\MakeShortVerb{\|}
\def\cmd#1{\texttt{\char'134#1}}
\title{How does \acro{HTML} handle mathematics?}
\author{Malcolm Clark}
\begin{Article}
\section{Introduction}
There is a very short answer to the question posed in
the title: not at all. However, as an instantiation of
SGML (see, for example, \cite{EvH}, \cite{Goossens} and \cite{Bryan}),
we can look first at how mathematics is handled in other
\acro{SGML} \acro{DTD}s; and then examine how the expired \acro{HTML}3 draft
proposed to include mathematics.
The \acro{DTD}s already available which are designed to
handle mathematics include \acro{ISO}\,9573 (\cite{ISO9573})
(confusingly, also known as \acro{ISO}\,12083),
which is part of \acro{CALS}, \acro{AAP} (Association of
American Publishers) (\cite{AAP}), and the \acro{HTML}3 draft (\cite{math}).
The Euromath \acro{DTD} might also be relevant (\cite{Euromath}),
but since its status is closer to that
of proprietary it is too awkward to consider here.
Van Herwijnen (\cite{EvH}) comments on the first two,
comparing them to \TeX\ and |eqn|.
A longer and more detailed examination of the \acro{AAP}, Euromath and
\acro{ISO}\,12083\slash \acro{ISO}\,9573 is given in Poppelier, van
Herwijnen and Rowley (\cite{PvHR}).
Van Herwijnen provides an example from physics for the decay of a particle
together with representations in
\TeX, |eqn|, \acro{ISO}\,9573 and \acro{AAP}.
The equation is:
\begin{displaymath}
\Gamma(J/\psi\rightarrow\eta_c\gamma)
= \frac{\alpha Q_c^2}{24}
\left\vert A(J/\psi\rightarrow\eta_c\gamma)\right\vert^2
\frac{m_\psi^2}{m_{\eta_c}^2}
\left(1-\frac{m_{\eta_c}^2}{m_\psi^2}\right)^3
\end{displaymath}
The entire expression is too extensive to compare here, but
the left hand side of the equation in \acro{ISO}\,9573 may be
given as
\begin{Verbatim}
Γ J/ψ → η
_{ c} γ
\end{Verbatim}
while using the AAP dtd, it could be
\begin{Verbatim}
G(Jy &ar; h
cg)
\end{Verbatim}
For the sake of completeness, the |eqn| alternative is
\begin{Verbatim}
Gamma(J/psi rarrow eta sub c gamma)
\end{Verbatim}
Eric's comments are interesting.
He comments that the two \acro{SGML} representations are cumbersome
and difficult to read, especially when contrasted to \TeX\ and |eqn|.
He also rails against the obsession with representation.
For example $\Gamma$ `means' decay width, but as far as the \acro{DTD}s are
concerned we have \texttt{\Γ} and \texttt{G}.
To be fair, \TeX\ and |eqn| hardly
fare better, but at least we do know that we could have
provided a more meaningful command. The second representation (\acro{AAP}) is
particularly unfortunate, since instead of treating the symbol
as a symbol, it treats it as a Greek letter. Of course, Eric is deeply
imbued with the basic notions of \acro{SGML}, and would be very
sensitive to this.
His contention is that someone who already knew \TeX\ or |eqn| would
have no motivation for learning or using these rather baroque alternatives.
There is a point to be made about the rather cumbersome nature of
the \acro{SGML}. Writing it by hand will be cumbersome, but surely no-one
ever wants to write in this way. Structure editors are available.
In the \TeX\ world, Scientific Word gives a structure editor
for \LaTeX. This can be done since it is possible
to hold an equation as elements in a tree structure, so that modification
or correction to an element can be managed quite simply, and changes can
propagate down the tree. The same sort of thing exists within the \acro{SGML}
world. Euromath uses the Grif (\cite{Grif} and \cite{GrifSA})
editor for just this, and it would
be easy to see other similar editors, like Chamberlin's
Quill (\cite{Chamberlin})
maintaining the information. There is a question lurking whether
mathematicians would actually like to input in this way.
