Control.Monad.Codensity

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 Voightländer 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

MonadReader r m => MonadState r (Codensity m)

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

Monad (Codensity f)

Functor (Codensity k)

MonadPlus v => MonadPlus (Codensity v)

Applicative (Codensity f)

Alternative v => Alternative (Codensity v)

MonadIO m => MonadIO (Codensity m)

Plus v => Plus (Codensity v)

Alt v => Alt (Codensity v)

Apply (Codensity f)

MonadSpec (Codensity m)

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