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

[W Naylor]

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
> > 
> > ________________________________________________________________________
> > This e-mail has been scanned for all viruses by Star Internet. The
> > service is powered by MessageLabs. For more information on a proactive
> > anti-virus service working around the clock, around the globe, visit:
> > http://www.star.net.uk
> > ________________________________________________________________________
> 
> ________________________________________________________________________
> This e-mail has been scanned for all viruses by Star Internet. The
> service is powered by MessageLabs. For more information on a proactive
> anti-virus service working around the clock, around the globe, visit:
> http://www.star.net.uk
> ________________________________________________________________________
> 
> 
>