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Extensible datatypes

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:Define a type <tt>A</tt> such that for any type <tt>B</tt> you can define
 
:Define a type <tt>A</tt> such that for any type <tt>B</tt> you can define
   
up :: B -> A
+
<haskell>
down :: A -> Maybe B
+
up :: B -> A
  +
down :: A -> Maybe B
  +
</haskell>
   
 
:such that
 
:such that
   
down . up = Just
+
<haskell>
  +
down . up = Just
  +
</haskell>
   
You can do this quite easily in Java or C++, mutatis mutandis. You can't do this in Haskell, I don't think. You can't actually do this in [[O'Haskell]] either, it seems the O' essentially amounts to syntactic sugar.
+
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 <tt>[[Dynamic]]</tt>, where you only have to deal with Bs that are instances of <tt>Typeable</tt>. But even with that, note that <tt>Dynamic</tt>/<tt>Typeable</tt>/<tt>TypeRep</tt> are a bit messy, with instances for <tt>Typeable</tt> defined for a wide range of known types.
+
You can do a weaker form of this with Haskell's <tt>[[Dynamic]]</tt>, where you only have to deal with <tt>B</tt>s that are instances of <tt>Typeable</tt>. But even with that, note that <tt>Dynamic</tt>/<tt>Typeable</tt>/<tt>TypeRep</tt> are a bit messy, with instances for <tt>Typeable</tt> defined for a wide range of known types.
   
 
An alternative approach would be to identify your <tt>B</tt> within <tt>A</tt> not per-<tt>B</tt> but per-(up,down). This would allow for instance separate (up,down) for the same <tt>B</tt> such that
 
An alternative approach would be to identify your <tt>B</tt> within <tt>A</tt> not per-<tt>B</tt> but per-(up,down). This would allow for instance separate (up,down) for the same <tt>B</tt> such that
   
down1 . up2 = Nothing
+
<haskell>
down2 . up1 = Nothing
+
down1 . up2 = Nothing
  +
down2 . up1 = Nothing
  +
</haskell>
   
 
Of course this can be done with <tt>Dynamic</tt> too, by defining dummy types. But it's ugly.
 
Of course this can be done with <tt>Dynamic</tt> too, by defining dummy types. But it's ugly.
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== Extensible datatypes ==
 
== Extensible datatypes ==
   
'''Extensible data-types''' 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:
+
'''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
+
<haskell>
data A = ..
+
module P where
   
module Q where
+
-- define open datatype
import P
+
open data A :: *
   
A |= MkB B
+
module Q where
  +
import P
   
up = MkB
+
-- add constructor to A
down (MkB b) = Just b
+
MkB :: B -> A
down _ = Nothing
+
  +
up = MkB
  +
down (MkB b) = Just b
  +
down _ = Nothing
  +
</haskell>
   
 
== Deriving Dynamic ==
 
== Deriving Dynamic ==
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It's possible to define [[Dynamic]] using extensible datatypes. Here's a naive attempt:
 
It's possible to define [[Dynamic]] using extensible datatypes. Here's a naive attempt:
   
data Dynamic = ..
+
<haskell>
  +
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
  +
</haskell>
  +
  +
  +
This attempt however doesn't allow easy creation of <tt>Typeable1</tt>, <tt>Typeable2</tt> etc. A better way is to use type-constructor parameters:
   
class Typeable' a where
+
<haskell>
toDyn :: a -> Dynamic
+
open data Dynamic0 :: (* -> *) -> *
fromDynamic :: Dynamic -> Maybe a
 
