Control.Comonad.Cofree

free-4.12.4: Monads for free

Copyright | (C) 2008-2013 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | MPTCs, fundeps
Safe Haskell | Safe
Language | Haskell2010

Control.Comonad.Cofree

Contents

- Lenses into cofree comonads

Description

Cofree comonads

Synopsis

- data Cofree f a = a :< (f (Cofree f a))
- class (Functor f, Comonad w) => ComonadCofree f w | w -> f where
  - unwrap :: w a -> f (w a)
- section :: Comonad f => f a -> Cofree f a
- coiter :: Functor f => (a -> f a) -> a -> Cofree f a
- coiterW :: (Comonad w, Functor f) => (w a -> f (w a)) -> w a -> Cofree f a
- unfold :: Functor f => (b -> (a, f b)) -> b -> Cofree f a
- unfoldM :: (Traversable f, Monad m) => (b -> m (a, f b)) -> b -> m (Cofree f a)
- hoistCofree :: Functor f => (forall x. f x -> g x) -> Cofree f a -> Cofree g a
- _extract :: Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a)
- _unwrap :: Functor f => (g (Cofree g a) -> f (g (Cofree g a))) -> Cofree g a -> f (Cofree g a)
- telescoped :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (a -> f a) -> Cofree g a -> f (Cofree g a)

# Documentation

data Cofree f a

The `Cofree` `Comonad` of a functor `f`.

Formally

A `Comonad` `v` is a cofree `Comonad` for `f` if every comonad homomorphism from another comonad `w` to `v` is equivalent to a natural transformation from `w` to `f`.

A `cofree` functor is right adjoint to a forgetful functor.

Cofree is a functor from the category of functors to the category of comonads that is right adjoint to the forgetful functor from the category of comonads to the category of functors that forgets how to `extract` and `duplicate`, leaving you with only a `Functor`.

In practice, cofree comonads are quite useful for annotating syntax trees, or talking about streams.

A number of common comonads arise directly as cofree comonads.

For instance,

- ``Cofree` `Maybe`` forms the a comonad for a non-empty list.
- ``Cofree` (`Const` b)` is a product.
- ``Cofree` `Identity`` forms an infinite stream.
- ``Cofree` ((->) b)'` describes a Moore machine with states labeled with values of type a, and transitions on edges of type b.

Furthermore, if the functor `f` forms a monoid (for example, by being an instance of `Alternative`), the resulting `Comonad` is also a `Monad`. See Monadic Augment and Generalised Shortcut Fusion by Neil Ghani et al., Section 4.3 for more details.

In particular, if `f a ≡ [a]`, the resulting data structure is a Rose tree. For a practical application, check Higher Dimensional Trees, Algebraically by Neil Ghani et al.

Constructors

a :< (f (Cofree f a)) infixr 5 |

Instances

ComonadHoist Cofree |
ComonadTrans Cofree | This is not a true `Comonad` transformer, but this instance is convenient.
ComonadTraced m w => ComonadTraced m (Cofree w) |
ComonadStore s w => ComonadStore s (Cofree w) |
ComonadEnv e w => ComonadEnv e (Cofree w) |
Functor f => ComonadCofree f (Cofree f) |
Alternative f => Monad (Cofree f) |
Functor f => Functor (Cofree f) |
Alternative f => Applicative (Cofree f) |
Foldable f => Foldable (Cofree f) |
Traversable f => Traversable (Cofree f) |
(Alternative f, MonadZip f) => MonadZip (Cofree f) |
Functor f => Comonad (Cofree f) |
ComonadApply f => ComonadApply (Cofree f) |
Distributive f => Distributive (Cofree f) |
(Functor f, Eq1 f) => Eq1 (Cofree f) |
(Functor f, Ord1 f) => Ord1 (Cofree f) |
(Functor f, Show1 f) => Show1 (Cofree f) |
(Functor f, Read1 f) => Read1 (Cofree f) |
Traversable1 f => Traversable1 (Cofree f) |
Apply f => Apply (Cofree f) |
Functor f => Extend (Cofree f) |
Foldable1 f => Foldable1 (Cofree f) |
(Eq (f (Cofree f a)), Eq a) => Eq (Cofree f a) |
(Ord (f (Cofree f a)), Ord a) => Ord (Cofree f a) |
(Read (f (Cofree f a)), Read a) => Read (Cofree f a) |
(Show (f (Cofree f a)), Show a) => Show (Cofree f a) |

class (Functor f, Comonad w) => ComonadCofree f w | w -> f where

Allows you to peel a layer off a cofree comonad.

