# Existential type

## Contents

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

## Introduction to existential types

### Overview

Normally when creating a new type using `type`

, `newtype`

, `data`

, etc., every type variable that appears on the right-hand side must also appear on the left-hand side. Existential types are a way of turning this off.

### Basics

Existential types can be *used* for several different purposes. But what they *do* is to 'hide' a type variable on the right-hand side.

Normally, any type variable appearing on the right must also appear on the left:

```
data Worker x y = Worker {buffer :: b, input :: x, output :: y}
```

This is an error, since the type of the buffer isn't specified on the right (it's a type variable rather than a type) but also isn't specified on the left (there's no 'b' in the left part). In Haskell98, you would have to write

```
data Worker b x y = Worker {buffer :: b, input :: x, output :: y}
```

That may or may not be an actual problem.

Usually there is no problem at all with this state of affairs (which is why Haskell98 works this way). However, suppose that a `Worker`

can use *any* type 'b' so long as it belongs to some particular class. Then every function that uses a `Worker`

will have a type like

```
foo :: (Buffer b) => Worker b Int Int
```

or something. (In particular, failing to write an explicit type signature will invoke the dreaded monomorphism restriction.) Using existential types, we can avoid this:

```
data Worker x y = forall b. Buffer b => Worker {buffer :: b, input :: x, output :: y}
foo :: Worker Int Int
```

The type of the buffer now does *not* appear in the `Worker`

type at all.

This has a number of consequences. First of all, it is now impossible for a function to demand a `Worker`

having a specific type of buffer. Second, the type of `foo`

can now be derived automatically without needing an explicit type signature. (No monomorphism restriction.) Thirdly, since code now has *no idea* what type the `buffer`

function returns, you are more limited in what you can do to it.

In general, when you use a 'hidden' type in this way, you will usually want that type to belong to a specific class, or you will want to pass some functions along that can work on that type. Otherwise you'll have some value belonging to a random unknown type, and you won't be able to *do* anything to it!

Note: You can use existential types to convert a more specific type into a less specific one. (See the examples below.) There is *no way* to perform the reverse conversion!

## Examples

### 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]
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 doesn't 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)

### Existentials in terms of "forall"

It is also possible to express existentials as type expressions directly (without a `data`

declaration) with RankNTypes. Taking the above example:

```
data Obj = forall a. (Show a) => Obj a
```

the type `Obj`

is equivalent to:

```
forall r. (forall a. Show a => a -> r) -> r
```

(the leading `forall r.`

is optional unless the expression is part of another expression). The conversions are:

```
fromObj :: Obj
-> forall r. (forall a. Show a => a -> r) -> r
fromObj (Obj x) k = k x
toObj :: (forall r. (forall a. Show a => a -> r) -> r)
-> Obj
toObj f = f Obj
```

## 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
- Another mailing list discussion (functional vs OO approaches): http://www.haskell.org/pipermail/haskell/2005-June/016058.html
- Just another one: http://www.haskell.org/pipermail/haskell-cafe/2008-January/037950.html "type question again"

### Trac

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