Control.Monad.Trans.Free

free-4.12.4: Monads for free

Copyright | (C) 2008-2013 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.Trans.Free

Contents

- The base functor
- The free monad transformer
- The free monad
- Operations
- Operations of free monad
- Free Monads With Class

Description

The free monad transformer

Synopsis

- data FreeF f a b
  - = Pure a
  - | Free (f b)
- newtype FreeT f m a = FreeT {
  - runFreeT :: m (FreeF f a (FreeT f m a))
  }
- type Free f = FreeT f Identity
- free :: FreeF f a (Free f a) -> Free f a
- runFree :: Free f a -> FreeF f a (Free f a)
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iterT :: (Functor f, Monad m) => (f (m a) -> m a) -> FreeT f m a -> m a
- iterTM :: (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FreeT f m a -> t m a
- hoistFreeT :: (Monad m, Functor f) => (forall a. m a -> n a) -> FreeT f m b -> FreeT f n b
- transFreeT :: (Monad m, Functor g) => (forall a. f a -> g a) -> FreeT f m b -> FreeT g m b
- joinFreeT :: (Monad m, Traversable f) => FreeT f m a -> m (Free f a)
- cutoff :: (Functor f, Monad m) => Integer -> FreeT f m a -> FreeT f m (Maybe a)
- partialIterT :: Monad m => Integer -> (forall a. f a -> m a) -> FreeT f m b -> FreeT f m b
- intersperseT :: (Monad m, Functor f) => f a -> FreeT f m b -> FreeT f m b
- intercalateT :: (Monad m, MonadTrans t, Monad (t m)) => t m a -> FreeT (t m) m b -> t m b
- retractT :: (MonadTrans t, Monad (t m), Monad m) => FreeT (t m) m a -> t m a
- retract :: Monad f => Free f a -> f a
- iter :: Functor f => (f a -> a) -> Free f a -> a
- iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> Free f a -> m a
- class Monad m => MonadFree f m | m -> f where

# The base functor

data FreeF f a b #

The base functor for a free monad.

