free-5.1.7: Monads for free

Copyright | (C) 2008-2015 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | MPTCs, fundeps
Safe Haskell | Safe
Language | Haskell2010

Control.Monad.Free

Description

Monads 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))
- retract :: Monad f => Free f a -> f a
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iter :: Functor f => (f a -> a) -> Free f a -> a
- iterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a
- iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a
- hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b
- foldFree :: Monad m => (forall x. f x -> m x) -> Free f a -> m a
- toFreeT :: (Functor f, Monad m) => Free f a -> FreeT f m a
- cutoff :: Functor f => Integer -> Free f a -> Free f (Maybe a)
- unfold :: Functor f => (b -> Either a (f b)) -> b -> Free f a
- unfoldM :: (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 g m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (g (Free g a))) -> p (Free f a) (m (Free g a))

# 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 #

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 #

#### Instances

Instances details

(Functor f, MonadFree f m) => MonadFree f (ListT m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ListT m a) -> ListT m 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 #
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 (Free f) # |
Instance details Defined in Control.Monad.Free 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 #
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 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, Error e) => MonadFree f (ErrorT e m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ErrorT e m a) -> ErrorT 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, 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 (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) => MonadFree f (ReaderT e m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ReaderT e m a) -> ReaderT e m a #
(Applicative f, Applicative m, 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 #
(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 #
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 (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 a #

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

Formally

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

Why Free?

Every "free" functor is left adjoint to some "forgetful" functor.

If we define a forgetful functor `U` from the category of monads to the category of functors that just forgets the `Monad`, leaving only the `Functor`. i.e.

    U (M,return,join) = M

then `Free` is the left adjoint to `U`.

`Free` being left adjoint to `U` means that there is an isomorphism between

``Free` f -> m` in the category of monads and `f -> U m` in the category of functors.

Morphisms in the category of monads are `Monad` homomorphisms (natural transformations that respect `return` and `join`).

Morphisms in the category of functors are `Functor` homomorphisms (natural transformations).

Given this isomorphism, every monad homomorphism from ``Free` f` to `m` is equivalent to a natural transformation from `f` to `m`

Showing that this isomorphism holds is left as an exercise.

In practice, you can just view a ``Free` f a` as many layers of `f` wrapped around values of type `a`, where `(`>>=`)` performs substitution and grafts new layers of `f` in for each of the free variables.

This can be very useful for modeling domain specific languages, trees, or other constructs.

This instance of `MonadFree` is fairly naive about the encoding. For more efficient free monad implementation see Control.Monad.Free.Church, in particular note the `improve` combinator. You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

A number of common monads arise as free monads,

- Given `data Empty a`, ``Free` Empty` is isomorphic to the `Identity` monad.
- ``Free` `Maybe`` can be used to model a partiality monad where each layer represents running the computation for a while longer.

