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

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

Add a layer.

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

default wrap :: (m ~ t n, MonadTrans t, MonadFree f n, Functor f) => f (m a) -> m a Source #

Instances

Instances details

Monad m => MonadFree Identity (IterT m) Source #
Instance details Defined in Control.Monad.Trans.Iter Methods wrap :: Identity (IterT m a) -> IterT m a Source #
Functor f => MonadFree f (Free f) Source #
Instance details Defined in Control.Monad.Free Methods wrap :: f (Free f a) -> Free f a Source #
Applicative f => MonadFree f (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods wrap :: f (Free f a) -> Free f a Source #
Functor f => MonadFree f (F f) Source #
Instance details Defined in Control.Monad.Free.Church Methods wrap :: f (F f a) -> F f a Source #
(Functor f, MonadFree f m) => MonadFree f (MaybeT m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (MaybeT m a) -> MaybeT m a Source #
(Functor f, Monad m) => MonadFree f (FreeT f m) Source #
Instance details Defined in Control.Monad.Trans.Free Methods wrap :: f (FreeT f m a) -> FreeT f m a Source #
(Applicative f, Monad m) => MonadFree f (FreeT f m) Source #
Instance details Defined in Control.Monad.Trans.Free.Ap Methods wrap :: f (FreeT f m a) -> FreeT f m a Source #
MonadFree f (FT f m) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods wrap :: f (FT f m a) -> FT f m a Source #
(Functor f, MonadFree f m) => MonadFree f (ExceptT e m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ExceptT e m a) -> ExceptT e m a Source #
(Functor f, MonadFree f m) => MonadFree f (IdentityT m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (IdentityT m a) -> IdentityT m a Source #
(Functor f, MonadFree f m) => MonadFree f (ReaderT e m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ReaderT e m a) -> ReaderT e m a Source #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (StateT s m a) -> StateT s m a Source #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (StateT s m a) -> StateT s m a Source #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (WriterT w m a) -> WriterT w m a Source #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (WriterT w m a) -> WriterT w m a Source #
(Functor f, MonadFree f m) => MonadFree f (ContT r m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ContT r m a) -> ContT r m a Source #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (RWST r w s m a) -> RWST r w s m a Source #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) Source #
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (RWST r w s m a) -> RWST r w s m a Source #

data Free f a Source #

A free monad given an applicative

Constructors

Pure a
Free (f (Free f a))

