Control.Monad.Trans.Free

The free monad transformer

The base functor

data FreeF f a b

The base functor for a free monad.

Constructors

Pure a

Free (f b)

Instances

Traversable f => Bitraversable (FreeF f)

Functor f => Bifunctor (FreeF f)

Foldable f => Bifoldable (FreeF f)

Eq1 f => Eq2 (FreeF f)

Ord1 f => Ord2 (FreeF f)

Show1 f => Show2 (FreeF f)

Read1 f => Read2 (FreeF f)

Functor f => Functor (FreeF f a)

Foldable f => Foldable (FreeF f a)

Traversable f => Traversable (FreeF f a)

(Eq1 f, Eq a) => Eq1 (FreeF f a)

(Ord1 f, Ord a) => Ord1 (FreeF f a)

(Show1 f, Show a) => Show1 (FreeF f a)

(Read1 f, Read a) => Read1 (FreeF f a)

Typeable ((* -> *) -> * -> * -> *) FreeF

(Eq a, Eq (f b)) => Eq (FreeF f a b)

(Ord a, Ord (f b)) => Ord (FreeF f a b)

(Read a, Read (f b)) => Read (FreeF f a b)

(Show a, Show (f b)) => Show (FreeF f a b)

The free monad transformer

newtype FreeT f m a

The "free monad transformer" for a functor f

Constructors

FreeT

Fields

runFreeT :: m (FreeF f a (FreeT f m a))

Instances

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

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

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

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

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

MonadTrans (FreeT f)

(Functor f, MonadPlus m) => Alternative (FreeT f m)

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

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

(Functor f, MonadPlus m) => MonadPlus (FreeT f m)

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

(Foldable m, Foldable f) => Foldable (FreeT f m)

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

(Functor f, MonadIO m) => MonadIO (FreeT f m)

(Functor f, MonadCont m) => MonadCont (FreeT f m)

(Functor f, Eq1 f, Functor m, Eq1 m) => Eq1 (FreeT f m)

(Functor f, Ord1 f, Functor m, Ord1 m) => Ord1 (FreeT f m)

(Functor f, Show1 f, Functor m, Show1 m) => Show1 (FreeT f m)

(Functor f, Read1 f, Functor m, Read1 m) => Read1 (FreeT f m)

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

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

Eq (m (FreeF f a (FreeT f m a))) => Eq (FreeT f m a)

Ord (m (FreeF f a (FreeT f m a))) => Ord (FreeT f m a)

Read (m (FreeF f a (FreeT f m a))) => Read (FreeT f m a)

Show (m (FreeF f a (FreeT f m a))) => Show (FreeT f m a)

The free monad

type Free f = FreeT f Identity

The "free monad" for a functor f.

free :: FreeF f a (Free f a) -> Free f a

Pushes a layer into a free monad value.

runFree :: Free f a -> FreeF f a (Free f a)

Evaluates the first layer out of a free monad value.

Operations

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

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

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

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

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

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

hoistFreeT :: (Monad m, Functor f) => (m ~> n) -> FreeT f m ~> FreeT f n

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

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

cutoff :: (Functor f, Monad m) => Integer -> FreeT f m a -> FreeT f m (Maybe a)

Cuts off a tree of computations at a given depth. If the depth is 0 or less, no computation nor monadic effects will take place.

Some examples (n ≥ 0):

cutoff 0     _        ≡ return Nothing
cutoff (n+1) . return ≡ return . Just
cutoff (n+1) . lift   ≡ lift . liftM Just
cutoff (n+1) . wrap   ≡ wrap . fmap (cutoff n)

Calling retract . cutoff n is always terminating, provided each of the steps in the iteration is terminating.

partialIterT :: Monad m => Integer -> (forall a. f a -> m a) -> FreeT f m b -> FreeT f m b

partialIterT n phi m interprets first n layers of m using phi. This is sort of the opposite for cutoff.

Some examples (n ≥ 0):

partialIterT 0 _ m              ≡ m
partialIterT (n+1) phi . return ≡ return
partialIterT (n+1) phi . lift   ≡ lift
partialIterT (n+1) phi . wrap   ≡ join . lift . phi

intersperseT :: (Monad m, Functor f) => f a -> FreeT f m b -> FreeT f m b

intersperseT f m inserts a layer f between every two layers in m.

intersperseT f . return ≡ return
intersperseT f . lift   ≡ lift
intersperseT f . wrap   ≡ wrap . fmap (iterTM (wrap . (<$ f) . wrap))

intercalateT :: (Monad m, MonadTrans t, Monad (t m), Functor (t m)) => t m a -> FreeT (t m) m b -> t m b

intercalateT f m inserts a layer f between every two layers in m and then retracts the result.

intercalateT f ≡ retractT . intersperseT f

retractT :: (MonadTrans t, Monad (t m), Monad m) => FreeT (t m) m a -> t m a

Tear down a free monad transformer using Monad instance for t m.

Operations of free monad

retract :: Monad f => Free f a -> f a

retract is the left inverse of liftF

retract . liftF = id

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

Tear down a Free Monad using iteration.

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

Minimal complete definition

Nothing

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)