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some broad statements



I'm posting this note to restate and make a little more explicit
(perhaps only in my own mind) what some of our issues are.  This isn't
meant to sidetrack current conversation, and I'm trying to keep the
text below to a minimum.  This is also not meant to reinterpret
previous position statements or override them.  I'm only providing
another framework which is understandable to me, and attempts to place
the 4 "positions" with which we have some relation --- the Wolfram
proposal, Ka-Ping's MINSE system, Roy Pike's Math DTD work, and the
OpenMath Consortium's efforts --- in a common context.  This is only
meant to be a set of statements off which others may or may not
bounce, not a position paper for the HTML-Math ERB.  I welcome
corrections, adjustments, contrary opinions, ...

-Ron

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BROADLY
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We (the HTML-Math ERB) are attempting to devise a notation for
mathematics (I recognize Ping's interest in broadening our notational
target, but here will not move beyond mathematics) which can be
automatically rendered to visual and audio formats (at least one of
each) and for use by various computational engines as well.  The
visual format is the sine qua non here, audio seems achievable as long
as we avoid some pitfalls of systems designed purely for visual
rendering, and more in doubt is the degree to which we can support
rendering to computer algebra systems, theorem provers, and other more
specialized software for dealing with symbolic, numeric, and
knowledge-retrieval problems.

My understanding is that we are attempting to solve a set of problems
somewhat smaller in scope than those addressed in the OpenMath
Consortium's Objectives document.  (This OpenMath document does give a
good overview of problems and objectives.  My emphasis here is
slightly different, and I don't consider that this little description
summarizes the views of the OpenMath Consortium.)  There may well be
questions regarding the degree to which we can break off a part of the
mathematics interchange problem, and we should be open to discussion
on this.  We certainly don't want the efforts here to be in conflict
with those of OpenMath.  Consistency and symbiosis, if not identity,
should be important in our considerations.

It is apparent that, although visually-oriented notational systems
(such as printed form, or code, such as TeX, to produce print form)
are efficient carriers of information, their effectivity is highly
dependent upon human facility at interpretation, and such notational
systems are quite ambiguous to software written now (and probably
within the next many years).  When software is not capable of
disambiguation, humans must do so in the notation, to a degree seen as
necessary.  Thus, printed mathematical notation must be disambiguated
and regularized to some degree in order to serve the purpose of machine
readability and interchange.

On the other hand, ambiguity has its place (and very seriously so) in
the guise of abstraction from details which do not require further
specification for purposes at hand or foreseen.  This is simply in the
name of prioritization and efficiency.  The free market of printed
mathematics decides on its own when efficiencies of notational
commonality are merited (this occurs within individual papers and
within fields of common discourse).  As mathematical ideas are
generated and sifted, notation is introduced and settles to more
stable form over time.  As new ideas enter, the mode of expression for
old problems and theorems also changes.  And there is great cost in
developing software which deals with some set of mathematical issues,
using a certain concrete syntax and semantics in these particular
areas of mathematics.  Although computerized realms of mathematics are
ever-widening, there will always be advantage in speaking in an
informal, abstract and machine-ambiguous way about realms not yet
machine treated.  Thus, I do think there are avenues of mathematical
discourse where there is a strong need to "speak" with the simple
printed (implying also "spoken") word, to not require the high level
of disambiguation necessary for machine treatment.  The path of
disambiguation has a cost which must be considered.

And even aside from poles of traditional, printed (machine-ambiguous)
mathematical notation and machine-processable notation, there are
problems of inconsistency among those systems which treat the same
material on the same level and might be intertranslatable.  These
inconsistencies in syntax and semantics block or increase the cost of
moving between the systems automatically.

OpenMath sets out to solve the intercommunication issues on all
levels.  HTML-Math is trying to present a notation sufficient for
visual and audio display, to which it is possible to translate from
other visual display languages (such as TeX and the existing ISO
12083), and from which it is possible to derive other notations
sufficient for computation.

In speaking of solutions to interchange problems, we should keep in
mind (at least in background) the scale on which a solution works
effectively.  All parties recognize the patched and non-global natures
of notational systems.  OpenMath speaks of lexicons, Pike of domains.
Notations and their conventions or "contexts" range from a formula (or
subformula) to paragraphs, articles, books, and beyond (lexicons).
Insofar as the OpenMath and Pike approaches aim for large-scale
intertranslatability, it seems not unreasonable to me that HTML-Math
differentiate itself to a smaller scale.


