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RFC 1003 - Issues in defining an equations representation standa


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Network Working Group                                          Alan Katz
Request for Comments: 1003                                       USC/ISI
                                                              March 1987

        Issues in Defining an Equations Representation Standard

Status of This Memo

    This memo is intended to identify and explore issues in defining a
    standard for the exchange of mathematical equations.  No attempt is
    made at a complete definition and more questions are asked than are
    answered.  Questions about the user interface are only addressed to
    the extent that they affect interchange issues.  Comments are
    welcome.  Distribution of this memo is unlimited.

I.  Introduction

    Since the early days of the Arpanet, electronic mail has been in
    wide use and many regard it as an essential tool.  Numerous mailing
    lists and newsgroups have sprung up over the years, allowing large
    numbers of people all over the world to participate remotely in
    discussions on a variety of topics.  More recently, multimedia mail
    systems have been developed which allow users to not only send and
    receive text messages, but also those containing voice, bitmaps,
    graphics, and other electronic media.

    Most of us in the Internet community take electronic mail for
    granted, but for the rest of the world, it is a brand new
    capability.  Many are not convinced that electronic mail will be
    useful for them and may also feel it is just an infinite time sink
    (as we all know, this is actually true).  In particular, most
    scientists (apart from computer scientists) do not yet use, or are
    just beginning to use, electronic mail.

    The current NSF supercomputer initiative may change this.  Its
    primary purpose is to provide remote supercomputer access to a much
    greater number of scientists across the country.  However, doing
    this will involve the interconnection of many university-wide
    networks to NSF supercomputer sites and therefore to the NSF
    backbone network.  Thus, in the very near future we will have a
    large number of scientists in the country suddenly able to
    communicate via electronic mail.

    Generally, text-only mail has sufficed up until now.  One can dream
    of the day (not so far in the future) when everyone will have
    bitmapped display workstations with multimedia mail systems, but we
    can get by without it for now.  I believe, however, that the new NSF
    user community will find one other capability almost essential in
    making electronic mail useful to them, and that is the ability to

    include equations in messages.

    A glance through any scientific journal will demonstrate the
    importance of equations in scientific communication.  Indeed, papers
    in some fields seem to contain more mathematics than English.  It is
    hard to imagine that when people in these fields are connected into
    an electronic mail community they will be satisfied with a mail
    system which doesn't allow equations.  Indeed, with the advent of
    the NSF's Experimental Research in Electronic Submission (EXPRESS)
    project, scientists will begin submitting manuscripts and project
    proposals directly through electronic mail and the ability to handle
    equations will be essential.

    Currently, there exists no standard for the representation of
    equations.  In fact, there is not even agreement on what it is that
    ought to be represented.  Users of particular equation systems (such
    as LaTex or EQN) sometimes advocate just including source files of
    that system in messages, but this may not be a good long-term
    solution.  With the new NSF community coming on line in the near
    future, I feel the time is now right to try to define a standard
    which will meet the present and future needs of the user community.

    Such a standard should allow the interchange of equations via
    electronic mail as well as be compatible with as many existing
    systems as possible.  It should be as general as possible, but still
    efficiently represent those aspects of equations which are most
    commonly used.  One point to be kept in mind is that most equations
    typesetting is currently being done by secretaries and professional
    typesetters who do not know what the equations mean, only what they
    look like.  Although this is mainly a user interface consideration,
    any proposed standard must not require the user to understand an
    equation in order to type it in.  We are not interested here in
    representing mathematics, only displayed equations.

    In this memo, I will try to raise issues that will need to be
    considered in defining such a standard and to get a handle on what
    it is that needs to be represented.  Hopefully, this  will form the
    basis of a discussion leading eventually to a definition.  Before
    examining what it is that could be or should be represented in the
    standard, we will first review the characteristics of some existing
    systems.

2.  Existing Systems

    There currently exist many incompatible systems which can handle
    equations to a certain extent. Most of these are extensions to text
    formatting systems to allow the inclusion of equations.  As such,
    general representation and standards considerations were not a major
    concern when these systems were initially designed.  We will examine
    the three main types of systems: Directive systems, Symbolic
    Language systems, and Full Display systems.

    Some text editing facilities simply allow an expanded font set which
    includes those symbols typically used in mathematics.  I do not
    consider these systems as truly able to handle equations since much
    of mathematics cannot be represented.  It takes more than the Greek
    alphabet and an integral and square root symbol to make an equations
    system.

