Control.Monad.Free.Ap

free-5.2: Monads for free

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

- Applicative Effects in Free Monads
- More on Applicative Effects in Free Monads

Synopsis

- class Monad m => MonadFree (f :: Type -> Type) (m :: Type -> Type) | m -> f where
  - wrap :: f (m a) -> m a
- data Free (f :: Type -> Type) 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 :: forall (f :: Type -> Type) (m :: Type -> Type) a. (Applicative f, Monad m) => Free f a -> FreeT f m a
- cutoff :: forall (f :: Type -> Type) a. 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 :: Type -> Type) 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 :: Type -> Type) (m :: Type -> Type) | 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 #

Add a layer.

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

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

#### Instances

Instances details

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

data Free (f :: Type -> Type) a #

A free monad given an applicative

Constructors

Pure a |
Free (f (Free f a)) |

#### Instances

Instances details

MonadTrans Free # | 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 #
Applicative f => MonadFree f (Free f) # |
Instance details Defined in Control.Monad.Free.Ap Methods wrap :: f (Free f a) -> Free f a #
Functor f => Generic1 (Free f :: Type -> Type) # |
Instance details Defined in Control.Monad.Free.Ap Associated Types
type Rep1 (Free f :: Type -> Type) |
Instance details Defined in Control.Monad.Free.Ap type Rep1 (Free f :: Type -> Type) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-80KVzMloHSZ6giRVM2rsod" '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))))
Methods from1 :: Free f a -> Rep1 (Free f) a # to1 :: Rep1 (Free f) a -> Free f a #
MonadError e m => MonadError e (Free m) # |
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) # |
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) # |
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) # |
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 #
Foldable1 f => Foldable1 (Free f) # |
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) # |
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) # |
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) # |
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) # |
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 #
Alternative v => Alternative (Free v) # | 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) # |
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) # |
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) # |
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) # | 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 #
Applicative f => MonadFix (Free f) # |
Instance details Defined in Control.Monad.Free.Ap Methods mfix :: (a -> Free f a) -> Free f a #
Foldable f => Foldable (Free f) # |
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 #
Traversable f => Traversable (Free f) # |
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) #
MonadCont m => MonadCont (Free m) # |
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) # |
Instance details Defined 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 details Defined 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 #
Traversable1 f => Traversable1 (Free f) # |
Instance details Defined in Control.Monad.Free.Ap Methods traverse1 :: Apply f0 => (a -> f0 b) -> Free f a -> f0 (Free f b) # sequence1 :: Apply f0 => Free f (f0 b) -> f0 (Free f b) #
(Typeable f, Data a, Data (f (Free f a))) => Data (Free f a) # |
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) # |
Instance details Defined in Control.Monad.Free.Ap Associated Types
type Rep (Free f a) |
Instance details Defined in Control.Monad.Free.Ap type Rep (Free f a) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-80KVzMloHSZ6giRVM2rsod" '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)))))
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) # |
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) # |
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) # |
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) # |
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) # |
Instance details Defined in Control.Monad.Free.Ap type Rep1 (Free f :: Type -> Type) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-80KVzMloHSZ6giRVM2rsod" '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 details Defined in Control.Monad.Free.Ap type Rep (Free f a) = D1 ('MetaData "Free" "Control.Monad.Free.Ap" "free-5.2-80KVzMloHSZ6giRVM2rsod" '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 #

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

Given an applicative homomorphism, you get a monad homomorphism.

toFreeT :: forall (f :: Type -> Type) (m :: Type -> Type) a. (Applicative f, 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 :: forall (f :: Type -> Type) a. 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, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a) #

Unfold a free monad from a seed, monadically.

_Pure :: forall (f :: Type -> Type) 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))