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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
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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.
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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.
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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
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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.
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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
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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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