Mappings between presentation markup and semantic markup for variable size objects

Bill Naylor
Ontario Research Centre for Computer Algebra
The University of Western Ontario
London Ontario CANADA N6A 5B7
bill@orcca.on.ca

Abstract:

In this paper we give a brief overview of the paper "meta stylesheets for the Conversion of Mathematical Documents into Multiple forms" [1]. We propose an extension to the techniques proposed in [1] in order to overcome the problem of specifying notation for objects of a varying size, e.g. matrices or vectors. The extensions proposed in this paper are based on regular expression or XML Schema like ideas. We discuss the basics of using XSLT for performing the calculations defined by the template functions. We discuss how we may use the document element to access external processes which may be required to perform calculations for which XSLT is ill-suited. Finally, we suggest a way in which a user of the extended MathML we propose may specify different notation styles at different levels within an object.

1 Introduction

It is an unfortunate fact that due to the varied history of mathematics, mathematical notation is ambiguous. There is a many to many relationship between the meaning of a mathematical object and the notation which mathematicians use to write them down (as noted in chapter 4 of the MathML specification [2]). A document marked up using the OpenMath ([4] and [3]) markup language is a possible alternative approach which does not suffer from this problem. However, the OpenMath concept does not address the issue of presentation markup. Ideally, we want some markup which gives flexible presentation markup for semantically unambiguous mathematics.
We discuss a possible solution to this quandary in our paper [1] in which a method is proposed whereby a set of ntn files (which may be held in a parallel directory structure to the OpenMath content dictionaries, in a similar way to the sts small type system) hold information which declares a mapping between an OpenMath symbol and a variety of different presentations for that symbol, marked up using presentation MathML. These different presentations will be matched against a semantic-template, which will provide a prototype for the mathematical object. The prototype will typically be an OMA element, where the first child is the symbol for which we are giving notations. The ordering of the parameters for a symbol are conserved by means of xref and id attributes on the presentation and the semantic side of the mapping respectively.
Example: 1   We give an example of the entries to the notation element for the BesselJ symbol, which detail the notations given in the traditional style; $J_\nu(z)$ , and that used by Maple 7; BesselJ(n,m): 

      <Notation>
        <version precedence="10">    <!-- The default style -->
          <math><mrow>
            <msub><mi>J</mi>
              <mrow>
                <value xref="arg1">&nu;</value>
              </mrow>
            </msub>
            <mo>&ApplyFunction;</mo>
            <mfenced>
              <value xref="arg2">z</value>
            </mfenced>
          </mrow></math>
        </version>
        <version precedence="10" style="maple"> <!-- The Maple Notation -->
          <math><mrow>
            <mi>BesselJ</mi><mo>&ApplyFunction;</mo>
            <mfenced>
              <value xref="arg1">n</value>
              <value xref="arg2">z</value>
            </mfenced>
          </mrow></math>
        </version>
        <semantic_template><OMOBJ>
          <OMA><OMS cd="bessel" name="BesselJ"/>
            <OMV name="nu" id="arg1"/>
            <OMV name="x" id="arg2"/>
          </OMA>
        </OMOBJ></semantic_template>
      </Notation>


Occasionally it occurs that part of the presentation for a notation for a particular symbol do not correspond to a particular part of the prototype of the semantic template, for example the superscript n in the partial derivative presentation:

$\frac{\partial^n}{\partial x^{n_1}\partial y^{n_2}}f(x,y)$

is redundant, (implicitly $n=n_1+n_2$), and so has no corresponding part in an efficient scheme for a semantic template. As a solution to this problem, we have proposed that a template function, marked up in OpenMath markup, should be present in the ntn file. The function may have parameters, the example above however would have a nullary template function. A calculation represented in the body of the function, will be an (OpenMath) expression involving its parameters and possible references (via xref pointers) to the prototype. For the example above the template function would be $\{\}\rightarrow n_1+n_2$, where $n_1$ and $n_2$ would be references to objects in the prototype (the XML markup for this particular case is given in Appendix 1). The particular presentation which is required in a specific document will be specified either by a particular attribute, or via a set of defaulting mechanisms (which are detailed in [1]). This mechanism allows an author freedom to use the notation he (or she) wants, whilst retaining precise mathematical meaning.

