Control.Monad.Codensity

Documentation

newtype Codensity (m :: k -> TYPE rep) 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.

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)

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)

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)

Defined in Control.Monad.Codensity

Methods

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

MonadTrans (Codensity :: (Type -> Type) -> Type -> Type)

Defined in Control.Monad.Codensity

Methods

lift :: Monad m => m a -> Codensity m a

Monad (Codensity f)

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

Functor (Codensity k)

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)

Defined in Control.Monad.Codensity

Methods

fail :: String -> Codensity f a

Applicative (Codensity f)

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)

Defined in Control.Monad.Codensity

Methods

liftIO :: IO a -> Codensity m a

Alternative v => Alternative (Codensity v)

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)

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)

Defined in Control.Monad.Codensity

Methods

zero :: Codensity v a

Alt v => Alt (Codensity v)

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)

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

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

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

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

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

Delimited continuations

reset :: Monad m => Codensity m a -> Codensity m a

reset m delimits the continuation of any shift inside m.

• reset (return m) = return m

shift :: Applicative m => (forall b. (a -> m b) -> Codensity m b) -> Codensity m a

shift f captures the continuation up to the nearest enclosing reset and passes it to f:

• reset (shift f >>= k) = reset (f (lowerCodensity . k))