rfc1019.txt

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Network Working Group                                           D. Arnon
Request for Comments: 1019                                    Xerox PARC
                                                          September 1987



  Report of the Workshop on Environments for Computational Mathematics
                                July 30, 1987
                          ACM SIGGRAPH Conference
              Anaheim Convention Center, Anaheim, California


Status of This Memo

   This memo is a report on the discussion of the representation of
   equations in a workshop at the ACM SIGGRAPH Conference held in
   Anaheim, California on 30 July 1987.  Distribution of this memo is
   unlimited.

Introduction

   Since the 1950's, many researchers have worked to realize the vision
   of natural and powerful computer systems for interactive mathematical
   work.  Nowadays this vision can be expressed as the goal of an
   integrated system for symbolic, numerical, graphical, and
   documentational mathematical work.  Recently the development of
   personal computers (with high resolution screens, window systems, and
   mice), high-speed networks, electronic mail, and electronic
   publishing, have created a technological base that is more than
   adequate for the realization of such systems.  However, the growth of
   separate Mathematical Typesetting, Multimedia Electronic Mail,
   Numerical Computation, and Computer Algebra communities, each with
   its own conventions, threatens to prevent these systems from being
   built.

   To be specific, little thought has been given to unifying the
   different expression representations currently used in the different
   communities.  This must take place if there is to be interchange of
   mathematical expressions among Document, Display, and Computation
   systems. Also, tools that are wanted in several communities (e.g.,
   WYSIWYG mathematical expression editors), are being built
   independently by each, with little awareness of the duplication of
   effort that thereby occurs.  Worst of all, the ample opportunities
   for cross-fertilization among the different communities are not being
   exploited.  For example, some Computer Algebra systems explicitly
   associate a type with a mathematical expression (e.g.,   3 x 3 matrix
   of polynomials with complex number coefficients), which could enable
   automated math proofreaders, analogous to spelling checkers.

   The goal of the Workshop on Environments for Computational
   Mathematics was to open a dialogue among representatives of the



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RFC 1019                                                  September 1987


   Computer Algebra, Numerical Computation, Multimedia Electronic Mail,
   and Mathematical Typesetting communities.  In July 1986, during the
   Computers and Mathematics Conference at Stanford University, a subset
   of this year's participants met at Xerox PARC to discuss User
   Interfaces for Computer Algebra Systems.  This group agreed to hold
   future meetings, of which the present Workshop is the first.  Alan
   Katz's recent essay, "Issues in Defining an Equations Representation
   Standard", RFC-1003, DDN Network Information Center, March 1987
   (reprinted in the ACM SIGSAM Bulletin May 1987, pp. 19-24),
   influenced the discussion at the Workshop, especially since it
   discusses the interchange of mathematical expressions.

   This report does not aim to be a transcript of the Workshop, but
   rather tries to extract the major points upon which (in the Editor's
   view) rough consensus was reached.  It is the Editor's view that the
   Workshop discussion can be summarized in the form of a basic
   architecture for "Standard Mathematical Systems", presented in
   Section II below.  Meeting participants seemed to agree that: (1)
   existing mathematical systems should be augmented or modified to
   conform to this architecture, and (2) future systems should be built
   in accordance with it.

   The Talks and Panel-Audience discussions at the Workshop were
   videotaped.  Currently, these tapes are being edited for submission
   to the SIGGRAPH Video Review, to form a "Video Proceedings".  If
   accepted by SIGGRAPH, the Video Proceedings will be publicly
   available for a nominal distribution charge.

   One aspect of the mathematical systems vision that we explicitly left
   out of this Workshop is the question of "intelligence" in
   mathematical systems.  This has been a powerful motivation to systems
   builders since the early days.  Despite its importance, we do not
   expect intelligent behavior in mathematical systems to be realized in
   the short term, and so we leave it aside.  Computer Assisted
   Instruction for mathematics also lies beyond the scope of the
   Workshop.  And although it might have been appropriate to invite
   representatives of the Spreadsheets and Graphics communities, we did
   not.  Many of those who were at the Workshop have given considerable
   thought to Spreadsheets and Graphics in mathematical systems.