    Directive systems are those which represent equations and formating
    information in terms of directives embedded in the text.  LaTex and
    EQN are two examples.  LaTex is a more friendly version of Knuth's
    Tex system, while EQN is a preprocessor for Troff, a document
    preparation system available under Unix.

    With these Directive systems, it is usually necessary to actually
    print out the document to see what the equations and formatted text
    will look like, although there are on-screen previewers which run on
    workstations such as the Sun.  Directive systems have the advantage
    that the source files are just text and can be edited with standard
    text editors (such as Emacs) and transferred as text in standard
    electronic messages (a big advantage considering existing mail
    interconnectivity of the various user communities).  Also, it is
    relatively easy to make global changes with the help of your
    favorite text editor (for example, to change all Greek letter
    alpha's to beta's or all integrals to summation signs in a document.
    This is generally impossible with the other types of systems
    described below).

    The primary disadvantage of these systems is that writing an
    equation corresponds to writing a portion of a computer program.
    The equations are sometimes hard to read, generally hard to edit,
    and one may make syntax errors which are hard to identify.  Also,
    people who are not used to programming, and typesetters who do not
    actually know what an equation means, only what it should look like,
    find specifying an equation in this language very difficult and may
    not be willing to put up with it.

    Full Display Systems are those such as Xerox STAR and VIEWPOINT.
    The user enters an equation using the keyboard and sees exactly that
    equation displayed as it is typed.  At all times, what is displayed
    is exactly how things will look when it is printed out.
    Unfortunately, VIEWPOINT does not allow the user to place any symbol
    anywhere on the page.  There are many things (such as putting dots
    on indices) which are not possible.  For those things which are
    implemented, it works rather nicely.

    Hockney's Egg is a display system which was developed at the UCLA
    Physics Department and runs on the IBM PC.  It has the advantage of
    being able to put any character of any font anywhere on the screen,
    thus allowing not only equations, but things like chemical diagrams.

    Interleaf's Workstation Publishing Software system is not strictly
    speaking an equations system, but equations may be entered via a cut
    and paste method.  At all times, what one sees is what will be
    printed out and one may put any symbol anywhere on the page.  The
    problem with this system is that one HAS TO put everything in a
    certain place.  It sometimes takes an enormous amount of work to get
    things to be positioned correctly and to look nice.

    Generally, Full Display Systems are specific to a particular piece
    of hardware and the internal representation of the equations is not
    only hidden from the user, but is in many cases proprietary.

    Symbolic Language systems, such as Macsyma and Reduce, also allow
    the entry of equations.  These are in the form of program function
    calls.  These are systems that actually know some mathematics.  One
    can only enter the particular type of mathematics that the system
    knows.

    We next will look at what should be represented in an equations
    system.  We will want a representation standard general enough to
    allow (almost) anything which comes up to be represented, but does
    not require vast amounts of storage.

3.  What Could be Represented?

    We will first examine what it is that could be represented.  At the
    most primative level, one could simply store a bitmap of each
    printed equation (expensive in terms of storage).  At the other end
    of the spectrum, one could represent the actual mathematical
    information that the equation itself represents (as in the input to
    Macsyma).  In between, one could represent the mathematical symbols
    and where they are, or represent a standard set of mathematical
    notation, as in EQN.

    It is useful to think of an analogy with printed text.  Suppose we
    have text printed in a certain font.  How could it be represented?
    Well, we could store a bitmap of the printed text, store characters
    and fonts, store words, or at the most abstract, we could store the
    meaning behind the words.

    What we actually do, of course, is store characters (in ordinary
    text) and sometimes fonts (in text intended to be printed).  We do
    not attempt to represent the meaning of words, or even represent the
    notion of a word.  We generally only have characters, separated by
    spaces or carriage returns (which are also characters).  Even when
    we specify fonts, if a slightly different one happened to be printed
    out it would not matter greatly.

    Equations may be considered an extension of ordinary text, together
    with particular fonts.  However, the choice of font may be extremely
    important.  If the wrong font happens to be printed out, the meaning

    of the equation may be completely changed.  There are also items,
    such as growing parentheses, fractions, and matrices, which are
    particular to equations.

    We are not interested in representing the meaning of an equation,
    even if we knew how to in general, but in representing a picture of
    the equation.  Thus, we will not further consider the types of
    representations made in the Symbolic Language systems.  We still
    have Directive systems and the Full Display systems.  We shall
    assume that both of these will continue to exist and that the
    defined standard should be able to deal with existing systems of
    either type.

