A sample revised prelude for numeric classes

William Lee Irwin III wli@holomorphy.com
Sun, 11 Feb 2001 22:46:15 -0800

At 2001-02-11 21:18, Tom Pledger wrote:
>>The main complication is that the type system needs to deal with
>>integer exponents of dimensions, if it's to do the job well.

On Sun, Feb 11, 2001 at 10:16:02PM -0800, Ashley Yakeley wrote:
> Very occasionally non-integer or 'fractal' exponents of dimensions are 
> useful. For instance, geographic coastlines can be measured in km ^ n, 
> where 1 <= n < 2. This doesn't stop the CIA world factbook listing all 
> coastline lengths in straight kilometres, however.

This is pretty rare, and it's also fairly tough to represent points in
spaces of fractional dimension. I'll bet the sorts of complications
necessary to do so would immediately exclude it from consideration in
the design of a standard library, but nevertheless would be interesting
to hear about. Can you comment further on this?

On Sun, Feb 11, 2001 at 10:16:02PM -0800, Ashley Yakeley wrote:
> More unit weirdness occurs with logarithms. For instance, if y and x are 
> distances, log (y/x) = log y - log x. Note that 'log x' is some number + 
> log (metre). Strange, huh?

If you (or anyone else) could comment on what sorts of units would be
appropriate for the result type of a logarithm operation, I'd be glad to
hear it. I don't know what the result type of this example is supposed
to be if the units of a number are encoded in the type.

On Sun, Feb 11, 2001 at 10:16:02PM -0800, Ashley Yakeley wrote:
> Interestingly, in C++ you can parameterise types by values. For instance:
	[interesting C++ example elided]
> Can you do this sort of thing in Haskell?

No, in general I find it necessary to construct some sort of set of
types parallel to the actual data type, define some sort of existential
data type encompassing the set of all types which can represent one of
those appropriate values, and "lift" things to that type by means of
sample arguments. I usually like ensuring that the types representing
things like integers never actually have any sort of data manifest,
i.e. the sample arguments are always undefined. This is a bit awkward.

I think Okasaki's work on square matrices and perhaps some other ideas
should be exploited for this sort of thing, as there is quite a bit of
opposition to the usage of sample arguments. I'd like to see a library
for vector spaces based on similar ideas. I seem to be caught up in
other issues caused by mucking with fundamental data types' definitions,
my working knowldedge of techniques like Okasaki's is insufficient for
the task, and my design concepts are probably too radical for general
usage, so I'm probably not the man for the job, though I will very
likely take a stab at such a beast for my own edification.