[Haskell-cafe] Context for type parameters of type constructors

[Henning Thielemann]

the next lap ...

On Tue, 30 Mar 2004, Dylan Thurston wrote:

> I recommend you use multi-parameter type classes, with a type of the
> scalars and the type of the vectors.  For the method you're using, you
> need to add a 'Num a' context.  You say that you 'cannot catch all
> requirements that instances may need', but certainly any instance will
> need that context. 


Following your advice I tried to solve the problem
with a multi-parameter type classes.
Btw. I need this stuff for computations with physical values,
i.e. numeric values equipped with physical units.
Pure computation with physical values is no problem at all
only converting the values to strings is what causes all the trouble!

I compiled this text with
 ghc -fglasgow-exts -c VectorSpace.lhs
  (I like to omit  -fallow-undecidable-instances
   before knowing what it means)

> module VectorSpace
>    where
>
> import Data.Complex

Here is the new vector space class.
Now 'v' is the vector type
and 'a' is a compliant scalar type.

> -- a classical linear space
> class VectorSpace v a where
>    zero  :: v
>    add   :: v -> v -> v
>    scale :: a -> v -> v
>
> instance Num a => VectorSpace a a where
>    zero  = 0
>    add   = (+)
>    scale = (*)

Here the compiler complains the first time:

VectorSpace.lhs:27:
    Illegal instance declaration for `VectorSpace a a'
        (There must be at least one non-type-variable in the instance head
         Use -fallow-undecidable-instances to permit this)
    In the instance declaration for `VectorSpace a a'

> instance RealFloat a => VectorSpace (Complex a) a where
>    zero           = 0
>    add            = (+)
>    scale s (x:+y) = (s*x) :+ (s*y)
>
> instance Num a => VectorSpace [a] a where
>    zero    = repeat 0
>    add     = zipWith (+)
>    scale s = map (s*)
>
> instance Num a => VectorSpace (b -> a) a where
>    zero      _ = 0
>    add   f g x = (f x) + (g x)
>    scale s f x = s*(f x)


To stay conform to mathematical systematics
I separated the definition of the norm
from the 'VectorSpace' definition.

> -- a vector space equipped with a norm
> class VectorSpace v a => Normed v a where
>    norm :: v -> a
>
> instance Num a => Normed a a where
>    norm = abs
>
> instance RealFloat a => Normed (Complex a) a where
>    norm = magnitude
>
> instance Num a => Normed [a] a where
>    -- fails for infinite lists
>    norm = sum.(map abs)


Now I introduce a new datatype for a vector valued quantity.
The 'show' function in this simplified example
may show the vector with the magnitude separated
from the vector components.

> data Quantity v = Quantity v
>
> instance (Show v, Fractional a, Normed v a) =>
>         Show (Quantity v) where
>    show (Quantity v) =
>        let nv::a = norm v
>        in  (show (scale (1/nv) v)) ++ "*" ++
>            (show nv)

The problem which arises here is that the type 'a'
is used for internal purposes of 'show' only.
Thus the compiler can't decide which instance of 'Normed'
to use if I call 'show':

Prelude VectorSpace> show (Quantity [1,2,3])

<interactive>:1:
    No instance for (Normed [t] a)
      arising from use of `show' at <interactive>:1
    In the definition of `it': it = show (Quantity [1, 2, 3])




So I tried the approach which is more similar
to what I tried before with a single-parameter type class:
I use a type constructor 'v'
instead of a vector type 'v'
but now by the two-parameter type class
I mention the type 'a' explicitly
which allows for context restrictions
on instantation later.

> class VectorSpaceC v a where
>    zeroC  :: v a
>    addC   :: v a -> v a -> v a
>    scaleC :: a -> v a -> v a

One consequence is now that I cannot use
the scalar type 'a' as vector type, too.
Instead I need some type constructor 'Identity'
which I can make an instance of class 'VectorSpace'.
So let's immediately switch to the complex numbers.

> instance RealFloat a => VectorSpaceC Complex a where
>    zeroC  = 0
>    addC   = (+)
>    scaleC s (x:+y) = (s*x) :+ (s*y)
>
> class VectorSpaceC v a => NormedC v a where
>    normC :: v a -> a
>
> instance RealFloat a => NormedC Complex a where
>    normC = magnitude

But the 'Show' instance causes new trouble:

> data QuantityC v a = QuantityC (v a)
>
> instance (Fractional a, NormedC v a, Show (v a)) =>
>         Show (QuantityC v a) where
>    show (QuantityC v) =
>        let nv = normC v
>        in  (show (scaleC (1/nv) v)) ++ "*" ++
>            (show nv)

It lead the compiler eventually fail with:
VectorSpace.lhs:138:
    Non-type variables in constraint: Show (v a)
    (Use -fallow-undecidable-instances to permit this)
    In the context: (Fractional a, NormedC v a, Show (v a))
    While checking the context of an instance declaration
    In the instance declaration for `Show (QuantityC v a)'


Does exist a clean solution for the problem at all?