Constructors

Pure a |
Free (f b) |

Instances

Functor f => Bifunctor (FreeF f) # |
Methods bimap :: (a -> b) -> (c -> d) -> FreeF f a c -> FreeF f b d # first :: (a -> b) -> FreeF f a c -> FreeF f b c # second :: (b -> c) -> FreeF f a b -> FreeF f a c #
Traversable f => Bitraversable (FreeF f) # |
Methods bitraverse :: Applicative f => (a -> f c) -> (b -> f d) -> FreeF f a b -> f (FreeF f c d) #
Foldable f => Bifoldable (FreeF f) # |
Methods bifold :: Monoid m => FreeF f m m -> m # bifoldMap :: Monoid m => (a -> m) -> (b -> m) -> FreeF f a b -> m # bifoldr :: (a -> c -> c) -> (b -> c -> c) -> c -> FreeF f a b -> c # bifoldl :: (c -> a -> c) -> (c -> b -> c) -> c -> FreeF f a b -> c #
Eq1 f => Eq2 (FreeF f) # |
Methods (==##) :: (Eq a, Eq b) => FreeF f a b -> FreeF f a b -> Bool #
Ord1 f => Ord2 (FreeF f) # |
Methods compare2 :: (Ord a, Ord b) => FreeF f a b -> FreeF f a b -> Ordering #
Show1 f => Show2 (FreeF f) # |
Methods showsPrec2 :: (Show a, Show b) => Int -> FreeF f a b -> ShowS # showList2 :: (Show a, Show b) => [FreeF f a b] -> ShowS #
Read1 f => Read2 (FreeF f) # |
Methods readsPrec2 :: (Read a, Read b) => Int -> ReadS (FreeF f a b) # readList2 :: (Read a, Read b) => ReadS [FreeF f a b] #
Functor f => Functor (FreeF f a) # |
Methods fmap :: (a -> b) -> FreeF f a a -> FreeF f a b # (<$) :: a -> FreeF f a b -> FreeF f a a #
Foldable f => Foldable (FreeF f a) # |
Methods fold :: Monoid m => FreeF f a m -> m # foldMap :: Monoid m => (a -> m) -> FreeF f a a -> m # foldr :: (a -> b -> b) -> b -> FreeF f a a -> b # foldr' :: (a -> b -> b) -> b -> FreeF f a a -> b # foldl :: (b -> a -> b) -> b -> FreeF f a a -> b # foldl' :: (b -> a -> b) -> b -> FreeF f a a -> b # foldr1 :: (a -> a -> a) -> FreeF f a a -> a # foldl1 :: (a -> a -> a) -> FreeF f a a -> a # toList :: FreeF f a a -> [a] # null :: FreeF f a a -> Bool # length :: FreeF f a a -> Int # elem :: Eq a => a -> FreeF f a a -> Bool # maximum :: Ord a => FreeF f a a -> a # minimum :: Ord a => FreeF f a a -> a # sum :: Num a => FreeF f a a -> a # product :: Num a => FreeF f a a -> a #
Traversable f => Traversable (FreeF f a) # |
Methods traverse :: Applicative f => (a -> f b) -> FreeF f a a -> f (FreeF f a b) # sequenceA :: Applicative f => FreeF f a (f a) -> f (FreeF f a a) # mapM :: Monad m => (a -> m b) -> FreeF f a a -> m (FreeF f a b) # sequence :: Monad m => FreeF f a (m a) -> m (FreeF f a a) #
(Eq1 f, Eq a) => Eq1 (FreeF f a) # |
Methods (==#) :: Eq a => FreeF f a a -> FreeF f a a -> Bool #
(Ord1 f, Ord a) => Ord1 (FreeF f a) # |
Methods compare1 :: Ord a => FreeF f a a -> FreeF f a a -> Ordering #
(Show1 f, Show a) => Show1 (FreeF f a) # |
Methods showsPrec1 :: Show a => Int -> FreeF f a a -> ShowS # showList1 :: Show a => [FreeF f a a] -> ShowS #
(Read1 f, Read a) => Read1 (FreeF f a) # |
Methods readsPrec1 :: Read a => Int -> ReadS (FreeF f a a) # readList1 :: Read a => ReadS [FreeF f a a] #
(Eq a, Eq (f b)) => Eq (FreeF f a b) # |
Methods (==) :: FreeF f a b -> FreeF f a b -> Bool # (/=) :: FreeF f a b -> FreeF f a b -> Bool #
(Ord a, Ord (f b)) => Ord (FreeF f a b) # |
Methods compare :: FreeF f a b -> FreeF f a b -> Ordering # (<) :: FreeF f a b -> FreeF f a b -> Bool # (<=) :: FreeF f a b -> FreeF f a b -> Bool # (>) :: FreeF f a b -> FreeF f a b -> Bool # (>=) :: FreeF f a b -> FreeF f a b -> Bool # max :: FreeF f a b -> FreeF f a b -> FreeF f a b # min :: FreeF f a b -> FreeF f a b -> FreeF f a b #
(Read a, Read (f b)) => Read (FreeF f a b) # |
Methods readsPrec :: Int -> ReadS (FreeF f a b) # readList :: ReadS [FreeF f a b] # readPrec :: ReadPrec (FreeF f a b) # readListPrec :: ReadPrec [FreeF f a b] #
(Show a, Show (f b)) => Show (FreeF f a b) # |
Methods showsPrec :: Int -> FreeF f a b -> ShowS # show :: FreeF f a b -> String # showList :: [FreeF f a b] -> ShowS #

# The free monad transformer

newtype FreeT f m a #

The "free monad transformer" for a functor `f`

Constructors

FreeT |
Fields runFreeT :: m (FreeF f a (FreeT f m a))

