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+\newcommand{\mod}{\mathop{\rm mod}\nolimits}
+\newcommand\dtd{\acro{DTD}}
+\newcommand\SGML{\acro{SGML}\xspace}
+\newcommand\ISO{\acro{ISO}\xspace}
+
+\title{Standard DTDs and scientific publishing}
+\author{N. A. F. M. Poppelier (\texttt{n.poppelier@elsevier.nl}),\\
+E. van Herwijnen (\texttt{eric@vanherwijnen.org}), and \\
+C.A. Rowley (\texttt{C.A.Rowley@open.ac.uk})}
+
+\date{7 August 1992}
+
+\let\Tub\TUB
+
+\begin{Article}
+
+\section{Abstract}
+
+ This paper has two parts.
+In the first part we argue that scientific publishing
+needs \textsl{one} standard \dtd{} for each class of documents
+that is published. For example one for all research
+papers and one for all books. In the second part
+we apply this reasoning to mathematical formulas, and
+we outline some design requirements for a document
+type definition for mathematical formulas. In the
+appendices we discuss and compare existing document
+type definitions for mathematical formulas.
+
+\section{Introduction}
+
+In the preface to \cite{one} Charles Goldfarb wrote that the
+Standard Generalized Markup Language can be described
+as many things, and that \SGML is all that -- and more. In
+the introduction to \cite{one} Yuri Rubinsky wrote:
+\begin{quote}
+\ISO~8870 never describes \SGML as a meta-language, but
+everything about its system of declarations and notations
+implies that a developer has the tools to build exactly what
+is required to indicate the internal structure of any type of
+information in a common tool independent manner.
+\end{quote}
+Indeed, a strong point of \SGML is that it can be regarded as
+a meta-language, a tool with which one can define the syntax
+of many languages, very much similar to context-free grammars.
+In \SGML terminology these `languages' are called \textsl{document type
+definitions}, called \textsl{\dtd{}} for short. \dtd{}s can he written
+for any type of information, research papers, books and music. A
+\dtd{} can be used for many purposes, of which two important ones
+are storage and exchange of information coded according to this
+\dtd{}.
+
+The premise of this paper is that the exchange of information,
+if it is based on \SGML, needs a single common \dtd{}, agreed upon
+by all parties involved, for each class of documents that is
+exchanged
+
+Suppose two parties, $A$ and~$B$, exchange information in the
+form of one class of documents. and that they each have a \dtd{},
+$D(A)$ and $D(B)$, with $D(A)$ not identical to $D(B)$. If~$A$ sends a
+document to~$B$ then~$A$ can include the document type
+definition $D(A)$. for that document (instance) at the beginning of the
+document. This enables~$B$ to use an \SGML parser to check the
+validity of the document he received. However, there is nothing more~$B$
+can do with the document: the \dtd{} $D(A)$ contains no information about the
+meaning of the coding scheme that $D(A)$ defines, and a mapping of the document
+from $D(A)$ to $D(B)$ is a procedure that cannot be automated. The problem
+becomes even more difficult when a third party, $C$, is introduced, who
+accepts material from both~$A$ and~$B$. How is~$C$ going to
+handle material with two different coding schemes?
+
+This is where we encounter one of the weaknesses of \SGML \textsl{as it is being
+used currently}, namely that it enables every party involved in this process to
+define and use a different \dtd{}.
+
+\section{Scientific publishing}\label{sci-pub}
+
+In the rest of this paper we concentrate on the exchange of information that
+occurs in scientific publishing, in particular on the exchange of papers that
+contain mathematical formulas and are published in research journals. Recent
+developments in this area formed the main reason for writing this paper. A few
+standards for encoding of mathematical formulas have already emerged, of which a
+well-known one is the \acro{AAP} Standard or Electronic Manuscript Standard
+\cite{two}. A \dtd{} for mathematical formulas accompanies this
+standard, but it is not part of it. Another standard for mathematical
+formulas is the one adopted by CALS \cite{three}, and others are
+under development \cite{four},
+\cite{five}.
