[Haskell-cafe] vector-space and standard API for vectors

[wren ng thornton]

On 10/26/10 8:51 AM, Alexey Khudyakov wrote:
> On 24.10.2010 03:38, wren ng thornton wrote:
>> I don't care much about the name of the class, I'd just like support for
>> monoids, semirings,... when they lack a group, ring,... structure.
>>
> Then what about following type class hierarchy? I think it supports
> those structures. Only restriction is that it forces one to have both
> left and right modules. It's possible to split them but I think it will
> be to painful for vector spaces over R and C.
>
> class Module v where
> type Scalar v :: *
> (*^) :: Scalar v → v → v
> (^*) :: v → Scalar v → v
> (^*) = flip (*^)

Is there any good reason for forcing them together? Why not, use the 
hierarchy I proposed earlier? If you want to reduce the clutter in type 
signatures for real and complex vector spaces then just add to my previous

     -- Or just call it "Module" if preferred.
     class (LeftModule v, RightModule v) => AssociativeModule v where
         -- Law: (^*) == flip (*^)

This way, when (not if) people want nonassociative modules the classes 
are already there. The additional overhead in defining an associative 
module is only three lines when using default implementation; two lines 
otherwise:

     type instance Scalar Foo = Bar
     instance AssociativeModule Foo where
     instance RightModule Foo where
         (^*) = flip (^*)
     instance LeftModule Foo where
         (*^) = ...

vs

     instance Module Foo where
         type Scalar Foo = Bar
         (*^) = ...

And once it's defined, the usage is the same: just require 
AssociativeModule and you'll pull in both (*^) and (^*).

We already know that there are noncommutative modules/vectorspaces of 
interest (e.g., modules over quaternions and modules over graph paths), 
why not support them from the beginning? It seems like you're going out 
of your way to exclude things that would be trivial to include. This is 
exactly why this is my standard complaint against the various proposals 
out there for new numeric hierarchies. People who are used to only using 
R^n think the proposals are just fine, but none of the proposals capture 
the structures I work with daily. Which means the new proposals are no 
better than the Prelude for me.

-- 
Live well,
~wren