Version 19 (modified by john@…, 7 years ago)

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# Existential Quantification

## Brief Explanation

Existential quantification hides a type variable within a data constructor. The current syntax uses a forall before the data constructor, as in the following example , which packs a value together with operations on that value:

```data Accum a = forall s. MkAccum s (a -> s -> s) (s -> a)
```

Constraints are also allowed.

When a value of this type is constructed, s will be instantiated to some concrete type. When such a value is analysed, s is abstract.

The forall hints at the polymorphic type of the data constructor:

```MkAccum :: s -> (a -> s -> s) -> (s -> a) -> Accum a
```

but see below for alternatives.

When pattern matching, the type variable s is instantiated to a Skolem variable, which cannot be unified with any other type and cannot escape the scope of the match.

#26

## Syntax of existentials

Several possibilities have been proposed:

implicit quantification
```data Accum a = MkAccum s (a -> s -> s) (s -> a)
```
As the type variable s does not appear on the left hand side, it is considered to be existentially quantified. The main counterargument is that one can easily introduce an existential type by forgetting an argument to the data type. Error messages might confuse the novice programmer.
forall before the constructor
```data Accum a = forall s. MkAccum s (a -> s -> s) (s -> a)
```
This is the currently implemented syntax, motivated by the type of the data constructor, but it can be confused with PolymorphicComponents.
exists before the constructor
```data Accum a = exists s. MkAccum s (a -> s -> s) (s -> a)
```
which reinforces the connection to existential types: when analysing such a value, you know only that there exists some type s such that the arguments have these types. This syntax is used by  Tim Sheard's Omega language, which is based on Haskell.

Reserves an extra word.

```data Accum :: * -> * where
MkAccum :: s -> (a -> s -> s) -> (s -> a) -> Accum a
```
use ExistentialQuantifier

if exists quantifiers are allowed in any type, existential data types are

simply a special case of this.

```data Foo = Foo (exists a . a -> a)
```

and they may be declared via type synonym as well

```type Any = exists a . Num a => a
data Foo = Foo Any
```

if you with the quantifier to scope over multiple components of a data type,

you pull it in front of the constructor:

```data Accum a = exists s. MkAccum s (a -> s -> s) (s -> a)
```

Note that this is the same syntax one would use when using existential

quantifiers in type signatures with the exists coming before the constructor:

```foo :: Int -> (exists a . Either (a -> Char) a) -> Char
foo = ...
```

This is the syntax supported by jhc.

## Variations

• Hugs allows existential quantification for newtype declarations, as long as there are no class constraints.
```newtype T = forall a. C a
```
GHC and Nhc98 do not and jhc can not.
• Hugs and Nhc98 allow matching on an existentially quantified constructor in a pattern binding declaration, except at the top level.
```data T = forall a. C (a -> Int) a
foo t = let C f x = t in f x
```
GHC does not allow such matching.
• None of the implementations can derive instances for existentially quantified types, but this could be relaxed, e.g
```data T = forall a. Show a => C [a]
deriving Show
```
• GHC 6.5 allows fields with existentially quantified types, though selectors may only be used if their type does not include the quantified variable.
```data T = forall a. C { f1 :: a, f2 :: Int }
```

## Pros

• offered by GHC, Hugs and Nhc98 for years, HBC even longer.
• typing rules well understood.
• quite handy for representations that delay composition or application, e.g. objects, various parser combinator libraries from Utrecht.

## Cons

• tricky to implement on implementations without a common runtime representation of values such as jhc. (however, John Meacham believes any trickiness is worth it due to the value of this extension, at least for jhc)

# Extension: better syntactic support

Discussion on the list in late January focussed on a  proposal from Johannes Waldmann to add syntactic support for the use of existential types.

## Background example

The status quo is illustrated by the concrete example of geometrical figures, each of which is an instance of an interface given by the Haskell class

```class Figure f ...
```

with instances Circle r and Rectangle h w, say. The existential built on the Figure class is given by

```data EFigure = exists f . Figure f => EFigure f
```

Elements of this type have the form EFigure (Circle r) and so the values of the existential type are explicitly 'boxed' by the EFigure constructor when they are built and need to be unboxed when they are used. Unboxing can be achieved by an additional instance declaration:

```instance Figure EFigure ...
```

but boxing needs to be explicit.

## First proposal

The extension proposed here is to hide the existence of the data type and constructor EFigure, just referring to Figure instead. The issue becomes one of conversion between the figure types Circle and Rectangle and the type Figure. A concrete example is given by

```f :: Integer -> Circle
g :: Figure -> Blah
```

in order for the expression g . f to be well typed, a conversion needs to be inserted thus: g . EFigure . f. Conversions become more complicated in case that

```f :: Integer -> [Circle]
g :: [Figure] -> Blah
```

for instance.

## an existential for each single parameter class

A proposal based on the discussion is automatically to introduce an existential type for each single parameter type class. The type and class would have the same name, as in

```class Figure f ...
data Figure = exists f . Figure f => Figure f
instance Figure Figure ...
```

Discussion focussed on whether there should or should not be automatic conversion from figure types into Figure; it was agreed that the Figure constructor could be used to do this, with the Figure instance providing the unboxing required.

This is a signifigantly more difficult problem than this proposal makes it seem.