# Extensible datatypes

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## Revision as of 22:31, 7 May 2006

## Contents |

## 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 `B`s 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 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

## 3 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...

## 4 Open functions

*This article is a stub. You can help by expanding it.*

## 5 References

- Andres Löh and Ralf Hinze. Open data types and open functions