+
+The handling of mathematical formulas in scientific publishing is part of the
+bigger whole of information exchange within a (the) scientific community, with
+the publisher as intermediary, as is shown below:
+
+\begin{picture}(100,80)(-70,0)
+
+\put(40,50){\oval(80,40)}
+\put(30,60){$C$}
+
+\put(59,50){\oval(20,10)}
+\put(55,46){$P$}
+\put(65,50){\vector(-1,-2){20}}
+
+\put(40,10){\oval(20,10)}
+\put(36,6){$G$}
+
+\put(34,10){\vector(-1,2){20}}
+
+
+\end{picture}
+
+\noindent The authors of research
+papers are the providers, $P$. The publishers are the gatherers of information,
+$G$. They accept information from many providers, gather this in the form of a
+journal issue, and distribute this. In this process, the publisher provides a
+quality check via the system of peer reviewing, makes notation consistent, and
+in some cases improves the prose. The information is distributed to a group of
+consumers, $C$, with the set~$C$ a superset of the set~$P$. In this process, two
+sorts of information can be exchanged:
+\begin{itemize}
+\item material that is structured in the sense
+of being encoded according to, and checked against, some formal structural
+specification such as a \dtd{};
+\item material that is not structured.
+\end{itemize}
+At present most of the material exchanged in the process of scientific
+publishing is of the unstructured type. We expect that this will remain the
+situation in the near future. As soon as authors get the possibility of using
+more sophisticated tools, we expect that publishers will receive increasing
+numbers of papers of the structured type.
+
+Several scientific publishers, among whom Elsevier Science Publishers, have
+adopted \SGML as the future main tool for the process of publishing scientific
+articles \cite{six}, and several other publishers have made, or are
+expected to make, the same choice. The European Laboratory for
+Particle Physics (\acro{CERN}), a large community of information providers,
+are using \SGML to automate the loading of bibliographic information
+in their library's database \cite{seven}. For both authors and
+publishers it would be advantageous to agree on one \dtd{} for the
+encoding of research papers. There are several reasons for this:
+ \begin{itemize}
+\item Most authors do not submit all their articles to one and the same
+publisher every time me. At present they are confronted with `Instructions to
+Authors' that differ significantly from publisher to publisher.
+\item A recent trend is that authors prepare their papers with text-processing
+software on some computer. This enables them to send the paper in electronic
+form (electronic manuscript or `compuscript') to the publisher. Publishers are
+confronted with a variety of text-processing software on a variety of computer
+systems \cite{eight}, \cite{nine}. Moreover, every field of science
+appears to have its own `Top Ten' of most used text processing
+packages.
+\item Bibliographic information about all research papers in all (or most)
+scientific journals is stored in bibliographic databases.
+In an ideal world, authors would still be able to use their favourite text-processing system, which would
+generate \SGML `behind the screens', so to speak. All publishers would
+accept one standard \dtd{}, and all text-processing systems would be
+able to generate documents prepared according to this \dtd{}, and all
+bibliographic databases would be able to store this material.
+\end{itemize}
+
+An example of activities towards achieving this ideal situation: the European
+Working Group on \SGML (\acro{EWS}) and the European Physical Society (\acro{EPS}) have taken
+the Electronic Manuscript Standard and are trying to develop it into a complete
+\dtd{}, which should be acceptable to information providers, information gatherers
+and information consumers. The Electronic Manuscript Standard is now a Draft
+International Standard, \ISO/\acro{DIS} 12083. The \acro{EWS} and \acro{EPS} hope that the final
+standard will include their work.
+
+\section{Encoding of mathematical formulas}
+
+In Annex A of \ISO~8879~\cite{ten} we find the following:
+ \begin{quotation}
+Generalized markup is based on two novel postulates:
+\begin{itemize}
+\item Markup should describe a document's structure and other
+attributes rather than specify processing to be performed on it, as descriptive
+markup need be done only once and will suffice for all future processing.
+\item Markup should be rigorous so that the techniques available for
+processing rigorously defined objects like programs and databases can be used
+for processing documents as well.
+\end{itemize}
+\end{quotation}
+
+There is no reason why this should not be
+valid for mathematical formulas. We need to delimit the kind of mathematical
+formulas we are trying to describe if we want an unambiguous structure. The
+field of mathematics is so vast, that it may be impossible to design a single
+\dtd{} that covers every kind of mathematical formula. If we concentrate on those
+sciences which use mathematics as a tool, for example physics, we see that the
+mathematics used in many physics papers can be described as ``advanced
+calculus'' This definition can be made more precise by referring to some standard
+textbooks containing these types of formulas, e.g.\ \textsl{Handbook of
+Mathematical Functions} \cite{eleven} and the \textsl{Table of
+integrals, series and products} \cite{twelve}.