Example: 2   In this example we assume the availability of the Notation element given in example 1. An author may select the Maple style of notation by giving the following markup. It should be noted that the ordering of the parameters will be as given in the respective content dictionary. The following shall be displayed as BesselJ(1,2):1 

<xm:apply><xm:BesselJ style="maple"/>
  <xm:mn> 1 </xm:mn> <xm:mn> 2 </xm:mn>
</xm:apply>

It would appear that these mechanisms are sufficient for providing a presentation/semantics mapping for mathematical objects which take a specific number of parameters. However the situation is more complex in the case of mathematical operators which take a variable, or arbitrary number of parameters, for example nary functions like plus which in OpenMath terminology is represented by the symbol: 

<OMS cd="arith1" name="plus"/>
or a constructor of objects with no predetermined size, for example the basic vector constructor, in OpenMath this is represented by the symbol: 

<OMS cd="linalg2" name="vector"/>
Some attempt has been made to address this problem in the OMDoc [5] specification with the introduction of the presentation element. The style of the presentation may be controlled via a number of attributes which control various aspects of the presentation, e.g. the bracketing style, fixity etc. However notations exist for which there is no obvious way to include the relevant presentation markup in a presentation element apart from including XSLT in an XML CDATA section in the OMDoc. This is discouraged even in the OMDoc specification, which recognizes that hand-coding XSLT is "a tedious and error-prone process". An example of such a presentation is the standard presentation for a simple continued fraction:
$a_0+\frac{\displaystyle 1}{\displaystyle a_1+\frac{\displaystyle 1}{\displaystyle a_2+\frac{\displaystyle 1}{\displaystyle a_3+\cdots}}}$

2 Specification Languages for classes of documents or texts

A regular grammar allows specification of classes of sentences in a string. For example:

It is matches of the final type with which we shall be primarily interested.
In a similar way XML Schema [6] allows specification of a class of objects in an XML document in an XML format. One may specify sequences of elements with particular types, containing children of particular types. It is also possible in XML Schema to restrict the names and values of attributes belonging to a specific element. Text content of an element may be restricted in an exact manner. The subset of the regular expression grammar that we are especially interested in when expressed in the XML Schema framework, is shown by the following example:2 

<xsd:sequence minOccurs="1" maxOccurs="5">
  <xsd:element name="mn" type="xsd:decimal"/>
</xsd:sequence>
This fragment would match a sequence of elements which consisted of between 1 to 5 occurrences of the element "<mn> nn.nn </mn>", where nn.nn denotes some decimal content.
We incorporate some of the ideas from XML Schema in the extended MathML language we propose. However we have necessarily made some changes for the following reason. In order to use the template function technology outlined above, it is not sufficient to merely have a way of specifying the sets of objects we would like to operate on (as with XML Schema or regular grammars), we must also have a handle on the individual instances of the objects. We also realise that we do not require the full power of a regular grammar (or XML Schema) for our purposes, for example, we do not require the ability to choose members of a class.


3 Extension to the Specification Presentation MathML

With the considerations of the previous section in mind we propose a slight change to the part of the XML Schema language that we shall utilise. A dtd fragment detailing this is shown below: 
<!ELEMENT sequence (minOccurs,maxOccurs,particle)>
  <!ATTLIST sequence var NMTOKEN #REQUIRED>
<!ELEMENT minOccurs (#PCDATA)>
  <!ATTLIST minOccurs val NMTOKEN #IMPLIED
                      xref NMTOKEN #IMPLIED>
<!ELEMENT maxOccurs (#PCDATA)>
  <!ATTLIST maxOccurs val NMTOKEN #IMPLIED
                      xref NMTOKEN #IMPLIED>
<!ELEMENT particle ANY>
The meaning and reasons for including these different elements follows:

Example: 3   we give an example of a notation element for the nary function plus.
  <Notation name="plus">
    <version precedence="10">
      <math>
        <value xref="itheltPlus">1</value>
        <sequence var="i">
          <minOccurs val="2"/><maxOccurs xref="sizePlus"/>
          <particle>
            <mo>+</mo><value xref="itheltPlus">i</value>
          </particle>
        </sequence>
      </math>
    </version>
    <semantic_template>
      <OMOBJ><OMA>
        <OMS cd="arith1" name="plus"/>
        <OMV name="seq" id="sequencePlus"/>
      </OMA></OMOBJ>
      <OMBIND id="sizePlus"><!-- return the length of a sequence -->
$\{\}\rightarrow {\tt\#sequencePlus.i}$ 
      </OMBIND>
      <OMBIND id="itheltPlus"> <!-- return the i'th element of a sequence -->
${\tt i}\rightarrow {\tt\sequencePlus.i}$ 
      </OMBIND>
    </semantic_template> 
  </Notation>

The template functions of the preceding, perform utility operations like selecting the i'th parameter to an nary operator, or selecting the i'th element of a vector.
More complex situations, for example notation specification for the structured matrices, specified in the content dictionary linalg5 involve complicated mappings between the presentation and semantics, these often involve a two dimensional nesting of the XML Schema (like) expressions.