   Financial support from the Xerox Corporation for AudioVisual
   equipment rental at SIGGRAPH is gratefully acknowledged.  Thanks are
   due to Kevin McIsaac for serving as chief cameraman, providing
   critical comments on this report, and contributing in diverse other
   ways to the Workshop.  Thanks also to Richard Fateman, Michael
   Spivak, and Neil Soiffer for critical comments on this report.
   Subhana Menis and Erin Foley have helped with logistics and
   documentation at several points along the way.

   Information on the Video Proceedings, and any other aspect of the
   Workshop can be obtained from the author of this report.



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RFC 1019                                                  September 1987


I. Particulars of the meeting

   The Workshop had four parts: (1) Talks, (2) Panel Discussion, (3)
   Panel and Audience discussion, (4) and Live demos.  Only a few of the
   systems presented in the talks were demonstrated live. However, many
   of the talks contained videotapes of the systems being discussed.

   The talks, each 15 minutes in length, were:

   1. "The MathCad System: a Graphical Interface for Computer
      Mathematics", Richard Smaby, MathSOFT Inc.

   2. "MATLAB - an Interactive Matrix Laboratory", Cleve Moler,
      MathWorks Inc.

   3. "Milo: A Macintosh System for Students", Ron Avitzur, Free Lance
      Developer, Palo Alto, CA.

   4. "MathScribe: A User Interface for Computer Algebra systems", Neil
      Soiffer, Tektronix Labs.

   5. "INFOR: an Interactive WYSIWYG System for Technical Text",
      William Schelter, University of Texas.

   6. "Iris User Interface for Computer Algebra Systems", Benton Leong,
      University of Waterloo.

   7. "CaminoReal: A Direct Manipulation Style User Interface for
      Mathematical Software", Dennis Arnon, Xerox PARC.

   8. "Domain-Driven Expression Display in Scratchpad II", Stephen
      Watt, IBM Yorktown Heights.

   9. "Internal and External Representations of Valid Mathematical
      Reasoning", Tryg Ager, Stanford University.

   10. "Presentation and Interchange of Mathematical Expressions in the
       Andrew System", Maria Wadlow, Carnegie-Mellon University.

   The Panel discussion lasted 45 minutes.  The panelists were:

      Richard Fateman, University of California at Berkeley
      Richard Jenks, IBM Yorktown Heights
      Michael Spivak, Personal TeX
      Ronald Whitney, American Mathematical Society









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RFC 1019                                                  September 1987


   The panelists were asked to consider the following issues in planning
   their presentations:

   1. Should we try to build integrated documentation/computation
      systems?

   2. WYSIWYG editing of mathematical expressions.

   3. Interchange representation of mathematics.

   4. User interface design for integrated documentation/computation
      systems.

   5. Coping with large mathematical expressions.

   A Panel-Audience discussion lasted another 45 minutes, and the Demos
   lasted about one hour.

   Other Workshop participants, besides those named above, included:

      S. Kamal Abdali, Tektronix Labs
      George Allen, Design Science
      Alan Katz, Information Sciences Institute
      J. Robert Cooke, Cornell University and Cooke Publications
      Larry Lesser, Inference Corporation
      Tom Libert, University of Michigan
      Kevin McIsaac, Xerox PARC and University of Western Australia
      Elizabeth Ralston, Inference Corporation

II. Standard Mathematical Systems - a Proposed Architecture

   We postulate that there is an "Abstract Syntax" for any mathematical
   expression.  A piece of Abstract Syntax consists of an Operator and
   an (ordered) list of Arguments, where each Argument is (recursively)
   a piece of Abstract Syntax.  Functional Notation, Lisp SExpressions,
   Directed Acyclic Graphs, and N-ary Trees are equivalent
   representations of Abstract Syntax, in the sense of being equally
   expressive, although one or another might be considered preferable
   from the standpoint of computation and algorithms.  For example, the
   functional expression "Plus[Times[a,b],c]" represents the Abstract
   Syntax of an expression that would commonly be written "a*b+c".

   A "Standard Mathematical Component" (abbreviated SMC) is a collection
   of software and hardware modules, with a single function, which if it
   reads mathematical expressions, reads them as Abstract Syntax, and if
   it writes mathematical expressions, writes them as Abstract Syntax.
   A "Standard Mathematical System" (abbreviated SMS) is a collection of
   SMC's which are used together, and which communicate with each other
   in Abstract Syntax.

   We identify at least four possible types of components in an SMS.



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RFC 1019                                                  September 1987