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Re: [Axiom-math] Re: [Axiom-developer] Re: musings on notation
From: |
W Naylor |
Subject: |
Re: [Axiom-math] Re: [Axiom-developer] Re: musings on notation |
Date: |
Wed, 11 Aug 2004 15:07:56 +0100 (BST) |
Well you can check out my thesis from my web site if you like:
http://www.cs.bath.ac.uk/~wn/thesis.ps.gz
there is also a tar file of some of the code I wrote as well (can't
remember if it works though!!),
cheers,
Bill
On Wed, 11 Aug 2004, Mike Dewar wrote:
> 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
> >
> >
> >
> > _______________________________________________
> > Axiom-math mailing list
> > address@hidden
> > http://lists.nongnu.org/mailman/listinfo/axiom-math
> >
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