This is the third of a series of articles that illustrates functional programming (FP) concepts. As you go through these articles with code examples in Haskell (a very popular FP language), you gain the grounding for picking up any FP language quickly. Why should you care about FP? See this post.
In the last post, we learned about higher order functions, which dramatically increase one’s expressive power. In this post, we’ll learn more about types and typeclasses, and the foldable typeclass, which will help us learn folds, an advanced and important tool.
Typeclasses and polymorphism
Typeclasses are categories of types, and so the instances of a class are types (not values)! For the types to be in a typeclass, they have to have certain traits.
This graph shows the hierarchy of Haskell classes defined in the Prelude and the Prelude types that are instances of these classes:
As shown above, a typeclass typically contains more than one type. This is useful because it allows us to specify categories of types and still write polymorphic functions (functions that work on more than one type).
We’ve already seen the
Num typeclasses in the last couple posts! Let’s look at them in detail!
The ordering typeclass
In the first post, we saw the
Ord (ordering) typeclass. It was a pre-requisite for the sorting algorithm to be applicable. If we have a type that is not ordered (e.g, a datatype of colours or shapes) then we cannot sort it. Therefore, when a type is in the
Ord typeclass, it has a defined way of ordering the elements of that type.
The documentation shows that the
Ord typeclass is a subclass of the
Eq typeclass (everything in
Ord is also in
class Eq a => Ord a where
In fact, as shown in the above graph, all except
(->) are subclasses of the
Eq typeclass! This makes sense, if we don’t know how to tell when things are the same, we cannot order them or do much else.
The documentation also specifies the associated methods of this typeclass:
compare :: a -> a -> Ordering (<) :: a -> a -> Bool (<=) :: a -> a -> Bool (>) :: a -> a -> Bool (>=) :: a -> a -> Bool max :: a -> a -> a min :: a -> a -> a
For a type to be in this typeclass, these methods are defined which means we can apply them to the elements of this type. E.g., from the graph above we know that all except a few types are in the
Ord typeclass. So we can apply the function
compare to instances of the
compare takes two inputs, both of a type that is in the
Ord typeclass, and returns the
Ordering type. The
Ordering type has three constructors (possible instances),
GT (greater than),
Eq (equal), and
LT (less than). It may be unexpected but we can trust the documentation and run
Bool. Running compare in GHCi yields:
Prelude> compare True False GT Prelude> compare True True EQ Prelude> compare False True LT
Similarly for the other functions:
Prelude> max True False True Prelude> (<) True False False
The numeric typeclass
In the second post, we saw the
Num (numeric) typeclass. It was a pre-requisite for our function
g along with the
g :: (Ord a, Num a) => (t -> a) -> t -> Bool g f y = f y > 100
Note that we call the function
g. So we know that we must apply it to a type that is in the
Ord typeclass. (As we see above that
> is a method of the
Ord typeclass. Because
> takes two inputs of the same type and is applied to
a has to be an instance of the
Num typeclass too because
100 is, as we can see from GHCi:
Prelude> :t 100 100 :: Num p => p
Num typeclass includes instances such as
Double. For a full list see the “instances” section in the documentation.
The foldable typeclass
Recursion schemes are powerful tool functional programmers should master. One of the most useful recursion schemes is fold. The
Foldable typeclass has the methods
foldr (the function for folding) and its variants.
Prelude> :t foldr foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b
The documentation also shows instances of the typeclass:
Foldable  Foldable Maybe Foldable (Either a) Foldable ((,) a) Foldable (Proxy *) Foldable (Const m)
Note that the foldable typeclass instances are type constructors. They take a type to form a type. E.g.,
t a in the type signature of
foldr. This means that
t could be a list, or a
(Either a) etc. These are the possible structures over
a that we can fold. More on
folds in the next post!
Understanding types and typeclasses deeply is important for every FPer and you’ve done it! As we progress we will see more types and typeclasses. Once you get familiar with them you’ll have enough grounding to work with any type, and eventually create any type!
It’s also very important to understand that
folds can be used on many types. In the next post however, we’ll focus on folding a list, a common structure to fold over. I’m sure you’ll be impressed by the power of
folds, so stay tuned!