Just as experienced keyboarders find
\acro{GUI}s very difficult and slow to use, perhaps the same sort of
resistance would
be found. However, the real point is that humans should not be expected
to write \acro{SGML}. If they really must write \LaTeX, then an approach
like Scientific Word, which could be coerced into generating a tree structure
which could be mapped onto a \acro{DTD}, is potentially more valuable.
A general issue, which Eric raises implicitly, is that none of the \acro{DTD}s
offer a way of encoding meaning in a flexible way. Either the
element is present already, or it is not. There appears
to be no straightforward way of extending the range of elements. In the world
of high energy physics and mathematics this must be something
of a straightjacket. On the other hand, the bane of many editors'
lives is the ease with which individual authors
can `extend' \TeX\ or \LaTeX\ by adding a few new definitions.
\subsection{SGML Notations}
If we really did have existing equations, then one way to
handle them within \acro{SGML} is through a Notation (see also \cite{Bryan}).
A Notation permits
a document to include data which is not to be parsed. It is
therefore possible to include \TeX\ or \LaTeX\ and assume that
at that point a convenient processor will be magicked to deal with
it. He gives the example of the definition in a \acro{DTD}:
\begin{Verbatim}
\end{Verbatim}
which may then be used later with the |Formula| element as
\begin{Verbatim}
...
\end{Verbatim}
A scheme which already maps \acro{SGML} to \LaTeX\ (e.g.~\cite{Flynn} or
\cite{Goossens2}) would
find this a very easy way to absorb maths, provided all the
equations were in the same notation. The prospect of a
\cmd{newcommand} or \cmd{def} within the Notation could be
worrying.
Although this sort of expedient is plausible, it is not entirely
successful. One of the arguments behind the use of \acro{SGML} is that it codes
structure or meaning, rather than appearance (to echo one of Eric's points).
\TeX\ and \LaTeX\ sometimes code meaning,
but not in a consistent and reliable way.
How do we extract information? If we have \acro{SGML}, it is relatively
easy to find corresponding structural elements, which may then be extracted.
Once we start including a Notation, this chance is all but gone;
and if we include alternative Notations (say \TeX, \LaTeX\ \emph{and} |eqn|)
it becomes even more problematic.
It is not clear to me how these Notations, or even the \acro{DTD}s
differentiate between in-line and displayed equations. I assume
that an attribute could be included which specified the style.
On the other hand \acro{ISO}\,12083 distinguishes between
in line, displayed and `display formula groups' styles, through the
use of different elements.
There is perhaps a deeper question here,
which is this, should it be at all relevant?
Should the author be able to specify that
some equations are in-line and others are to be displayed?
It should make no difference at all to the content, although
it would make great changes to the appearance. But to deal with maths
is to deal with appearance, to a large extent. The display seems
to be a key issue. Since many equations are strongly two dimensional
(as opposed to the one dimensional nature of most text), it is a key question
whether it is reasonable to expect this aspect to be reflected in
any linearisation.
To give \acro{ISO}\,12083 its due, it says ``Since there is no consensus
on how to describe the semantics of formulas, it only describes
the presentational or visual structure.''
\section{\acro{HTML}3 (expired draft)}
\acro{HTML}3 supports a |_{|.\footnote{The notion of exponent and index may be welcome,
since we would expect \acro{HTML} to be concerned with the underlying
content. Perhaps \TeX\ does tend to overload the idea, since a superscript may
signify more things than just an exponent. A semantic separation
might be a very good
idea, but this is unlikely to be the intention, since they are
|shortref| characters for || and ||.
Poppelier \textsl{et al.}\ put this very succinctly:
``What is the function of the 2 in SU$_2$, $\log_2x$, $x_2$, $x^2$ $T_2^2$?
In SU$_2$ it is the number of dimensions in the Lie group; in $\log_2x$ it is
the base of the logarithm; if $x$ is a vector, the ${}_2$ in $x_2$ is an index;
the ${}^2$ in $x^2$ could be a power, but if $T$ is a tensor, the ${}^2$ in
$T_2^2$ is a contrainvariant tensor index.''