Instances

(Functor f, MonadError e m) => MonadError e (FreeT f m) # |
Methods throwError :: e -> FreeT f m a # catchError :: FreeT f m a -> (e -> FreeT f m a) -> FreeT f m a #
(Functor f, MonadReader r m) => MonadReader r (FreeT f m) # |
Methods ask :: FreeT f m r # local :: (r -> r) -> FreeT f m a -> FreeT f m a # reader :: (r -> a) -> FreeT f m a #
(Functor f, MonadState s m) => MonadState s (FreeT f m) # |
Methods get :: FreeT f m s # put :: s -> FreeT f m () # state :: (s -> (a, s)) -> FreeT f m a #
(Functor f, MonadWriter w m) => MonadWriter w (FreeT f m) # |
Methods writer :: (a, w) -> FreeT f m a # tell :: w -> FreeT f m () # listen :: FreeT f m a -> FreeT f m (a, w) # pass :: FreeT f m (a, w -> w) -> FreeT f m a #
(Functor f, Monad m) => MonadFree f (FreeT f m) # |
Methods wrap :: f (FreeT f m a) -> FreeT f m a #
MonadTrans (FreeT f) # |
Methods lift :: Monad m => m a -> FreeT f m a #
(Functor f, Monad m) => Monad (FreeT f m) # |
Methods (>>=) :: FreeT f m a -> (a -> FreeT f m b) -> FreeT f m b # (>>) :: FreeT f m a -> FreeT f m b -> FreeT f m b # return :: a -> FreeT f m a # fail :: String -> FreeT f m a #
(Functor f, Monad m) => Functor (FreeT f m) # |
Methods fmap :: (a -> b) -> FreeT f m a -> FreeT f m b # (<$) :: a -> FreeT f m b -> FreeT f m a #
(Functor f, Monad m) => Applicative (FreeT f m) # |
Methods pure :: a -> FreeT f m a # (<*>) :: FreeT f m (a -> b) -> FreeT f m a -> FreeT f m b # (*>) :: FreeT f m a -> FreeT f m b -> FreeT f m b # (<*) :: FreeT f m a -> FreeT f m b -> FreeT f m a #
(Foldable m, Foldable f) => Foldable (FreeT f m) # |
Methods fold :: Monoid m => FreeT f m m -> m # foldMap :: Monoid m => (a -> m) -> FreeT f m a -> m # foldr :: (a -> b -> b) -> b -> FreeT f m a -> b # foldr' :: (a -> b -> b) -> b -> FreeT f m a -> b # foldl :: (b -> a -> b) -> b -> FreeT f m a -> b # foldl' :: (b -> a -> b) -> b -> FreeT f m a -> b # foldr1 :: (a -> a -> a) -> FreeT f m a -> a # foldl1 :: (a -> a -> a) -> FreeT f m a -> a # toList :: FreeT f m a -> [a] # null :: FreeT f m a -> Bool # length :: FreeT f m a -> Int # elem :: Eq a => a -> FreeT f m a -> Bool # maximum :: Ord a => FreeT f m a -> a # minimum :: Ord a => FreeT f m a -> a # sum :: Num a => FreeT f m a -> a # product :: Num a => FreeT f m a -> a #
(Monad m, Traversable m, Traversable f) => Traversable (FreeT f m) # |
Methods traverse :: Applicative f => (a -> f b) -> FreeT f m a -> f (FreeT f m b) # sequenceA :: Applicative f => FreeT f m (f a) -> f (FreeT f m a) # mapM :: Monad m => (a -> m b) -> FreeT f m a -> m (FreeT f m b) # sequence :: Monad m => FreeT f m (m a) -> m (FreeT f m a) #
(Functor f, MonadIO m) => MonadIO (FreeT f m) # |
Methods liftIO :: IO a -> FreeT f m a #
