free-4.2: Monads for free

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

Control.MonadPlus.Free

Description

left-distributive MonadPlus for free.

Synopsis

- class Monad m => MonadFree f m | m -> f where
  - wrap :: f (m a) -> m a
- data Free f a
  - = Pure a
  - | Free (f (Free f a))
  - | Plus [Free f a]
- retract :: MonadPlus f => Free f a -> f a
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iter :: Functor f => (f a -> a) -> ([a] -> a) -> Free f a -> a
- iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> ([m a] -> m a) -> Free f a -> m a
- hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g 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 Free f a Source

The `Free` `MonadPlus` for a `Functor` `f`.

Formally

A `MonadPlus` `n` is a free `MonadPlus` for `f` if every monadplus homomorphism from `n` to another MonadPlus `m` is equivalent to a natural transformation from `f` to `m`.

We model this internally as if left-distribution holds.

Constructors

Pure a |
Free (f (Free f a)) |
Plus [Free f a] |

Instances

MonadTrans Free | This is not a true monad transformer. It is only a monad transformer "up to `retract`".
(Functor m, MonadPlus m, MonadError e m) => MonadError e (Free m) |
(Functor m, MonadPlus m, MonadReader e m) => MonadReader e (Free m) |
(Functor m, MonadState s m) => MonadState s (Free m) |
(Functor m, MonadPlus m, MonadWriter e m) => MonadWriter e (Free m) |
Functor f => MonadFree f (Free f) |
Functor f => Monad (Free f) |
Functor f => Functor (Free f) |
Typeable1 f => Typeable1 (Free f) |
Functor f => MonadPlus (Free f) |
Functor f => Applicative (Free f) |
Foldable f => Foldable (Free f) |
Traversable f => Traversable (Free f) |
Functor f => Alternative (Free f) |
(Functor m, MonadPlus m, MonadCont m) => MonadCont (Free m) |
Functor f => Apply (Free f) |
Functor f => Bind (Free f) |
(Eq (f (Free f a)), Eq a) => Eq (Free f a) |
(Typeable1 f, Typeable a, Data a, Data (f (Free f a))) => Data (Free f a) |
(Ord (f (Free f a)), Ord a) => Ord (Free f a) |
(Read (f (Free f a)), Read a) => Read (Free f a) |
(Show (f (Free f a)), Show a) => Show (Free f a) |
Functor f => Monoid (Free f a) |
Functor f => Semigroup (Free f a) |

retract :: MonadPlus f => Free f a -> f aSource

`retract` is the left inverse of `lift` and `liftF`

     retract . lift = id
     retract . liftF = id

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

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

iter :: Functor f => (f a -> a) -> ([a] -> a) -> Free f a -> aSource

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

iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> ([m a] -> m a) -> Free f a -> m aSource

Like iter for monadic values.

hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g bSource

Lift a natural transformation from `f` to `g` into a natural transformation from ``FreeT` f` to ``FreeT` g`.