subsequent work of the consortium, resulting in a situation where the two languages are visibly complementary rather than in competition. In summary, new elements were added to MathML 2 which align it quite closely with OpenMath, and the OpenMath Content Dictionaries were re-organised so that symbols that correspond to MathML 2 constructs are explicitly grouped together. MathML provides a natural mechanism for rendering OpenMath expressions and conversely OpenMath, and in particular the OpenMath Content Dictionaries, provide a natural target for the extension mechanisms in MathML. We give some more specific details and examples below.
Unlike OpenMath in which XML is but one possible encoding of an abstract model of the mathematical expression, MathML is by definition an XML vocabulary. MathML consists of two essentially separate languages.
Presentation MathML consists of XML constructs specifying the layout forms traditionally required for mathematics (superscripts, subscripts, fractions, tables, etc.) It also provides access to a repertoire of around 2000 Mathematical symbols. In MathML 1.x these symbols were accessed in XML as unofficial ``Private Use'' Characters, however MathML 2 has been able to take advantage of the proposed unicode extensions for mathematics in Unicode 3.1 and 3.2.
Content MathML contains XML elements denoting constructors such as function application, together with a fixed set of (typically empty) elements denoting mathematical functions. The intention is to provide the functions typically encountered in High School or an introductory University course. Content MathML also has mechanisms for extending beyond this fixed range, however in contrast to OpenMath's Content Dictionary mechanism there is no specified mechanism for defining the semantics of such extensions. In MathML one just gives a URI at which an external definition may be found in some format. In fact the majority of extension examples in the MathML specification use OpenMath Content Dictionaries as the target of these DefinitionURL links.
OpenMath is designed to encode the semantics of a mathematical expression, however it is often necessary to display such an expression in a conventional form. By specifying, for each symbol in an OpenMath dictionary, a fragment of Presentation MathML that describes the symbols and layout forms suitable for that function, it is possible to build a stylesheet that may convert arbitrary (XML encoded) OpenMath expressions built from those symbols into Presentation MathML. This transform could be encoded in any of several languages although XSLT, being a W3C specified language specifically for XML transforms, is currently the most popular. Given such a stylesheet it is possible to display OpenMath expressions using a MathML enabled Browser (For example Mozilla or Amaya, or a standard browser using the Techexplorer plugin or WebEq applet). Microsoft internet explorer, and some builds of Mozilla include an XSLT transform engine in the client browser. With such a system it is possible to serve OpenMath (rather than MathML or HTML derived from OpenMath sources) from a web server, and have it displayed directly in a client browser.
XSL stylesheets which provide this transformation have been produced for the majority of Content Dictionaries developed by the consortium. They will be made available from the OpenMath Society web site.
As an alternative to using stylesheets, or as an ``override'' mechanism
for special circumstances it is also possible directly to embed the
Presentation form in the document. Either the OpenMath expression
may use an OpenMath attribution object giving the MathML form, or
perhaps more usefully for a browser that may render MathML, a MathML
expression may use the MathML <semantics> element to annotate the
presentation form with further semantics given in OpenMath.
<sin/>) that cover a fixed but useful range of ``elementary''
mathematics. This Consortium and the W3C Math Working group have ensured
that the development of the Content elements in MathML2, and the
development of the core OpenMath Content Dictionaries (which are now
known as the MathML Content Dictionary Group) occurred in parallel.
As far as possible, symbols in the MathML CD group correspond directly
to specified attribute/element combinations in MathML, and vice-versa.
There are some
small areas (especially in the treatment of complex and floating point
numbers) where there is not a simple one-to-one correspondence between
the OpenMath and MathML constructs, but in general
the translation in both directions is fully specified, and is being
implemented, again using XSLT.
Thus the OpenMath syntax, being more regular and more naturally extensible, is perhaps more suitable for more advanced uses and users; whereas the more compact and limited Content MathML syntax may be more suitable for uses within its target audience of High School Mathematics. So for example the MathML
<apply><sin/> <ci>x</ci> </apply>may be seen as a convenient shorthand for
<OMA><OMS name="sin" cd="transc1"/> <OMV name="x"/> </OMA>However for a more advanced topic one sees that the OpenMath syntax remains regular, but Content MathML requires a different syntax to refer to extended functionality, and has no single method of encoding the semantics. We use an OpenMath CD here, but MathML does not enforce any particular format for the documents referred to by definitionURL attributes.
<csymbol definitionURL="http://www.openmath.org/cd/group1.ocd" encoding="OpenMath"> DerivedSubgroup </csymbol>this is a Content MathML expression equivalent to the OpenMath
<OMS name="DerivedSubgroup" cd="group1"/>
The details of the mapping between Content MathML and the OpenMath MathML CD Group are given in the document Conversion between MathML and OpenMath.
In addition to the csymbol element used above (and which has been added
to MathML 2 intentionally to mirror the facilities of the OpenMath
Symbol construct) MathML has two other extension mechanisms (which were
both available in MathML 1). Firstly most Content MathML elements take
the definitionURL attribute which, as in csymbol, may point to further
semantics (in any format). This may be used to override the default
semantics of a MathML element. So for example <times/> has
default semantics being commutative multiplication on subsets of the
Complex numbers. However <times definitionURL="...."/>
would denote a (presumably related) symbol, but with a definition
modified by the text at the specified URI. This technique has advantages
in the MathML context that the construct may inherit some properties, in
particular a default presentation, from the built in <times/>
element. However if inheriting such defaults is not an issue
is usually preferable to use the new csymbol construct.
The final extension mechanism is the MathML semantics element.
This is very similar to the OpenMath attribution mechanism.
It allows an entire subtree of an expression to be ``wrapped''
as the first child of a semantics element. Subsequent children
may annotate this base in various ways. In particular one might annotate
a Presentation MathML expression with its OpenMath or Content MathML
equivalent, or to give equivalent expressions in computer algebra
systems, etc.
We believe that the emergence of MathML at this time is a great advantage for OpenMath. Not only does it offer a presentation mechanism suitable for use in browsers and interactive environments, but being a W3C Recommendation it is likely to be quite widely implemented whether natively or in plug-ins. We hope that MathML is a success and that OpenMath can exploit this to gain exposure and popularity.