   
-- for each type...
+
open data Dynamic1 :: ((* -> *) -> *) -> *
   
Dynamic |= MkBool Bool
+
type Dynamic = Dynamic0 Identity
   
instance Typeable' Bool where
+
data Type a = MkType
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:
+
type TypeRep = Dynamic0 Type
   
data Dynamic0 (f :: * -> *) = ..
+
class Typeable0 a where
  +
toDyn0 :: f a -> Dynamic0 f
  +
fromDynamic0 :: Dynamic0 f -> Maybe (f a)
   
data Dynamic1 (g :: (* -> *) -> *) = ..
+
class Typeable1 p where
  +
toDyn1 :: g p -> Dynamic1 g
  +
fromDynamic1 :: Dynamic1 g -> Maybe (g p)
   
type Dynamic = Dynamic0 Identity
+
data Compose p q a = MkCompose (p (q a))
  +
data Compose1 d0 f p = MkCompose1 (d0 (Compose f p))
   
data Type a = MkType
+
MkDynamic1 :: (Dynamic1 (Compose1 Dynamic0 f)) -> Dynamic0 f
   
type TypeRep = Dynamic0 Type
+
unDynamic1 :: Dynamic0 f -> Maybe (Dynamic1 (Compose1 Dynamic0 f))
  +
unDynamic1 (MkDynamic1 xx) = Just xx
  +
unDynamic1 _ = Nothing
   
class Typeable0 a where
+
instance (Typeable1 p,Typeable0 a) => Typeable0 (p a)
toDyn0 :: f a -> Dynamic0 f
+
-- toDyn0 :: f (p a) -> Dynamic0 f
fromDynamic0 :: Dynamic0 f -> Maybe (f a)
+
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
   
class Typeable1 p where
+
-- for each type
toDyn1 :: g p -> Dynamic1 g
 
fromDynamic1 :: Dynamic1 g -> Maybe (g p)
 
   
data Compose p q a = MkCompose (p (q a))
+
MkInt :: (f Int) -> Dynamic0 f
data Compose1 d0 f p = MkCompose1 (d0 (Compose f p))
 
   
Dynamic0 f |= MkDynamic1 (Dynamic1 (Compose1 Dynamic0 f))
+
instance Typeable0 Int where
  +
toDyn0 = MkInt
  +
fromDynamic0 (MkInt fi) = Just fi
  +
fromDynamic0 _ = Nothing
   
unDynamic1 :: Dynamic0 f -> Maybe (Dynamic1 (Compose1 Dynamic0 f))
+
MkMaybe :: (g Maybe) -> Dynamic1 g
unDynamic1 (MkDynamic1 xx) = Just xx
 
unDynamic1 _ = Nothing
 
   
instance (Typeable1 p,Typeable0 a) => Typeable0 (p a)
+
instance Typeable1 Maybe where
-- toDyn0 :: f (p a) -> Dynamic0 f
+
toDyn1 = MkMaybe
toDyn0 = MkDynamic1 . toDyn1 . MkCompose1 . toDyn0 . MkCompose
+
fromDynamic1 (MkMaybe gm) = Just gm
-- fromDynamic0 :: Dynamic0 f -> Maybe (f (p a))
+
fromDynamic1 _ = Nothing
fromDynamic0 dyn = do
+
</haskell>
dcdf <- unDynamic1 dyn
 
(MkCompose1 dcfp) <- fromDynamic1 dcdf
 
(MkCompose fpa) <- fromDynamic0 dcfp
 
return fpa
 
   
-- for each type
+
I submit that this is "hairy" rather than "ugly", but I suspect the Type-Constructors Of Unusual Kind (TCOUKs) get even hairier for <tt>Typeable2</tt>, <tt>Typeable3</tt> etc...
   
Dynamic0 f |= MkInt (f Int)
+
== Open functions ==
   
instance Typeable0 Int where
+
{{stub}}
toDyn0 = MkInt
 
fromDynamic0 (MkInt fi) = Just fi
 
fromDynamic0 _ = Nothing
 
   
Dynamic1 g |= MkMaybe (g Maybe)
+
== References ==
   
instance Typeable1 Maybe where
+
* Andres Löh and Ralf Hinze. [http://people.cs.uu.nl/andres/OpenDatatypes.pdf Open Data Types and Open Functions]
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...
+
[[Category:Proposals]]

Latest revision as of 07:43, 16 May 2009

Contents

[edit] 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.

[edit] 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

[edit] 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...

[edit] 4 Open functions

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

[edit] 5 References