+
+If we aim for rigorous encoding of mathematical formulas (the second postulate), we must develop a
+system of descriptive markup of mathematical formulas that enables us to:
+\begin{itemize}
+\item convert the formulas between different word processors;
+\item store the formulas in and extract them from a database;
+\item allow programs to input or output formulas in descriptive markup.
+\end{itemize}
+An example of the first application would be the conversion of mathematical
+formulas coded in \LaTeX\ to, say, Word\footnote{Word is a registered
+trademark of MicroSoft.} via \SGML. The benefits of using \SGML as an intermediate
+language for conversion are described in \cite{thirteen}. Note,
+for example, that the number of programs required for pairwise
+conversion between~$n$ languages is proportional to $n^2-n$ without
+an intermediate language, but to
+$2n$ with an intermediate language.
+
+An example of the second application would be encoding and storing the complete
+contents of the above mentioned \textsl{Handbook of Mathematical Functions}
+\cite{eleven} and \textsl{Table of integrals, series and products}
+\cite{twelve} in a database, so that this information can be accessed
+on-line by, say, mathematicians and physicists. Many articles have
+mathematical formulas in their titles, so any program that extracts
+bibliographic data should be able to handle mathematics as well.
+
+An example of the third application would be the extraction and subsequent use
+in a computer program, written in an ordinary programming language or, for
+example, in Mathematica.\footnote{Mathematica is a registered trademark of
+Wolfram Research.}
+
+At this point we come back to the ideal world for scientific publishing we
+sketched earlier. In this world, publishers would use one standard \dtd{} for
+scientific papers, which enables them to prepare a primary publication -- in
+paper and (or) in some electronic form - and to store the information in
+databases for various secondary purposes.
+
+The question now is: what should a \dtd{} for mathematical formulas look like, if
+it is going to be used for these purposes?
+
+There are two choices for a \dtd{} for mathematics:
+\begin{itemize}
+\item P-type: the \dtd{} reflects the Presentation or visual structure; examples
+of this type are discussed in the appendices.
+\item S-type: the \dtd{} reflects the Semantics or logical structure; at present no
+\dtd{}s of this type exist.
+\end{itemize}
+The quotation from Annex~A of \ISO~8879 \cite{ten} indicates
+the preference of the creator(s) of \SGML: markup of a formula should be of
+S-type, it should describe the logical structure of the formula, rather than
+the way it is represented on a certain medium, say the page of a traditional
+(non-electronic) book.
+
+Let us suppose, for the sake of the argument, that an information gatherer, a
+publisher, chooses a \dtd{} of S-type. This raises two further questions:
+\begin{enumerate}
+\item Is descriptive markup of mathematical material possible?
+\item If it is possible, who can use it and for which purposes?
+\end{enumerate}
+The second question needs some explanation. As discussed in section
+\ref{sci-pub}, in the process of scientific publishing two sorts of
+information can be exchanged. mathematical material that is structured
+according to a formal structural specification, and material that is not
+structured. This means that there are two possible scenarios.
+
+Scenario 1: an author submits a paper in the form of a manuscript
+(paper), i.e.\ with unstructured formulas, or a compuscript with
+mathematical formulas in P-type notation (\TeX, WordPerfect, \dots).
+
+Scenario 2: an author submits a paper with mathematical formulas in S-type
+notation. In scenario 1 it is the task of the publisher to convert from paper
+or P-type notation to S-type notation. Before we discuss the feasibility of
+this conversion, we will first look at some characteristics of mathematical
+notation.
+
+\subsection{Characteristics of mathematical notation}\label{character}
+
+Mathematical notation is designed to create the correct ideas in the mind of
+the reader. It is \textsl{deliberately} ambiguous and incomplete: indeed, it is
+almost meaningless to all other readers. Or, more technically: the intrinsic
+information content of any mathematical formula is very low. A formula gets its
+meaning, i.e. its information content, only when used to communicate between
+two minds which share a large collection of concepts and assumptions, together
+with an agreed language for communicating the associated ideas.
+
+The ambiguity encountered in mathematical notation can be of two types
+\cite{fourteen}
+\begin{enumerate}
+\item A generic notation uses the same symbols to
+represent similar but different functions, for example `$+$' or `$\times$'. In
+the case of addition this is not really a problem, but multiplication is a
+problem since, multiplication of numbers is commutative, whereas matrix
+multiplication is non-commutative!
+\item A more fundamental ambiguity is posed
+by the same notation being used in different fields in different ways. For
+example: $f'$ stands for the first derivative of~$f$ in calculus, but can mean
+`any other entity different from $f$' in other areas.
