*2017 update - I think this article doesn't really get to the point regarding
Monads. Specifically, I don't think it goes beyond Functor. You are probably
best looking somewhere else.*

*An attempt to explain monads and their usefulness in Haskell, assuming only
some simple knowledge of Python.*

List comprehensions are a great Python feature that have been borrowed from Haskell/ML, and ultimately from set theory. As you probably know, they provide a convenient syntax for building lists from other lists or sequences:

>>> lst = [1, 2, 3] >>> [x*2 for x in lst] [2, 4, 6]

The alternative to the list comprehension is something like this:

>>> newlist = [] >>> for x in lst: ... newlist.append(x*2)

List comprehensions remove this annoying bit of boilerplate, helping to give Python the conciseness we love. The list comprehension knows all about creating an empty initial list and appending values etc, so we don't have to do that each time.

While learning Haskell, I found list comprehensions a reassuring plateau of
familiarity amidst the otherwise rather treacherous, alien landscape, but
they also provide a way in to the impenetrable forest of 'Monads'. In
Haskell, the above Python code looks very similar (at a GHCi interactive
prompt, where `Prelude>` is the standard prompt):

Prelude> let lst = [1,2,3] Prelude> [ x*2 | x <- lst ] [2,4,6]

That should be pretty easy to understand if you know Python, but here is a full 'translation' of the list comprehension into English: unpack each value in the list 'lst' in turn, give it the label 'x', and for each one return 'x * 2' as a value in a new list.

Haskell has another way of implementing the same thing, though — using its 'do' notation:

Prelude> do { x <- lst; return (x*2) } [2,4,6]

The similarity with the above should be pretty obvious, and in fact it does exactly the same thing. You'll notice that the order is reversed so it is more like our English version, and matches an imperative style of thinking (do this, then that), whereas the list comprehension syntax reflects its origins in the mathematical notations of set theory.

However, there is one difference. While list comprehension are specific to creating lists, 'do' notation has no knowledge of lists at all. In fact, the only list in this expression is the input value 'lst'. So, to prove a point, let's eliminate even that bit, by creating a function out of this expression:

Prelude> let double val = do { x <- val; return (x*2) }

This is a definition of the function 'double', that takes a single value 'val'. (At the interactive prompt, we have to add the keyword 'let', which wouldn't be present in normal Haskell source code). We can now use our function as before:

Prelude> double [0,1,2] [0,2,4]

GHCi doesn't complain about any of this, even though Haskell is statically
typed and we haven't told it what type of thing `val` is. So what type of
thing is `val`? We can ask GHCi using `:type`, but we'll have to do it
indirectly, by asking it the type of our function `double`:

[WARNING: the next few paragraphs are a bit tricky, but it gets easier again soon - hang in there!]

Prelude> :type double double :: (Monad m, Num a) => m a -> m a

Woah! What on earth does all *that* mean? Well, let's ignore the bit in
brackets, and look at the end first:

m a -> m a

The arrow tells us we have a function (as we expected), and it takes values of
type `m a` and returns values of type `m a`. `m a` is a 'parameterised
type' — as a simplification, we can say that 'm' is a placeholder for a
container type, and 'a' is a placeholder for the type of thing it contains. In
the case of our input `lst`, `m` is a list (represented as `[]` usually)
and `a` is an integer. This is normally written `[Integer]` (a list of
integers), rather than `[] Integer`. In our instance, `double` took a list
of integers and returned a list of integers.

But GHC knew that our function was more generic than simply operating on
lists of integers. From the 'do' notation, and the lack of anything about
lists, it deduced that 'm' can actually be any 'Monad'. From the use of
multiplication in '* 2', it deduced that 'a' has to be restricted to numbers
— more specifically, a type which implements the 'Num' interface (which
includes integers, fractional numbers etc). The `(Monad m, Num a) =>` bit
is just indicating these restrictions.

So, how did our 'double' function achieve the feat of unpacking the data from the list, doubling each item, and returning the data in a new list, all without knowing anything about lists? Well, by using the 'do' notation, it was implicitly using a set of methods that are defined in the 'Monad' interface. Since lists are instances of the Monad interface, and defines all the methods correctly, it just worked.

Are there other Monads that we could use instead of lists? Well, of course,
or no-one would have bothered to create an interface if there was only a
single instance of it. One example would be the `Maybe` monad, that
contains either `Nothing`, or an actual value, written `Just somevalue`.
The `Maybe` monad encapsulates a value that might be there, but might not,
and the logic that if it contains `Nothing`, then any function operating on
it will have to return `Nothing` too. So we can now do this:

Prelude> double Nothing Nothing Prelude> double (Just 1.5) Just 3.0

Magic! Trying to do `Nothing * 2` will give you a type error, but by using
the Maybe monad, and our function that was generic over monads, we did it
easily, with no extra work. Impressed?