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 Methods lift :: Monad m => m a -> Free m a #
(Functor m, MonadWriter e m) => MonadWriter e (Free m) # |
Instance details Defined in Control.Monad.Free 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 #
(Functor m, MonadState s m) => MonadState s (Free m) # |
Instance details Defined in Control.Monad.Free Methods get :: Free m s # put :: s -> Free m () # state :: (s -> (a, s)) -> Free m a #
(Functor m, MonadReader e m) => MonadReader e (Free m) # |
Instance details Defined in Control.Monad.Free Methods ask :: Free m e # local :: (e -> e) -> Free m a -> Free m a # reader :: (e -> a) -> Free m a #
(Functor m, MonadError e m) => MonadError e (Free m) # |
Instance details Defined in Control.Monad.Free Methods throwError :: e -> Free m a # catchError :: Free m a -> (e -> Free m a) -> Free m a #
Functor f => MonadFree f (Free f) # |
Instance details Defined in Control.Monad.Free Methods wrap :: f (Free f a) -> Free f a #
Functor f => Monad (Free f) # |
Instance details Defined in Control.Monad.Free 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 #
Functor f => Functor (Free f) # |
Instance details Defined in Control.Monad.Free Methods fmap :: (a -> b) -> Free f a -> Free f b # (<$) :: a -> Free f b -> Free f a #
Functor f => MonadFix (Free f) # |
Instance details Defined in Control.Monad.Free Methods mfix :: (a -> Free f a) -> Free f a #
Functor f => Applicative (Free f) # |
Instance details Defined in Control.Monad.Free 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 #
Foldable f => Foldable (Free f) # |
Instance details Defined in Control.Monad.Free 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 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) #
Eq1 f => Eq1 (Free f) # |
Instance details Defined in Control.Monad.Free Methods liftEq :: (a -> b -> Bool) -> Free f a -> Free f b -> Bool #
Ord1 f => Ord1 (Free f) # |
Instance details Defined in Control.Monad.Free Methods liftCompare :: (a -> b -> Ordering) -> Free f a -> Free f b -> Ordering #
Read1 f => Read1 (Free f) # |
Instance details Defined in Control.Monad.Free 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 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 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] #
(Functor v, MonadPlus v) => MonadPlus (Free v) # | This violates the MonadPlus laws, handle with care.
Instance details Defined in Control.Monad.Free Methods mzero :: Free v a # mplus :: Free v a -> Free v a -> Free v a #
(Functor m, MonadCont m) => MonadCont (Free m) # |
Instance details Defined in Control.Monad.Free Methods callCC :: ((a -> Free m b) -> Free m a) -> Free m a #
Traversable1 f => Traversable1 (Free f) # |
Instance details Defined in Control.Monad.Free 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) #
Functor f => Apply (Free f) # |
Instance details Defined in Control.Monad.Free 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 #
Functor f => Bind (Free f) # |
Instance details Defined in Control.Monad.Free Methods (>>-) :: Free f a -> (a -> Free f b) -> Free f b # join :: Free f (Free f a) -> Free f a #
Foldable1 f => Foldable1 (Free f) # |
Instance details Defined in Control.Monad.Free Methods fold1 :: Semigroup m => Free f m -> m # foldMap1 :: Semigroup m => (a -> m) -> Free f a -> m # toNonEmpty :: Free f a -> NonEmpty a #
Functor f => Generic1 (Free f :: Type -> Type) # |
Instance details Defined in Control.Monad.Free 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 #
FunctorWithIndex i f => FunctorWithIndex [i] (Free f) # |
Instance details Defined in Control.Monad.Free Methods imap :: ([i] -> a -> b) -> Free f a -> Free f b #
FoldableWithIndex i f => FoldableWithIndex [i] (Free f) # |
Instance details Defined in Control.Monad.Free Methods ifoldMap :: Monoid m => ([i] -> a -> m) -> Free f a -> m # ifoldMap' :: Monoid m => ([i] -> a -> m) -> Free f a -> m # ifoldr :: ([i] -> a -> b -> b) -> b -> Free f a -> b # ifoldl :: ([i] -> b -> a -> b) -> b -> Free f a -> b # ifoldr' :: ([i] -> a -> b -> b) -> b -> Free f a -> b # ifoldl' :: ([i] -> b -> a -> b) -> b -> Free f a -> b #
TraversableWithIndex i f => TraversableWithIndex [i] (Free f) # |
Instance details Defined in Control.Monad.Free Methods itraverse :: Applicative f0 => ([i] -> a -> f0 b) -> Free f a -> f0 (Free f b) #
(Eq1 f, Eq a) => Eq (Free f a) # |
Instance details Defined in Control.Monad.Free Methods (==) :: Free f a -> Free f a -> Bool # (/=) :: Free f a -> Free f a -> Bool #
(Typeable f, Data (f (Free f a)), Data a) => Data (Free f a) # |
Instance details Defined in Control.Monad.Free 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) #
(Ord1 f, Ord a) => Ord (Free f a) # |
Instance details Defined in Control.Monad.Free 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 #
(Read1 f, Read a) => Read (Free f a) # |
Instance details Defined in Control.Monad.Free 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 Methods showsPrec :: Int -> Free f a -> ShowS # show :: Free f a -> String # showList :: [Free f a] -> ShowS #
Generic (Free f a) # |
Instance details Defined in Control.Monad.Free 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 #
type Rep1 (Free f :: Type -> Type) # |
Instance details Defined in Control.Monad.Free type Rep1 (Free f :: Type -> Type) = D1 ('MetaData "Free" "Control.Monad.Free" "free-5.1.7-9nI9X2IK8ka44DLAkRlzg5" '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 type Rep (Free f a) = D1 ('MetaData "Free" "Control.Monad.Free" "free-5.1.7-9nI9X2IK8ka44DLAkRlzg5" '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 :: Functor f => (f a -> a) -> Free f a -> a #

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

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

Like `iter` for applicative values.

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

Like `iter` for monadic values.

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

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

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

The very definition of a free monad is that given a natural transformation you get a monad homomorphism.

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

Convert a `Free` monad from Control.Monad.Free to a `FreeT` monad from Control.Monad.Trans.Free.

cutoff :: Functor 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 :: Functor f => (b -> Either a (f b)) -> b -> Free f a #

Unfold a free monad from a seed.

unfoldM :: (Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a) #

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)) #

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 g m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (g (Free g a))) -> p (Free f a) (m (Free g a)) #

This is `Prism (Free f a) (Free g a) (f (Free f a)) (g (Free g a))` in disguise

    >>> preview _Free (review _Free (Just (Pure 3)))
    Just (Just (Pure 3))

    >>> review _Free (Just (Pure 3))
    Free (Just (Pure 3))