[SOLVED] Monad Transformer missing parameter (?)

Monad Transformer missing parameter (?)

I've got this type alias:

type Board a = ReaderT String (StateT String IO) a

I know that StateT has the kind * -> (* -> *) -> * -> * so it should get three parameters. But in the above example, StateT only receives String and the IO-Monad. Now I'm wondering where the missing parameter is passed to StateT.

Same goes for IO which should get one parameter but doesn't get any.

Solution

If you look at the definition of ReaderT, you will see that its second parameter is itself applied to its last parameter:

newtype ReaderT r m a = ReaderT { runReaderT :: r -> m a }
                                                     ^^^
                                                  right here

Which means that its second parameter m must have kind m :: * -> *. And this is exactly what you get if you apply only two, not three, parameters to StateT:

StateT :: * -> (* -> *) -> * -> *
StateT x y :: * -> *
(where x :: * and y :: * -> *)

When substituting the definition of ReaderT in your type Board, you'll get:

(1) type Board a = ReaderT String (StateT String IO) a

-- Substituting ReaderT definition (pseudocode)
(2) type Board a = { runReaderT :: String -> StateT String IO a }
                                             ^^^^^^^^^^^^^^^^^^
                                Look, StateT has all its three parameters!