Learning Haskell through Maybe - Functors

Types

You have a Type, Integer. Integer is an integer, representing intergers, just like real life.

Other types are Char, which are characters such as 'a', 'b' etc. The one we will look at in detail is Maybe.

Type variable, constructors

A data of type Maybe can only be made if a type variable is specified. This means that you can make a Maybe Integer, Maybe Char, but a Maybe isn’t a complete type.

The way to make a data of type Maybe is to use a type constructor. Maybe has two type constructors, Just and Nothing. Just is like Maybe, it can only be made if a type variable is specified. Nothing can be made simply on its own, it is a complete type.

Let’s try making Maybes in GHCI:

> let best_friends = Just 1
> best_friends
Just 1

You have just used the type constructor Just, passed in the Int 1, to make a type of Maybe Int. To verify the type of best_friends:

> :t best_friends
best_friends :: Maybe Integer

:: can be read as is of type. How about make a Maybe using Nothing?

> let girlfriend = Nothing
> girlfriend
Nothing

Functor

Suppose you have twice as many best friends as I do, how can I find out how many best friends you have?

> let your_best_friends = 1 * 2 --hard coded value for the number best friends I have
> your_best_friends
2
> let your_best_friends = best_friends * 2 --fails terribly

You get a crazy error message. It means that you cannot multiply 2 with best_friends, because best_friends is not a type that can be multiplied with 2. Clearly there is something going on here, because best_friends can contain an Integer, and we can definitely multiply Integers, what we can do is to define a function that helps us do so.

> :{
| let calc_friends :: Maybe Integer -> Maybe Integer
|       calc_friends Nothing = Nothing
|           calc_friends (Just x) = Just (x * 2)
> :}
> let your_best_friends = calc_friends best_friends
> your_best_friends
Just 2

What we did was to define a function that can calculate the number of best friends you have, given the number of best friends I have. By applying this function to best_friends, you get the number of best friends you have, wrapped in a Just. What if I told you that you had twice as many girlfriends as I do?

> let your_girlfriends = calc_friends girlfriend
> your_girlfriends
Nothing

Hahaha.

Perhaps you grow more popular, and you actually had 3 times more friends than I do. Since we hard coded the value 2, we have to make another function with the value 3, or we can actually generalize the function as such.

> :{
| let calc_friends :: (Integer -> Integer) -> Maybe Integer -> Maybe Integer
|     calc_friends _ Nothing = Nothing
|     calc_friends f (Just x) = Just (f x)
> :}
> let your_best_friends = calc_friends (* 3) best_friends
> your_best_friends
Just 3
> let your_girlfriends = calc_friends (* 2) girlfriend
> your_girlfriends
Nothing

We define calc_friends as a function that takes in another function, f, and this function f takes in an Integer and returns another Integer. An example of this function would be (* 3). calc_friends also takes in a Maybe Integer, which it will apply the function f to, and returning a resut of type Maybe Integer

What if instead of number of best friends, an Integer, we wanted to compare height, a Double, and you are 1 metre taller than me, this looks pretty similar to calc_friends, so let’s reuse that:

> let height = Just 1.7
> let calc_height = calc_friends (+ 1)
> calc_height height

Throws you an error about incompatible types. This is becase we defined calc_height in terms of calc_friends, which only worked with Maybe Integer, but our height was a Maybe Double. So let’s generalize our function using type variables, just like how Maybe is defined.

> :{
| let calc_stuff :: (a -> a) -> Maybe a -> Maybe a
|     calc_stuff _ Nothing = Nothing
|     calc_stuff f (Just x) = Just (f x)
> :}
> calc_stuff (* 3) best_friends
Just 9
> calc_stuff (+ 1) height
Just 2.7
> calc_stuff (* 2) girlfriend
Nothing

The last bit to understanding Functor is a method called fmap, all types that are instances of the typeclass Functor have to implement a method called fmap, which has the signature:

fmap              :: (a -> b) -> f a -> f b 
calc_stuff :: (a -> a) -> Maybe a -> Maybe a

I’m putting them one above the other to let you see how similar they are. fmap is just calc_stuff generalized - generalizing the function that calc_stuff takes in and - generaizing the type Maybe to other types that take in a single type variable

And the f we see in the fmap is the Functor, a typeclass of which Maybe is an instance of.