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Network Working Group                                          Alan KatzRequest for Comments: 1003                                       USC/ISI                                                              March 1987Issues in Defining an Equations Representation StandardStatus 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 toKatz                                                            [Page 1]

RFC 1003                                                      March 1987    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.Katz                                                            [Page 2]

RFC 1003                                                      March 1987    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.Katz                                                            [Page 3]

RFC 1003                                                      March 1987    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 meaningKatz                                                            [Page 4]

RFC 1003                                                      March 1987    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.Katz                                                            [Page 5]

RFC 1003                                                      March 1987    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 isKatz                                                            [Page 6]

RFC 1003                                                      March 1987            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.Katz                                                            [Page 7]