} Since \acro{HTML} syntax and \LaTeX\
syntax are rather different, the ease of \LaTeX's sub- and super-scripting has
to be abandoned. The \acro{HTML} tag has to be terminated. It is
unfortunate that there
was not a way of employing an implied end tag.
For example
\begin{small}
\begin{displaymath}
a^{23}_n
\end{displaymath}
\end{small}
is given from
\begin{Verbatim}
$a^23^\_n\_$
\end{Verbatim}
Should you need to subscript a
subscript, the |shortref| form cannot be used. Although |$|\; does\; support\; a\; grouping\; operator,\; \backslash begin\{displaymath\}\; a\_\{b\_c\}\; \backslash end\{displaymath\}\; is\; obtained\; from\; \backslash begin\{Verbatim\}$ a$$bc
\end{Verbatim}
Perhaps cleverly, superscripting an expression with a binary operator results
in the expression being placed over the operator, like \cmd{stackrel}.
For example
\begin{displaymath}
A \stackrel{\alpha'}{\longrightarrow} B \stackrel{\beta'}{\longleftarrow} C
\end{displaymath}
would be
\begin{Verbatim}
$A\; \to ^\alpha \prime ^\; B\; \leftarrow ^\beta \prime ^\; C$
\end{Verbatim}
In passing note that the use of the `prime' operator is different from
\TeX\slash \LaTeX\ use, and the use of space. I am not entirely clear about the
use of space in ||. It is certainly not ignored, as in \TeX, and
the draft does comment on the use of different horizontal white space
within the equation. The draft runs
\begin{quote}
Spacing between constants, variables and operators is determined
automatically. Additional spacing can be inserted with entities such as
| |, |&sp;| and |&quadsp;|. White space in the markup
is used only to delimit adjacent variables or constants. You don't need
spaces before or after binary operators or other special symbols as these
are recognised by the \acro{HTML} math tokeniser. White space can be
useful, though, for increased legibility while authoring.
\end{quote}
This does imply a rather different use of space.
This use of space does have the effect that a `prescript' can be made
quite unambiguously simply by ensuring it is preceded by a space.
It would imply
a string |sin| is recognised as `some sort of function'. The string |xyz|,
would also presumably imply `function', while in \LaTeX\ it would imply
the three variables $x$, $y$ and $z$.
It does leave unclear how
\(\sin^2\theta \) and
\begin{displaymath}
\max_{i=1}^n x_i
\end{displaymath}
would be handled with ease. If the $\sin^2\theta$ uses ||
but the
\begin{displaymath}
\max_{i=1}^n x_i
\end{displaymath}
requires || and || then
we have an interesting inconsistency.
|$|\; does\; adopt\; \backslash TeX/\backslash LaTeX\text{'}s\; notion\; of\; binary\; operators,\; and\; in\; general\; claims\; to\; reflect\; the\; assumptions\; of\; \backslash TeX/\backslash LaTeX.\; It\; does\; not\; however\; provide\; support\; for\; multi-line\; equations,\; stating\; that\; `this\; can\; be\; effectively\; handled\; by\; combining\; math\; with\; the\; |TABLE|\; element\text{'}.\; To\; me\; this\; wanders\; far\; from\; the\; basic\; concepts\; of\; \backslash acro\{SGML\}.\; However,\; what\; it\; appears\; to\; mean\; is\; that\; the\; ||\; tag\; uses\; the\; same\; sort\; of\; syntax\; as\; |$|, not that
an array uses the table tags.
