[om] Indexed vars
Andreas Strotmann
Strotmann at rrz.uni-koeln.de
Wed Dec 3 15:49:26 CET 2003
I know there's been some discussion on this topic before (namely, the
problems involved in the current proposal for indexed variables), but I
think there is an interesting point that I haven't seen discussed yet.
We haven't really considered the way that MathML treats the issue, at
least not on the mailing list, and I don't recall any mention from the
Bremen workshop, where this topic was also discussed.
If I understand MathML correctly, it also supports "indexed" variables.
In fact, it supports *very* complex ways of writing variables. But: the
important thing is that content MathML says that the internal structure
of a variable is *presentation* MathML -- i.e., no semantics must be
attached to any internal component of a variable. You can have
<ci><msub><mi>x</mi><mi>i</mi></msub></ci>, but *by design* there is no
way to claim reliably that that i in there is related to any <ci>i</ci>
that might be used outside.
Supposing OpenMath wants to do as MML-C does. The problem of course is
that OpenMath, unlike MathML, has no way of saying "OM presentation, not
OM content". So what could we do to mirror MML-Content?
The current proposal says that variables may have OpenMath objects as
content. These, by definition, are objects with a semantics, and even in
the examples in the proposal they may be variables. This is *not*
compatible with MathML.
Indeed, we have only two possibilities that I can see to match MathML
(again, should this be our goal):
a) restrict content of a variable to foreign objects (e.g. MathML
presentation XML) -- these are non-semantic by nature, anyway.
b) do not allow content of a variable, but use attribution (the
non-semantic variety). This has been considered a prototypical use of
non-semantic attribution from the start. In this case, one could argue
that an OpenMath object that is a value of a non-sematic attribute is
non-semantic by nature, and a formal semantics could be made to conform
to that interpretation.
c) specify that content of variables is non-semantic even though
arbitrary OpenMath objects are allowed
I think that some of David's comments correspond (roughly) to b)
("backing out", I think he phrased it) and c) , but a) has not been
considered.
The problem with both b) and c) is that I will almost guarantee that at
least 50% of the phrasebooks would go and interpret the contents of
either a variable or its non-sematic attribute in spite of any
definitions in OpenMath to the contrary. Thus, an index "i" within a
variable or within an attribute value of a non-semantic attribution
would almost certainly be interpreted to be bound by a surrounding
binding element with a bound variable "i". In fact, I'm not even sure
whether or not this is correct behaviour from an OpenMath perspective in
the case of b). But I'm certain that from a MathML Content perspective,
that <mi>i</mi> inside a content identifier is not bound by any
surrounding apply with <bvar>i</bvar>.
So, perhaps a combination of a) and b) plus a CD that contains a
MathML-Presentation non-semantic attribute symbol in the MathML CD group
might be the safest bet, with the MathML Presentation attribute value
generally being used as in a), that is as a foreign (XML) object.
Just my 0.02 Euros ;-)
-- Andreas
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