Classes for monad transformers.

A monad transformer makes new monad out of an existing monad, such that computations of the old monad may be embedded in the new one. To construct a monad with a desired set of features, one typically starts with a base monad, such as Identity, [] or IO, and applies a sequence of monad transformers.

Most monad transformer modules include the special case of applying the transformer to Identity. For example, State s is an abbreviation for StateT s Identity.

Each monad transformer also comes with an operation runXXX to unwrap the transformer, exposing a computation of the inner monad.

One might define a parsing monad by adding a state (the String remaining to be parsed) to the [] monad, which provides non-determinism:

 import Control.Monad.Trans.State

 type Parser = StateT String []

Then Parser is an instance of MonadPlus: monadic sequencing implements concatenation of parsers, while mplus provides choice. To use parsers, we need a primitive to run a constructed parser on an input string:

 runParser :: Parser a -> String -> [a]
 runParser p s = [x | (x, "") <- runStateT p s]

Finally, we need a primitive parser that matches a single character, from which arbitrarily complex parsers may be constructed:

 item :: Parser Char
 item = do
     c:cs <- get
     put cs
     return c

In this example we use the operations get and put from Control.Monad.Trans.State, which are defined only for monads that are applications of StateT. Alternatively one could use monad classes from other packages, which contain methods get and put with types generalized over all suitable monads.

We can define a parser that also counts by adding a WriterT transformer:

 import Control.Monad.Trans.Class
 import Control.Monad.Trans.State
 import Control.Monad.Trans.Writer
 import Data.Monoid

 type Parser = WriterT (Sum Int) (StateT String [])

The function that applies a parser must now unwrap each of the monad transformers in turn:

 runParser :: Parser a -> String -> [(a, Int)]
 runParser p s = [(x, n) | ((x, Sum n), "") <- runStateT (runWriterT p) s]

To define item parser, we need to lift the StateT operations through the WriterT transformers.

 item :: Parser Char
 item = do
     c:cs <- lift get
     lift (put cs)
     return c

In this case, we were able to do this with lift, but operations with more complex types require special lifting functions, which are provided by monad transformers for which they can be implemented. If you use one of packages of monad classes, this lifting is handled automatically by the instances of the classes, and you need only use the generalized methods get and put.

We can also define a primitive using the Writer:

 tick :: Parser ()
 tick = tell (Sum 1)

Then the parser will keep track of how many ticks it executes.