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Functors, Applicatives, And Monads In Pictures (In ReasonML)

kevanstannard profile image Kevan Stannard ・8 min read

This is a translation of Functors, Applicatives, And Monads In Pictures from Haskell into ReasonML.

I take no credit for this work, and if you enjoy this post be sure to say thanks to Aditya Bhargava (@_egonschiele) the author of the original version.


Here's a simple value:

And we know how to apply a function to this value:

Simple enough. Let's extend this by saying that any value can be in a context. For now you can think of a context as a box that you can put a value in:

Now when you apply a function to this value, you'll get different results depending on the context. This is the idea that Functors, Applicatives, Monads, Arrows etc are all based on.

Let's create a Maybe data type that defines two related contexts:


module Maybe = {
  type t('a) =
    | Nothing
    | Just('a);
}

In a second we'll see how function application is different when something is a Just(a) versus a Nothing. First let's talk about Functors!

Functors

When a value is wrapped in a context, you can't apply a normal function to it:

This is where fmap comes in. fmap is from the street, fmap is hip to contexts. fmap knows how to apply functions to values that are wrapped in a context.

Suppose you want to apply (+3) to Just(2). We can implement fmap:

Maybe.fmap((+)(3), Just(2));
// Just(5)

Bam! fmap shows us how it's done!

Just what is a Functor, really?

Functor is a class of functions that define an fmap function.

In Haskell they are defined as a type class.

In ReasonML we don't have type classes yet, but they are being worked on in OCaml, and therefore ReasonML.

Here's the definition:

A Functor is any data type that defines how fmap applies to it. Here's how fmap works:

So we can do this:

Maybe.fmap((+)(3), Just(2));
// Just(5)

This specifies how fmap applies to Justs and Nothings:

Let's add fmap to our Maybe module:

let fmap = (f, m) => {
  switch (m) {
  | Nothing => Nothing
  | Just(a) => Just(f(a))
  };
};

Here's what is happening behind the scenes when we write Maybe.fmap((+)(3), Just(2));:

So then you're like, alright fmap, please apply (+3) to a Nothing?


Maybe.fmap((+)(3), Nothing)
// Nothing

Like Morpheus in the Matrix, fmap knows just what to do; you start with Nothing, and you end up with Nothing! fmap is zen. Now it makes sense why the Maybe data type exists. For example, here's how you work with a database record in a language without Maybe:

post = Post.find_by_id(1)
if post
  return post.title
else
  return nil
end

Let's create a simple Post module in ReasonML with these functions.

module Post = {
  type t = {
    id: int,
    title: "string,"
  };
  let make = (id, title) => {id, title};
  let fmap = (f, post) => f(post);
  let getPostTitle = post => post.title;
  let findPost = id => make(id, "Post #" ++ string_of_int(id));
};

Now we can write:

Post.(fmap(getPostTitle, findPost(1)));

If findPost returns a post, we will get the title with getPostTitle. If it returns Nothing, we will return Nothing! Pretty neat, huh?

In Haskell, <$> is the common infix version of fmap.

In ReasonML we can create an equivalent alias. Adding to our Post module:

let (<$>) = fmap;

So we can now write:

Post.(getPostTitle <$> findPost(1));

Here's another example: what happens when you apply a function to a list?

Lists can operate as functors too!

In ReasonML we can make use of the list map function:

List.map

Okay, okay, one last example: what happens when you apply a function to another function?

For this case we can define a Function module that has fmap:

module Function = {
  let fmap = (f, g, x) => f(g(x));
};

So we can now write:

Function.fmap((+)(3), (+)(1));

Here's a function:

Here's a function applied to another function:

The result is just another function!

let foo = Function.fmap((+)(3), (+)(2));
foo(10);
// 15

So functions can be Functors too!

When you use fmap on a function, you're just doing function composition!

Applicatives

Applicatives take it to the next level. With an applicative, our values are wrapped in a context, just like Functors:

But our functions are wrapped in a context too!

Yeah. Let that sink in. Applicatives don't kid around. They know how to apply a function wrapped in a context to a value wrapped in a context:

Applicatives define an apply function, also written as <*> in Haskell, which we can create an alias for in ReasonML.

Let's add the applicative functions to our Maybe module:

let apply = (mf, mv) => {
  switch (mv) {
  | Nothing => Nothing
  | Just(v) =>
    switch (mf) {
    | Nothing => Nothing
    | Just(f) => Just(f(v))
    }
  };
};

let (<*>) = apply;

Here's an example using them:

Maybe.(Just((+)(3)) <*> Just(2));
// Just(5)

Let's also define applicative functions for a list. We'll create a MyList module to avoid name clashes with the built in List module:

module MyList = {
  type apply('a, 'b) = (list('a => 'b), list('a)) => list('b);

  let apply: apply('a, 'b) =
    (fs, xs) => List.flatten(List.map(f => List.map(f, xs), fs));

  let (<*>) = apply;
};

Using <*> can lead to some interesting situations. For example:

let funList = [(*)(2), (+)(3)];
let valList = [1, 2, 3];
MyList.(funList <*> valList);
// [2, 4, 6, 4, 5, 6]

Here's something you can do with Applicatives that you can't do with Functors. How do you apply a function that takes two arguments to two wrapped values?

