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Extensible datatypes

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1 The problem

Here's a simple test for object orientation (for some reasonable definition):

Define a type A such that for any type B you can define
   up :: B -> A
   down :: A -> Maybe B
such that
   down . up = Just

You can do this quite easily in Java or C++, mutatis mutandis. You can't do this in Haskell (or O'Haskell either).

You can do a weaker form of this with Haskell's Dynamic, where you only have to deal with Bs that are instances of Typeable. But even with that, note that Dynamic/Typeable/TypeRep are a bit messy, with instances for Typeable defined for a wide range of known types.

An alternative approach would be to identify your B within A not per-B but per-(up,down). This would allow for instance separate (up,down) for the same B such that

 down1 . up2 = Nothing
 down2 . up1 = Nothing

Of course this can be done with Dynamic too, by defining dummy types. But it's ugly.

2 Extensible datatypes

Extensible datatypes allow a type to be defined as "open", which can later be extended by disjoint union. Here's a sample syntax that achieves the above OO test:

 module P where
 data A = ..
 module Q where
 import P
 A |= MkB B
 up = MkB
 down (MkB b) = Just b
 down _ = Nothing

3 Deriving Dynamic

It's possible to define Dynamic using extensible datatypes. Here's a naive attempt:

 data Dynamic = ..
 class Typeable' a where
   toDyn :: a -> Dynamic
   fromDynamic :: Dynamic -> Maybe a
 -- for each type...
 Dynamic |= MkBool Bool
 instance Typeable' Bool where
   toDyn = MkBool
   fromDynamic (MkBool b) = Just b
   fromDynamic _ = Nothing

This attempt however doesn't allow easy creation of Typeable1, Typeable2 etc. A better way is to use type-constructor parameters:

 data Dynamic0 (f :: * -> *) = ..
 data Dynamic1 (g :: (* -> *) -> *) = ..
 type Dynamic = Dynamic0 Identity
 data Type a = MkType
 type TypeRep = Dynamic0 Type
 class Typeable0 a where
   toDyn0 :: f a -> Dynamic0 f
   fromDynamic0 :: Dynamic0 f -> Maybe (f a)
 class Typeable1 p where
   toDyn1 :: g p -> Dynamic1 g
   fromDynamic1 :: Dynamic1 g -> Maybe (g p)
 data Compose p q a = MkCompose (p (q a))
 data Compose1 d0 f p = MkCompose1 (d0 (Compose f p))
 Dynamic0 f |= MkDynamic1 (Dynamic1 (Compose1 Dynamic0 f))
 unDynamic1 :: Dynamic0 f -> Maybe (Dynamic1 (Compose1 Dynamic0 f))
 unDynamic1 (MkDynamic1 xx) = Just xx
 unDynamic1 _ = Nothing
 instance (Typeable1 p,Typeable0 a) => Typeable0 (p a)
   -- toDyn0 :: f (p a) -> Dynamic0 f
   toDyn0 = MkDynamic1 . toDyn1 . MkCompose1 . toDyn0 . MkCompose
   -- fromDynamic0 :: Dynamic0 f -> Maybe (f (p a))
   fromDynamic0 dyn = do
     dcdf <- unDynamic1 dyn
     (MkCompose1 dcfp) <- fromDynamic1 dcdf
     (MkCompose fpa) <- fromDynamic0 dcfp
     return fpa
 -- for each type
 Dynamic0 f |= MkInt (f Int)
 instance Typeable0 Int where
    toDyn0 = MkInt
    fromDynamic0 (MkInt fi) = Just fi
    fromDynamic0 _ = Nothing
 Dynamic1 g |= MkMaybe (g Maybe)
 instance Typeable1 Maybe where
    toDyn1 = MkMaybe
    fromDynamic1 (MkMaybe gm) = Just gm
    fromDynamic1 _ = Nothing

I submit that this is "hairy" rather than "ugly", but I suspect the Type-Constructors Of Unusual Kind (TCOUKs) get even hairier for Typeable2, Typeable3 etc...