Friday, January 22, 2010

A monad non-tutorial

...or why you shouldn't ask what a monad is

"What is a monad?" is one of the most common question when you're learning Haskell. And it's there when troubles start, because it is the wrong question. Ask a wrong question, and you'll get the wrong answer. The only right answer to this question is a mathematic definition:

[...] A monad is a triple (M, unitM, bindM) consisting in a type constructor M and a pair of polymorphic functions {unitM :: a -> M a} and {bindM :: M a -> (a -> M b) -> M b}

These functions must satisfy three laws [...]

(unitM a) `bindM` k = k a
m `bindM` unitM = m
m `bindM` (\a -> (k a) `bindM` (\b -> h b)) =
(m `bindM` \a -> (k a)) `bindM` (\b -> h b)

After a reverential minute of silence, you start to figure it out that maybe you shouldn't be asking that. Your mind is working fast to come out with a phrase to fool your interlocutor into thinking that you have understood. But there's nothing to understand.

You'll see, that's the way definitions work. They are use to label objects, that's the only thing they are good for. So if the definition of homo sapiens is "an animal, mammal, of the order of primates, family hominidae and genus homo" and if you know what is an animal, a mammal, a primate... etc. then you can put objects into classes: "my friend Ed, homo sapiens; mi dog Buch, no".

It is why, not what

The least interesting thing about monads is its definition. The real question is "why?". You only make a definition if it is useful, if you use it frequently. So, let's see how this definition is implemented in Haskell:
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
return a :: m a
So, a monad is an Haskell class. Or, if you are a Java guy (and let's confess it, who isn't?), then you can think in a monad as an interface. But, again, why? When you see FileishInterface with its methods open, close, read, write and seek, you say: "Oh, so FileishInterface represents objects that behave like a file, I get it". You can think in a FileishInterface instance using a TCP connection or a byte array. With monads, there is nothing in its interface that make you think in a concrete example.

An that is a good thing. After all, FileishInterface is so concrete that it could be used only in a very few cases. The interface of Monad is so generic that it could be used in a wide range of objects. Paradoxically, that make it hard to think in a use when you are learning the concept.


The other problem when you are learning monad is that you want to use them. You open your editor and say "I'll make a program and I'm gonna create my own monads and I'll show it in the Haskell Café and I'll become a member of the Secret Category Cabal and I'm gonna spend my evenings talking about Kleili arrows and functors and maybe Paul Hudak will friend me in facebook". Curiously, you have never say "Mmm, today I feel like I'm gonna make a program that uses hashtables". You don't try to impose data structures to your imperative programming, so why are you trying to force monads in your haskell programs? Don't go after monads, let the monads find you.

You'll know when you need them. One day you will make a program and you'll write a type constructor:
type M a = ....
(did you notice how your type constructor's name is "M" as in "Monad"? This is a clever, subtle way in which the author -- me -- hints that you've started to write a monad).

If you use your type constructor M to create only one type, say M Integer, then you don't need monads. But, after a few hours of programming, types start to appear and you have M Char, M String, M Bool and others. After a few coups of coffee you have functions all over the place with types like Integer -> M Integer, Integer -> M Bool, Bool -> M Char.

Then you need to chain these function together, but it's a pain in the functional ass. Each time you need to connect two functions, say Integer -> M Bool and Bool -> M Char, you need to invoke the first function, extract the value inside M and invoke the second one. And it is then --I hope-- that the "aha! moment" comes to you. You can write a function that extract the a from M a and use it to invoke a function with type a -> M b. You have invented >>=, the bind function!

Since you wrote the type constructor M, you are the one who know how to extract a from M a. And you know how to chain that value to a -> M b. Each monad has its own definition of >>=, adapted to its structure.

When is return used? If you see the >>= definition, it needs a M a to start. If you have a value of type a instead, you'll need a way to transform it into M a. That's what return does.

Now what?

Hopefully you know by now that the definition of monads is not the important part for a programmer. You only need to focus in the type constructor and the bind function. What does the type constructor represent? What is it used for? How they defined bind: how it extract the value inside one of those new types and how it chain it to the a -> m b function?

So, now you can go back to those tutorials that explain what are monads and give them a second look. You can review the do notation that help you to write your long chain of >>= expression in a clear way. Good luck.

The IO Monad

But wait! I can't end this post without talking about IO monads, can I? A monad is a monad is a monad. IO monads are not the exception. Since your understanding of monad is better now and my hands are tired, I'll cheat a little.

The IO a data constructor can be seen as a function that takes a value of type a and returns a C program. When that program is compiled and executed, it will return that value. That's the work of the IO bind. It compiles and executes the first argument (IO a), gets the a values and feed it to its second argument (a -> IO b) which returns a new C program that will return a value of type b. That's why the haskellers can see you right into the eyes and say that Haskell is a pure language: "Of course that putChar is a pure function. Giving the same input, it will always return the same C program!".

Thinking in this way is useful to understand why IO is a monad, but it is a waste of your attention. In your daily programming, you'll be better thinking that the IO values are an imperative language embedded in haskell and >>= is its interpreter.


Conal said...

I think you're right that when people as "What is a monad?", they're almost always asking the wrong question.

> The IO a data constructor can be seen as a function that takes a value of type a and returns a C program.

I think you mean: The IO a data type can be seen as a C program that yields a value of type a when executed.

For a similar point of view, see the blog post The C language is purely functional.

See also the discussion on Monad documentation hate and the wiki entry What a Monad is not (and discussion).

Jonno_FTW said...

An excellent stepping stone in my understanding of monads. After reading about a gajillion of them already, I still don't understand them fully. Although I cheated in my learning and didn't really bother with the whole defining data types and type construction thing.

Trevion said...

Just a nit... most mathematical definitions of monads use unit and join, not unit and bind...

Calvin Spealman said...


I still don't get it :-(

Aivar said...

Good post! But beginning was misleading, at first i thought the post is gonna be about mathematics.

christian said...

You're brilliant, thank you so much for this article. It's a shame that I didn't find this BEFORE my exam...