| Safe Haskell | Safe |
|---|---|
| Language | Haskell2010 |
Control.Monad.Free.Ap
Description
"Applicative Effects in Free Monads"
Often times, the '(<*>)' operator can be more efficient than ap.
Conventional free monads don't provide any means of modeling this.
The free monad can be modified to make use of an underlying applicative.
But it does require some laws, or else the '(<*>)' = ap law is broken.
When interpreting this free monad with foldFree,
the natural transformation must be an applicative homomorphism.
An applicative homomorphism hm :: (Applicative f, Applicative g) => f x -> g x
will satisfy these laws.
hm (pure a) = pure a
hm (f <*> a) = hm f <*> hm a
This is based on the "Applicative Effects in Free Monads" series of articles by Will Fancher
- class Monad m => MonadFree f m | m -> f where
- data Free f a
- retract :: (Applicative f, Monad f) => Free f a -> f a
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iter :: Applicative f => (f a -> a) -> Free f a -> a
- iterA :: (Applicative p, Applicative f) => (f (p a) -> p a) -> Free f a -> p a
- iterM :: (Applicative m, Monad m, Applicative f) => (f (m a) -> m a) -> Free f a -> m a
- hoistFree :: (Applicative f, Applicative g) => (forall a. f a -> g a) -> Free f b -> Free g b
- foldFree :: (Applicative f, Applicative m, Monad m) => (forall x. f x -> m x) -> Free f a -> m a
- toFreeT :: (Applicative f, Applicative m, Monad m) => Free f a -> FreeT f m a
- cutoff :: Applicative f => Integer -> Free f a -> Free f (Maybe a)
- unfold :: Applicative f => (b -> Either a (f b)) -> b -> Free f a
- unfoldM :: (Applicative f, Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a)
- _Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a))
- _Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a))
Documentation
class Monad m => MonadFree f m | m -> f where Source #
Monads provide substitution (fmap) and renormalization (join):
m>>=f =join(fmapf 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]]
On the other hand, consider:
data Tree a = Bin (Tree a) (Tree a) | Tip a
instanceMonadTree wherereturn= 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:
instanceMonadFreePair Tree wherewrap(Pair l r) = Bin l r
Or we could choose to program with Free PairTree
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 Source #
Add a layer.
wrap (fmap f x) ≡ wrap (fmap return x) >>= f
wrap :: (m ~ t n, MonadTrans t, MonadFree f n, Functor f) => f (m a) -> m a Source #
Add a layer.
wrap (fmap f x) ≡ wrap (fmap return x) >>= f
Instances
A free monad given an applicative
Instances
liftF :: (Functor f, MonadFree f m) => f a -> m a Source #
A version of lift that can be used with just a Functor for f.
iter :: Applicative f => (f a -> a) -> Free f a -> a Source #
iterA :: (Applicative p, Applicative f) => (f (p a) -> p a) -> Free f a -> p a Source #
Like iter for applicative values.
iterM :: (Applicative m, Monad m, Applicative f) => (f (m a) -> m a) -> Free f a -> m a Source #
Like iter for monadic values.
hoistFree :: (Applicative f, Applicative g) => (forall a. f a -> g a) -> Free f b -> Free g b Source #
foldFree :: (Applicative f, Applicative m, Monad m) => (forall x. f x -> m x) -> Free f a -> m a Source #
Given an applicative homomorphism, you get a monad homomorphism.
toFreeT :: (Applicative f, Applicative m, Monad m) => Free f a -> FreeT f m a Source #
Convert a Free monad from Control.Monad.Free.Ap to a FreeT monad
from Control.Monad.Trans.Free.Ap.
WARNING: This assumes that liftF is an applicative homomorphism.
cutoff :: Applicative f => Integer -> Free f a -> Free f (Maybe a) Source #
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.
unfold :: Applicative f => (b -> Either a (f b)) -> b -> Free f a Source #
Unfold a free monad from a seed.
unfoldM :: (Applicative f, Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a) Source #
Unfold a free monad from a seed, monadically.
_Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a)) Source #
This is Prism' (Free f a) a in disguise
>>>preview _Pure (Pure 3)Just 3
>>>review _Pure 3 :: Free Maybe IntPure 3
_Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a)) Source #
This is Prism' (Free f a) (f (Free f a)) in disguise
>>>preview _Free (review _Free (Just (Pure 3)))Just (Just (Pure 3))
>>>review _Free (Just (Pure 3))Free (Just (Pure 3))