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	Haskell98

Control.Monad.Codensity

Description

Synopsis

Documentation

newtype Codensity m a #

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.

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

Constructors

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

Instances

MonadTrans Codensity #
Methods lift :: Monad m => m a -> Codensity m a #
MonadReader r m => MonadState r (Codensity m) #
Methods get :: Codensity m r # put :: r -> Codensity m () # state :: (r -> (a, r)) -> Codensity m a #
MonadReader r m => MonadReader r (Codensity m) #
Methods ask :: Codensity m r # local :: (r -> r) -> Codensity m a -> Codensity m a # reader :: (r -> a) -> Codensity m a #
(Functor f, MonadFree f m) => MonadFree f (Codensity m) #
Methods wrap :: f (Codensity m a) -> Codensity m a #
Monad (Codensity f) #
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) #
Methods fmap :: (a -> b) -> Codensity k a -> Codensity k b # (<$) :: a -> Codensity k b -> Codensity k a #
MonadFail f => MonadFail (Codensity f) #
Methods fail :: String -> Codensity f a #
Applicative (Codensity f) #
Methods pure :: a -> Codensity f a # (<>) :: 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 #
MonadIO m => MonadIO (Codensity m) #
Methods liftIO :: IO a -> Codensity m a #
Alternative v => Alternative (Codensity v) #
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) #
Methods mzero :: Codensity v a # mplus :: Codensity v a -> Codensity v a -> Codensity v a #
Plus v => Plus (Codensity v) #
Methods zero :: Codensity v a #
Alt v => Alt (Codensity v) #
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) #
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 #

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

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) #

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 #

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

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 #

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

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 #

composedRepToCodensity = ranToCodensity . composedRepToRan

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

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