# Existential type

### From HaskellWiki

## 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

## 1 Introduction to existential types

### 1.1 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'"`

### 1.2 Expanded example - rendering objects in a raytracer

#### 1.2.1 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 ... -}

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.

#### 1.2.2 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 {- ... -}

[ AnyRenderable x , AnyRenderable y , AnyRenderable z ]

### 1.3 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).

### 1.4 Generalised algebraic datatype

The type of the## 2 Alternate methods

### 2.1 Concrete data types

#### 2.1.1 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 makeinstance Shape SHAPE where draw (SHAPE d t) = d translate (SHAPE d t) = t

So SHAPE is a sort of universal instance.

#### 2.1.2 Using constructors and combinators

Why bother with classdata 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

### 2.2 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 ())

## 3 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 ) may be confusing. In Essential Haskell, existential types can occur not only in data declarations, and a separate keywordforallis used for their notation.exists
- Essential Haskell Compiler overview
- Examples

## 4 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

### 4.1 Trac

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