+\end{enumerate}
+
+More examples of ambiguity are:
+\begin{itemize}
+\item Does~$\bar x$ represent a mean, a conjugation or a negation?
+\item Is~$i$ an integer variable, e.g.\ the index of a matrix, or is it
+$\sqrt{-1}$?
+\item The other way around: is $\sqrt{-1}$ denoted by~$i$ or
+by~$j$?\footnote{There are examples of authors actually writing something like
+$[L_i,L_j] =\frac{i}{2}L_k$, where the first~$i$ is an
+index, and the second~$i$ stands for~$\sqrt{-1}$.}
+
+\item What is the function of the~2 in $\textrm{SU}_2$ $\log_2x$, $x^2$,
+$T_2^2$?\footnote{In $\textrm{SU}_2$ it is the number of dimensions of 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.}
+\item Is $|X|$ the absolute value of a real (complex) number~$X$
+or the polyhedron of a simplicial complex~$X$ \cite{fifteen}?
+\end{itemize}
+The inverse problem, which is equally common, arises when different typographical
+constructs have the same mathematical meaning. For example, the meanings of
+both the following two lines would be coded identically
+\begin{eqnarray*}
+3 &+& 4 (\mod 5)\\
+3 &+_5& 4
+\end{eqnarray*}
+and this would lead to great difficulty if an author wanted to write:
+\begin{quote}
+We shall often write, for example, $3 + 4 (\mod 5)$ in the shorter form $3 +_5
+4$, or even as simply $3+4$ when this will not lead to confusion.
+\end{quote}
+
+
+Of course, natural languages are similarly ambiguous and incomplete, but no one
+we know is suggesting that in an \SGML document each word should be coded such
+that it reflects the full dictionary definition of the meaning which that
+particular use of the word is intended to have!
+
+\subsection{Who performs the markup of math?}
+How does one convert P-type mathematical material, which an author has
+produced, to S-type notation, which the publisher uses?
+In \cite{one}, (p.9) Goldfarb gives a three-step model for document
+processing:
+\begin{enumerate}
+\item recognition of part of a document (adding a generic identifier
+for the appropriate element);\label{first}
+\item mapping (associating a processing function
+with each element);\label{second}
+\item processing (e.g.\ translating elements into word
+processor commands).\label{third}
+\end{enumerate}
+ In the publishing of scientific papers and books steps~\ref{second}
+and~\ref{third} are the responsibility of the publisher.
+Traditionally, step~\ref{first} was also their responsibility: the
+technical editor adds markup signs in the margin of the manuscript,
+depending on the text and the visual representation that the house
+style dictates. It is, however, unlikely that a technical editor is
+capable of identifying the precise function of every part of a
+mathematical formula, for several reasons, most of which were
+discussed in the previous subsection, namely that mathematical
+notation:
+\begin{itemize}
+\item is not unambiguous,
+\item is not completely standardized,
+\item is not a closed system.
+\end{itemize}
+ Even if the technical editor were capable of identifying every
+part of a formula, this would be too time- consuming -- and therefore too costly.
+However, under certain conditions \cite{sixteen}, automatic
+translation from visual structure to logical structure of
+mathematical material is simplified greatly.
+
+This, and what we discussed in section~\ref{character}, leads us to
+conclude the following. A publisher has no choice but to use a
+P-type \dtd{} for mathematical material that is submitted in
+unstructured form or in P-type notation. Even if S-type markup of a
+mathematical formula would be possible, conversion from P-type to
+S-type would be difficult or even impossible. Conclusion: the tags
+for S-type markup should not be added by the information gatherer,
+but by the information providers, i.e. the authors, who should be
+able to identify each part of their formulas.
+
+\subsection{Feasibility of S-type notation}
+ In our second scenario, authors
+would submit papers with mathematical formulas in S-type notation. This would
+enable the publisher to `down translate'\footnote{`Down' because information is
+lost in the process; we borrowed the terminology of translating `up' and
+`down' from Exoterica OmniMark.} to any mathematics typesetting language
+(P-type notation). However, the same reasoning as in section 3.1 leads us to
+the following conjecture:
+
+Conjecture. It is impossible to create an S-type \dtd{} for all of mathematics.
+
+Representing the ``full meaning'' of a mathematical formula, if such a notion
+exists, will almost certainly lead to attempts to pack more and more
+unnecessary information into the representation until it becomes useless for
+any purpose. This is rather like Russell and Whitehead reducing ``simple
+arithmetic'' to logic and taking several pages of symbols to represent the
+``true meaning of $2+2=4$''.
