Control.Monad.Codensity

kan-extensions-5.0.1: Kan extensions, Kan lifts, various forms of the Yoneda lemma, and (co)density (co)monads

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

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

# 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 |
MonadReader r m => MonadState r (Codensity m) |
MonadReader r m => MonadReader r (Codensity m) |
(Functor f, MonadFree f m) => MonadFree f (Codensity m) |
Monad (Codensity f) |
Functor (Codensity k) |
Applicative (Codensity f) |
Alternative v => Alternative (Codensity v) |
Alternative v => MonadPlus (Codensity v) |
MonadIO m => MonadIO (Codensity m) |
Plus v => Plus (Codensity v) |
Alt v => Alt (Codensity v) |
Apply (Codensity f) |

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