Maybe.((+) <$> Just(5));
// Just((+)(5))

Maybe.(Just((+)(5)) <$> Just(4));
// ERROR ??? WHAT DOES THIS EVEN MEAN WHY IS THE FUNCTION WRAPPED IN A JUST

Applicatives:

Maybe.((+) <$> Just(5));
// Just((+)(5))

Maybe.(Just((+)(5)) <*> Just(3));
// Just(8)

Applicative pushes Functor aside. "Big boys can use functions with any number of arguments," it says. "Armed with <$> and <*>, I can take any function that expects any number of unwrapped values. Then I pass it all wrapped values, and I get a wrapped value out! AHAHAHAHAH!"

Maybe.((*) <$> Just(5) <*> Just(3));

Monads

How to learn about Monads:

  1. Get a PhD in computer science.
  2. Throw it away because you don't need it for this section!

Monads add a new twist.

Functors apply a function to a wrapped value:

Applicatives apply a wrapped function to a wrapped value:

Monads apply a function that returns a wrapped value to a wrapped value.

Monads have a function bind, or the operator alias >>= to do this.

Let's see an example.

First, we'll need to add bind to our good ol' Maybe module:

let bind = (mv, f) => {
  switch (mv) {
  | Nothing => Nothing
  | Just(v) => f(v)
  };
};

let (>>=) = bind;

Suppose half is a function that only works on even numbers:

Let's write half in ReasonML (we'll also need to define even and odd functions):

/*
 Mutually recursive function
 https://ocaml.org/learn/tutorials/labels.html
 */
let rec even = x =>
  if (x <= 0) {
    true;
  } else {
    odd(x - 1);
  }
and odd = x =>
  if (x <= 0) {
    false;
  } else {
    even(x - 1);
  };

let half = x =>
  if (even(x)) {
    Maybe.Just(x / 2);
  } else {
    Nothing;
  };

What if we feed it a wrapped value?

We need to use >>= to shove our wrapped value into the function. Here's a photo of >>=:

Here's how it works:

Maybe.(Just(3) >>= half);
// Nothing

Maybe.(Just(4) >>= half);
// Just(2)

Maybe.(Nothing >>= half);
// Nothing

What's happening inside? Monad defines a bind (or >>=) function:

Let's make our Maybe into a monad by adding the bind functions.

let bind = (mv, f) => {
  switch (mv) {
  | Nothing => Nothing
  | Just(v) => f(v)
  };
};

let (>>=) = bind;

Here it is in action with a Just(3)!

And if you pass in a Nothing it's even simpler:

You can also chain these calls:

Maybe.(Just(20) >>= half >>= half >>= half);

Cool stuff! So now we have implemented Maybe to be a Functor, an Applicative, and a Monad.

Now let's mosey on over to another example and create an IO monad:

The IO monad exists in Haskell, but we will declare our own in ReasonML.

Specifically three functions. getLine takes no arguments and gets user input:

readFile takes a string (a filename) and returns that file's contents:

putStrLn takes a string and prints it:

Our IO module might look something like this (leaving out implementation details of the helper functions):

module IO = {
  type t = Js.Promise.t(string);

  type bind('a, 'b) = (t, string => t) => t;
  let bind: bind('a, 'b) = (pa, f) => pa |> Js.Promise.then_(a => f(a));

  let (>>=) = bind;

  type getLine = unit => t;
  let getLine = ...

  type readFile = string => t;
  let readFile = ...

  type putStrLn = string => t;
  let putStrLn = ...
};

If you're interested, the full source code is available.

All three functions take a regular value (or no value) and return a wrapped Promise value. We can chain all of these using >>=!


IO.(getLine() >>= readFile >>= putStrLn);

Aw yeah! Front row seats to the monad show!

Conclusion

  1. A functor is a data type that implements the fmap function.
  2. An applicative is a data type that implements the apply function.
  3. A monad is a data type that implements the bind function.

The Maybe module in our examples implements all three, so it is a functor, an applicative, and a monad.

What is the difference between the three?

Functors: you apply a function to a wrapped value using fmap or <$>.

Applicatives: you apply a wrapped function to a wrapped value using apply or <*>.

Monads: you apply a function that returns a wrapped value, to a wrapped value using bind or >>=.

So, dear friend (I think we are friends by this point), I think we both agree that monads are easy and a SMART IDEA(tm). Now that you've wet your whistle on this guide, why not pull a Mel Gibson and grab the whole bottle. Check out LYAH's section on Monads. There's a lot of things I've glossed over because Miran does a great job going in-depth with this stuff.


If you feel there is something I could change to improve this translation to ReasonML please let me know.

Thanks again to Aditya Bhargava for writing the original version of this post ❤️

Discussion

markdown guide
 

Thank you for such a fun and understandable explanation!

 

Thank you for this.
One remark: I would add a note that the Maybe type is called Option in ReasonML, Just is equivalent to Some and Nothing to None.

Edit: And bind is (sometimes) called flatMap.