+
+Even if it were possible to define an S-type \dtd{} for a certain
+branch of mathematics, this still gives problems. Supposing an
+S-type \dtd{} contains an element for a ``derivative'' of a function.
+Since the S-type \dtd{} will not contain any presentational attributes,
+a decision will have to be made to represent the derivative of
+$f(x)$ on paper as $f'(x)$ or $\frac{\text{\fontfamily{cmr}\selectfont
+ d}f(x)}{\text{\fontfamily{cmr}\selectfont d}x}$.
+There are, however, times (such as in this article) that both
+representations are required for the same semantic object, and that
+the author will need other notation in addition to that defined by
+the S-type \dtd{}.
+
+A likely reason for the belief that an S-type \dtd{} is possible, is
+that many people in the worlds of document processing or computer
+science are convinced that each symbol has at most a few possible
+uses and that mathematical notation is as straightforward to analyse
+as, for example, a piece of code for a somewhat complicated
+programming language. The reality is that mathematical notation is
+more akin to natural language: it is ambiguous and incomplete, as we
+pointed out earlier.
+
+\subsection{Some problems with existing languages}
+To show that it is not obvious to capture mathematical syntax in a
+\dtd{}, let alone its semantics, consider the example of a limit
+\[
+\lim_{x\to a}f(x)
+\]
+The syntactic structure of a limit is:
+\begin{itemize}
+\item The limit operator
+\item The part containing the variable and its limit value
+\item The expression of which the limit is to be taken
+\end{itemize}
+The first part could:
+\begin{itemize}
+\item always be ``lim'', in which case it is just a part of the
+presentation of the formula and it should be left out.
+\item be one of a finite list of alternatives, indicating the type
+of limit($\liminf$, $\sup$, $\max$, etc.). In this case it should be
+an attribute.
+\item be any expression.
+\item be any text.
+\end{itemize}
+We think the second possibility comes closest to the syntax of the
+limit construct. Th second and third parts can be any mathematical
+expression.
+
+Now let's look at the way this formula is coded with the \dtd{}s from
+\ISO \acro{TR}~9573, \acro{AAP} math and Euromath respectively. Using the
+mathematics \dtd{} from \ISO \acro{TR}~9573 there are three possibilities:
+\begin{itemize}
+\item \verb|lim <sub pos=mid> x &rarr; a </sub> f(x)|
+\item \verb|<plex><operator>lim</operator><from>x &darr;|
+\verb|a</from> <of>f(x)</of></plex>|
+\item \verb|<mfn name=lim><sub pos=mid>x &rarr;|\\
+\verb|a</ll><opd>f(x)</opd></lim>|
+\end{itemize}
+whereas with the Euromath \dtd{} we would have:
+\begin{verbatim}
+<lim.cst><l.part.c limitop=lim><range>
+<relation>x\&rarr; a </relation></range>
+</l.part.c><r.part.c><textual>f(x)</textual>
+\end{verbatim}
+
+
+We see that the \acro{AAP} and Euromath expressions are closest to the limit syntax.
+The best solution from \ISO \acro{TR}~9573 involves a more general ``plex''
+construct, which can be used for integrals, sums, products, set
+unions, limits and others. When the plex construct contains the
+actual lower and upper bounds it may even give semantic
+information.
+
+Some mathematicians, however, are not satisfied with
+this solution \cite{seventeen}. The plex operation is probably a
+notation for an iterated application of a binary operation (e.g.\
+sums and products), while limits are of a different nature. In many
+cases only the from part will be used, and there the whole range of
+the bound variable will be indicated, as an interval or a more
+general set. How does one go about extracting the bound variable?
+
+This supports our conjecture from the previous section, namely that it is very
+hard to capture the semantics for all mathematics. it also suggests that some
+redundancy is required to select whichever notation is most appropriate in a
+certain context.
+
+\section{Re-using mathematical formulas}
+ There are two important uses for a
+generically coded mathematical formula. The first one is in a mathematical
+manipulation -- or computer algebra -- system (\acro{MMS}), such as Mathematica
+\cite{eighteen} or Maple \cite{nineteen}. Computer programs for the
+numerical evaluation of formulas, for example written in
+\textsc{Fortran} or Modula-2, can also be regarded as mathematical
+manipulation programs.
