Re: [Axiom-math] Re: [Axiom-developer] Re: musings on notation

[Mike Dewar]

I don't know if Bill Naylor subscribes to this list, but his PhD (at
Bath, supervised by James Davenport) involved using straight-line
programs to represent polynomials in Axiom.  Just as with other
mathematical objects you could do arithmetic with them, perform
operations such as GCD computations etc., however their representation
was as an explicit program.  These programs were represented in Axiom as
instances of domains in the usual way - if I remember rightly the
infrastructure he created was quite extensive.  I don't know if this
work really addresses Tim's original thoughts about notation which
started off this thread but it might be worth looking at or even
reviving.  

Mike.

On Wed, Aug 11, 2004 at 12:54:36PM +0000, Martin Rubey wrote:
> root writes:
>  > The problem that needs to be attacked, however, is that there doesn't
>  > appear to be a notation that I could write by hand for a "thing" that
>  > has the properties of a program (including the notion of process) as
>  > well as the properties of a mathematical object. (Or the "thing" that
>  > has the properties of a closure as well as a mathematical object).
> 
> Sorry, but I still do not understand. In fact, I don't see the need for such a
> notation. I'd say that "programs" are just "mathematical objects"... After 
> all,
> a polynomial for example, or better, the cosine is definitely a mathematical
> object, but it's also a "program".
> 
>  > Let me try an example. Consider the simple case of trying to raise a
>  > square matrix to an integer power:
>  >  
>  >  P = 3
>  >  M:SquareMatrix(2) = matrix([[1,2],[3,4]])
>  >  M^P
>  > 
>  > which we know how to do. 
> 
> OK.
> 
>  > The harder case is to assume we don't know the actual value of P but
>  > we know its Category. So if an IndefiniteInteger which have the
>  > property of integers but we don't say which one. IndefiniteInteger is
>  > a type we understand so we can say:
>  > 
>  >  P = IndefiniteInteger()
>  >  M = SquareMatrix(2)
>  >  M^P
> 
> Well, we do not yet have reached a conclusion what an IndefiniteInteger should
> be, do we? There is the possibility described by Davenport and Faure, and
> certainly there are others. In the above I also have trouble determining the
> type of M^P. I don't think you meant to have an exponentiation of domains? So
> it should probably read
> 
>  P : IndefiniteInteger()
>  M : SquareMatrix(2) = matrix([[1,2],[3,4]])
>  M^P
> 
> or
> 
>  P : IndefiniteInteger()
>  M : IndefiniteSquareMatrix(2)
>  M^P
> 
> or something like that. I'm not sure whether we want to modify the domain
> SquareMatrix to allow for exponentiation with an IndefiniteInteger, but on the
> other hand, why not? The result would be an IndefiniteSquareMatrix (or the 
> zero
> matrix or the identity -- oops, bug report on the way), that's for sure...
> 
>  > The notational case is even harder. So I'd like to be able
>  > to say:
>  > 
>  >  P = Program(foo)
>  >  M:SquareMatrix(2) = matrix([[1,2],[3,4]])
>  >  M^P
> 
> What do you mean by that? Is M^P a program, that evaluates to a
> SquareMatrix(2)? I don't think that there is a notational problem here.
> I don't really know whether an operator that delays execution of a program
> would be useful. Its consequences for the type-system are -- I admit -- not
> easy to foresee. However, I have the feeling that we do not have the userbase
> yet to explore these fields. I have the feeling, that it disperses our 
> "energy"
> a little, however.
> 
> I think it would be good to continue the discussion on indefinite things, but
> one such topic is enough -- for me at least. One suggestion: could we have a
> wishlist on the savannah website? Maybe registered users could even vote for
> priorities there?
> 
> All the best,
> 
> Martin
> 
> 
> 
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