"Ch. A. Herrmann" <herrmann(a)infosun.fmi.uni-passau.de> writes:
> moved to haskell-cafe
No, but *now* it is. (Does haskell@ strip Reply-To? Bad list! Bad!)
> the problem is that the --majority, I suppose?-- of mathematicians
> tend to overload operators. They use "*" for matrix-matrix
> multiplication as well as for matrix-vector multiplication etc.
Yes, obviously. On the other hand, I think you could get far by
defining (+) as an operator in a Group, (*) in a Ring, and so forth.
Another problem is that the mathematical constructs include properties
not easily encoded in Haskell, like commutativity, associativity, etc.
> I don't think that it is acceptable for a language like Haskell
> to permit the user to overload predefined operators, like "*".
Depends on your definition of overloading. Is there a difference
between overloading and instantiating a class? :-)
> A cheap solution could be to define a type MathObject and operators like
> :*: MathObject -> MathObject -> MathObject
> Then, the user can implement:
> a :*: b = case (a,b) of
> (Matrix x, Matrix y) -> foo
> (Matrix x, Vector y) -> bar
Yes. If it is useful to have a fine granularity of classes, you can
imagine doing:
class Multiplicative a b c where
(*) :: a -> b -> c
now I can do
instance Multiplicative (Vector a) (Vector a) (Vector a) where
x * y = ...
but also scalar multiplication
instance Multiplicative a (Vector a) (Vector a) where
a * x = ....
Also, I think I can define Group a to be
Additive a a a => class Group a where
-- inherits plus from "Additive"
zero :: a
instance Group Int where
(+) = built_in_int_addition
zero = 0::Int
Long qualifier lists might be countered by having Classes -- Num, say
-- that just serve to include other classes in reasonable collections.
Funny mathematical names would - at least to some extent - be avoided
by having simple names for the classes actually defining the
operators, so that errors will warn you about missing "Multiplicative"
rather than Field or Ring or what have you.
>From experience, I guess there are probably issues that haven't
crossed my mind. :-)
-kzm
--
If I haven't seen further, it is by standing in the footprints of giants