+
+The second form of re-usage is in a mathematical typesetting system, for
+formatting the formula on paper or on screen; examples of this are \TeX\
+\cite{twenty} and eqn/troff \cite{twentyone}, \cite{twentytwo}.
+
+For computer algebra systems the notation for the formula should be such that a
+particular type of manipulation on a particular system is possible, given a
+`background' of concepts and assumptions that enables the system to interpret
+the input as a mathematical statement.
+
+The coding of a formula that is adequate for document formatting, for example the
+\TeX\ notation \verb|f^{(2)}(x)|, is very unlikely to contain much of the
+information required for a manipulation system to make use of it. However, for
+a limited held of discourse it is feasible to use the same coding for both types
+of system \cite{sixteen}.
+
+Some examples: the square of $\sin x$ is typographically represented as
+$\sin^2x$, but a system like Mathematics or Maple would probably prefer
+something like $(\sin x)^2$ as input. Typesetting the inverse of $\sin x$ as
+$\sin^{-1}x$, however, could be confusing: does it mean $1/(\sin x)$ or $\arcsin
+x$?
+
+An \acro{MMS} would probably require the second derivative of a function~$f$ with
+respect to its argument~$x$ to be coded as $(D,x)((D,x)f(x)))$ but
+on paper this would be represented as $f''(x)$, or $f^{(2)}(x)$, or
+$\frac{\displaystyle\text{\fontfamily{cmr}\selectfont d}^2f(x)}%
+ {\displaystyle\text{\fontfamily{cmr}\selectfont d}x^2}$.
+
+On the output side of a \acro{MMS} there are other problems since some of the coding
+necessary for typographically acceptable output cannot be automatically derived
+by the system from the coding used by the \acro{MMS}.
+
+The Euromath view \cite{seventeen} is that a common interface should
+be designed together with the manufacturer of a \acro{MMS}. Perhaps an
+\acro{MMS}-type \dtd{} will be required.
+
+\section{Related problems}
+Another problem is, of course, that mathematics is by its nature extensible, so
+there will always be new types of manipulations to be done. Notations are
+changed or new notations are invented almost every day, figuratively speaking.
+Normally these new subjects will use existing typographic representations, but
+the computer algebra system will not know what formatting to use! Occasionally
+a new typographic convention will be needed. And although there is agreement
+on the notation for most mathematical concepts, authors of books on mathematics
+tend to introduce alternative notations, for instance when they feel this is
+necessary for didactic reasons. Mathematical notation is not standardized, and
+it is open -- anyone can use it, and add to it, in any way they wish.
+
+If we consider a given \dtd{} at any time, we have to ask ourselves: can an author
+add elements when the need for this arises? Theoretically the answer is `Yes,
+he can' \cite{twentythree}, (p.71), although it is not
+straightforward to include the new elements in the content models of
+existing elements.
+
+Are such modification by the author desirable? A \dtd{} which is locally modified by
+an author will quickly give rise to the situation described in the introduction
+to this paper, and this should therefore probably be discouraged. Others,
+however, have also noticed a need for private elements, as described in \acro{EPSIG}
+News 3, no.~4; one of the challenging aspects of using \SGML being encountered by
+the Text Encoding Initiative is that the guidelines
+need to be extensible by researchers. They need to be able to extend
+the \dtd{} in a disciplined way.
+
+This problem, however, may not be a serious one. The collection of style
+elements is almost a closed set, since the number of fonts, symbols and ways
+to combine them is limited. In fact, most notation is not syntactically new,
+since the limited number of constructs works well as a notation. The multitude
+of notations is obtained by combinations of fonts, symbols and positions (left
+or right subscript, left or right superscript, atop, below, \dots), and by
+giving one notation more than one meaning. This again seems to support our view
+that only a P-type \dtd{} can be constructed for \emph{all} of mathematics.
+
+An \SGML \dtd{}, of whatever type, also doesn't solve the problems of new atomic or
+composite symbols, which occur frequently in mathematics. As with new elements,
+an author can add entities for these new symbols. There is no method to add the
+name of a new symbol, whether atomic or composite, to an existing set of entity
+definitions for symbols, other than to contact the owner of the set and wait for
+an update.
+
+Although there is now a standard method to describe that symbol's glyph
+(shape) \cite{twentyfive}, it is not practical for an author to
+include it. A compromise solution seems to be to extend an existing
+set, such as the one from \ISO \cite{twentysix}, as much as
+possible, and try to standardize its use.
