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diff --git a/usergrps/uktug/baskervi/7_1/dtd.tex b/usergrps/uktug/baskervi/7_1/dtd.tex new file mode 100644 index 0000000000..21493e25a1 --- /dev/null +++ b/usergrps/uktug/baskervi/7_1/dtd.tex @@ -0,0 +1,785 @@ +\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 → a </sub> f(x)| +\item \verb|<plex><operator>lim</operator><from>x ↓| +\verb|a</from> <of>f(x)</of></plex>| +\item \verb|<mfn name=lim><sub pos=mid>x →|\\ +\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\→ 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]. |