Functors, Applicatives, And Monads In Pictures
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This is a translation of Functors, Applicatives, And Monads In Pictures from Haskell into Python. Hopefully this should make the article much easier to understand for people who don't know Haskell. All code samples uses the Python OSlash library (Python 3 only).
How to make the examples work with Python:
$ pip3 install oslash python3 >>> from oslash import *
Here’s a simple value:
And we know how to apply a function to this value:
Simple enough. Lets 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. The Maybe data type defines two related contexts:
class Maybe(Monad, Monoid, Applicative, Functor, metaclass=ABCMeta): """The Maybe type encapsulates an optional value. A value of type Maybe a either contains a value of (represented as Just a), or it is empty (represented as Nothing). Using Maybe is a good way to deal with errors or exceptional cases without resorting to drastic measures such as error. """ ... class Just(Maybe): """Represents a value of type Maybe that contains a value (represented as Just a). """ ... class Nothing(Maybe): """Represents an empty Maybe that holds nothing (in which case it has the value of Nothing). """ ...
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!
When a value is wrapped in a context, you can’t apply a normal function to it:
This is where
map comes in (
fmap in Haskell).
map is from the street,
map is hip to contexts.
map knows how to apply functions to values that are wrapped in a context. For example, suppose you want to apply
Just 2. Use
>>> Just(2).map(lambda x: x+3) Just 5
map shows us how it’s done! But how does
map know how to apply the function?
Just what is a Functor, really?
Functor is an Abstract Base Class (typeclass in Haskell). Here’s the definition:
class Functor(metaclass=ABCMeta): @abstractmethod def map(self, func) -> "Functor": return NotImplemented
A Functor is any data type that defines how
map applies to it. Here’s how
So we can do this:
>>> Just(2).map(lambda x: x+3) Just 5
map magically applies this function, because
Maybe is a
Functor. It specifies how
fmap applies to
class Just(Maybe): def map(self, mapper) -> Maybe: result = ... return Just(result)
class Nothing(Maybe): def map(self, _) -> Maybe: return Nothing()
Here’s what is happening behind the scenes when we write
Just(2).map(lambda x: x+3):
So then you’re like, alright
map, please apply
lambda x: x+3 to a Nothing?
>>> Nothing().map(lambda x: x+3) Nothing
Bill O’Reilly being totally ignorant about the Maybe functor
Like Morpheus in the Matrix,
map knows just what to do; you start with
Nothing, and you end up with
map 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
post = Post.find_by_id(1) if post return post.title else return nil end
But in Python:
find_post() returns a post, we will get the title with
get_post_title. If it returns
Nothing, we will return
Nothing! Pretty neat, huh?
<$> in Haskell) is the infix version of
map, so you will often see this instead:
get_post_title % find_post(1)
Here’s another example: what happens when you apply a function to a list?
Lists are functors too! Here’s the definition:
class List(Monad, Monoid, Applicative, Functor, list)
Okay, okay, one last example: what happens when you apply a function to another function?
map(lambda x: x+2, lambda y: y+3)
Here’s a function applied to another function:
The result is just another function!
>>> def fmap(f, g): ... return lambda x: g(f(x)) ... >>> foo = fmap(lambda x: x+3, lambda y: y+2) >>> foo(10) 15
So functions are Functors too!
It was actually quite easy to define an
fmapfunction in Python that made it possible for us to compose functions.
When you use fmap on a function, you’re just doing function composition!
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. Applicative defines
<*> in Haskell), which knows how to apply a function wrapped in a context to a value wrapped in a context:
>>> Just(lambda x: x+3) * Just(2) == Just(5) True
* can lead to some interesting situations. For example:
>> List([lambda x: x*2, lambda y: y+3]) * List([1, 2, 3]) [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?
>>> (lambda x,y: x+y) % Just(5) Just functools.partial(<function <lambda> at 0x1003c1bf8>, 5) >>> Just(lambda x: x+5) % Just(5) Traceback (most recent call last): File "<stdin>", line 1, in <module> TypeError: unsupported operand type(s) for <<: 'Just' and 'Just'
>>> (lambda x,y: x+y) % Just(5) Just functools.partial(<function <lambda> at 0x1003c1bf8>, 5) >>> Just(lambda x: x+5) * Just(5) Just 10
Functor aside. “Big boys can use functions with any number of arguments,” it says. “Armed with
*, 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!”
>>> (lambda x,y: x*y) % Just(5) * Just(3) Just 15
And hey! There’s a method called
lift_a2 that does the same thing:
>>> Just(5).lift_a2(lambda x,y: x*y, Just(3)) Just 15
How to learn about Monads:
- Get a PhD in computer science.
- 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
| (>>= in Haskell) (pronounced “bind”) to do this.
Let’s see an example. Good ol’ Maybe is a monad:
Just a monad hanging out
half is a function that only works on even numbers:
half = lambda x: Just(x // 2) if (x % 2== 0) else Nothing()
What if we feed it a wrapped value?
We need to use
>>= in Haskell) to shove our wrapped value into the function. Here’s a photo of
Here’s how it works:
>>> Just(3) | half Nothing >>> Just(4) | half Just 2 >>> Nothing() | half Nothing
What’s happening inside? Monad is another Abstract Base Class (typeclass in Haskell). Here’s a partial definition:
class Monad(metaclass=ABCMeta): @abstractmethod def bind(self, func) -> "Monad": """ :param Monad[A] self: :param Callable[[A], Monad[B]] func: :rtype: Monad[B] :returns: New Monad wrapping B """
Where bind (
| in Python,
>>= in Haskell) is:
Maybe is a Monad:
class Maybe(Monad, Monoid, Applicative, Functor, metaclass=ABCMeta):
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:
>>> Just(20) | half | half | half Nothing
Cool stuff! So now we know that Maybe is a Functor, an Applicative, and a Monad.
Now let’s mosey on over to another example: the
Specifically three functions.
getLine in Haskell) takes no arguments and gets user input:
def get_line() -> IO: return Get(lambda s: IO(s))
readFile in Haskell) takes a string (a filename) and returns that file’s contents:
def read_file(filename) -> IO: return ReadFile(filename, lambda s: IO(s))
putStrLn in Haskell) takes a string and prints it:
def put_line(string) -> IO: return Put(string, IO(()))
All three functions take a regular value (or no value) and return a wrapped value. We can chain all of these using
get_line() | read_file | put_line
Aw yeah! Front row seats to the monad show!
Haskell also provides us with some syntactical sugar for monads, called do notation:
foo = do filename <- getLine contents <- readFile filename putStrLn contents
Python does not have a do notation, so we have to write things a bit differently:
foo = get_line() | (lambda filename: read_file(filename) | (lambda contents: put_line(contents)))
- A functor is a data type that implements the
Functorabstract base class.
- An applicative is a data type that implements the
Applicativeabstract base class.
- A monad is a data type that implements the
Monadabstract base class.
Maybeimplements 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
applicatives: you apply a wrapped function to a wrapped value using
monads: you apply a function that returns a wrapped value, to a wrapped value using ´|´ 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.