Control.Monad.Trans.Free.Church

Church-encoded free monad transformer.

The free monad transformer

newtype FT f m a

The "free monad transformer" for a functor f

Constructors

FT

Fields

runFT :: forall r. (a -> m r) -> (f (m r) -> m r) -> m r

Instances

(Functor f, MonadError e m) => MonadError e (FT f m)

(Functor f, MonadReader r m) => MonadReader r (FT f m)

(Functor f, MonadState s m) => MonadState s (FT f m)

(Functor f, MonadWriter w m) => MonadWriter w (FT f m)

Functor f => MonadFree f (FT f m)

MonadTrans (FT f)

Monad (FT f m)

Functor (FT f m)

MonadPlus m => MonadPlus (FT f m)

Applicative (FT f m)

(Foldable f, Foldable m, Monad m) => Foldable (FT f m)

(Monad m, Traversable m, Traversable f) => Traversable (FT f m)

Alternative m => Alternative (FT f m)

MonadIO m => MonadIO (FT f m)

MonadCont m => MonadCont (FT f m)

Apply (FT f m)

Bind (FT f m)

(Functor f, Monad m, Eq (FreeT f m a)) => Eq (FT f m a)

(Functor f, Monad m, Ord (FreeT f m a)) => Ord (FT f m a)

The free monad

type F f = FT f Identity

The "free monad" for a functor f.

free :: Functor f => (forall r. (a -> r) -> (f r -> r) -> r) -> F f a

Wrap a Church-encoding of a "free monad" as the free monad for a functor.

runF :: Functor f => F f a -> forall r. (a -> r) -> (f r -> r) -> r

Unwrap the Free monad to obtain it's Church-encoded representation.

Operations

toFT :: (Monad m, Functor f) => FreeT f m a -> FT f m a

Generate a Church-encoded free monad transformer from a FreeT monad transformer.

fromFT :: (Monad m, Functor f) => FT f m a -> FreeT f m a

Convert to a FreeT free monad representation.

iterT :: (Functor f, Monad m) => (f (m a) -> m a) -> FT f m a -> m a

Tear down a free monad transformer using iteration.

iterTM :: (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FT f m a -> t m a

Tear down a free monad transformer using iteration over a transformer.

hoistFT :: (Monad m, Monad n, Functor f) => (forall a. m a -> n a) -> FT f m b -> FT f n b

Lift a monad homomorphism from m to n into a monad homomorphism from FT f m to FT f n

hoistFT :: (Monad m, Monad n, Functor f) => (m ~> n) -> FT f m ~> FT f n

transFT :: (Monad m, Functor g) => (forall a. f a -> g a) -> FT f m b -> FT g m b

Lift a natural transformation from f to g into a monad homomorphism from FT f m to FT g n

Operations of free monad

improve :: Functor f => (forall m. MonadFree f m => m a) -> Free f a

Improve the asymptotic performance of code that builds a free monad with only binds and returns by using F behind the scenes.

This is based on the "Free Monads for Less" series of articles by Edward Kmett:

http://comonad.com/reader/2011/free-monads-for-less/ http://comonad.com/reader/2011/free-monads-for-less-2/

and "Asymptotic Improvement of Computations over Free Monads" by Janis Voightländer:

http://www.iai.uni-bonn.de/~jv/mpc08.pdf

fromF :: (Functor f, MonadFree f m) => F f a -> m a

Convert to another free monad representation.

toF :: Functor f => Free f a -> F f a

Generate a Church-encoded free monad from a Free monad.

retract :: (Functor f, Monad f) => F f a -> f a

retract is the left inverse of liftF

 retract . liftF = id

iter :: Functor f => (f a -> a) -> F f a -> a

Tear down an F Monad using iteration.

iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> F f a -> m a

Like iter for monadic values.

Free Monads With Class

class Monad m => MonadFree f m | m -> f where

Monads provide substitution (fmap) and renormalization (join):

m >>= f = join (fmap f m)

A free Monad is one that does no work during the normalization step beyond simply grafting the two monadic values together.

[] is not a free Monad (in this sense) because join [[a]] smashes the lists flat.

On the other hand, consider:

 data Tree a = Bin (Tree a) (Tree a) | Tip a

 instance Monad Tree where
   return = Tip
   Tip a >>= f = f a
   Bin l r >>= f = Bin (l >>= f) (r >>= f)

This Monad is the free Monad of Pair:

 data Pair a = Pair a a

And we could make an instance of MonadFree for it directly:

 instance MonadFree Pair Tree where
    wrap (Pair l r) = Bin l r

Or we could choose to program with Free Pair instead of Tree and thereby avoid having to define our own Monad instance.

Moreover, Control.Monad.Free.Church provides a MonadFree instance that can improve the asymptotic complexity of code that constructs free monads by effectively reassociating the use of (>>=). You may also want to take a look at the kan-extensions package (http://hackage.haskell.org/package/kan-extensions).

See Free for a more formal definition of the free Monad for a Functor.

Methods

wrap :: f (m a) -> m a

Add a layer.

 wrap (fmap f x) ≡ wrap (fmap return x) >>= f

Instances

(Functor f, MonadFree f m) => MonadFree f (ListT m)

(Functor f, MonadFree f m) => MonadFree f (IdentityT m)

(Functor f, MonadFree f m) => MonadFree f (MaybeT m)

Functor f => MonadFree f (Free f)

Functor f => MonadFree f (F f)

Monad m => MonadFree Identity (IterT m)

(Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m)

(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m)

(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m)

(Functor f, MonadFree f m) => MonadFree f (ContT r m)

(Functor f, MonadFree f m) => MonadFree f (StateT s m)

(Functor f, MonadFree f m) => MonadFree f (StateT s m)

(Functor f, MonadFree f m) => MonadFree f (ReaderT e m)

(Functor f, Monad m) => MonadFree f (FreeT f m)

Functor f => MonadFree f (FT f m)

(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m)

(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m)