Why are side-effects modeled as monads in Haskell?

Could anyone give some pointers on why the unpure computations in Haskell are modeled as monads?

This question contains a widespread misunderstanding. Impurity and Monad are independent notions. Impurity is not modeled by Monad. Rather, there are a few data types, such as IO, that represent imperative computation. And for some of those types, a tiny fraction of their interface corresponds to the interface pattern called "Monad". Moreover, there is no known pure/functional/denotative explanation of IO (and there is unlikely to be one, considering the "sin bin" purpose of IO), though there is the commonly told story about World -> (a, World) being the meaning of IO a. That story cannot truthfully describe IO, because IO supports concurrency and nondeterminism. The story doesn't even work when for deterministic computations that allow mid-computation interaction with the world.

For more explanation, see this answer.

Edit: On re-reading the question, I don't think my answer is quite on track. Models of imperative computation do often turn out to be monads, just as the question said. The asker might not really assume that monadness in any way enables the modeling of imperative computation.

Suppose a function has side effects. If we take all the effects it produces as the input and output parameters, then the function is pure to the outside world.

So, for an impure function

f' :: Int -> Int

we add the RealWorld to the consideration

f :: Int -> RealWorld -> (Int, RealWorld)
-- input some states of the whole world,
-- modify the whole world because of the side effects,
-- then return the new world.

then f is pure again. We define a parametrized data type type IO a = RealWorld -> (a, RealWorld), so we don't need to type RealWorld so many times, and can just write

f :: Int -> IO Int

To the programmer, handling a RealWorld directly is too dangerous—in particular, if a programmer gets their hands on a value of type RealWorld, they might try to copy it, which is basically impossible. (Think of trying to copy the entire filesystem, for example. Where would you put it?) Therefore, our definition of IO encapsulates the states of the whole world as well.

Composition of "impure" functions

These impure functions are useless if we can't chain them together. Consider

getLine     :: IO String            ~            RealWorld -> (String, RealWorld)
getContents :: String -> IO String  ~  String -> RealWorld -> (String, RealWorld)
putStrLn    :: String -> IO ()      ~  String -> RealWorld -> ((),     RealWorld)

We want to

• get a filename from the console,
• read that file, and
• print that file's contents to the console.

How would we do it if we could access the real world states?

printFile :: RealWorld -> ((), RealWorld)
printFile world0 = let (filename, world1) = getLine world0
                       (contents, world2) = (getContents filename) world1 
                   in  (putStrLn contents) world2 -- results in ((), world3)

We see a pattern here. The functions are called like this:

...
(<result-of-f>, worldY) = f               worldX
(<result-of-g>, worldZ) = g <result-of-f> worldY
...

So we could define an operator ~~~ to bind them:

(~~~) :: (IO b) -> (b -> IO c) -> IO c

(~~~) ::      (RealWorld -> (b,   RealWorld))
      ->                    (b -> RealWorld -> (c, RealWorld))
      ->      (RealWorld                    -> (c, RealWorld))
(f ~~~ g) worldX = let (resF, worldY) = f worldX
                   in g resF worldY

then we could simply write

printFile = getLine ~~~ getContents ~~~ putStrLn

without touching the real world.

"Impurification"

Now suppose we want to make the file content uppercase as well. Uppercasing is a pure function

upperCase :: String -> String

But to make it into the real world, it has to return an IO String. It is easy to lift such a function:

impureUpperCase :: String -> RealWorld -> (String, RealWorld)
impureUpperCase str world = (upperCase str, world)

This can be generalized:

impurify :: a -> IO a

impurify :: a -> RealWorld -> (a, RealWorld)
impurify a world = (a, world)

so that impureUpperCase = impurify . upperCase, and we can write

printUpperCaseFile = 
    getLine ~~~ getContents ~~~ (impurify . upperCase) ~~~ putStrLn

_{(Note: Normally we write getLine ~~~ getContents ~~~ (putStrLn . upperCase))}

We were working with monads all along

Now let's see what we've done:

1. We defined an operator (~~~) :: IO b -> (b -> IO c) -> IO c which chains two impure functions together
2. We defined a function impurify :: a -> IO a which converts a pure value to impure.

Now we make the identification (>>=) = (~~~) and return = impurify, and see? We've got a monad.

---

Technical note

To ensure it's really a monad, there's still a few axioms which need to be checked too:

1. return a >>= f = f a

    impurify a                =  (\world -> (a, world))
   (impurify a ~~~ f) worldX  =  let (resF, worldY) = (\world -> (a, world )) worldX 
                                 in f resF worldY
                              =  let (resF, worldY) =            (a, worldX)       
                                 in f resF worldY
                              =  f a worldX
   

2. f >>= return = f

   (f ~~~ impurify) worldX  =  let (resF, worldY) = f worldX 
                               in impurify resF worldY
                            =  let (resF, worldY) = f worldX      
                               in (resF, worldY)
                            =  f worldX
   

3. f >>= (\x -> g x >>= h) = (f >>= g) >>= h

   Left as exercise.