From the draft, it is anticipated that chemistry could be set from within
the || tag. I would view this as a mistake. It may be (almost) defensible
from within \LaTeX\ to use math structures, although the various chemistry
packages at least try to separate the notions. It seems unfortunate that HTML3
should not attempt something similar.\footnote{And of course this
emphasises the
inadequacy of referring to a subscript as an index and a superscript as an
exponent. The terms are pretty meaningless for chemical notation.} An example
might be
\begin{Verbatim}
Fe_2_^2+^+Cr_2_O_4_
\end{Verbatim}
for
\begin{displaymath}
\mathrm{Fe}_2^{2+}+\mathrm{Cr}^{}_2\mathrm{O}^{}_4
\end{displaymath}
where the different notational style of chemistry is tackled, notably
its use of an upright font and consistent baselines for subscripts.
Some hints on appearance are provided: it is expected that functions
(operators), numbers and other constants are portrayed in an upright font, and
variables are italic. Unlike \TeX/\LaTeX, limits for integrals and summation
signs are said to be placed directly above or below, or to the immediate right
(depending on the symbol). Unfortunately, the draft does not indicate quite
what this ambiguous term means. I suppose it does not mean `emulate' the
\TeX/\LaTeX\ mode, though that is obviously plausible, and from the point of view
of a browser author could be a reasonable path.
What does it look like?
\begin{Verbatim}
$\int \_a\_^b^\{f(x)1+x\}\; d\; x$
\end{Verbatim}
for
\begin{displaymath}
\int_a^b\frac{f(x)}{1+x}dx
\end{displaymath}
Note that the sub- and
super-scripts, like \TeX/\LaTeX\ also denote limits.
Some maths accents are available: ||, ||, ||, ||, ||
and ||. There are no explicit equivalents for \cmd{check}, \cmd{breve},
\cmd{acute} and \cmd{grave}, although they could be created with ||.
Another borrowing from \TeX/\LaTeX\ is the notion of grouping:
\acro{HTML}3 uses a || element where \TeX/\LaTeX\ would use parentheses.
|| can be replaced by a |shortref| form of |{|
and |}|, which greatly aids brevity and comprehension.\footnote{Should
you need the symbols themselves, they are obtained by
the entities \texttt{\{} and \texttt{\}}.}
Although \TeX nically a braced group is a sort of `box',
perhaps || might have
been a better, though less concise term, in the context.
It is perhaps an unfortunate choice, since `box' carries overtones for
many \TeX\
users. Still, it does ensure that all the power of grouping is present (fairly
essential in view of the || element). To overload slightly, one of the
attributes of the |$|\; element\; is\; |box|,\; which\; causes\; a\; box\; to\; be\; drawn\; around\; the\; formulae.\; The\; ||\; element\; is\; used\; in\; a\; number\; of\; ways;\; it\; is\; used,\; for\; example\; with\; the\; ||\; and\; ||\; commands\; for\; delimiters\; which\; grow.\; This\; leads\; to\; a\; rather\; strange\; construction:\; \backslash begin\{Verbatim\}$ f(x)=(1+xsin\; x)\; <\backslash box>$\backslash end\{Verbatim\}\; where\; |(|\; gives\; a\; left\; parenthesis\; of\; appropriate\; size\; and\; |)|\; gives\; the\; corresponding\; right\; parenthesis.\; As\; with\; \backslash TeX,\; it\; is\; recognised\; that\; sometimes\; it\; may\; be\; necessary\; to\; have\; a\; delimiter\; larger\; than\; `default\text{'}.\; ||\; therefore\; has\; a\; |size|\; attribute\; to\; enable\; this\; to\; happen.\; The\; permitted\; values\; are\; |normal|,\; |medium|,\; |large|\; and\; |huge|.\; The\; |shortref|\; form\; cannot\; take\; attributes.\; Integrals\; (and\; other\; large\; operands\; which\; are\; stretchy)\; also\; need\; the\; use\; of\; ||,\; without\; any\; corresponding\; ||.\; For\; example\; \backslash begin\{Verbatim\}$ \int \_-\&inf;\_^\&inf;^f(x,y)x^2^+y^2^d\; x\hspace{0.17em}d\; y$\backslash end\{Verbatim\}\; should\; give\; \backslash begin\{displaymath\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\backslash frac\{f(x,y)\}\{x^2+y^2\}dx\backslash ,dy\; \backslash end\{displaymath\}\; Although\; |$ |\; is\; said\; to\; recognise\; functions\; as\; maths\; entities,\; there\; is\; no\; list\; of\; all\; the\; entities\; given\; in\; the\; draft.