Haskell/Polymorphism

Parametric Polymorphism

Section goal = short, enables reader to read code (ParseP) with ∀ and use libraries (ST) without horror. Question Talk:Haskell/The_Curry-Howard_isomorphism#Polymorphic types would be solved by this section.

Link to the following paper: Luca Cardelli: On Understanding Types, Data Abstraction, and Polymorphism.

forall a

A polymorphic function is a function that works for many different types. For instance,

  length :: [a] -> Int

can calculate the length of any list, be it a string String = [Char] or a list of integers [Int]. The type variable a indicates that length accepts any element type. Other examples of polymorphic functions are:

  fst :: (a, b) -> a
  snd :: (a, b) -> b
  map :: (a -> b) -> [a] -> [b]

Type variables always begin in lowercase whereas concrete types like Int or String always start with an uppercase letter. This allows us to tell them apart.

There is a more explicit way to indicate that a can be any type

 length :: forall a. [a] -> Int

In other words, "for all types a, the function length takes a list of elements of type a and returns an integer". You should think of the old signature as an abbreviation for the new one with the forall keyword. That is, the compiler will internally insert any missing forall for you. Another example: the types signature for fst is really a shorthand for:

  fst :: forall a. forall b. (a,b) -> a

or equivalently:

  fst :: forall a b. (a,b) -> a

Similarly, the type of map is really:

  map :: forall a b. (a -> b) -> [a] -> [b]

This notion, that something is applicable to every type or holds for everything, is called universal quantification. In mathematical logic, the symbol ∀ (an upside-down A, read as "for all") is commonly used for that; it is called the universal quantifier.

Higher rank types

With explicit forall, it now becomes possible to write functions that expect polymorphic arguments, like for instance

  foo :: (forall a. a -> a) -> (Char,Bool)
  foo f = (f 'c', f True)

Here, f is a polymorphic function, it can be applied to anything. In particular, foo can apply it to both the character 'c' and the boolean True.

It is not possible to write a function like foo in Haskell98, the type checker will complain that f may only be applied to values of either the type Char or the type Bool and reject the definition. The closest we could come to the type signature of foo would be

  bar :: (a -> a) -> (Char, Bool)

which is the same as

  bar :: forall a. ((a -> a) -> (Char, Bool))

But this is very different from foo. The forall at the outermost level means that bar promises to work with any argument f as long as f has the shape a -> a for some type a unknown to bar. Contrast this with foo, where it's the argument f which promises to be of shape a -> a for all types a at the same time , and it's foo which makes use of that promise by choosing both a = Char and a = Bool.

Concerning nomenclature, simple polymorphic functions like bar are said to have a rank-1 type while the type foo is classified as rank-2 type. In general, a rank-n type is a function that has at least one rank-(n-1) argument but no arguments of any higher rank.

The theoretical basis for higher rank types is System F, also known as the second-order lambda calculus. We will detail it in the section System F in order to better understand the meaning of forall and its placement like in foo and bar.

Haskell98 is based on the Hindley-Milner type system, which is a restricted version of System F and does not support forall and rank-2 types or types of even higher rank. You have to enable the RankNTypes language extension to make use of the full power of System F.

There is a good reason that Haskell98 does not support higher rank types: type inference for the full System F is undecidable; the programmer would have to write down all type signatures. So, the early versions of Haskell have adopted the Hindley-Milner type system which only offers simple polymorphic functions but enables complete type inference in exchange. Recent advances in research have reduced the burden of writing type signatures and made rank-n types practical in current Haskell compilers.

runST

For the practical Haskell programmer, the ST monad is probably the first example of a rank-2 type in the wild. Similar to the IO monad, it offers mutable references

  newSTRef   :: a -> ST s (STRef s a)
  readSTRef  :: STRef s a -> ST s a
  writeSTRef :: STRef s a -> a -> ST s ()

and mutable arrays. The type variable s represents the state that is being manipulated. But unlike IO, these stateful computations can be used in pure code. In particular, the function

  runST :: (forall s. ST s a) -> a

sets up the initial state, runs the computation, discards the state and returns the result. As you can see, it has a rank-2 type. Why?

The point is that mutable references should be local to one runST. For instance,

  v   = runST (newSTRef "abc")
  foo = runST (readSTRef v)

is wrong because a mutable reference created in the context of one runST is used again in a second runST. In other words, the result type a in (forall s. ST s a) -> a may not be a reference like STRef s String in the case of v. But the rank-2 type guarantees exactly that! Because the argument must be polymorphic in s, it has to return one and the same type a for all states s; the result a may not depend on the state. Thus, the unwanted code snippet above contains a type error and the compiler will reject it.

You can find a more detailed explanation of the ST monad in the original paper Lazy functional state threads.

Impredicativity

• predicative = type variables instantiated to monotypes. impredicative = also polytypes. Example: length [id :: forall a . a -> a] or Just (id :: forall a. a -> a). Subtly different from higher-rank.

• relation of polymorphic types by their generality, i.e. `isInstanceOf`.
• haskell-cafe: RankNTypes short explanation.

Bounded parametric polymorphism

Type classes

TODO

System F

Section goal = a little bit lambda calculus foundation to prevent brain damage from implicit type parameter passing.

• System F = Basis for all this ∀-stuff.
• Explicit type applications i.e. map Int (+1) [1,2,3]. ∀ similar to the function arrow ->.
• Terms depend on types. Big Λ for type arguments, small λ for value arguments.

Examples

Section goal = enable reader to judge whether to use data structures with ∀ in his own code.

• Church numerals, Encoding of arbitrary recursive types (positivity conditions): &forall x. (F x -> x) -> x
• Continuations, Pattern-matching: maybe, either and foldr

I.e. ∀ can be put to good use for implementing data types in Haskell.

Other forms of Polymorphism

So far we talked about primarily parametric polymorphism. There are however two more forms of polymorphism that are predominantly employed in other programming languages:

• Ad-hoc polymorphism
• Subtype Polymorphism

Ad-hoc polymorphism

In C++, ad-hoc polymorphism can be seen as equivalent to function overloading:

int square(int x);
float square(float x);

We can do something similar in Haskell using type classes:

class Square a where
    square :: a -> a
    
instance Square Int where
    square x = x * x

instance Square Float where
    square x = x * x

The main thing to take away with ad-hoc polymorphism is there will always be types that the function cannot accept, though the number of types the function can accept may be infinite. Contrast this with parametric polymorphism, equivalent in C++ to template functions:

template <class T>
T id(T a) {
    return a;
}

template <class A, class B>
A const_(A first, B second) {
    return first;
}

template <class T>
int return10 (T a) {
    return 10;
}

Which is equivalent to the following in Haskell:

id :: a -> a
id a = a

const :: a -> b -> a
const a _ = a

return10 :: a -> Int
return10 _ = 10

The main take-away with parametric polymorphism is that any type must be accepted as an input to the function, regardless of its return type.

Note, with both forms of polymorphism, it is not possible to have two identically named functions that differ only in their return type.

For example, the following C++ is not valid:

void square(int x);
int square(int x);

Nor the Haskell version:

class Square a where
    square :: a -> a
    
instance Square Int where
    square x = x*x

instance Square Int where
    square x = ()

Since the compiler would have no way to determine which version to use given an arbitrary function call.

Subtype polymorphism

TODO = contrast polymorphism in OOP and stuff.

Free Theorems

Section goal = enable reader to come up with free theorems. no need to prove them, intuition is enough.

• free theorems for parametric polymorphism.

See also

• Luca Cardelli. On Understanding Types, Data Abstraction, and Polymorphism.