In other languages you can create functions that work with different types of
collections, by using, for example, the iterator protocol in Python, or the
IEnumerable interface in C#. But here we have taken it to a higher level —
the Monad interface is like an abstraction of *any* kind of container.

Alternatively, `Maybe` and lists can both be thought of as encapsulating
different strategies for computing values. `Maybe` handles the case of 0 or
1 values, while list can handle any number of values, and it tries them all.
This in turn leads to the concept that a monadic value represents a
*computation* — a method for computing a value, bound together with its
input value. This becomes especially important when you move on to 'State
transformation' monads, such as the famous IO monad. In these cases, the
container is actually a function (but just don't think about that if it hurts
your head!).

So, in Haskell Monads are simply an interface for a very generic container,
but an interface that is so important that it has special syntactic support
in the language, similarly to how the iterator protocol and lists in Python
have various bits of syntactic support (such as `for`, `in`, list
comprehensions, etc). The interface is more abstract than that of
collections, which makes it more difficult to understand, but more powerful
too, and the syntactic sugar all the more sweet, as it is much more broadly
applicable.

For instance, monads were used to write the Parsec parser library which ends up with a syntax that allows a pretty direct translation from BNF, and is in fact more readable. The parser monads know all about applying constraints, backtracking etc, in the same way that the list monad knows about how to take each element and apply the function to each one. Writing monads is hard, but it pays off as using them in Haskell is surprisingly easy, and allows you to do some very powerful things.

I hope that begins to explain why monads are useful. The next difficulty is understanding the methods that make up the Monad interface, which is beyond the scope of this article really, but I'll try to give an introduction.

You can already guess something about the Monad methods. One of them you have
seen explicitly — it's the 'return' method, responsible for packing things
up into the monad. The other is called 'bind' or '>>=', and it does the
'unpacking' involved with the `<-` arrow in the do notation.

Actually, the 'bind' method doesn't really unpack and return the data.
Instead, it is defined in such a way that it handles all unpacking
'internally', and you have to provide functions that always have to return
data inside the monad. Why is this important? Well, some uses of monads,
especially the IO monad, need to ensure that data can't 'escape', in order to
be able to make certain guarantees that keep your program working as
expected. Monads like `Maybe` and list are much less possessive — they are
quite happy for you to get the data out. But by defining the Monad interface
in this way, it can handle both cases, and it turns out to be quite
convenient for both.

What is the `<-` in the `do` notation then? It is simply some syntactic
sugar that allows you to define the right kind of functions easily and
painlessly. It looks very much like 'unpack this data from the monad so I can
use it', so it helps conceptually. In fact, together with the rest of the
body of the 'do' block it forms an anonymous lambda function, and we could
write our `double` function something pretty much like this in Python:

def double(val): return val.bind(lambda x: val.return_(x*2))

(I've had to use `return_` to avoid the clash with Python's 'return'
keyword). Haskell's `do` notation eliminates the explicit call to `bind`,
and the `lambda`, making it quite a bit easier to use. This becomes
especially important when you have long 'do' blocks, and functions with
multiple monadic input values. One final thing to say - you will be pleased
to learn that in Haskell you can actually use whitespace (newlines and
indentation) instead of semi-colons and braces! (OK, OK, calm down you
Pythonistas at the back, that's enough cheering now ;-).

I've done a complete example implementation of the List and Maybe monads in Python, along with the double function as above, trying to stay close to how it works in Haskell. You can't really translate Haskell's type system, but Python can do a pretty good job of implementing this, partly due to the fact that non-instance methods can be polymorphic, unlike many other OOP languages. The code is also a nice example of how succinct functional style code often is — there isn't a function more than 4 lines long.

Has this helped at all? I'd be interested in any feedback, or corrections. I'm a Haskell newbie myself, so I may have got some things wrong, in which case I'll pretend that I was simplifying things for the sake of clarity :-) .

## Further reading

- Why monads are important, and an alternative explanation of how monads work, by Shannon Behrens: Everything Your Professor Failed to Tell You About Functional Programming
- To understand the IO monad, I've found this IO inside article the most helpful. It starts from a completely different angle, though, so don't be surprised. Eventually the different concepts do converge :-)

## Updates

- 2006-07-28: added a translation of the 'double' function into Python, to explain the do notation's implicit lambda.
- 2006-08-15: added some Python code that implements the same thing in Python.
- 2006-09-11: fixed some bugs in the downloadable Python code, and adding 'lifting' examples to it.