Copyright	(C) 2008-2014 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	non-portable (rank-2 polymorphism, MTPCs)
Safe Haskell	Safe
Language	Haskell2010

Control.Monad.Trans.Free.Church

Contents

The free monad transformer
The free monad
Operations
Operations of free monad
Free Monads With Class

Description

Church-encoded free monad transformer.

Synopsis

newtype FT (f :: Type -> Type) (m :: Type -> Type) a = FT {
- runFT :: forall r. (a -> m r) -> (forall x. (x -> m r) -> f x -> m r) -> m r
}
type F (f :: Type -> Type) = FT f Identity
free :: (forall r. (a -> r) -> (f r -> r) -> r) -> F f a
runF :: Functor f => F f a -> forall r. (a -> r) -> (f r -> r) -> r
improveT :: forall (f :: Type -> Type) (m :: Type -> Type) a. (Functor f, Monad m) => (forall (t :: (Type -> Type) -> Type -> Type). MonadFree f (t m) => t m a) -> FreeT f m a
toFT :: forall (m :: Type -> Type) (f :: Type -> Type) a. Monad m => FreeT f m a -> FT f m a
fromFT :: forall (m :: Type -> Type) (f :: Type -> Type) a. (Monad m, Functor f) => FT f m a -> FreeT f m a
iterT :: (Functor f, Monad m) => (f (m a) -> m a) -> FT f m a -> m a
iterTM :: forall f (m :: Type -> Type) t a. (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FT f m a -> t m a
hoistFT :: forall m n (f :: Type -> Type) b. (Monad m, Monad n) => (forall a. m a -> n a) -> FT f m b -> FT f n b
transFT :: forall f g (m :: Type -> Type) b. (forall a. f a -> g a) -> FT f m b -> FT g m b
joinFT :: forall m (f :: Type -> Type) a. (Monad m, Traversable f) => FT f m a -> m (F f a)
cutoff :: forall (f :: Type -> Type) (m :: Type -> Type) a. (Functor f, Monad m) => Integer -> FT f m a -> FT f m (Maybe a)
improve :: forall (f :: Type -> Type) a. Functor f => (forall (m :: Type -> Type). MonadFree f m => m a) -> Free f a
fromF :: forall (f :: Type -> Type) m a. (Functor f, MonadFree f m) => F f a -> m a
toF :: forall (f :: Type -> Type) a. Free f a -> F f a
retract :: Monad f => F f a -> f a
retractT :: forall t (m :: Type -> Type) a. (MonadTrans t, Monad (t m), Monad m) => FT (t m) m a -> t m a
iter :: Functor f => (f a -> a) -> F f a -> a
iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> F f a -> m a
class Monad m => MonadFree (f :: Type -> Type) (m :: Type -> Type) | m -> f where
- wrap :: f (m a) -> m a
liftF :: (Functor f, MonadFree f m) => f a -> m a

The free monad transformer

newtype FT (f :: Type -> Type) (m :: Type -> Type) a Source #

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

Instances details

MonadFree f (FT f m) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods wrap :: f (FT f m a) -> FT f m a Source #
(Functor f, MonadError e m) => MonadError e (FT f m) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods lift :: Monad m => m a -> FT f m a #
MonadFail m => MonadFail (FT f m) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods fail :: String -> FT f m a #
MonadIO m => MonadIO (FT f m) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods catch :: (HasCallStack, Exception e) => FT f m a -> (e -> FT f m a) -> FT f m a #
MonadThrow m => MonadThrow (FT f m) Source #
Instance details Defined in Control.Monad.Trans.Free.Church Methods throwM :: (HasCallStack, Exception e) => e -> FT f m a #
MonadCont m => MonadCont (FT f m) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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) Source #
Instance details 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 :: Type -> Type) = FT f Identity Source #

The "free monad" for a functor f.

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

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 Source #

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

Operations

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

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 :: forall (m :: Type -> Type) (f :: Type -> Type) a. Monad m => FreeT f m a -> FT f m a Source #

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

fromFT :: forall (m :: Type -> Type) (f :: Type -> Type) a. (Monad m, Functor f) => FT f m a -> FreeT f m a Source #

Convert to a FreeT free monad representation.

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

Tear down a free monad transformer using iteration.

iterTM :: forall f (m :: Type -> Type) t a. (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FT f m a -> t m a Source #

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

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

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 f g (m :: Type -> Type) b. (forall a. f a -> g a) -> FT f m b -> FT g m b Source #

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

joinFT :: forall m (f :: Type -> Type) a. (Monad m, Traversable f) => FT f m a -> m (F f a) Source #

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

cutoff :: forall (f :: Type -> Type) (m :: Type -> Type) a. (Functor f, Monad m) => Integer -> FT f m a -> FT f m (Maybe a) Source #

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 :: forall (f :: Type -> Type) a. Functor f => (forall (m :: Type -> Type). MonadFree f m => m a) -> Free f a Source #

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 :: forall (f :: Type -> Type) m a. (Functor f, MonadFree f m) => F f a -> m a Source #

Convert to another free monad representation.

toF :: forall (f :: Type -> Type) a. Free f a -> F f a Source #

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

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

retract is the left inverse of liftF

retract . liftF = id

retractT :: forall t (m :: Type -> Type) a. (MonadTrans t, Monad (t m), Monad m) => FT (t m) m a -> t m a Source #

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

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

Tear down an F Monad using iteration.

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

Like iter for monadic values.

Free Monads With Class

class Monad m => MonadFree (f :: Type -> Type) (m :: Type -> Type) | m -> f where Source #

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 Source #

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 Source #

Instances

Instances details

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

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

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