free-4.2: Monads for free

Portability | MPTCs, fundeps
Stability | provisional
Maintainer | Edward Kmett <ekmett@gmail.com>
Safe Haskell | None

Control.Monad.Trans.Iter

Description

Based on Capretta's Iterative Monad Transformer

Unlike `Free`, this is a true monad transformer.

Synopsis

- class Monad m => MonadFree f m | m -> f where
  - wrap :: f (m a) -> m a
- data IterF a b
  - = Pure a
  - | Iter b
- data IterT m a = IterT {
  - runIterT :: m (IterF a (IterT m a))
  }
- delay :: (Monad f, MonadFree f m) => m a -> m a
- retract :: Monad m => IterT m a -> m a
- iter :: Monad m => (m a -> a) -> IterT m a -> a
- hoistIterT :: Monad n => (forall a. m a -> n a) -> IterT m b -> IterT n b

# Documentation

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

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 aSource

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) |

data IterF a b Source

Constructors

Pure a |
Iter b |

Instances

Typeable2 IterF |
Bitraversable IterF |
Bifunctor IterF |
Bifoldable IterF |
Functor (IterF a) |
Foldable (IterF a) |
Traversable (IterF a) |
(Eq a, Eq b) => Eq (IterF a b) |
(Ord a, Ord b) => Ord (IterF a b) |
(Read a, Read b) => Read (IterF a b) |
(Show a, Show b) => Show (IterF a b) |

data IterT m a Source

The monad supporting iteration based over a base monad `m`.

     IterT ~ FreeT Identity

Constructors

IterT |

Fields

runIterT :: m (IterF a (IterT m a))

Instances

MonadTrans IterT | This is not a true monad transformer. It is only a monad transformer "up to `retract`".
(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) |
(Monad m, Traversable1 m) => Traversable1 (IterT m) |
Foldable1 m => Foldable1 (IterT m) |
Monad m => Apply (IterT m) |
Monad m => Bind (IterT m) |
Eq (m (IterF a (IterT m a))) => Eq (IterT m a) |
(Typeable1 m, Typeable a, Data (m (IterF a (IterT m a))), Data a) => Data (IterT m a) |
Ord (m (IterF a (IterT m a))) => Ord (IterT m a) |
Read (m (IterF a (IterT m a))) => Read (IterT m a) |
Show (m (IterF a (IterT m a))) => Show (IterT m a) |

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

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

`retract` is the left inverse of `lift`

     retract . lift = id

iter :: Monad m => (m a -> a) -> IterT m a -> aSource

Tear down a `Free` `Monad` using iteration.

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

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