(Functor f, MonadPlus m) => Alternative (FreeT f m) # |
Methods empty :: FreeT f m a # (<|>) :: FreeT f m a -> FreeT f m a -> FreeT f m a # some :: FreeT f m a -> FreeT f m [a] # many :: FreeT f m a -> FreeT f m [a] #
(Functor f, MonadPlus m) => MonadPlus (FreeT f m) # |
Methods mzero :: FreeT f m a # mplus :: FreeT f m a -> FreeT f m a -> FreeT f m a #
(Functor f, MonadThrow m) => MonadThrow (FreeT f m) # |
Methods throwM :: Exception e => e -> FreeT f m a #
(Functor f, MonadCatch m) => MonadCatch (FreeT f m) # |
Methods catch :: Exception e => FreeT f m a -> (e -> FreeT f m a) -> FreeT f m a #
(Functor f, MonadCont m) => MonadCont (FreeT f m) # |
Methods callCC :: ((a -> FreeT f m b) -> FreeT f m a) -> FreeT f m a #
(Functor f, Eq1 f, Functor m, Eq1 m) => Eq1 (FreeT f m) # |
Methods (==#) :: Eq a => FreeT f m a -> FreeT f m a -> Bool #
(Functor f, Ord1 f, Functor m, Ord1 m) => Ord1 (FreeT f m) # |
Methods compare1 :: Ord a => FreeT f m a -> FreeT f m a -> Ordering #
(Functor f, Show1 f, Functor m, Show1 m) => Show1 (FreeT f m) # |
Methods showsPrec1 :: Show a => Int -> FreeT f m a -> ShowS # showList1 :: Show a => [FreeT f m a] -> ShowS #
(Functor f, Read1 f, Functor m, Read1 m) => Read1 (FreeT f m) # |
Methods readsPrec1 :: Read a => Int -> ReadS (FreeT f m a) # readList1 :: Read a => ReadS [FreeT f m a] #
(Functor f, Monad m) => Apply (FreeT f m) # |
Methods (<.>) :: FreeT f m (a -> b) -> FreeT f m a -> FreeT f m b # (.>) :: FreeT f m a -> FreeT f m b -> FreeT f m b # (<.) :: FreeT f m a -> FreeT f m b -> FreeT f m a #
(Functor f, Monad m) => Bind (FreeT f m) # |
Methods (>>-) :: FreeT f m a -> (a -> FreeT f m b) -> FreeT f m b # join :: FreeT f m (FreeT f m a) -> FreeT f m a #
Eq (m (FreeF f a (FreeT f m a))) => Eq (FreeT f m a) # |
Methods (==) :: FreeT f m a -> FreeT f m a -> Bool # (/=) :: FreeT f m a -> FreeT f m a -> Bool #
Ord (m (FreeF f a (FreeT f m a))) => Ord (FreeT f m a) # |
Methods compare :: FreeT f m a -> FreeT f m a -> Ordering # (<) :: FreeT f m a -> FreeT f m a -> Bool # (<=) :: FreeT f m a -> FreeT f m a -> Bool # (>) :: FreeT f m a -> FreeT f m a -> Bool # (>=) :: FreeT f m a -> FreeT f m a -> Bool # max :: FreeT f m a -> FreeT f m a -> FreeT f m a # min :: FreeT f m a -> FreeT f m a -> FreeT f m a #
Read (m (FreeF f a (FreeT f m a))) => Read (FreeT f m a) # |
Methods readsPrec :: Int -> ReadS (FreeT f m a) # readList :: ReadS [FreeT f m a] # readPrec :: ReadPrec (FreeT f m a) # readListPrec :: ReadPrec [FreeT f m a] #
Show (m (FreeF f a (FreeT f m a))) => Show (FreeT f m a) # |
Methods showsPrec :: Int -> FreeT f m a -> ShowS # show :: FreeT f m a -> String # showList :: [FreeT f m a] -> ShowS #

