Control.Monad.Trans.Free.Church

Church-encoded free monad transformer.

The free monad transformer

newtype FT f m a

The "free monad transformer" for a functor f

Constructors

FT

Fields

• runFT :: forall r. (a -> m r) -> (forall x. (x -> m r) -> f x -> m r) -> m r

Instances

MonadFree f (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

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

(Functor f, MonadError e m) => MonadError e (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

throwError :: e -> FT f m a

catchError :: FT f m a -> (e -> FT f m a) -> FT f m a

MonadReader r m => MonadReader r (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

ask :: FT f m r

local :: (r -> r) -> FT f m a -> FT f m a

reader :: (r -> a) -> FT f m a

MonadState s m => MonadState s (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

get :: FT f m s

put :: s -> FT f m ()

state :: (s -> (a, s)) -> FT f m a

(Functor f, MonadWriter w m) => MonadWriter w (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

writer :: (a, w) -> FT f m a

tell :: w -> FT f m ()

listen :: FT f m a -> FT f m (a, w)

pass :: FT f m (a, w -> w) -> FT f m a

MonadTrans (FT f)

Defined in Control.Monad.Trans.Free.Church

Methods

lift :: Monad m => m a -> FT f m a

MonadFail m => MonadFail (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

fail :: String -> FT f m a

MonadIO m => MonadIO (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

liftIO :: IO a -> FT f m a

(Foldable f, Foldable m, Monad m) => Foldable (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

fold :: Monoid m0 => FT f m m0 -> m0

foldMap :: Monoid m0 => (a -> m0) -> FT f m a -> m0

foldMap' :: Monoid m0 => (a -> m0) -> FT f m a -> m0

foldr :: (a -> b -> b) -> b -> FT f m a -> b

foldr' :: (a -> b -> b) -> b -> FT f m a -> b

foldl :: (b -> a -> b) -> b -> FT f m a -> b

foldl' :: (b -> a -> b) -> b -> FT f m a -> b

foldr1 :: (a -> a -> a) -> FT f m a -> a

foldl1 :: (a -> a -> a) -> FT f m a -> a

toList :: FT f m a -> [a]

null :: FT f m a -> Bool

length :: FT f m a -> Int

elem :: Eq a => a -> FT f m a -> Bool

maximum :: Ord a => FT f m a -> a

minimum :: Ord a => FT f m a -> a

sum :: Num a => FT f m a -> a

product :: Num a => FT f m a -> a

(Functor f, Monad m, Eq1 f, Eq1 m) => Eq1 (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

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

(Functor f, Monad m, Ord1 f, Ord1 m) => Ord1 (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

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

(Monad m, Traversable m, Traversable f) => Traversable (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

traverse :: Applicative f0 => (a -> f0 b) -> FT f m a -> f0 (FT f m b)

sequenceA :: Applicative f0 => FT f m (f0 a) -> f0 (FT f m a)

mapM :: Monad m0 => (a -> m0 b) -> FT f m a -> m0 (FT f m b)

sequence :: Monad m0 => FT f m (m0 a) -> m0 (FT f m a)

Alternative m => Alternative (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

empty :: FT f m a

(<|>) :: FT f m a -> FT f m a -> FT f m a

some :: FT f m a -> FT f m [a]

many :: FT f m a -> FT f m [a]

Applicative (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

pure :: a -> FT f m a

(<*>) :: FT f m (a -> b) -> FT f m a -> FT f m b

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

(*>) :: FT f m a -> FT f m b -> FT f m b

(<*) :: FT f m a -> FT f m b -> FT f m a

Functor (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

fmap :: (a -> b) -> FT f m a -> FT f m b

(<$) :: a -> FT f m b -> FT f m a

Monad (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

(>>=) :: FT f m a -> (a -> FT f m b) -> FT f m b

(>>) :: FT f m a -> FT f m b -> FT f m b

return :: a -> FT f m a

MonadPlus m => MonadPlus (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

mzero :: FT f m a

mplus :: FT f m a -> FT f m a -> FT f m a

(Functor f, MonadCatch m) => MonadCatch (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

catch :: Exception e => FT f m a -> (e -> FT f m a) -> FT f m a

MonadThrow m => MonadThrow (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

throwM :: Exception e => e -> FT f m a

MonadCont m => MonadCont (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

callCC :: ((a -> FT f m b) -> FT f m a) -> FT f m a

Apply (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

(<.>) :: FT f m (a -> b) -> FT f m a -> FT f m b

(.>) :: FT f m a -> FT f m b -> FT f m b

(<.) :: FT f m a -> FT f m b -> FT f m a

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

Bind (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

(>>-) :: FT f m a -> (a -> FT f m b) -> FT f m b

join :: FT f m (FT f m a) -> FT f m a

(Functor f, Monad m, Eq1 f, Eq1 m, Eq a) => Eq (FT f m a)

