[om] Indexed vars

[Andreas Strotmann]

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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