4 Different styles at different levels

It is sometimes necessary to specify different styles at different nesting levels within the notation for some mathematical objects, for example, if it was required to print a matrix with style="round" which had entries which where rational numbers and it was required that they appear with style="display_style". This is no problem if all the content of the matrix appears in the Extended MathML markup, for example, in order to markup a matrix which was to appear as:
(we are assuming that this is the display intended by the options style="round" for matrix and style="display_style" for rationals.)

$\left ( \begin{array}{ccc} \frac{\displaystyle 1}{\displaystyle 2}&\frac{\displaystyle 1}{\displaystyle 3}\\ \frac{\displaystyle 1}{\displaystyle 5}&\frac{\displaystyle 1}{\displaystyle 7} \end{array} \right )$

one could use the following Extended MathML markup:
<xm:apply>
  <xm:matrix style="round"/>
  <xm:apply>
    <xm:matrixrow/>
    <xm:apply>
      <xm:rational style="display_style"/>
      <xm:mn>1</xm:mn><xm:mn>2</xm:mn>
    </xm:apply>
    <xm:apply>
      <xm:rational style="display_style"/>
      <xm:mn>1</xm:mn><xm:mn>3</xm:mn>
    </xm:apply>
  </xm:apply>
  <xm:apply>
    <xm:matrixrow/>
    <xm:apply>
      <xm:rational style="display_style"/>
      <xm:mn>1</xm:mn><xm:mn>5</xm:mn>
    </xm:apply>
    <xm:apply>
      <xm:rational style="display_style"/>
      <xm:mn>1</xm:mn><xm:mn>7</xm:mn>
    </xm:apply>
  </xm:apply>
</xm:apply>

However we do have a problem if the elements of the output are implicit in the style for that object. For example, the markup for a 3x3 identity matrix with style="square" in Extended MathML would be: 

<xm:apply>
  <xm:identity style="square"/>
  <cn>3</cn>    <!-- N.B. this element does not occur in the presentation, -->
                <!--      so we use content MathML. -->
</xm:apply>
This gives us no freedom to specialise the specification of the '0's and '1's which will appear in this markup. One could give a different version associated with the identity symbol for each different style of '0' or '1', but this was thought undesirable as it would mean that the number of version elements necessary to describe every presentation for a symbol would be exponential in the depth of implicit features in the notation. Another reason why do not proceed in this direction is that it is conceivable that there may exist notations, which have some recursive nature to their implicit features, and this would imply an infinite nature to the number of versions which where necessary. Clearly it is impossible to provide these. In order to avoid this problem we propose supplying this information in the value of the style attribute which must be supplied to specify anything but a default style.
The value of the style element should be a concatenation of the value used to specify the style for the outermost element, the '[' character, a sequence of strings to specify styles for inner elements separated by the ';' character and the ']' character. The strings used to specify inner elements should be modeled on the string "name:val", where name is the name of the element and val is the value used to specify the style attribute for this element (which may have internal elements, in which case the same scheme is used recursively to provide a value for val). i.e.

style="val[name1:val1;name2:val2 \cdots ]"

where val is the name of the outermost element, and name1, name2, etc are the names of the inner elements, which have values val1, val2, etc.

For example if it was required to display a 3x3 identity matrix as:


$\left [\begin{array}{ccc}{\bf 1}&{\it 0}&{\it 0}\\{\it 0}&{\bf 1}&{\it 0}\\{\it 0}&{\it 0}&{\bf 1}\end{array} \right ]$

as opposed to

$\left [\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right ]$

we would use the following markup: 
<xm:apply>
  <xm:identity style="square[zero:italics;one:bold]"/>
  <cn> 3 </cn>
</xm:apply>

5 Meta Stylesheets

The main theme of the paper [1] concerns the process of producing meta stylesheets to construct XSLT stylesheets [7] for converting documents written in an extended MathML into another form, for example presentation MathML. Now, one of the most difficult problems encountered in the construction of these meta stylesheets, is how to deal with the template functions. It is clear that for different template functions one may (not always, as certain mathematical objects are structurally similar with respect to their components, e.g. consider a vector and a list object) require different meta stylesheets in order to perform the desired translation. This could be seen as a never ending job if one is allowed total freedom in the OpenMath symbols used in the template functions (especially since OpenMath is an extensible markup mechanism). It is therefore necessary to specify a restricted vocabulary for use within the template functions. It may also occur that XSLT is ill-suited for performing some of the calculations implied by the template functions. We may use the XSLT document function, element in this case. One use of the document element would be to execute a Java method designed to perform the required calculation via a servlet. The Java method could take XML elements as its parameters and return XML as its return value, e.g. it might operate on an OpenMath expression and return a minimal form.