Defined in Control.Monad.Trans.Free.Church

Methods

(==) :: FT f m a -> FT f m a -> Bool

(/=) :: FT f m a -> FT f m a -> Bool

(Functor f, Monad m, Ord1 f, Ord1 m, Ord a) => Ord (FT f m a)

Defined in Control.Monad.Trans.Free.Church

Methods

compare :: FT f m a -> FT f m a -> Ordering

(<) :: FT f m a -> FT f m a -> Bool

(<=) :: FT f m a -> FT f m a -> Bool

(>) :: FT f m a -> FT f m a -> Bool

(>=) :: FT f m a -> FT f m a -> Bool

max :: FT f m a -> FT f m a -> FT f m a

min :: FT f m a -> FT f m a -> FT f m a

The free monad

type F f = FT f Identity

The "free monad" for a functor f.

free :: (forall r. (a -> r) -> (f r -> r) -> r) -> F f a

Wrap a Church-encoding of a "free monad" as the free monad for a functor.

runF :: Functor f => F f a -> forall r. (a -> r) -> (f r -> r) -> r

Unwrap the FreeF monad to obtain it's Church-encoded representation.

Operations

improveT :: (Functor f, Monad m) => (forall t. MonadFree f (t m) => t m a) -> FreeT f m a

Improve the asymptotic performance of code that builds a free monad transformer with only binds and returns by using FT behind the scenes.

Similar to improve.

toFT :: Monad m => FreeT f m a -> FT f m a

Generate a Church-encoded free monad transformer from a FreeT monad transformer.

fromFT :: (Monad m, Functor f) => FT f m a -> FreeT f m a

Convert to a FreeT free monad representation.

iterT :: (Functor f, Monad m) => (f (m a) -> m a) -> FT 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) -> FT f m a -> t m a

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

hoistFT :: (Monad m, Monad n) => (forall a. m a -> n a) -> FT f m b -> FT f n b

Lift a monad homomorphism from m to n into a monad homomorphism from FT f m to FT f n

hoistFT :: (Monad m, Monad n, Functor f) => (m ~> n) -> FT f m ~> FT f n

transFT :: (forall a. f a -> g a) -> FT f m b -> FT g m b

Lift a natural transformation from f to g into a monad homomorphism from FT f m to FT g n

joinFT :: (Monad m, Traversable f) => FT f m a -> m (F f a)

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

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

Operations of free monad

improve :: Functor f => (forall m. MonadFree f m => m a) -> Free f a

Improve the asymptotic performance of code that builds a free monad with only binds and returns by using F behind the scenes.

This is based on the "Free Monads for Less" series of articles by Edward Kmett:

https://ekmett.github.io/reader/2011/free-monads-for-less/ https://ekmett.github.io/reader/2011/free-monads-for-less-2/

and "Asymptotic Improvement of Computations over Free Monads" by Janis Voightländer:

http://www.iai.uni-bonn.de/~jv/mpc08.pdf

fromF :: (Functor f, MonadFree f m) => F f a -> m a

Convert to another free monad representation.

toF :: Free f a -> F f a

Generate a Church-encoded free monad from a FreeF monad.

retract :: Monad f => F f a -> f a

retract is the left inverse of liftF

retract . liftF = id

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

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

iter :: Functor f => (f a -> a) -> F f a -> a

Tear down an F Monad using iteration.

iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> F 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.

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

Monad m => MonadFree Identity (IterT m)

Defined in Control.Monad.Trans.Iter

Methods

wrap :: Identity (IterT m a) -> IterT m a

Functor f => MonadFree f (Free f)

Defined in Control.Monad.Free

Methods

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

Applicative f => MonadFree f (Free f)

Defined in Control.Monad.Free.Ap

Methods

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

Functor f => MonadFree f (F f)

Defined in Control.Monad.Free.Church

Methods

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

(Functor f, MonadFree f m) => MonadFree f (ListT m)

Defined in Control.Monad.Free.Class

Methods

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

(Functor f, MonadFree f m) => MonadFree f (MaybeT m)

Defined in Control.Monad.Free.Class

Methods

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

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

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)

Defined in Control.Monad.Trans.Free.Ap

Methods

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

MonadFree f (FT f m)

Defined in Control.Monad.Trans.Free.Church

Methods

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

(Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m)

Defined in Control.Monad.Free.Class

Methods

wrap :: f (ErrorT e m a) -> ErrorT e m a

(Functor f, MonadFree f m) => MonadFree f (ExceptT e m)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Defined in Control.Monad.Free.Class

Methods

wrap :: f (RWST r w s m a) -> RWST r w s m a

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

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