wiki:ExistentialQuantification

Version 11 (modified by ross@…, 8 years ago) (diff)

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Existential Quantification

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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.

References

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. Reserves an extra word.
GADT syntax
data Accum :: * -> * where
    MkAccum :: s -> (a -> s -> s) -> (s -> a) -> Accum a
Existentials are subsumed by GADTs.

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 }
    
  • The Omega language based on Haskell has an 'exists' keyword to denote an existential type.

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.