This is pretty cool. It's a bit abstract though. I can imagine people who don't know what monads are already get confused due to the lack of real examples.
So let me try to comply, and just to be really clear I'll do an example in C#, even though it will look ugly. I'll add the equivalent Haskell at the end and show you the cool Haskell syntactic sugar which is where, IMO, monads really start getting useful.
Okay, so one of the easiest Monads it called the "Maybe monad" in Haskell. In C# the Maybe type is called Nullable<T>. It's basically a tiny class that just encapsulates the concept of a value that is either valid and has a value, or is "null" and has no value.
A useful thing to stick inside a monad for combining values of this type is the notion of failure. I.e. we want to be able to look at multiple nullable values and return "null" as soon as any one of them is null. This could be useful if you, for example, look up lots of keys in a dictionary or something, and at the end you want to process all of the results and combine them somehow, but if any of the keys are not in the dictionary, you want to return "null" for the whole thing. It would be tedious to manually have to check each lookup for "null" and return, so we can hide this checking inside the bind operator (which is sort of the point of monads, we hide book-keeping in the bind operator which makes the code easier to use since we can forget about the details).
Here's the program that motivates the whole thing (I'll define the Bind later, this just to show you why it's nice).
class Program
{
static Nullable<int> f(){ return 4; }
static Nullable<int> g(){ return 7; }
static Nullable<int> h(){ return 9; }
static void Main(string[] args)
{
Nullable<int> z =
f().Bind( fval =>
g().Bind( gval =>
h().Bind( hval =>
new Nullable<int>( fval + gval + hval ))));
Console.WriteLine(
"z = {0}", z.HasValue ? z.Value.ToString() : "null" );
Console.WriteLine("Press any key to continue...");
Console.ReadKey();
}
}
Now, ignore for a moment that there already is support for doing this for Nullable in C# (you can add nullable ints together and you get null if either is null). Let's pretend that there is no such feature, and it's just a user-defined class with no special magic. The point is that we can use the Bind function to bind a variable to the contents of our Nullable value and then pretend that there's nothing strange going on, and use them like normal ints and just add them together. We wrap the result in a nullable at the end, and that nullable will either be null (if any of f, g or h returns null) or it will be the result of summing f, g, and h together. (this is analogous of how we can bind a row in a database to a variable in LINQ, and do stuff with it, safe in the knowledge that the Bind operator will make sure that the variable will only ever be passed valid row values).
You can play with this and change any of f, g, and h to return null and you will see that the whole thing will return null.
So clearly the bind operator has to do this checking for us, and bail out returning null if it encounters a null value, and otherwise pass along the value inside the Nullable structure into the lambda.
Here's the Bind operator:
public static Nullable<B> Bind<A,B>( this Nullable<A> a, Func<A,Nullable<B>> f )
where B : struct
where A : struct
{
return a.HasValue ? f(a.Value) : null;
}
The types here are just like in the video. It takes an "M a" (Nullable<A> in C# syntax for this case), and a function from "a" to "M b" (Func<A, Nullable<B>> in C# syntax), and it returns an "M b" (Nullable<B>).
The code simply checks if the nullable contains a value and if so extracts it and passes it onto the function, else it just returns null. This means that the Bind operator will handle all the null-checking logic for us. If and only if the value that we call Bind on is non-null then that value will be "passed along" to the lambda function, else we bail out early and the whole expression is null.
This allows the code that we write using the monad to be entirely free of this null-checking behaviour, we just use Bind and get a variable bound to the value inside the monadic value (fval, gval and hval in the example code) and we can use them safe in the knowledge that Bind will take care of checking them for null before passing them along.
There are other examples of things you can do with a monad. For example you can make the Bind operator take care of an input stream of characters, and use it to write parser combinators. Each parser combinator can then be completely oblivious to things like back-tracking, parser failures etc., and just combine smaller parsers together as if things would never go wrong, safe in the knowledge that a clever implmenetation of Bind sorts out all the logic behind the difficult bits. Then later on maybe someone adds logging to the monad, but the code using the monad doesn't change, because all the magic happens in the definition of the Bind operator, the rest of the code is unchanged.
Finally, here's the implemenation of the same code in Haskell (-- begins a comment line).
-- Here's the data type, it's either nothing, or "Just" a value
-- this is in the standard library
data Maybe a = Nothing | Just a
-- The bind operator for Nothing
Nothing >>= f = Nothing
-- The bind operator for Just x
Just x >>= f = f x
-- the "unit", called "return"
return = Just
-- The sample code using the lambda syntax
-- that Brian showed
z = f >>= ( \fval ->
g >>= ( \gval ->
h >>= ( \hval -> return (fval+gval+hval ) ) ) )
-- The following is exactly the same as the three lines above
z2 = do
fval <- f
gval <- g
hval <- h
return (fval+gval+hval)
As you can see the nice "do" notation at the end makes it look like straight imperative code. And indeed this is by design. Monads can be used to encapsulate all the useful stuff in imperative programming (mutable state, IO etc.) and used using this nice imperative-like syntax, but behind the curtains, it's all just monads and a clever implementation of the bind operator!
The cool thing is that you can implement your own monads by implemnting >>= and return. And if you do so those monads will also be able to use the do notation, which means you can basically write your own little languages by just defining two functions!