Methods

unwrap :: w a -> f (w a)

Remove a layer.

Instances

ComonadCofree [] Tree |
ComonadCofree Maybe NonEmpty |
ComonadCofree f w => ComonadCofree f (IdentityT w) |
Functor f => ComonadCofree f (Cofree f) |
Comonad w => ComonadCofree Identity (CoiterT w) |
(ComonadCofree f w, Monoid m) => ComonadCofree f (TracedT m w) |
ComonadCofree f w => ComonadCofree f (StoreT s w) |
ComonadCofree f w => ComonadCofree f (EnvT e w) |
(Functor f, Comonad w) => ComonadCofree f (CofreeT f w) |
ComonadCofree (Const b) ((,) b) |

section :: Comonad f => f a -> Cofree f a

    lower . section = id

coiter :: Functor f => (a -> f a) -> a -> Cofree f a

Use coiteration to generate a cofree comonad from a seed.

    coiter f = unfold (id &&& f)

coiterW :: (Comonad w, Functor f) => (w a -> f (w a)) -> w a -> Cofree f a

Like coiter for comonadic values.

unfold :: Functor f => (b -> (a, f b)) -> b -> Cofree f a

Unfold a cofree comonad from a seed.

unfoldM :: (Traversable f, Monad m) => (b -> m (a, f b)) -> b -> m (Cofree f a)

Unfold a cofree comonad from a seed, monadically.

hoistCofree :: Functor f => (forall x. f x -> g x) -> Cofree f a -> Cofree g a

# Lenses into cofree comonads

_extract :: Functor f => (a -> f a) -> Cofree g a -> f (Cofree g a)

This is a lens that can be used to read or write from the target of `extract`.

Using (^.) from the `lens` package:

    foo ^. _extract == extract foo

For more on lenses see the `lens` package on hackage

    _extract :: Lens' (Cofree g a) a

_unwrap :: Functor f => (g (Cofree g a) -> f (g (Cofree g a))) -> Cofree g a -> f (Cofree g a)

This is a lens that can be used to read or write to the tails of a `Cofree` `Comonad`.

Using (^.) from the `lens` package:

    foo ^. _unwrap == unwrap foo

For more on lenses see the `lens` package on hackage

    _unwrap :: Lens' (Cofree g a) (g (Cofree g a))

telescoped :: Functor f => [(Cofree g a -> f (Cofree g a)) -> g (Cofree g a) -> f (g (Cofree g a))] -> (a -> f a) -> Cofree g a -> f (Cofree g a)

Construct an `Lens` into a ``Cofree` g` given a list of lenses into the base functor. When the input list is empty, this is equivalent to `_extract`. When the input list is non-empty, this composes the input lenses with `_unwrap` to walk through the ``Cofree` g` before using `_extract` to get the element at the final location.

For more on lenses see the `lens` package on hackage.

    telescoped :: [Lens' (g (Cofree g a)) (Cofree g a)]      -> Lens' (Cofree g a) a

    telescoped :: [Traversal' (g (Cofree g a)) (Cofree g a)] -> Traversal' (Cofree g a) a

    telescoped :: [Getter (g (Cofree g a)) (Cofree g a)]     -> Getter (Cofree g a) a

    telescoped :: [Fold (g (Cofree g a)) (Cofree g a)]       -> Fold (Cofree g a) a

    telescoped :: [Setter' (g (Cofree g a)) (Cofree g a)]    -> Setter' (Cofree g a) a