Control.Monad.Free

free-4.12.4: Monads for free

Copyright | (C) 2008-2015 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.Monad.Free

Description

Monads for free

Synopsis

- class Monad m => MonadFree f m | m -> f where
  - wrap :: f (m a) -> m a
- data Free f a
  - = Pure a
  - | Free (f (Free f a))
- retract :: Monad f => Free f a -> f a
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iter :: Functor f => (f a -> a) -> Free f a -> a
- iterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a
- iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a
- hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b
- foldFree :: Monad m => (forall x. f x -> m x) -> Free f a -> m a
- toFreeT :: (Functor f, Monad m) => Free f a -> FreeT f m a
- cutoff :: Functor f => Integer -> Free f a -> Free f (Maybe a)
- unfold :: Functor f => (b -> Either a (f b)) -> b -> Free f a
- unfoldM :: (Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a)
- _Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a))
- _Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a))

# Documentation

class Monad m => MonadFree f m | m -> f where

Monads provide substitution (`fmap`) and renormalization (`join`):

    m >>= f = join (fmap f m)

A free `Monad` is one that does no work during the normalization step beyond simply grafting the two monadic values together.

`[]` is not a free `Monad` (in this sense) because ``join` [[a]]` smashes the lists flat.

On the other hand, consider:

    data Tree a = Bin (Tree a) (Tree a) | Tip a

    instance Monad Tree where
      return = Tip
      Tip a >>= f = f a
      Bin l r >>= f = Bin (l >>= f) (r >>= f)

This `Monad` is the free `Monad` of Pair:

    data Pair a = Pair a a

And we could make an instance of `MonadFree` for it directly:

    instance MonadFree Pair Tree where
       wrap (Pair l r) = Bin l r

Or we could choose to program with ``Free` Pair` instead of `Tree` and thereby avoid having to define our own `Monad` instance.

Moreover, Control.Monad.Free.Church provides a `MonadFree` instance that can improve the asymptotic complexity of code that constructs free monads by effectively reassociating the use of (`>>=`). You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

See `Free` for a more formal definition of the free `Monad` for a `Functor`.

Minimal complete definition

Nothing

Methods

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

Add a layer.

    wrap (fmap f x) ≡ wrap (fmap return x) >>= f

Instances

(Functor f, MonadFree f m) => MonadFree f (ListT m) |
(Functor f, MonadFree f m) => MonadFree f (IdentityT m) |
(Functor f, MonadFree f m) => MonadFree f (MaybeT m) |
Functor f => MonadFree f (Free f) |
Functor f => MonadFree f (F f) |
Monad m => MonadFree Identity (IterT m) |
(Functor f, MonadFree f m) => MonadFree f (ExceptT e m) |
(Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m) |
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) |
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) |
(Functor f, MonadFree f m) => MonadFree f (ContT r m) |
(Functor f, MonadFree f m) => MonadFree f (StateT s m) |
(Functor f, MonadFree f m) => MonadFree f (StateT s m) |
(Functor f, MonadFree f m) => MonadFree f (ReaderT e m) |
(Functor f, Monad m) => MonadFree f (FreeT f m) |
MonadFree f (FT f m) |
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) |
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) |

data Free f a

The `Free` `Monad` for a `Functor` `f`.

Formally

A `Monad` `n` is a free `Monad` for `f` if every monad homomorphism from `n` to another monad `m` is equivalent to a natural transformation from `f` to `m`.

Why Free?

Every "free" functor is left adjoint to some "forgetful" functor.

If we define a forgetful functor `U` from the category of monads to the category of functors that just forgets the `Monad`, leaving only the `Functor`. i.e.

    U (M,return,join) = M

then `Free` is the left adjoint to `U`.

Being `Free` being left adjoint to `U` means that there is an isomorphism between

``Free` f -> m` in the category of monads and `f -> U m` in the category of functors.

Morphisms in the category of monads are `Monad` homomorphisms (natural transformations that respect `return` and `join`).

Morphisms in the category of functors are `Functor` homomorphisms (natural transformations).

Given this isomorphism, every monad homomorphism from ``Free` f` to `m` is equivalent to a natural transformation from `f` to `m`

Showing that this isomorphism holds is left as an exercise.

In practice, you can just view a ``Free` f a` as many layers of `f` wrapped around values of type `a`, where `(`>>=`)` performs substitution and grafts new layers of `f` in for each of the free variables.

This can be very useful for modeling domain specific languages, trees, or other constructs.

