# Rank-N types

### From HaskellWiki

## Contents |

## 1 About

Normal Haskell '98 types are considered Rank-1 types. A Haskell '98 type signature such as

a -> b -> a

implies that the type variables are universally quantified like so:

forall a b. a -> b -> a

forall a. a -> (forall b. b -> a)

is also a Rank-1 type because it is equivalent to the previous signature.

However, aRank-N type reconstruction is undecidable in general, and some explicit type annotations are required in their presence.

Rank-2 or Rank-N types may be specifically enabled by the language extensions

## 2 Church-encoded Lists

Church-encoded lists use RankNTypes too, as seen in a StackOverflow answer by sacundim:

-- | Laws: -- -- > runList xs cons nil == xs -- > runList (fromList xs) f z == foldr f z xs -- > foldr f z (toList xs) == runList xs f z newtype ChurchList a = ChurchList { runList :: forall r. (a -> r -> r) -> r -> r } -- | Make a 'ChurchList' out of a regular list. fromList :: [a] -> ChurchList a fromList xs = ChurchList $ \k z -> foldr k z xs -- | Turn a 'ChurchList' into a regular list. toList :: ChurchList a -> [a] toList xs = runList xs (:) [] -- | The 'ChurchList' counterpart to '(:)'. Unlike 'DList', whose -- implementation uses the regular list type, 'ChurchList' abstracts -- over it as well. cons :: a -> ChurchList a -> ChurchList a cons x xs = ChurchList $ \k z -> k x (runList xs k z) -- | Append two 'ChurchList's. This runs in O(1) time. Note that -- there is no need to materialize the lists as @[a]@. append :: ChurchList a -> ChurchList a -> ChurchList a append xs ys = ChurchList $ \k z -> runList xs k (runList ys k z) -- i.e., nil = {- fromList [] = ChurchList $ \k z -> foldr k z [] = -} ChurchList $ \k z -> z singleton x = {- cons x nil = ChurchList $ \k z -> k x (runList nil k z) = -} ChurchList $ \k z -> k x z snoc xs x = {- append xs $ singleton x = ChurchList $ \k z -> runList xs k (runList (singleton x) k z) = -} ChurchList $ \k z -> runList xs k (k x z)

## 3 Relation to Existentials

In order to unpack an existential type, you need a polymorphic function that works on any type that could be stored in the existential. This leads to a natural relation between higher-rank types and existentials; and an encoding of existentials in terms of higher rank types in continuation-passing style.

In general, you can replace

data T a1 .. ai = forall t1 .. tj. constraints => Constructor e1 .. ek

Constructor exp1 .. expk -- application of the constructor

case e of (Constructor pat1 .. patk) -> res

with

data T' a1 .. ai = Constructor' (forall b. (forall t1..tj. constraints => e1 -> e2 -> ... -> ek -> b) -> b)

Constructor' (\f -> f exp1 .. expk)

case e of (Constructor' f) -> let k pat1 .. patk = res in f k

## 4 See also

- Rank-N types on the Haskell' website.
- The GHC User's Guide on higher-ranked polymorphism.