    Assuming we do not want to just store a bitmap of the equation
    (which would not allow any easy editing or interfacing with existing
    systems), we are now left with the following possibilities:

         1.   Store characters, fonts and positions only.  Allow
              anything to be anywhere (this is what Interleaf does).

         2.   Store characters, fonts, and positions, but only allow
              discrete positions.  This makes it easier to place
              subscripts and superscripts correctly (this is what
              Hockney's Egg does).

         3.   Use a language similar to EQN or LaTex, which has ideas
              such as subscripts, superscripts, fractions, and growing
              parentheses.  Generally positioning is done automatically
              when the typesetting occurs, but it is possible to do a
              sort of relative positioning of symbols with some work.

         4.   Use a language such as Troff or Tex, which is what EQN and
              Latex is translated into.

         5.   Some combination of the above.

    In the next section, I will argue for a particular combination of
    the above as a tentative choice.  It may turn out, with more
    information and experience, that this choice should be modified.

4.  What I Think Should be Represented

    Let us now take a stab at what sort of standard we should have.
    First of all, we would like our standard if at all possible to be
    compatible with all of the existing systems described previously.
    If the standard becomes widely accepted, it should be general enough
    not to constrain severely the design of new user interfaces.  Thus,
    while we should provide for efficiently representing those aspects
    of equations which are commonly used (subscripts, parentheses, etc.)
    we would like extensions to be possible which enable the
    representation of any symbol anywhere.

    We would like standard mathematical symbols, as well as all Greek
    and Latin letters to be available.  We would also like any required
    typesetting knowledge to be in programs and not required of the
    user.

    I feel that the exact position of a subscript or superscript should
    not have to be specified by the user or be represented (unless the
    user specifically wants it to be).  It is nice to be able to place
    any symbol anywhere (and indeed the standard ought to allow for
    this), but having to do this for everything is not good.  The
    standard should be able to represent the idea of a subscript,
    superscript, or growing fraction with no more specification.

    My suggestion, therefore, is for something like EQN, but with
    extensions to allow positioning of symbols in some kind of absolute
    coordinates as well as relative positioning (EQN does allow some
    positioning relative to where the next symbol would normally go).
    This has the advantage that the representation is in ordinary text,
    which can be sent in messages, the Directive systems can map almost
    directly into it, and it should allow representation for Full
    Display systems.  The ideas of subscript and superscripts (without
    having to specify a position), growing parentheses, fractions, and
    matrices, and special fonts are already there.

    Most equations can be specified very compactly within EQN, and if
    positioning is provided as an extension, exceptions can be handled.
    (The same could be said for LaTex, however, I consider the syntax
    there to be somewhat unreadable and prefer EQN.  Essentially, either
    will do).

    User interfaces should be able to be easily constructed which would
    allow one to type in an EQN style specification and have the
    equation appear immediately on the screen.  For non-specialists, it
    may be better to use existing Full Display systems which are then
    translated in this EQN like standard (perhaps using a lot of the
    absolute positioning facility).

5.  Conclusions

    In summary:

       1.   A standard for the efficient representation of mathematical
            equations should be defined as soon as possible in order to
            allow the interchange of equations in documents and mail
            messages and the transfer of equations between various
            existing internal representations.

       2.   Most equations entry is currently done by people who do not
            know what the equations mean, and are not programmers.  It
            may be that the optimal user interface for these people is

            different than for those who do know mathematics and/or are
            programmers.  An equations standard should not preclude
            this.

       3.   The standard should easily handle those aspects of equations
            which are common, such as the set of things provided in EQN.

       4.   It should also be possible, however, to place any defined
            symbol anywhere and the standard should allow this type of
            specification when needed.

       5.   As many of the existing systems (all of them if possible)
            should be able to be translated into the standard.

       6.   The standard should not make requirements on the user
            interface such that the user must have much typesetting
            knowledge.  This knowledge should be in the user interface
            or printing routines.

       7.   Full Display systems may be best for non-specialists and for
            non-programmers.  Directive systems, perhaps with the
            ability to preview the final equation on one's screen, may
            be best for the rest.

       8.   A distinction should be made between the representation of
            an equation (which we are dealing with here) and the
            mathematical knowledge it represents.

    I suggest something like EQN as a standard with extensions to allow
    positioning of symbols in some kind of absolute coordinates as well
    as relative positioning.  This has the advantage that the
    representation is in ordinary text, which can be sent in messages,
    the Directive systems can map almost directly into it, and it should
    allow representation for Full Display systems.  The ideas of
    subscript and superscripts (without having to specify a position),
    growing parentheses, fractions, and matrices, and special fonts are
    already there.

 

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