\; However,\; straightforward\; elements\; like\; ||\; and\; ||\; exist\; which\; work\; the\; same\; as\; their\; \backslash TeX\backslash \; counterparts.\; The\; \backslash LaTeX\backslash \; notation\; for\; \$\backslash root\; n\; \backslash of\; x\$\; however\; is\; \backslash cmd\{sqrt\}\; with\; an\; optional\; argument:\; |\backslash sqrt[n]\{x\}|,\; not\; as\; HTML3:\; \backslash begin\{Verbatim\}$ nx$\backslash end\{Verbatim\}\; which\; is\; rather\; closer\; to\; \backslash TeX.\; What\; the\; draft\; terms\; `Greek\; letters\text{'}\; are\; available\; in\; a\; similar\; way\; to\; \backslash TeX/\backslash LaTeX\backslash \; as\; entity\; references\; (and\; includes\; omicron).\; Some\; control\; over\; font\; styles\; is\; available\; through\; |$$| and ||. The first
emboldens, while the latter makes upright (Times?). You may even combine
the two as ||. Note that these changes apply to variables and constants,
and not to numbers, delimiters, operators and other symbols. An interesting
attribute is |class|, so that we could identify a vector as
\begin{Verbatim}
$$a=A′
\end{Verbatim}
Arrays or matrices are quite verbose, but broadly similar to \LaTeX. They are
introduced by ||, while each row starts with || and each cell with
an ||. This is in line with HTML's || model. Adapting some aspects
of \LaTeX, column definitions can be added, for example
|coldef="CCCC"|, the default, where columns are centred. The alternatives are
|R| and |L|. This is one of the few instances in HTML where case is vital. If the
attributes are separated by |+|, |-| or |=| this will propagate down the array
as a separator. For example
\begin{Verbatim}
\end{Verbatim}
Other attributes include |ldelim| and |rdelim| to specify the right and left
delimiters of the array or matrix. Unlike other instances where a name is used,
the symbol itself appears to be used in this context.
\subsection{Summary}
I tend to think that this makes it more difficult for someone with a \LaTeX\
background to interpret ||. When two languages are quite different, there
is rarely confusion in flipping from one to the other, but when they share
many similarities it can be frustratingly simple to converge at all the wrong
times. Note also that || uses the ISO entity names for
symbols (\cite{SandS}) rather than
the \TeX\ names. In a few cases this sows potential confusion.
The confusion which exists within the draft between \TeX\ and \LaTeX\
is not of itself a problem, except that people coming to HTML3, being told
it is `like \LaTeX', would find some key differences.
The main divergences are
\begin{enumerate}
\item the interpretation of space\label{space}
\item need to close most \acro{SGML} elements (e.g.\ |^| and |_|)
\item ||
\item interpretation of functions (a consequence of \ref{space})
\item entity names similar but not identical to corresponding commands
\item use of |′|
\item missing commands
\item poor support for cross-referencing
\end{enumerate}
\section{Future of maths in HTML}
At this point, it may be worth considering the extent to which any maths
expression can divorce the semantic component from the form on the page. Often
the way equations are portrayed can assist in their interpretation. Both tables
and maths seem to be examples where the meaning and the appearance are very
closely intertwined. There are instances where an author changes notation in
order to pursue an argument. One assumes that the essential meaning does not
change between changes in notation, and that perhaps a markup system might not
even note the change, except perhaps as an attribute.
The draft document which forms the basis of this discussion
expired in September, 1995. Some of the new structures which
it introduced, notably tables, form part of most browsers now,
but mathematics didn't make it (although || and ||, without
|| did) (see \cite{html32}). An email from Dave Raggett
(to David Carlisle), who wrote the draft, notes that \acro{W}3\acro{C}
\begin{quote}
has set up a small working party on math, and we expect
to publish a detailed proposal by early Summer. The March'95 spec
will provide a starting point, but we may end up with something
rather different.