+
+\section{Conclusions}
+We have argued as follows:
+\begin{itemize}
+\item That a logical \dtd{} in the sense of describing the structure of
+the mathematical meaning is as impossible for maths as it is for natural
+language, and also it is useless for formatting since the same mathematical
+structure can be visually represented in many different ways. The correct one
+for any given occurrence of that structure cannot be determined automatically,
+but must be specified by the author.
+\item That what needs to be encoded for formatting purposes, is information that
+enables a particular set of detailed rules for maths typesetting to be applied.
+This could he described as a `generic-visual encoding' or `encoding the logic
+of the visual structure'. To establish exactly what these code?, should
+be will require an expert analysis (probably involving expertise from
+mathematicians, particularly editors, and from typographers aware of the
+traditions of mathematical typesetting).
+\item That this is different to what
+needs to be encoded for use in mathematical manipulation software. Since neither
+of these encodings can be deduced automatically from the other, a useful
+database will need to store both. Perhaps a separate \dtd{} will be required to
+enable this communication.
+\end{itemize}
+Possible solutions are
+\begin{itemize}
+\item A \dtd{} based on a hybrid of visual structure and logical structure
+\item Two \dtd{}s, one for visual structure and one for logical structure, that
+are linked in some fashion
+\item Two concurrent \dtd{}s, one for visual structure and one for logical
+structure.
+\end{itemize}
+
+The simplest solution is probably to have a basic visual structure which is
+ described as an \SGML entity, supplemented with a (redundant) logical
+structure, described by a second \SGML entity. This solution avoids any special
+\SGML features and gives the user all flexibility for mixing and matching as
+required. We believe that similar reasoning can be applied to tables and
+chemical formulas, where the problem of separation form from content is just as
+complex, or even more.
+
+\begin{thebibliography}{10}
+
+\bibitem{one}
+Charles Goldfarb.
+\newblock {\em The {\SGML} Handbook}.
+\newblock Oxford University Press, Oxford, 1990.
+
+\bibitem{two}
+Standard for electronic manuscript preparation and markup version 2.0.
+\newblock Technical Report Z39.59-1988, {\acro{ANSI}/\acro{NISO}}, 1987.
+
+\bibitem{three}
+Techniques for using {\SGML}.
+\newblock Technical Report 9573, {\ISO}, 1988.
+
+\bibitem{four}
+American~Chemical Society.
+\newblock {\acro{ACS}} journal \dtd{}.
+
+\bibitem{five}
+Bj{\"{o}}rn von Sydow.
+\newblock On the \texttt{math} type in {E}uromath.
+
+\bibitem{six}
+N.~A. F.~M. Poppelier.
+\newblock {\SGML} and {\TeX} in scientific publishing.
+\newblock {\em \TUB}, 12:105--109, 1991.
+
+\bibitem{seven}
+E.~van Herwijnen, N.~A. F.~M. Poppelier, and J.C. Sens.
+\newblock Using the electronic manuscript standard for document conversion.
+\newblock {\em EPSIG News}, 1(14), 1992.
+
+\bibitem{eight}
+E.~van Herwijnen.
+\newblock The use of text interchange standards for submitting physics articles
+ to journals.
+\newblock {\em Comp. Phys. Comm.}, 57:244--250, 1989.
+
+\bibitem{nine}
+E.~van Herwijnen and J.C. Sens.
+\newblock Streamlining publishing procedures.
+\newblock {\em Europhysics News}, pages 171--174, November 1989.
+
+\bibitem{ten}
+Standard generalized markup language ({\SGML}).
+\newblock Technical Report 8879, {\ISO}, l986.
+
+\bibitem{eleven}
+M.~Abramovitz and I.~Stegun.
+\newblock {\em Handbook of mathematical functions}.
+\newblock Dover, New York, 1972.
+
+\bibitem{twelve}
+I.S. Gradshteyn and I.M. Ryzhik.
+\newblock {\em Tables of integrals, series, and products}.
+\newblock Academic Press, New York, 1980.
+
+\bibitem{thirteen}
+S.A. Mamrak, C.S. O'Connell, and J.~Barnes.
+\newblock Technical documentation for the integrated chameleon architecture.
+\newblock Technical report, March 1992.
+
+\bibitem{fourteen}
+Neil~M. Soiffer.
+\newblock {\em The design of a user interface for computer algebra systems}.
+\newblock PhD thesis, Computer Science Division ({\acro{EECS}}), University of
+ California, Berkeley, 1991.