6 Conclusion

By the use of the extensions suggested in section 3 to the scheme suggested in [1] we may specify effective presentation/semantic mappings for mathematical objects which are not of a fixed size. With respect to a database of notation/content dictionary mappings written using this scheme, it will then be possible to specify mathematical notation which has an unambiguous meaning. We also suggest an effective markup which allows specification of different styles for implicit objects appearing at any level within a notation. With respect to implementing some conversions of this extended markup to some other form (for example, presentation MathML), we suggest the use of the XSLT document element, in conjunction with servlets to communicate with external processes for performing calculations which are ill-suited to XSLT.

1 Appendix: Partial Differentiation example

3 We display a notation element for the partialdiff symbol. The version of the notation used here is that which will display as $\frac{\displaystyle\partial^n}{\displaystyle\partial x^{n_1}\partial y^{n_2}}f(x,y)$. Where the function f is differentiated with respect to x and y. The order of the differentiation is $n_1$ and $n_2$ with respect to x and y respectively and $n=n_1+n_2$. 
<Notation>
  <version precedence="100">
  <image src="partialDiffDefault.gif"/>
  <tex>
\frac{\partial^
     ...
  </tex>
  <math>
    <mrow>
      <mfrac>
        <msup><mo>&part;</mo><mi xref="sum">n</mi></msup>
        <mrow><msup>
          <mrow><mo>&part;</mo>
            <mi xref="x">x</mi>
          <mrow>
          <msub xref="n1"><mi>n</mi><mn>1</mn></msub></msup>
          <mrow><mo>&part;</mo>
            <mi xref="y">y</mi>
          <mrow>
          <msub xref="n2"><mi>n</mi><mn>2</mn></msub></msup>
        </mrow>
      </mfrac>
      <mi>f</mi>
      <mo>&ApplyFunction;</mo>
      <mfenced>
        <mi xref="x">x</mi>
        <mi xref="y">y</mi>
      </mfenced>
    </mrow>
  </math>
  </version>
  <semantic_template>
    <OMOBJ> <!-- this OMOBJ is the prototype -->
      <OMA><OMS cd="calculus2" name="partialdiff"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMA><OMS cd="list1" name="list"/>
            <OMI id='x'> 1 </OMI>
            <OMV name="n1" id="n1"/>
          </OMA>
          <OMA><OMS cd="list1" name="list"/>
            <OMI id='y'> 2 </OMI>
            <OMV name="n2" id="n2"/>
          </OMA>
        </OMA>
        <OMV name="f" id="functionName"/> <!-- this is the function -->
      </OMA>
    </OMOBJ>
<!-- the following function specification calculates the sum of the the two 
     orders located in the prototype -->
    <OMBIND id="sum">
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
      </OMBVAR>
      <OMA>
        <OMS cd="arith1" name="plus"/>
        <OMV name="n1" xref="n1"/>
        <OMV name="n2" xref="n2"/>
      </OMA>
    </OMBIND>
  </semantic_template>
</Notation>

Definition of some Non Standard Symbols

We give some definitions of symbols which have been used in the preceding, but which are not part of the standard OpenMath content dictionaries at present.
partialdiff:
This symbol represents the partial-differentiation of a function with respect to a number of variables. The arguments should be given in the following form:
first
argument - A list of pairs, where the first element of each pair is an index to the variable with respect to which the differentiation is taken, the second element is the order of that differentiation.
second
argument - The function on which the partial differentiation is taking place.

Bibliography

1
Bill Naylor, Stephen Watt: Meta Stylesheets for the Conversion of Mathematical Documents into Multiple Forms, Proceedings of the first International Workshop on Mathematical Knowledge Management, September 2001
2
MathML: http://www.w3.org/TR/MathML2
3
O.Caprotti, D.P.Carlisle, A.M.Cohen: The OpenMath Standard, February 2000
4
The OpenMath Society: http://www.openmath.org/
5
OMDoc, A standard for Open Mathematical Documents:
http://www.mathweb.org/omdoc/
6
XML Schema: http://www.w3.org/XML/Schema
7
XSLT: http://www.w3.org/TR/xslt

Footnotes

...BesselJ(1,2):1
in this markup it is assumed that the prefix xm has been bound (in an enclosing element) to a URI indicating the Extended MathML namespace.
... example:2
The xsd prefixes which appear on some elements will be declared to imply the XML Schema namespace; http:www.w3.org/2001/XMLSchema. This declaration should appear in some ancestor element.
... 3
We note here that the symbol partialdiff from the content dictionary calculus2 are not part of the OpenMath standard at present.
Bill Naylor 2002-03-30