Monomorphism, Polymorphism, and Higher Ranked Types.

Let’s get started with a monomorphic function -

1incr :: Int -> Int
2incr = (+1)

The function incr works on values of type Int and nothing else. We can make our function polymorphic by using type parameters in place of concrete types -

1incr' :: Num a => a -> a
2incr' = (+1)

The modified function incr' can work with any type that has a Num instance. Now let us move on and look at a simple higher order function -

1foo :: (a -> a) -> a -> a
2foo g x = g x
3
4intval  = foo (+1) 1       -- 2
5listval = foo (++[1]) [1]  -- [1, 1]

The first argument to foo is a function of type a -> a bound to g. Is the function g polymorphic here? The answer is No! The a type is being used polymorphically here, however, a is bound to whatever is the type of x when g is applied to x i.e., g is monomorphic!

You don’t believe me? Try this

1badfoo :: Show a => (a -> a) -> String
2badfoo g = (show (g 1000)) ++ (g "? what!")

Or this function -

1badbar :: Show a => (a -> a) -> IO ()
2badbar g = do
3    print $ g 100
4    print $ g "why?"

The compiler will not be happy with this code. If g were polymorphic, we could apply it to both numeric and string values!

Polymorphic Functions

We make our functions polymorphic by using type parameters instead of fixing the types. When the function is s applied to a value, the type parameter(s) are bound to actual type(s). We say the function is instantiated with the given types. The actual types (of type variables) are set to the types of values passed to the function.

Actual type of a polymorphic function is decided where the function is used, not where the function is defined, i.e., the user of the function decides the type.

Same applies to higher order functions. If the types of the argument function are parameterised, the actual types are decided when the argument function is used.

Coming back to badfoo function, the type of the argument function is (a -> a). The type for type variable a in (a -> a) is bound when this function is applied to a value (when we call g 1). This means, for the reminder of this function body, g is monomorphic.

What we wanted, though, is for g to be polymorphic. We want the type of g to be bound when user calls badfoo - not when g is used within badfoo.

Rank

Whenever we want to pass a polymorphic function g as an argument to another function f, we want the type of g to be decided not within the body of f but at the caller site - one level up so to say. We say, f is ranked one level higher than g.

All monomorphic functions are Rank 0. All polymorphic functions are Rank

1. A rank-2 polymorphic function takes as an argument a rank-1 polymorphic function. A rank-3 functions takes as an argument a rank-2 polymorphic function, and so on.

2. 2. If a higher order function takes as argument a rank (N-1) function, then it is said to be Rank-N polymorphic function.

RankNTypes GHC Extension

Haskell is based on the Hindley-Milner type system - and while it allows us to write polymorphic functions, it doesn’t allow us to write functions which take polymorphic functions as arguments.

Higher Ranked Types are provided in GHC as a language extension. It can be enabled via the RankNTypes extension. Practically speaking, enabling higher- ranked types lets us introduce new variable scopes inside of type signatures.

Here’s the implementation we want:

1{-# LANGUAGE RankNTypes #-}
2goodfoo :: (forall a. Show a => a -> a) -> String
3goodfoo g = (show (g 1)) ++ (g "Works!")

Higher Ranked Types make polymorphic Haskell functions first class.