Description

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

Synopsis

# Documentation

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 #

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 #

wrap (fmap f x) ≡ wrap (fmap return x) >>= f

Instances

data Free f a #

A free monad given an applicative

Constructors

 Pure a Free (f (Free f a))
Instances
 # This is not a true monad transformer. It is only a monad transformer "up to retract". Instance detailsDefined in Control.Monad.Free.Ap Methodslift :: Monad m => m a -> Free m a # (Applicative m, MonadWriter e m) => MonadWriter e (Free m) # Instance detailsDefined in Control.Monad.Free.Ap Methodswriter :: (a, e) -> Free m a #tell :: e -> Free m () #listen :: Free m a -> Free m (a, e) #pass :: Free m (a, e -> e) -> Free m a # (Applicative m, MonadState s m) => MonadState s (Free m) # Instance detailsDefined in Control.Monad.Free.Ap Methodsget :: Free m s #put :: s -> Free m () #state :: (s -> (a, s)) -> Free m a # (Applicative m, MonadReader e m) => MonadReader e (Free m) # Instance detailsDefined in Control.Monad.Free.Ap Methodsask :: Free m e #local :: (e -> e) -> Free m a -> Free m a #reader :: (e -> a) -> Free m a # (Applicative m, MonadError e m) => MonadError e (Free m) # Instance detailsDefined in Control.Monad.Free.Ap MethodsthrowError :: e -> Free m a #catchError :: Free m a -> (e -> Free m a) -> Free m a # Applicative f => MonadFree f (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodswrap :: f (Free f a) -> Free f a # Applicative f => Monad (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methods(>>=) :: Free f a -> (a -> Free f b) -> Free f b #(>>) :: Free f a -> Free f b -> Free f b #return :: a -> Free f a #fail :: String -> Free f a # Functor f => Functor (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodsfmap :: (a -> b) -> Free f a -> Free f b #(<\$) :: a -> Free f b -> Free f a # Applicative f => MonadFix (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodsmfix :: (a -> Free f a) -> Free f a # Applicative f => Applicative (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodspure :: a -> Free f a #(<*>) :: Free f (a -> b) -> Free f a -> Free f b #liftA2 :: (a -> b -> c) -> Free f a -> Free f b -> Free f c #(*>) :: Free f a -> Free f b -> Free f b #(<*) :: Free f a -> Free f b -> Free f a # Foldable f => Foldable (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodsfold :: Monoid m => Free f m -> m #foldMap :: Monoid m => (a -> m) -> Free f a -> m #foldr :: (a -> b -> b) -> b -> Free f a -> b #foldr' :: (a -> b -> b) -> b -> Free f a -> b #foldl :: (b -> a -> b) -> b -> Free f a -> b #foldl' :: (b -> a -> b) -> b -> Free f a -> b #foldr1 :: (a -> a -> a) -> Free f a -> a #foldl1 :: (a -> a -> a) -> Free f a -> a #toList :: Free f a -> [a] #null :: Free f a -> Bool #length :: Free f a -> Int #elem :: Eq a => a -> Free f a -> Bool #maximum :: Ord a => Free f a -> a #minimum :: Ord a => Free f a -> a #sum :: Num a => Free f a -> a #product :: Num a => Free f a -> a # Traversable f => Traversable (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodstraverse :: Applicative f0 => (a -> f0 b) -> Free f a -> f0 (Free f b) #sequenceA :: Applicative f0 => Free f (f0 a) -> f0 (Free f a) #mapM :: Monad m => (a -> m b) -> Free f a -> m (Free f b) #sequence :: Monad m => Free f (m a) -> m (Free f a) # Eq1 f => Eq1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap MethodsliftEq :: (a -> b -> Bool) -> Free f a -> Free f b -> Bool # Ord1 f => Ord1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap MethodsliftCompare :: (a -> b -> Ordering) -> Free f a -> Free f b -> Ordering # Read1 f => Read1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap MethodsliftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Free f a) #liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Free f a] #liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Free f a) #liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Free f a] # Show1 f => Show1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap MethodsliftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Free f a -> ShowS #liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Free f a] -> ShowS # Alternative v => Alternative (Free v) # This violates the Alternative laws, handle with care. Instance detailsDefined in Control.Monad.Free.Ap Methodsempty :: Free v a #(<|>) :: Free v a -> Free v a -> Free v a #some :: Free v a -> Free v [a] #many :: Free v a -> Free v [a] # (Applicative v, MonadPlus v) => MonadPlus (Free v) # This violates the MonadPlus laws, handle with care. Instance detailsDefined in Control.Monad.Free.Ap Methodsmzero :: Free v a #mplus :: Free v a -> Free v a -> Free v a # (Applicative m, MonadCont m) => MonadCont (Free m) # Instance detailsDefined in Control.Monad.Free.Ap MethodscallCC :: ((a -> Free m b) -> Free m a) -> Free m a # Traversable1 f => Traversable1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodstraverse1 :: Apply f0 => (a -> f0 b) -> Free f a -> f0 (Free f b) #sequence1 :: Apply f0 => Free f (f0 b) -> f0 (Free f b) # Foldable1 f => Foldable1 (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methodsfold1 :: Semigroup m => Free f m -> m #foldMap1 :: Semigroup m => (a -> m) -> Free f a -> m #toNonEmpty :: Free f a -> NonEmpty a # Apply f => Apply (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methods(<.>) :: Free f (a -> b) -> Free f a -> Free f b #(.>) :: Free f a -> Free f b -> Free f b #(<.) :: Free f a -> Free f b -> Free f a #liftF2 :: (a -> b -> c) -> Free f a -> Free f b -> Free f c # Apply f => Bind (Free f) # Instance detailsDefined in Control.Monad.Free.Ap Methods(>>-) :: Free f a -> (a -> Free f b) -> Free f b #join :: Free f (Free f a) -> Free f a # Functor f => Generic1 (Free f :: Type -> Type) # Instance detailsDefined in Control.Monad.Free.Ap Associated Typestype Rep1 (Free f) :: k -> Type # Methodsfrom1 :: Free f a -> Rep1 (Free f) a #to1 :: Rep1 (Free f) a -> Free f a # (Eq1 f, Eq a) => Eq (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap Methods(==) :: Free f a -> Free f a -> Bool #(/=) :: Free f a -> Free f a -> Bool # (Ord1 f, Ord a) => Ord (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap Methodscompare :: Free f a -> Free f a -> Ordering #(<) :: Free f a -> Free f a -> Bool #(<=) :: Free f a -> Free f a -> Bool #(>) :: Free f a -> Free f a -> Bool #(>=) :: Free f a -> Free f a -> Bool #max :: Free f a -> Free f a -> Free f a #min :: Free f a -> Free f a -> Free f a # (Read1 f, Read a) => Read (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap MethodsreadsPrec :: Int -> ReadS (Free f a) #readList :: ReadS [Free f a] #readPrec :: ReadPrec (Free f a) #readListPrec :: ReadPrec [Free f a] # (Show1 f, Show a) => Show (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap MethodsshowsPrec :: Int -> Free f a -> ShowS #show :: Free f a -> String #showList :: [Free f a] -> ShowS # Generic (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap Associated Typestype Rep (Free f a) :: Type -> Type # Methodsfrom :: Free f a -> Rep (Free f a) x #to :: Rep (Free f a) x -> Free f a # type Rep1 (Free f :: Type -> Type) # Instance detailsDefined in Control.Monad.Free.Ap type Rep1 (Free f :: Type -> Type) = D1 (MetaData "Free" "Control.Monad.Free.Ap" "free-5.1-4UATSEhaXgTAvcVwsU7IL6" False) (C1 (MetaCons "Pure" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) Par1) :+: C1 (MetaCons "Free" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (f :.: Rec1 (Free f)))) type Rep (Free f a) # Instance detailsDefined in Control.Monad.Free.Ap type Rep (Free f a) = D1 (MetaData "Free" "Control.Monad.Free.Ap" "free-5.1-4UATSEhaXgTAvcVwsU7IL6" False) (C1 (MetaCons "Pure" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :+: C1 (MetaCons "Free" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (f (Free f a)))))

