I am a newbie in the Haskell land. I was lost but found some good maps and discovered there is a tradition in Haskell land : writing a monad tutorial.

There are so many monad tutorials that writing a new one is getting difficult. And writing a good one if even more difficult. So, I am just going to explain my own understanding.

The first thing to note is that monads are EASY !!

What's difficult is trying to understand what they have in common because they can look so different. I have identified three kinds of monads (not exclusive - a monad can belong to more than one kind):

• Monad as control of the sequencing ;
• Monad as control of side effects ;

# 1. Monad as control of the sequencing

In a lazy functional programming language like Haskell, the order of evaluation does not matter. It does not mean you cannot control the order of evaluation. It means you can abstract it and build your own sequencing, your own control.

In imperative languages (like C), you need to extend the language to support new control statements.

In less elegant functional languages (like LISP) you need to have special forms which do not follow the normal rules for evaluation.

In Haskell, you "just" build your own control operators. Let's see some examples:

## 1.1. Control in IO monad

repeatN 0 a = return ()
repeatN n a = a >> repeatN (n-1) a

test = repeatN 3 \$ do
putStrLn "TEST"

And, if you want to pass the loop index to the loop body, you may write:

repeatN 0 a = return ()
repeatN n a = (a n) >> repeatN (n-1) a

test = repeatN 3 \$ \i -> do
putStrLn \$ "TEST : " ++ (show i)

Another example of control of the sequencing is the indeterminism monad:

-- f is a function returning several possible results
f :: Int -> [Int]
f x = [1+x,2*x]

test :: IO ()
test = putStrLn . show \$ do
a <- return 5
b <- f a
return b

Here we apply a function f to the value 5. The function f is returning several possible results.

It is possible to chain indeterminate functions like f:

test2 :: IO ()
test2 = putStrLn . show \$ do
a <- return 5
b <- f a
c <- f b
return c

but we do not need to give a name to the intermediate results, so let's write it like:

test2 :: IO ()
test2 = putStrLn . show \$ return 5 >>= f >>= f

The Maybe and Either monads are special cases

# 2. Monad as control of side effects

It is the standard example so I won't write about it

-- The data type for my environment
data MyState = MyState { vara :: Int
, varb :: Int
}

-- The initial environment
initState = MyState { vara = 10
, varb = 20
}

-- Computation in the initial environment
test = do theVarA <- asks vara
lift . putStrLn \$ show theVarA

This state is available whenever we need it in the monad and we do not need to pass it as argument.

runReaderT and lift are explained later. They are not important to understand this example. You just have to know that the line with lift is used to display a value and the runReaderT is used to initialize the environment.

Now, we can temporarily change the value of one variable and work in this modified environment.

-- Increment vara from the environment
incrementVarA ::  Int -> MyState -> MyState
incrementVarA x p = p {vara = (vara p) + x}

test = do theVarA <- asks vara
lift . putStrLn \$ show theVarA
-- computation in the new modified environment
local (incrementVarA 5) \$ do
lift . putStrLn \$ show theVarA
lift . putStrLn \$ show theVarA

We have a side effect since the environment is modified and this change is visible in a non local way. But this change is nevertheless restricted by the local function.

You may use runReader. With runReader the type of test is no more IO () but Int:

test = do theVarA <- asks vara
return theVarA

So, an equivalent code (with IO) is:

test = putStrLn . show \$ do theVarA <- asks vara
return theVarA

runReaderT is just a bit more complex. It has type: ReaderT r m a -> r -> m a

So, when you're working in ReaderT r IO a, you need to specify if you are working with values of type ReaderT r IO a or IO a. The lift function is used for this. Its type is m a -> t m a. So it will transform IO a values to ReaderT r IO a.

A different way to look at this (probably a wrong way) is:

If you have a value v of type a, you use return v to inject it in the ReaderT r IO a monad.

return v would not work if v was of type IO a since you would get a value of type ReaderT r IO (IO a).

So, lift is used to inject the value in the monad.

In each monad, you have the return function which is injecting an element into the monad. So, any monad can be seen as a kind of container. For the List monad it is obvious. Seeing a monad as a container can be very useful.

Assume you want to add an integer to the result of a computation which could return no result. You may have to do something like that

result = Just 20

test = case result of
Just a -> Just (a + 10)
_ -> Nothing

So, you need to extract the value from the container (if there is something to extract), apply your function and package the result in the same container.

Or you can just write:

test = (+10) `fmap` result

fmap is a kind of generalization of map. map is lifting a function a -> b to the container [a] -> [b]

fmap is doing the same for a container m (a monad). So, fmap is transforming the type a -> b to m a -> m b

# 4. Deriving monad (you have to use -fglasgow-exts)

In a same code you may have to use different Reader monads even if they have the same type since they may be for different uses.

You may create a type synonym :

type MyEnvironment a = Reader Int a -- (here the environment is just an Int)

But it would not prevent from mixing two different Reader monads if they have the same type.

So, you need to create a new type:

newtype MyEnvironment a = MyEnvironment {runMyEnvironment :: Reader Int a}

Then, you want the same behavior. This is just a reader monad (from a behavior point of view) like newtype Meter = Meter Int is just a number (from a behavior point of view).

So, instead of having to write several instance declarations, you just write:

newtype MyEnvironment a = MyEnvironment {runMyEnvironment :: Reader Int a}

Then you create an environment . It is just a Reader monad contained in your new type

r :: MyEnvironment Int
r = do
r <- MyEnvironment \$ ask -- This is packaging the result of ask
-- in MyEnvironment. Hence the work is done
return r                  -- in the MyEnvironment monad and not in a
-- return r is a MyEnvironment Int and not

Then, you extract the reader monad and apply it to the initial state

test = putStrLn . show \$ (runMyEnvironment r) `runReader` 4

# 5. What's common ?

What do the previous monads have in common ? Nothing ! Or not a lot. Indeed, being a monad is a very general concept and focusing on the part they have in common (return, >>=) is not the interesting part nor the difficult one. What is interesting is how different they are : a Reader monad is providing ask and local functions ; an IO monad is providing putStrLn etc...

Each monad has its own personality. Of course, >>= will not be the same in each monad but from a user point of view, it will respect the same monadic laws :

return a >>= k  ==  k a                      -- return is a "neutral element" on left
m >>= return  ==  m                          -- return is a "neutral element" on right
m >>= (\x -> k x >>= h)  ==  (m >>= k) >>= h -- a kind of associativity of >>=