Control.Monad.Trans.Iter

Based on Capretta's Iterative Monad Transformer

Unlike Free, this is a true monad transformer.

The iterative monad transformer

newtype IterT m a

The monad supporting iteration based over a base monad m.

 IterT ~ FreeT Identity

Constructors

IterT

Fields

runIterT :: m (Either a (IterT m a))

Instances

MonadTrans IterT

(Functor m, MonadError e m) => MonadError e (IterT m)

(Functor m, MonadReader e m) => MonadReader e (IterT m)

(Functor m, MonadState s m) => MonadState s (IterT m)

Monad m => MonadFree Identity (IterT m)

Monad m => Monad (IterT m)

Monad m => Functor (IterT m)

Typeable1 m => Typeable1 (IterT m)

MonadFix m => MonadFix (IterT m)

MonadPlus m => MonadPlus (IterT m)

Monad m => Applicative (IterT m)

Foldable m => Foldable (IterT m)

(Monad m, Traversable m) => Traversable (IterT m)

MonadPlus m => Alternative (IterT m)

(Functor m, MonadIO m) => MonadIO (IterT m)

MonadCont m => MonadCont (IterT m)

(Monad m, Traversable1 m) => Traversable1 (IterT m)

Foldable1 m => Foldable1 (IterT m)

Monad m => Apply (IterT m)

Monad m => Bind (IterT m)

Eq (m (Either a (IterT m a))) => Eq (IterT m a)

(Typeable1 m, Typeable a, Data (m (Either a (IterT m a))), Data a) => Data (IterT m a)

Ord (m (Either a (IterT m a))) => Ord (IterT m a)

Read (m (Either a (IterT m a))) => Read (IterT m a)

Show (m (Either a (IterT m a))) => Show (IterT m a)

Capretta's iterative monad

type Iter = IterT Identity

iter :: Either a (Iter a) -> Iter a

runIter :: Iter a -> Either a (Iter a)

Operations

delay :: (Monad f, MonadFree f m) => m a -> m a

hoistIterT :: Monad n => (forall a. m a -> n a) -> IterT m b -> IterT n b

Lift a monad homomorphism from m to n into a Monad homomorphism from IterT m to IterT n.

Consuming iterative monads

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

retract is the left inverse of lift

 retract . lift = id

fold :: Monad m => (m a -> a) -> IterT m a -> a

Tear down a Free Monad using iteration.

foldM :: (Monad m, Monad n) => (m (n a) -> n a) -> IterT m a -> n a

Like fold with monadic result.

IterT ~ FreeT Identity

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 (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)