Instances

Instances details

MonadTrans Free Source #	This is not a true monad transformer. It is only a monad transformer "up to `retract`".
Instance details Defined in Control.Monad.Free.Ap Methods lift :: Monad m => m a -> Free m a #
Functor f => Generic1 (Free f :: Type -> Type) Source #
Instance details Defined in Control.Monad.Free.Ap Associated Types type Rep1 (Free f) :: k -> Type # Methods from1 :: forall (a :: k). Free f a -> Rep1 (Free f) a # to1 :: forall (a :: k). Rep1 (Free f) a -> Free f a #
Applicative f => MonadFree f (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods wrap :: f (Free f a) -> Free f a Source #
MonadError e m => MonadError e (Free m) Source #
Instance details Defined in Control.Monad.Free.Ap Methods throwError :: e -> Free m a # catchError :: Free m a -> (e -> Free m a) -> Free m a #
MonadReader e m => MonadReader e (Free m) Source #
Instance details Defined in Control.Monad.Free.Ap Methods ask :: Free m e # local :: (e -> e) -> Free m a -> Free m a # reader :: (e -> a) -> Free m a #
MonadState s m => MonadState s (Free m) Source #
Instance details Defined in Control.Monad.Free.Ap Methods get :: Free m s # put :: s -> Free m () # state :: (s -> (a, s)) -> Free m a #
MonadWriter e m => MonadWriter e (Free m) Source #
Instance details Defined in Control.Monad.Free.Ap Methods writer :: (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 f => MonadFix (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods mfix :: (a -> Free f a) -> Free f a #
Foldable f => Foldable (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods fold :: Monoid m => Free f m -> m # foldMap :: Monoid m => (a -> m) -> Free f a -> 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 #
Foldable1 f => Foldable1 (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods fold1 :: Semigroup m => Free f m -> m # foldMap1 :: Semigroup m => (a -> m) -> Free f a -> m # foldMap1' :: Semigroup m => (a -> m) -> Free f a -> m # toNonEmpty :: Free f a -> NonEmpty a # maximum :: Ord a => Free f a -> a # minimum :: Ord a => Free f a -> a # head :: Free f a -> a # last :: Free f a -> a # foldrMap1 :: (a -> b) -> (a -> b -> b) -> Free f a -> b # foldlMap1' :: (a -> b) -> (b -> a -> b) -> Free f a -> b # foldlMap1 :: (a -> b) -> (b -> a -> b) -> Free f a -> b # foldrMap1' :: (a -> b) -> (a -> b -> b) -> Free f a -> b #
Eq1 f => Eq1 (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods liftEq :: (a -> b -> Bool) -> Free f a -> Free f b -> Bool #
Ord1 f => Ord1 (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods liftCompare :: (a -> b -> Ordering) -> Free f a -> Free f b -> Ordering #
Read1 f => Read1 (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods liftReadsPrec :: (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) Source #
Instance details Defined in Control.Monad.Free.Ap Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Free f a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Free f a] -> ShowS #
Traversable f => Traversable (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods traverse :: 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) #
Alternative v => Alternative (Free v) Source #	This violates the Alternative laws, handle with care.
Instance details Defined in Control.Monad.Free.Ap Methods empty :: 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 f => Applicative (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods pure :: 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 #
Functor f => Functor (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods fmap :: (a -> b) -> Free f a -> Free f b # (<$) :: a -> Free f b -> Free f a #
Applicative f => Monad (Free f) Source #
Instance details Defined 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 #
MonadPlus v => MonadPlus (Free v) Source #	This violates the MonadPlus laws, handle with care.
Instance details Defined in Control.Monad.Free.Ap Methods mzero :: Free v a # mplus :: Free v a -> Free v a -> Free v a #
MonadCont m => MonadCont (Free m) Source #
Instance details Defined in Control.Monad.Free.Ap Methods callCC :: ((a -> Free m b) -> Free m a) -> Free m a #
Apply f => Apply (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods (<.>) :: Free f (a -> b) -> Free f a -> Free f b Source # (.>) :: Free f a -> Free f b -> Free f b Source # (<.) :: Free f a -> Free f b -> Free f a Source # liftF2 :: (a -> b -> c) -> Free f a -> Free f b -> Free f c Source #
Apply f => Bind (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods (>>-) :: Free f a -> (a -> Free f b) -> Free f b Source # join :: Free f (Free f a) -> Free f a Source #
Traversable1 f => Traversable1 (Free f) Source #
Instance details Defined in Control.Monad.Free.Ap Methods traverse1 :: Apply f0 => (a -> f0 b) -> Free f a -> f0 (Free f b) Source # sequence1 :: Apply f0 => Free f (f0 b) -> f0 (Free f b) Source #
(Typeable f, Data a, Data (f (Free f a))) => Data (Free f a) Source #
Instance details Defined in Control.Monad.Free.Ap Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Free f a -> c (Free f a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Free f a) # toConstr :: Free f a -> Constr # dataTypeOf :: Free f a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Free f a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Free f a)) # gmapT :: (forall b. Data b => b -> b) -> Free f a -> Free f a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Free f a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Free f a -> r # gmapQ :: (forall d. Data d => d -> u) -> Free f a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Free f a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Free f a -> m (Free f a) #
Generic (Free f a) Source #
Instance details Defined in Control.Monad.Free.Ap Associated Types type Rep (Free f a) :: Type -> Type # Methods from :: Free f a -> Rep (Free f a) x # to :: Rep (Free f a) x -> Free f a #
(Read1 f, Read a) => Read (Free f a) Source #
Instance details Defined in Control.Monad.Free.Ap Methods readsPrec :: 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) Source #
Instance details Defined in Control.Monad.Free.Ap Methods showsPrec :: Int -> Free f a -> ShowS # show :: Free f a -> String # showList :: [Free f a] -> ShowS #
(Eq1 f, Eq a) => Eq (Free f a) Source #
Instance details Defined 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) Source #
Instance details Defined in Control.Monad.Free.Ap Methods compare :: 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 #
type Rep1 (Free f :: Type -> Type) Source #
Instance details Defined in Control.Monad.Free.Ap type Rep1 (Free f :: Type -> Type) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-8qBzyCSUwFNB6VzdtqY2wi" '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) Source #
Instance details Defined in Control.Monad.Free.Ap type Rep (Free f a) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-8qBzyCSUwFNB6VzdtqY2wi" '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 :: Monad f => Free f a -> f a Source #

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 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 #

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 Source #

Like iter for applicative values.

iterM :: (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 #

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

foldFree :: (Applicative f, 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, 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, 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 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)) 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))