Extensible datatypes

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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.

Extensible datatypes

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

module P where

-- define open datatype
open data A :: *

module Q where
import P

-- add constructor to A
MkB :: B -> A

up = MkB
down (MkB b) = Just b
down _ = Nothing

Deriving Dynamic

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

open Dynamic :: *

class Typeable' a where
  toDyn :: a -> Dynamic
  fromDynamic :: Dynamic -> Maybe a

-- for each type...

MkBool :: Bool -> Dynamic

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:

open data Dynamic0 :: (* -> *) -> *

open data Dynamic1 :: ((* -> *) -> *) -> *

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))

MkDynamic1 :: (Dynamic1 (Compose1 Dynamic0 f)) -> 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

MkInt :: (f Int) -> Dynamic0 f

instance Typeable0 Int where
   toDyn0 = MkInt
   fromDynamic0 (MkInt fi) = Just fi
   fromDynamic0 _ = Nothing

MkMaybe :: (g Maybe) -> Dynamic1 g

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...

Open functions

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