[om] more comments on CDs

[Mike Dewar]

I think I agree with David that we don't have any notion of
"constructor" in OpenMath, certainly not in the sense that a language
like Java has.  What is the constructor of the expression "a+x", is it
plus?

As far as "rational(4,6)" goes the STS signature is (Z x Z/{0}) -> Q and
there are no side conditions specified in the CD so I think that it is
allowed.  We cannot define rational more tightly because we have to be
compatible with the MathML definition.

While we probably all agree what the normal form of the rationals is, in
many cases this won't be the case.  Whether an algebra system stores
"a+x" (as Maple) or "x+a" (as Axiom, Reduce) is a matter of taste.  For
most computations this is irrelevent but a big exception is the kind of
"positional" deconstructors that you propose (what is the leading term
of a+x?).  While we could in theory go down the road of defining
OpenMath normal forms for everything that doesn't seem like a good idea
to me, although I note that we can do this for polynomials (where it is
clearly important) by for example adding an ordering attribute to a SDMP
object.

I think if you try to define the notion of a constructor in OpenMath
then you will have to address the question of what is a normal form and
that will end up making it much harder to write a phrasebook.  While I
can easily construct the object "a+x" in Axiom I believe that the only
way to do it in Reduce is to change a global variable which will change
the bahaviour of all similar objects in the system.  I think its much
better to leave that kind of implementation detail to the systems
themselves.

Regards, Mike.

On Thu, Oct 24, 2002 at 08:36:36PM +0200, Arjeh Cohen wrote:
<snip> 
> IN GENERAL:
>   When a symbol is defined as a constructor, we should be clear about
>   the conditions on the arguments when it is used as a head in an
>   apply. For instance: is rational(4,6) allowed? If we require a
>   normal form for 2/3, the answer is no. If we do not care, the answer
>   is yes. But then, what does it mean to take "denominator" (not yet
>   defined...): 2 or 4?
>   To prevent such ambiguities I propose:
>   a) we are very precise about the requirements on the arguments;
>   b) we make clear whether we are dealing with a normalform for the
>      object under construction or not. In our example, we may want to
>   add "rational_NF" which requires that there are two arguments, each
>   of them integers, and furthermore that the second argument be stricly
>   positive and 1 if the first is zero, and that the gcd of the first
>   and second argument is 1 if the first argument is nonzero.
>   etc..
>   
> IN GENERAL: Wouldn't it be useful for a symbol that is used as a
>   constructor to have default deconstructors---that is, to be able to
>   grab the i-th argument? It is not needed for codec's, where such
>   operations are standard (getleaf, etc.), but it is useful for
>   formulating FMP's. Sometimes, but not always, such arguments can be
>   retrieved by symbols defined elsewhere in a CD.  A realization of
>   this idea could take place in a CD defining a symbol "arg" having
>   two arguments, the first is an OM object whose root is an OMA, the
>   second is an integer. Eg, if f is a symbol then arg(f(a,b,c,d),3) =
>   c. The symbol "arg" would not make sense when applied top a function
>   symbol instead of a constructor symbol.  To prevent such
>   difficulties arising when using "arg", one might take an alternative
>   approach by staying away from the "meta symbol" arg and requiring
>   that each argument of a constructor symbol A is accessible via a
>   name/symbol defined in the same CD where A is defined. This would
>   then come close to Michael Kohlhase's suggestion of introducing structs.
> 

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