This instance of `MonadFree` is fairly naive about the encoding. For more efficient free monad implementation see Control.Monad.Free.Church, in particular note the `improve` combinator. You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

A number of common monads arise as free monads,

- Given `data Empty a`, ``Free` Empty` is isomorphic to the `Identity` monad.
- ``Free` `Maybe`` can be used to model a partiality monad where each layer represents running the computation for a while longer.

Constructors

Pure a |
Free (f (Free f a)) |

Instances

MonadTrans Free | This is not a true monad transformer. It is only a monad transformer "up to `retract`".
(Functor m, MonadError e m) => MonadError e (Free m) |
(Functor m, MonadReader e m) => MonadReader e (Free m) |
(Functor m, MonadState s m) => MonadState s (Free m) |
(Functor m, MonadWriter e m) => MonadWriter e (Free m) |
Functor f => MonadFree f (Free f) |
Functor f => Monad (Free f) |
Functor f => Functor (Free f) |
Functor f => MonadFix (Free f) |
Functor f => Applicative (Free f) |
Foldable f => Foldable (Free f) |
Traversable f => Traversable (Free f) |
Alternative v => Alternative (Free v) | This violates the Alternative laws, handle with care.
(Functor v, MonadPlus v) => MonadPlus (Free v) | This violates the MonadPlus laws, handle with care.
(Functor m, MonadCont m) => MonadCont (Free m) |
(Functor f, Eq1 f) => Eq1 (Free f) |
(Functor f, Ord1 f) => Ord1 (Free f) |
(Functor f, Show1 f) => Show1 (Free f) |
(Functor f, Read1 f) => Read1 (Free f) |
Traversable1 f => Traversable1 (Free f) |
Functor f => Apply (Free f) |
Functor f => Bind (Free f) |
Foldable1 f => Foldable1 (Free f) |
(Eq (f (Free f a)), Eq a) => Eq (Free f a) |
(Ord (f (Free f a)), Ord a) => Ord (Free f a) |
(Read (f (Free f a)), Read a) => Read (Free f a) |
(Show (f (Free f a)), Show a) => Show (Free f a) |

retract :: Monad f => Free f a -> f a

`retract` is the left inverse of `lift` and `liftF`

    retract . lift = id
    retract . liftF = id

liftF :: (Functor f, MonadFree f m) => f a -> m a

A version of lift that can be used with just a Functor for f.

iter :: Functor f => (f a -> a) -> Free f a -> a

Tear down a `Free` `Monad` using iteration.

iterA :: (Applicative p, Functor f) => (f (p a) -> p a) -> Free f a -> p a

Like `iter` for applicative values.

iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m a

Like `iter` for monadic values.

hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g b

Lift a natural transformation from `f` to `g` into a natural transformation from ``FreeT` f` to ``FreeT` g`.

foldFree :: Monad m => (forall x. f x -> m x) -> Free f a -> m a

The very definition of a free monad is that given a natural transformation you get a monad homomorphism.

toFreeT :: (Functor f, Monad m) => Free f a -> FreeT f m a

Convert a `Free` monad from Control.Monad.Free to a `FreeT` monad from Control.Monad.Trans.Free.

cutoff :: Functor f => Integer -> Free f a -> Free f (Maybe a)

Cuts off a tree of computations at a given depth. If the depth is 0 or less, no computation nor monadic effects will take place.

Some examples (n ≥ 0):

    cutoff 0     _        == return Nothing

    cutoff (n+1) . return == return . Just

    cutoff (n+1) . lift   ==   lift . liftM Just

    cutoff (n+1) . wrap   ==  wrap . fmap (cutoff n)

Calling 'retract . cutoff n' is always terminating, provided each of the steps in the iteration is terminating.

unfold :: Functor f => (b -> Either a (f b)) -> b -> Free f a

Unfold a free monad from a seed.

unfoldM :: (Traversable f, Applicative m, Monad m) => (b -> m (Either a (f b))) -> b -> m (Free f a)

Unfold a free monad from a seed, monadically.

_Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a))

This is `Prism' (Free f a) a` in disguise

    >>> preview _Pure (Pure 3)
    Just 3

    >>> review _Pure 3 :: Free Maybe Int
    Pure 3

_Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a))

This is `Prism' (Free f a) (f (Free f a))` in disguise

    >>> preview _Free (review _Free (Just (Pure 3)))
    Just (Just (Pure 3))

    >>> review _Free (Just (Pure 3))
    Free (Just (Pure 3))