Copyright	(C) 2008-2016 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	non-portable (rank-2 polymorphism)
Safe Haskell	Trustworthy
Language	Haskell2010

Control.Monad.Codensity

Description

Synopsis

newtype Codensity (m :: k -> TYPE rep) a = Codensity {
- runCodensity :: forall b. (a -> m b) -> m b
}
lowerCodensity :: Applicative f => Codensity f a -> f a
codensityToAdjunction :: Adjunction f g => Codensity g a -> g (f a)
adjunctionToCodensity :: Adjunction f g => g (f a) -> Codensity g a
codensityToRan :: Codensity g a -> Ran g g a
ranToCodensity :: Ran g g a -> Codensity g a
codensityToComposedRep :: Representable u => Codensity u a -> u (Rep u, a)
composedRepToCodensity :: Representable u => u (Rep u, a) -> Codensity u a
wrapCodensity :: (forall a. m a -> m a) -> Codensity m ()
improve :: Functor f => (forall m. MonadFree f m => m a) -> Free f a

Documentation

newtype Codensity (m :: k -> TYPE rep) a Source #

Codensity f is the Monad generated by taking the right Kan extension of any Functor f along itself (Ran f f).

This can often be more "efficient" to construct than f itself using repeated applications of (>>=).

See "Asymptotic Improvement of Computations over Free Monads" by Janis Voigtländer for more information about this type.

https://www.janis-voigtlaender.eu/papers/AsymptoticImprovementOfComputationsOverFreeMonads.pdf

Constructors

Codensity
Fields runCodensity :: forall b. (a -> m b) -> m b

Instances

MonadReader r m => MonadReader r (Codensity m) Source #
Instance details Defined in Control.Monad.Codensity Methods ask :: Codensity m r # local :: (r -> r) -> Codensity m a -> Codensity m a # reader :: (r -> a) -> Codensity m a #
MonadReader r m => MonadState r (Codensity m) Source #
Instance details Defined in Control.Monad.Codensity Methods get :: Codensity m r # put :: r -> Codensity m () # state :: (r -> (a, r)) -> Codensity m a #
(Functor f, MonadFree f m) => MonadFree f (Codensity m) Source #
Instance details Defined in Control.Monad.Codensity Methods wrap :: f (Codensity m a) -> Codensity m a #
MonadTrans (Codensity :: (Type -> Type) -> Type -> Type) Source #
Instance details Defined in Control.Monad.Codensity Methods lift :: Monad m => m a -> Codensity m a #
Monad (Codensity f) Source #
Instance details Defined in Control.Monad.Codensity Methods (>>=) :: Codensity f a -> (a -> Codensity f b) -> Codensity f b # (>>) :: Codensity f a -> Codensity f b -> Codensity f b # return :: a -> Codensity f a # fail :: String -> Codensity f a #
Functor (Codensity k) Source #
Instance details Defined in Control.Monad.Codensity Methods fmap :: (a -> b) -> Codensity k a -> Codensity k b # (<$) :: a -> Codensity k b -> Codensity k a #
MonadFail f => MonadFail (Codensity f) Source #
Instance details Defined in Control.Monad.Codensity Methods fail :: String -> Codensity f a #
Applicative (Codensity f) Source #
Instance details Defined in Control.Monad.Codensity Methods pure :: a -> Codensity f a # (<>) :: Codensity f (a -> b) -> Codensity f a -> Codensity f b # liftA2 :: (a -> b -> c) -> Codensity f a -> Codensity f b -> Codensity f c # (>) :: Codensity f a -> Codensity f b -> Codensity f b # (<*) :: Codensity f a -> Codensity f b -> Codensity f a #
MonadIO m => MonadIO (Codensity m) Source #
Instance details Defined in Control.Monad.Codensity Methods liftIO :: IO a -> Codensity m a #
Alternative v => Alternative (Codensity v) Source #
Instance details Defined in Control.Monad.Codensity Methods empty :: Codensity v a # (<\|>) :: Codensity v a -> Codensity v a -> Codensity v a # some :: Codensity v a -> Codensity v [a] # many :: Codensity v a -> Codensity v [a] #
Alternative v => MonadPlus (Codensity v) Source #
Instance details Defined in Control.Monad.Codensity Methods mzero :: Codensity v a # mplus :: Codensity v a -> Codensity v a -> Codensity v a #
Plus v => Plus (Codensity v) Source #
Instance details Defined in Control.Monad.Codensity Methods zero :: Codensity v a #
Alt v => Alt (Codensity v) Source #
Instance details Defined in Control.Monad.Codensity Methods (<!>) :: Codensity v a -> Codensity v a -> Codensity v a # some :: Applicative (Codensity v) => Codensity v a -> Codensity v [a] # many :: Applicative (Codensity v) => Codensity v a -> Codensity v [a] #
Apply (Codensity f) Source #
Instance details Defined in Control.Monad.Codensity Methods (<.>) :: Codensity f (a -> b) -> Codensity f a -> Codensity f b # (.>) :: Codensity f a -> Codensity f b -> Codensity f b # (<.) :: Codensity f a -> Codensity f b -> Codensity f a # liftF2 :: (a -> b -> c) -> Codensity f a -> Codensity f b -> Codensity f c #

lowerCodensity :: Applicative f => Codensity f a -> f a Source #

This serves as the *left*-inverse (retraction) of lift.

lowerCodensity . lift ≡ id

In general this is not a full 2-sided inverse, merely a retraction, as Codensity m is often considerably "larger" than m.

e.g. Codensity ((->) s)) a ~ forall r. (a -> s -> r) -> s -> r could support a full complement of MonadState s actions, while (->) s is limited to MonadReader s actions.

codensityToAdjunction :: Adjunction f g => Codensity g a -> g (f a) Source #

The Codensity monad of a right adjoint is isomorphic to the monad obtained from the Adjunction.

codensityToAdjunction . adjunctionToCodensity ≡ id
adjunctionToCodensity . codensityToAdjunction ≡ id

adjunctionToCodensity :: Adjunction f g => g (f a) -> Codensity g a Source #

codensityToRan :: Codensity g a -> Ran g g a Source #

The Codensity Monad of a Functor g is the right Kan extension (Ran) of g along itself.

codensityToRan . ranToCodensity ≡ id
ranToCodensity . codensityToRan ≡ id

ranToCodensity :: Ran g g a -> Codensity g a Source #

codensityToComposedRep :: Representable u => Codensity u a -> u (Rep u, a) Source #

The Codensity monad of a representable Functor is isomorphic to the monad obtained from the Adjunction for which that Functor is the right adjoint.

codensityToComposedRep . composedRepToCodensity ≡ id
composedRepToCodensity . codensityToComposedRep ≡ id

codensityToComposedRep = ranToComposedRep . codensityToRan

composedRepToCodensity :: Representable u => u (Rep u, a) -> Codensity u a Source #

composedRepToCodensity = ranToCodensity . composedRepToRan

wrapCodensity :: (forall a. m a -> m a) -> Codensity m () Source #

Wrap the remainder of the Codensity action using the given function.

This function can be used to register cleanup actions that will be executed at the end. Example:

wrapCodensity (`finally` putStrLn "Done.")

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

Right associate all binds in a computation that generates a free monad

This can improve the asymptotic efficiency of the result, while preserving semantics.

See "Asymptotic Improvement of Computations over Free Monads" by Janis Voightländer for more information about this combinator.

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