+\newblock Report {\acro{UCB}/\acro{USD}} 91/626.
+
+\bibitem{fifteen}
+M.~Nakahara.
+\newblock {\em Geometry, Topology and Physics}.
+\newblock Adam Hilger, Bristol, 1990.
+
+\bibitem{sixteen}
+Dennis~S. Arnon and Sandra~A. Mamra.
+\newblock On the logical structure of mathematical notation.
+\newblock {\em \TUB}, 12:479--484, 1991.
+
+\bibitem{seventeen}
+Bj{\"{o}}rn von Sydow.
+\newblock private communication to EvH.
+
+\bibitem{eighteen}
+Stephen Wolfram.
+\newblock {\em Mathematica: a system for doing mathematics by computer}.
+\newblock Addison-Wesley, Reading, 1991.
+
+\bibitem{nineteen}
+Bruce~W. Char, Keith~O. Geddes, Gaston~H. Gonnet, and Stephen~M. Watt.
+\newblock {\em Maple User's Guide}.
+\newblock \acro{WATCOM} Publications Ltd., Waterloo, 1985.
+
+\bibitem{twenty}
+Donald~E. Knuth.
+\newblock {\em The {\TeX}book}.
+\newblock Addison-Wesley, Reading, 1984.
+
+\bibitem{twentyone}
+Joseph~E Osanna.
+\newblock Nroff/troff.
+\newblock In {\em {UNIX} Programmer's Manual (2b)}. Bell Laboratories, 1978.
+
+\bibitem{twentytwo}
+Brian~W. Kernighan and Linda Cherry.
+\newblock Typesetting mathematics.
+\newblock In {\em {UNIX} Programmer's Manual (2b)}. Bell Laboratories, 1978.
+
+\bibitem{twentythree}
+E.~van Herwijnen.
+\newblock {\em Practical {\SGML}}.
+\newblock Kluwer Academic Publishers, Dordrecht, 1990.
+
+\bibitem{twentyfive}
+Font information interchange.
+\newblock Technical Report 9541, \ISO, 1991.
+
+\bibitem{twentysix}
+Information processing -- {\SGML} support facilities -- techniques for using
+ {\SGML} -- part 13.
+\newblock Technical Report 9573, \ISO, 1991.
+\newblock Proposed Draft Technical Report.
+
+\end{thebibliography}
+
+%\begin{tabular}{ll}
+%N. A. F. M. Poppelier& E. van Herwijnen, \\
+%Elsevier Science Publishers,&CERN,\\
+%P.O. Box 2400,&1211-CH,\\
+%1000 CK Amsterdam,&Geneva 23,\\
+%the Netherlands&Switzerland\\
+%\texttt{n.poppelier@elsevier.nl}&%???
+%\end{tabular}
+
+%\noindent\qquad and\\
+%\begin{tabular}{l}
+%C.A. Rowley\\\texttt{C.A.Rowley@open.ac.uk}
+%\end{tabular}
+
+\end{Article}
+
+\endinput
+\section{References}
+
+
+
+\end{Article}
+\endinput
+
+
+A Existing mathematical notations
+
+A.1 Comparison of existing \dtd{}s
+
+In making comparisons between existing \dtd{}s we shall refer often to what is probably the best-known
+system for coding mathematical notation in documents. This is the version of TEX coding used in
+LaTeX 127] (which differs little from Knuth's Plain T~ notation described in [201), now a de facto
+standard in many areas. It is a mixture of visual and logical tagging, with a bias towards the visual
+which probably results from reasoning similar to that in this paper.
+
+The following document type definitions for mathematical formulas were investigated for this paper:
+AAP 128], ISO [29] and Euromath [51.
+
+We will try to give a few general characteristics of each of them:
+
+AAP This \dtd{} shows a hybrid of visual and logical tagging. It is quite similar to the mathematical
+notation of TEX 120].
+Integrals, sums and similar constructions have sub-elements tagged explicitly as lower limit, upper limit
+and integrand (summand,...).
+
+The same goes for fractions, roots, and limit-like constructions.
+
+All rectangular schemes of mathematical expressions, e.g.\ matrices and determinants, are tagged as
+'array in this \dtd{}. The delimiters are not part of the construction, although matrices are usually indicated
+by ( ) or as C ], and determinants as I ( Alignment of rows, columns and cells is indicated by attributes,
+even though they have nothing to do with function, but are in fact processing information. This idea
+also appears in the array notation of LaTeX~[27].