retract :: (Applicative f, Monad f) => Free f a -> f a #

retract is the left inverse of lift and liftF

retract . lift = id
retract . liftF = id


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

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

iter :: Applicative f => (f a -> a) -> Free f a -> a #

Given an applicative homomorphism from f to Identity, tear down a Free Monad using iteration.

iterA :: (Applicative p, Applicative f) => (f (p a) -> p a) -> Free f a -> p a #

Like iter for applicative values.

iterM :: (Applicative m, Monad m, Applicative f) => (f (m a) -> m a) -> Free f a -> m a #

Like iter for monadic values.

hoistFree :: (Applicative f, Applicative g) => (forall a. f a -> g a) -> Free f b -> Free g b #

Lift an applicative homomorphism from f to g into a monad homomorphism from Free f to Free g.

foldFree :: (Applicative f, Applicative m, Monad m) => (forall x. f x -> m x) -> Free f a -> m a #

Given an applicative homomorphism, you get a monad homomorphism.

toFreeT :: (Applicative f, Applicative m, Monad m) => Free f a -> FreeT f m a #

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) #

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 #

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) #

_Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a)) #

This is Prism' (Free f a) a in disguise

>>> preview _Pure (Pure 3)
Just 3

>>> review _Pure 3 :: Free Maybe Int
Pure 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)) #

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))