[Haskell-cafe] a problem defining a monad instance

[Henning Thielemann]

On Fri, 6 Nov 2009, Petr Pudlak wrote:

>   Hi all,
>
> (This is a literate Haskell post.)
>
> I've encountered a small problem when trying to define a specialized
> monad instance. Maybe someone will able to help me or to tell me that
> it's impossible :-).
>
> To elaborate: I wanted to define a data type which is a little bit
> similar to the [] monad. Instead of just having a list of possible
> outcomes of a computation, I wanted to have a probability associated
> with each possible outcome.

http://hackage.haskell.org/package/probability



> A natural way to define such a structure is to use a map from possible
> values to numbers, let's say Floats:
>
>> module Distribution where
>>
>> import qualified Data.Map as M
>>
>> newtype Distrib a = Distrib { undistrib :: M.Map a Float }
>
> Defining functions to get a monad instance is not difficult.
> "return" is just a singleton:
>
>> dreturn :: a -> Distrib a
>> dreturn k = Distrib (M.singleton k 1)
>
> Composition is a little bit more difficult, but the functionality is
> quite natural. (I welcome suggestions how to make the code nicer / more
> readable.) However, the exact definition is not so important.
>
>> dcompose :: (Ord b) => Distrib a -> (a -> Distrib b) -> Distrib b
>> dcompose (Distrib m) f = Distrib $ M.foldWithKey foldFn M.empty m
>>   where
>>      foldFn a prob umap = M.unionWith (\psum p -> psum + prob * p) umap (undistrib $ f a)
>
> The problem is the (Ord b) condition, which is required for the Map
> functions.  When I try to define the monad instance as

This won't work and is the common problem of a Monad instance for 
Data.Set.
   http://www.randomhacks.net/articles/2007/03/15/data-set-monad-haskell-macros

There is however an idea of how to solve this using existential 
quantification and type families:
   http://code.haskell.org/~thielema/category-constrained/src/Control/Constrained/Monad.hs