# The free monad

type Free f = FreeT f Identity #

The "free monad" for a functor `f`.

free :: FreeF f a (Free f a) -> Free f a #

Pushes a layer into a free monad value.

runFree :: Free f a -> FreeF f a (Free f a) #

Evaluates the first layer out of a free monad value.

# Operations

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

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

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

Tear down a free monad transformer using iteration.

iterTM :: (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FreeT f m a -> t m a #

Tear down a free monad transformer using iteration over a transformer.

hoistFreeT :: (Monad m, Functor f) => (forall a. m a -> n a) -> FreeT f m b -> FreeT f n b #

Lift a monad homomorphism from `m` to `n` into a monad homomorphism from ``FreeT` f m` to ``FreeT` f n`

    hoistFreeT :: (Monad m, Functor f) => (m ~> n) -> FreeT f m ~> FreeT f n

transFreeT :: (Monad m, Functor g) => (forall a. f a -> g a) -> FreeT f m b -> FreeT g m b #

Lift a natural transformation from `f` to `g` into a monad homomorphism from ``FreeT` f m` to ``FreeT` g m`

joinFreeT :: (Monad m, Traversable f) => FreeT f m a -> m (Free f a) #

Pull out and join `m` layers of ``FreeT` f m a`.

cutoff :: (Functor f, Monad m) => Integer -> FreeT f m a -> FreeT f m (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.

partialIterT :: Monad m => Integer -> (forall a. f a -> m a) -> FreeT f m b -> FreeT f m b #

`partialIterT n phi m` interprets first `n` layers of `m` using `phi`. This is sort of the opposite for ``cutoff``.

Some examples (`n ≥ 0`):

    partialIterT 0 _ m              ≡ m
    partialIterT (n+1) phi . return ≡ return
    partialIterT (n+1) phi . lift   ≡ lift
    partialIterT (n+1) phi . wrap   ≡ join . lift . phi

intersperseT :: (Monad m, Functor f) => f a -> FreeT f m b -> FreeT f m b #

`intersperseT f m` inserts a layer `f` between every two layers in `m`.

    intersperseT f . return ≡ return
    intersperseT f . lift   ≡ lift
    intersperseT f . wrap   ≡ wrap . fmap (iterTM (wrap . (<$ f) . wrap))

intercalateT :: (Monad m, MonadTrans t, Monad (t m)) => t m a -> FreeT (t m) m b -> t m b #

`intercalateT f m` inserts a layer `f` between every two layers in `m` and then retracts the result.

    intercalateT f ≡ retractT . intersperseT f

retractT :: (MonadTrans t, Monad (t m), Monad m) => FreeT (t m) m a -> t m a #

Tear down a free monad transformer using Monad instance for `t m`.

# Operations of free monad

retract :: Monad f => Free f a -> f a #

`retract` is the left inverse of `liftF`

    retract . liftF = id

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

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

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

Like `iter` for monadic values.

# Free Monads With Class

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`.

Methods

wrap :: f (m a) -> m a #

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 #

Add a layer.

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

Instances

(Functor f, MonadFree f m) => MonadFree f (ListT m) # |
Methods wrap :: f (ListT m a) -> ListT m a #
(Functor f, MonadFree f m) => MonadFree f (MaybeT m) # |
Methods wrap :: f (MaybeT m a) -> MaybeT m a #
Functor f => MonadFree f (Free f) # |
Methods wrap :: f (Free f a) -> Free f a #
Functor f => MonadFree f (F f) # |
Methods wrap :: f (F f a) -> F f a #
Monad m => MonadFree Identity (IterT m) # |
Methods wrap :: Identity (IterT m a) -> IterT m a #
(Functor f, MonadFree f m) => MonadFree f (ExceptT e m) # |
Methods wrap :: f (ExceptT e m a) -> ExceptT e m a #
(Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m) # |
Methods wrap :: f (ErrorT e m a) -> ErrorT e m a #
(Functor f, MonadFree f m) => MonadFree f (IdentityT * m) # |
Methods wrap :: f (IdentityT * m a) -> IdentityT * m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) # |
Methods wrap :: f (WriterT w m a) -> WriterT w m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) # |
Methods wrap :: f (WriterT w m a) -> WriterT w m a #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) # |
Methods wrap :: f (StateT s m a) -> StateT s m a #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) # |
Methods wrap :: f (StateT s m a) -> StateT s m a #
(Functor f, Monad m) => MonadFree f (FreeT f m) # |
Methods wrap :: f (FreeT f m a) -> FreeT f m a #
MonadFree f (FT f m) # |
Methods wrap :: f (FT f m a) -> FT f m a #
(Functor f, MonadFree f m) => MonadFree f (ContT * r m) # |
Methods wrap :: f (ContT * r m a) -> ContT * r m a #
(Functor f, MonadFree f m) => MonadFree f (ReaderT * e m) # |
Methods wrap :: f (ReaderT * e m a) -> ReaderT * e m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) # |
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) # |
Methods wrap :: f (RWST r w s m a) -> RWST r w s m a #