\end{quote}
\noindent He goes on to state:
\begin{quotation}
\noindent The \acro{W}3\acro{C} math group has the goal to develop an
open specification
for \acro{HTML} math that:
\begin{itemize}
\item[]Is suitable for teaching, and scientific publishing.
\item[]Works with symbolic and numerical math applications
\item[]Supports filters to/from other math formats, e.g.\ \TeX
\item[]Is easy to learn and to edit by hand
\item[]Is well suited to template and other math editing techniques
\item[]Can be rendered to:
\begin{itemize}
\item[]graphical displays
\item[]speech synthesisers
\item[]plain text displays e.g.\ \acro{VT}100 emulators
\item[]print media, including braille
\end{itemize}
\item[]Supports lengthy expressions via fold\slash unfold and
line breaking with author control.
\end{itemize}
This is shaping up as the need for simple macros and declarations
that define terms etc. for use across multiple || elements,
and parsing of |PCDATA| using ``models'' referenced by || elements.
These models define how to interpret stuff at a level sufficient to
support symbolic manipulation without having to make all these
distinctions explicit in the markup itself.
\end{quotation}
This is quite a bold extrapolation from the original specification.
Some of the suggestions seem to me to be incompatible. The inclusion of
symbolic and numerical applications is interesting and goes far
beyond the existing maths \acro{DTD}s.
The relevant \acro{W}3\acro{C} web page (\cite{W3C}) contains a
reference to \acro{QED}, an ambitious
programme to build a `single, distributed, computerized
repository that rigorously represents all important, established
mathematical knowledge'. If this genuinely represents part of
\acro{HTML}'s solution to maths, we will have some time to wait before
a system is available.
\section{Conclusion}
It is not clear to me at present whether \acro{HTML} will ever be rich enough
to do the sorts of things which mathematicians and physicists want to do
with maths. Of course, it could be that these are not the market at all.
I have long argued that one of \TeX's (and \LaTeX's) major problems is
that the population of users who benefit by it is small -- very much
a minority. And the expansion of the use of computers has made them
an even smaller minority. At best we are a niche market. Some of the
simpler problems are already tackled quite conveniently by word processors,
further eroding the niche. Why bother with mathematics at all? Is it
really worth the effort, compared with something sexy like ||s?
Having said that, it appears that Public Entities in \acro{ISO}\,8879
(\cite{SandS})
are sufficient
to encompass most of the symbols I have seen in \LaTeX\ and \AmSTeX. The
potential is there. What makes this especially intriguing is that there
is software around like Panorama from SoftQuad which is designed to
enable any \acro{SGML} document whose \acro{DTD} is known to be
rendered on the screen.
Therefore for truly `heavy' applications, this seems a much better
way to go. In fact, I would see it as an altogether better way to go.
Browsers which could examine the |Doctype|, find it on an appropriate
server and then render it would be much more flexible, and enable us to use
existing \acro{SGML} documents easily on the Web. \acro{HTML} would
simply be a lightweight
\acro{DTD} used because it had a lower overhead.
On the other hand, browsers like Netscape Navigator
are becoming larger, are starting
to include `plug-ins' and be Acrobat-aware.
If \acro{HTML}3 version 2, as outlined by Raggett does support filters
(hopefully to \LaTeX\ rather than \TeX), as well as symbolic
manipulation (Mathematica, Maple, etc.), it could be a very powerful
tool.
However, the latest information available through the World Wide Web Consortium
does not encourage belief that this is much more than a dream.
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\bibitem[\acro{W}3\acro{C}, 1996c]{W3C}
\acro{W}3\acro{C} (1996c).
\newblock Math markup in \acro{HTML}.
\newblock \url{www.w3.org/hypertext/WWW/MarkUp/Math/}
\end{thebibliography}
\end{Article}
}