-----------------
MORE SPECIFICALLY
-----------------

The Wolfram Proposal
--------------------

The Wolfram proposal for HTML-Math suggests that we use traditional
mathematical notation (i.e. something akin to code for printed
notation) as our basis, and parse this notation via operator
precedence tables and bracketing to an "expression tree" which is to
be the fundamental structure from which other forms are derived.  The
notation provides a means of altering precedences and adding new
identifiers and operators.  Visual and audio rendering should be
directly achievable from an expression tree.  Conversion of
expression-tree data to forms appropriate for computational engines
will be done on the basis of template- or pattern-matching maps.  (So
semantics may be "attached" to the notation by adjoining or pointing
to a collection of pattern matches.)

Bruce and Neil (and I believe Dave) have expressed confidence in the
fact that template-matching will achieve the semantical mapping we
want.  Ping has expressed doubts (I believe based on the fact that
ambiguities in expressions will get out of control as the range over
which templates are matched grows, and that template-matching will
become unreliable).  Raman also spoke about wanting to determine the
extensibility mechanism early on, and I believe he was concerned about
the degree to which notation would be transformable.  I find the issue
very difficult to judge myself.  I have a vague idea that the
template-matching is to be something akin to the various pattern
transformation schemes I see in the Mathematica Manual, but I don't
have a good idea of details or an idea of the scope of operation.  I
think more conversation in this area would help me, also recognizing
Bruce's recent statements regarding priorities.  I have neither the
confidence to say this is a "go", nor the knowledge to provide
counterexamples.  Perhaps we would all benefit if someone would throw
up a good set of concrete examples (say, with traditional notation
appropriate for translation to computational engines).  I think this
is at least appropriate by the time of our October meeting.

I do like the late semantical binding afforded by the Wolfram
proposal, and its emphasis on notation rather than semantics.


Ping's MINSE notation
---------------------

MINSE has much in common with the Wolfram proposal.  MINSE also starts
with traditional notation, but allows augmentation with "compounds"
which are locally defined operators.  Again the fundamental structure
is an expression tree parsed from the notation, but in this case,
ambiguities are removable through attentions of a "good" author who
incorporates appropriate compounds.  As Ping has noted, MINSE notation
can lead to cleaner translation because it may have been disambiguated
by the author.  The approach here allows a semantical element to enter
insofar as authors may specify their own sense of semantics via
compounds.

My own reservations about MINSE spring from what I think may be
analogous situations in TeX documents where "good" authors have
sometimes produced notations which are very hard to handle by third
parties.  Despite a listing of locally defined compounds and what they
"mean", one may still have some more difficulty adding to or mapping
from the new notation than one would have with some more commonly-seen
notational forms.  It is also unclear to me to what degree authors
will "be good".  AMS authors have had opportunity in TeX for quite
some time, and they have been terrible at it.  This is undoubtedly
because there is no compensatory reward for being "good".


Pike approach
-------------

The Pike and OpenMath approaches are more semantically oriented than
the current proposals for HTML-Math.

Pike is attempting to define DTDs for areas of discourse (perhaps
corresponding to the lexicons of OpenMath).  This provides uniform
notation within each area and may be achievable in certain areas, but
does not seem very suited to areas in which concepts and notation are
in flux or locally tuned.  So, e.g., I foresee little chance of AMS
authors using such DTDs for their journal articles.


OpenMath approach
-----------------

This is clearly the project of largest scope among those mentioned
here.  The problems with which HTML-Math is concerned are sub-problems
of the general one of full mathematical intercommunication.  OpenMath
proposes to solve the large problem by implementing layers of
communication protocols, the top layer comprising semantical realms,
and the next-to-top notational (or expression) realms.  I believe that
OpenMath views most expression interchange as being mediated by the
semantical realms.  I have little quarrel with the general list of
objectives or the OpenMath layered model, but the project does seem
vast to me and I wonder how achievable it is.  I also need to hear
more detail about the thinking of those involved.  As mentioned
earlier, I suspect we may define our problems and objectives in such a
way as to be of smaller scale but of greater tractability than those
of OpenMath, all the while listening to whatever progress and concerns
the Consortium has to report.