# Monad Transformers

# A monadic counter

An example on how to compose the reader, writer, and state monad using monad transformers. The source code can be found in this repository

We want to implement a counter, that increments its value by a given constant.

We start by defining some types, and functions:

newtype Counter = MkCounter {cValue :: Int}
  deriving (Show)

-- | 'inc c n' increments the counter by 'n' units.
inc :: Counter -> Int -> Counter
inc (MkCounter c) n = MkCounter (c + n)

Assume we want to carry out the following computation using the counter:

  • set the counter to 0
  • set the increment constant to 3
  • increment the counter 3 times
  • set the increment constant to 5
  • increment the counter 2 times

The state monad provides abstractions for passing state around. We can make use of the state monad, and define our increment function as a state transformer.

-- | CounterS is a monad.
type CounterS = State Counter

-- | Increment the counter by 'n' units.
incS :: Int-> CounterS ()
incS n = modify (\c -> inc c n)

This already enables us to express a computation in a more clear and succinct way:

-- | The computation we want to run, with the state monad.
mComputationS :: CounterS ()
mComputationS = do
  incS 3
  incS 3
  incS 3
  incS 5
  incS 5

But we still have to pass the increment constant at each invocation. We would like to avoid this.

# Adding an environment

The reader monad provides a convenient way to pass an environment around. This monad is used in functional programming to perform what in the OO world is known as dependency injection.

In its simplest version, the reader monad requires two types:

  • the type of the value being read (i.e. our environment, `r` below),
  • the value returned by the reader monad (`a` below). Reader r a
  • However, we need to make use of the state monad as well. Thus, we need to use the ReaderT transformer:

    newtype ReaderT r m a :: * -> (* -> *) -> * -> *

    Using ReaderT, we can define our counter with environment and state as follows:

    type CounterRS = ReaderT Int CounterS

    We define an incR function that takes the increment constant from the environment (using ask), and to define our increment function in terms of our CounterS monad we make use of the lift function (which belongs to the monad transformer class).

    -- | Increment the counter by the amount of units specified by the environment.
    incR :: CounterRS ()
    incR = ask >>= lift . incS

    Using the reader monad we can define our computation as follows:

    -- | The computation we want to run, using reader and state monads.
    mComputationRS :: CounterRS ()
    mComputationRS = do
      local (const 3) $ do
        local (const 5) $ do

    # The requirements changed: we need logging!

    Now assume that we want to add logging to our computation, so that we can see the evolution of our counter in time.

    We also have a monad to perform this task, the writer monad. As with the reader monad, since we are composing them, we need to make use of the reader monad transformer:

    newtype WriterT w m a :: * -> (* -> *) -> * -> *

    Here w represents the type of the output to accumulate (which has to be a monoid, which allow us to accumulate this value), m is the inner monad, and a the type of the computation.

    We can then define our counter with logging, environment, and state as follows:

    type CounterWRS = WriterT [Int] CounterRS

    And making use of lift we can define the version of the increment function which logs the value of the counter after each increment:

    incW :: CounterWRS ()
    incW = lift incR >> get >>= tell . (:[]) . cValue

    Now the computation that contains logging can be written as follows:

    mComputationWRS :: CounterWRS ()
    mComputationWRS = do
      local (const 3) $ do
        local (const 5) $ do

    # Doing everything in one go

    This example intended to show monad transformers at work. However, we can achieve the same effect by composing all the aspects (environment, state, and logging) in a single increment operation.

    To do this we make use of type-constraints:

    inc' :: (MonadReader Int m, MonadState Counter m, MonadWriter [Int] m) => m ()
    inc' = ask >>= modify . (flip inc) >> get >>= tell . (:[]) . cValue

    Here we arrive at a solution that will work for any monad that satisfies the constraints above. The computation function is defined thus with type:

    mComputation' :: (MonadReader Int m, MonadState Counter m, MonadWriter [Int] m) => m ()

    since in its body we make use of inc'.

    We could run this computation, in the ghci REPL for instance, as follows:

    runState ( runReaderT ( runWriterT mComputation' ) 15 )  (MkCounter 0)