Control.Monad.Free.Church

"Free Monads for Less"

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

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

Documentation

newtype F f a

The Church-encoded free monad for a functor f.

It is asymptotically more efficient to use (>>=) for F than it is to (>>=) with Free.

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

Constructors

F

Fields

runF :: forall r. (a -> r) -> (f r -> r) -> r

Instances

MonadTrans F

MonadReader e m => MonadReader e (F m)

MonadState s m => MonadState s (F m)

MonadWriter w m => MonadWriter w (F m)

Functor f => MonadFree f (F f)

Monad (F f)

Functor (F f)

MonadPlus f => MonadPlus (F f)

Applicative (F f)

Alternative f => Alternative (F f)

MonadCont m => MonadCont (F m)

Apply (F f)

Bind (F f)

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 :: MonadFree f m => F f a -> m a

Convert to another free monad representation.

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

Like iter for monadic values.

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

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

retract :: Monad m => F m a -> m a

retract is the left inverse of lift and liftF

 retract . lift = id
 retract . liftF = id

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

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

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

A version of lift that can be used with just a Functor for f.