Control.Monad.Trans.Free

The free monad transformer

The base functor

data FreeF f a b

The base functor for a free monad.

Constructors

Pure a

Free (f b)

Instances

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

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

Eq1 f => Eq2 (FreeF f)

Methods

liftEq2 :: (a -> b -> Bool) -> (c -> d -> Bool) -> FreeF f a c -> FreeF f b d -> Bool

Ord1 f => Ord2 (FreeF f)

Methods

liftCompare2 :: (a -> b -> Ordering) -> (c -> d -> Ordering) -> FreeF f a c -> FreeF f b d -> Ordering

Read1 f => Read2 (FreeF f)

Methods

liftReadsPrec2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> Int -> ReadS (FreeF f a b)

liftReadList2 :: (Int -> ReadS a) -> ReadS [a] -> (Int -> ReadS b) -> ReadS [b] -> ReadS [FreeF f a b]

liftReadPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec (FreeF f a b)

liftReadListPrec2 :: ReadPrec a -> ReadPrec [a] -> ReadPrec b -> ReadPrec [b] -> ReadPrec [FreeF f a b]

Show1 f => Show2 (FreeF f)

Methods

liftShowsPrec2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> Int -> FreeF f a b -> ShowS

liftShowList2 :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> (Int -> b -> ShowS) -> ([b] -> ShowS) -> [FreeF f a b] -> ShowS

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

liftEq :: (a -> b -> Bool) -> FreeF f a a -> FreeF f a b -> Bool

(Ord1 f, Ord a) => Ord1 (FreeF f a)

Methods

liftCompare :: (a -> b -> Ordering) -> FreeF f a a -> FreeF f a b -> Ordering

(Read1 f, Read a) => Read1 (FreeF f a)

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (FreeF f a a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [FreeF f a a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (FreeF f a a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [FreeF f a a]

(Show1 f, Show a) => Show1 (FreeF f a)

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> FreeF f a a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [FreeF f a a] -> ShowS

(Eq (f b), Eq a) => 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 (f b), Ord a) => 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 (f b), Read a) => 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 (f b), Show a) => 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, 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, 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, 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, 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, MonadBase b m) => MonadBase b (FreeT f m)

Methods

liftBase :: b α -> FreeT f m α

(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) => MonadFail (FreeT f m)

Methods

fail :: String -> 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

liftA2 :: (a -> b -> c) -> FreeT f m a -> FreeT f m b -> FreeT f m c

(*>) :: 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)

(Eq1 f, Eq1 m) => Eq1 (FreeT f m)

Methods

liftEq :: (a -> b -> Bool) -> FreeT f m a -> FreeT f m b -> Bool

(Ord1 f, Ord1 m) => Ord1 (FreeT f m)

Methods

liftCompare :: (a -> b -> Ordering) -> FreeT f m a -> FreeT f m b -> Ordering

(Read1 f, Read1 m) => Read1 (FreeT f m)

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (FreeT f m a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [FreeT f m a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (FreeT f m a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [FreeT f m a]

(Show1 f, Show1 m) => Show1 (FreeT f m)

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> FreeT f m a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [FreeT f m a] -> ShowS

(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, 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

liftF2 :: (a -> b -> c) -> FreeT f m a -> FreeT f m b -> FreeT f m c

(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

(Eq1 f, Eq1 m, Eq 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

(Ord1 f, Ord1 m, Ord 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

(Read1 f, Read1 m, Read 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]

(Show1 f, Show1 m, Show 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

foldFreeT :: (MonadTrans t, Monad (t m), Monad m) => (forall n x. Monad n => f x -> t n x) -> FreeT f m a -> t m a

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

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

Applicative f => MonadFree f (Free f)

Methods

wrap :: f (Free f a) -> Free f 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

(Applicative f, Applicative m, Monad m) => MonadFree f (FreeT f m)

Methods

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