# Existential type

## Contents

This is a extension of Haskell available in GHC. See the GHC documentation: http://www.haskell.org/ghc/docs/latest/html/users_guide/type-extensions.html

## Introduction to existential types

### A short example

This illustrates creating a heterogeneous list, all of whose members implement "Show", and progressing through that list to show these items:

```
data Obj = forall a. (Show a) => Obj a
xs = [Obj 1, Obj "foo", Obj 'c']
doShow :: [Obj] -> String
doShow [] = ""
doShow ((Obj x):xs) = show x ++ doShow xs
```

With output: `doShow xs ==> "1\"foo\"'c'"`

### Expanded example - rendering objects in a raytracer

#### Problem statement

In a raytracer, a requirement is to be able to render several different objects (like a ball, mesh or whatever). The first step is a type class for Renderable like so:

```
class Renderable a where
boundingSphere :: a -> Sphere
hit :: a -> [Fragment] -- returns the "fragments" of all hits with ray
{- ... etc ... -}
```

To solve the problem, the `hit`

function must apply to several objects (like a sphere and a polygon for instance).

```
hits :: Renderable a => [a] -> [Fragment]
hits xs = sortByDistance $ concatMap hit xs
```

However, this does not work as written since the elements of the list can be of **SEVERAL** different types (like a sphere and a polygon and a mesh etc. etc.) but
lists need to have elements of the same type.

#### The solution

Use 'existential types' - an extension to Haskell that can be found in most compilers.

The following example is based on GHC :

```
{-# OPTIONS -fglasgow-exts #-}
{- ...-}
data AnyRenderable = forall a. Renderable a => AnyRenderable a
instance Renderable AnyRenderable where
boundingSphere (AnyRenderable a) = boundingSphere a
hit (AnyRenderable a) = hit a
{- ... -}
```

Now, create lists with type `[AnyRenderable]`

, for example,

```
[ AnyRenderable x
, AnyRenderable y
, AnyRenderable z ]
```

where x, y, z can be from different instances of `Renderable`

.

### Dynamic dispatch mechanism of OOP

**Existential types** in conjunction with type classes can be used to emulate the dynamic dispatch mechanism of object oriented programming languages. To illustrate this concept I show how a classic example from object oriented programming can be encoded in Haskell.

```
class Shape_ a where
perimeter :: a -> Double
area :: a -> Double
data Shape = forall a. Shape_ a => Shape a
type Radius = Double
type Side = Double
data Circle = Circle Radius
data Rectangle = Rectangle Side Side
data Square = Square Side
instance Shape_ Circle where
perimeter (Circle r) = 2 * pi * r
area (Circle r) = pi * r * r
instance Shape_ Rectangle where
perimeter (Rectangle x y) = 2*(x + y)
area (Rectangle x y) = x * y
instance Shape_ Square where
perimeter (Square s) = 4*s
area (Square s) = s*s
instance Shape_ Shape where
perimeter (Shape shape) = perimeter shape
area (Shape shape) = area shape
--
-- Smart constructor
--
circle :: Radius -> Shape
circle r = Shape (Circle r)
rectangle :: Side -> Side -> Shape
rectangle x y = Shape (Rectangle x y)
square :: Side -> Shape
square s = Shape (Square s)
shapes :: [Shape]
shapes = [circle 2.4, rectangle 3.1 4.4, square 2.1]
```

(You may see other Smart constructors for other purposes).

### Generalised algebraic datatype

The type of the `parse`

function for this GADT is a good example to illustrate the concept of existential type.

## Alternate methods

### Concrete data types

#### Universal instance of a Class

Here one way to simulate existentials (Hawiki note: (Borrowed from somewhere...))

Suppose I have a type class Shape a

```
type Point = (Float,Float)
class Shape a where
draw :: a -> IO ()
translate :: a-> Point -> a
```

Then we can pack shapes up into a concrete data type like this:

```
data SHAPE = SHAPE (IO ()) (Point -> SHAPE)
```

with a function like this

```
packShape :: Shape a => a -> SHAPE
packShape s = SHAPE (draw s) (\(x,y) -> packShape (translate s (x,y)))
```

This would be useful if we needed a list of shapes that we would need to translate and draw.

In fact we can make `SHAPE`

an instance of `Shape`

:

```
instance Shape SHAPE where
draw (SHAPE d t) = d
translate (SHAPE d t) = t
```

So SHAPE is a sort of universal instance.

#### Using constructors and combinators

Why bother with class `Shape`

? Why not just go straight to

```
data Shape = Shape {
draw :: IO()
translate :: (Int, Int) -> Shape
}
```

Then you can create a library of shape constructors and combinators that each have defined "draw" and "translate" in their "where" clauses.

```
circle :: (Int, Int) -> Int -> Shape
circle (x,y) r =
Shape draw1 translate1
where
draw1 = ...
translate1 (x1,y1) = circle (x+x1, y+y1) r
shapeGroup :: [Shape] -> Shape
shapeGroup shapes = Shape draw1 translate1
where
draw1 = sequence_ $ map draw shapes
translate1 v = shapeGroup $ map (translate v) shapes
```

### Cases that really require existentials

There are cases where this sort of trick doesnt work. Here are two examples from a haskell mailing list discussion (from K. Claussen) that don't seem expressible without existentials. (But maybe one can rethink the whole thing :)

```
data Expr a = Val a | forall b . Apply (Expr (b -> a)) (Expr b)
```

and

```
data Action = forall b . Act (IORef b) (b -> IO ())
```

(Maybe this last one could be done as a `type Act (IORef b) (IORef b -> IO ())`

then we could hide the `IORef`

as above, that is go ahead and apply the second argument to the first)

## Examples from the Essential Haskell Compiler project

See the documentation on EHC, each paper at the *Version 4* part:

- Chapter 8 (EH4) of Atze Dijkstra's Essential Haskell PhD thesis (most recent version). A detailed explanation. It explains also that existential types can be expressed in Haskell, but their use is restricted to data declarations, and the notation (using keyword
`forall`

) may be confusing. In Essential Haskell, existential types can occur not only in data declarations, and a separate keyword`exists`

is used for their notation. - Essential Haskell Compiler overview
- Examples

## See also

- A mailinglist discussion: http://haskell.org/pipermail/haskell-cafe/2003-October/005231.html
- An example of encoding existentials using RankTwoPolymorphism : http://haskell.org/pipermail/haskell-cafe/2003-October/005304.html

### Trac

Existential Quantification is a detailed material on the topic. It has link also to the smaller Existential Quantifier page.