Church-encoded datatypes in Haskell, part 2

Posted on December 27, 2015

Last time we covered Haskells Maybe type encoded as a funciton. This post will introduce two recursively defined structures, lists and natural numbers. Here I give the normal ADT representation of a list, and a corresponding Church encoding.

data List a = Nil | Cons a (List a)

newtype ListC a = ListC {unList :: forall r. r -> (a -> r -> r) -> r}

Just like MaybeC, ListC is a funciton that takes a default value and another function. In the case of an empty list, it returns the default value. To understand the recursive (non-empty) case it is helpful to note the similarity between the type of ListC and the type of foldr on regular lists, and then to see how cons is implemented for Church lists:

foldr :: (a -> r -> r) -> r -> [a] -> r

In fact, the types of foldr and ListC are so similar because Church encoded lists are exactly functions which fold over the lists they encode. To see how, observe the cons function given here:

-- nil is a fold that does nothing but return the zero value supplied to it
nil :: ListC a
nil = ListC $ \z f -> z

-- cons take an element to add and a list, and build a new function which
-- folds over the new (longer) list
cons :: a -> ListC a -> ListC a
cons head (ListC tail) = ListC $ \z f -> head `f` (tail z f)

To give a concrete example, let’s look at the list [1, 2] and use some equational reasoning to figure out how its Church encoding works.

-- Ordinary representation, then with explicit (:) calls
list :: [Int]
list = [1, 2]
     -- With explicit constructors:
list = 1 : (2 : [])

-- Then the Church encoding, using substitution to expand the definitions
-- newtype wrapping and unwrapping removed for clarity
church :: ListC Int
church = 1 `cons` (2 `cons` nil)
       -- using the definition of cons:
       = 1 `cons` (\z f -> 2 `f` (nil z f))
       -- by def. of nil, with capture avoiding substitution:
       = 1 `cons` (\z f -> 2 `f` ((\z' f' -> z') z f))
       -- by applying the inner lambda:
       = 1 `cons` (\z f -> 2 `f` z)
       -- by def. of cons:
       = \z f -> 1 `f` ((\z' f' -> 2 `f'` z') z f
       -- by applying the inner lambda:
church = \z f -> 1 `f` (2 `f` z)

As the last line shows, the Church encoded version is a function which folds over the list from the right.

And we can observe the parallels between these constructors and the definition of the foldr function for lists in the Prelude:

foldr k z = go
        where
          go []     = z           --nil
          go (y:ys) = y `k` go ys --cons

So if we want to define the Foldable instance for ListC, we need only define foldr as follows:

instance Folable ListC where
    foldr f z (ListC l) = l z f

-- and a couple of useful instances follow from that directly:

instance Functor ListC where
    fmap f = foldr (cons . f) nil

instance Show a => Show (ListC a) 
    show = foldr (\head tail -> show head ++ "," ++ tail) "End"

Defining Nat

Having seen how ListC works to encode a recursive type, we can define our first primitive type (rather than a type constructor like ListC or MaybeC), NatC. As usual, we’ll look at the (recursive) ADT and the Church encoding side by side:

data Nat = Zero | Succ Nat

newtype NatC = NatC {unNat :: forall r. r -> (r -> r) -> r}

This looks exactly like the ListC type signature, but missing the type that goes inside the list. Looking at the constructors for NatC:

zero :: NatC
zero = NatC $ \x f -> x

succ :: NatC -> NatC -> NatC
succ (NatC n) = NatC $ \x f -> f (n x f) 

So zero applies the function f to x zero times, succ zero applies the function f to x one time, and so on. This can be directly shown be equational reasoning similar to the example of the cons function above.

To define the sum of two Church numerals, use one to apply succ to the other repeatedly, as follows:

(+) :: NatC -> NatC -> NatC
(NatC n) + m = n m succ

And multiplication can be defined similarly, in terms of addition:

(*) :: NatC -> NatC -> NatC
(NatC n) * m = n zero (+m)

Which can both be used to give Nat instances of Monoid, as necessary.

One last note is the interesting parallel between ListC and NatC. Observe their types:

newtype ListC a = ListC {unList :: forall r. r -> (a -> r -> r) -> r}

newtype NatC = NatC {unNat :: forall r. r -> (r -> r) -> r}

The only difference is that ListC has a type parameter a where NatC does not. Indeed NatC can be defined in terms of ListC by stating:

newtype NatC = ListC ()

Where we can then reuse some of the machinery for ListC to define operations on NatC, using concatenation of lists for addition, fmap and concatenation for multiplication, etc. Especially challenging is the definition of tail for ListC or pred for NatC.