Control.Monad.Trans.Free

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.Monad.Trans.Free

Contents

- The base functor
- The free monad transformer
- The free monad
- Operations
- Operations of free monad
- Free Monads With Class

Description

The free monad transformer

Synopsis

- data FreeF f a b
  - = Pure a
  - | Free (f b)
- newtype FreeT f m a = FreeT {
  - runFreeT :: m (FreeF f a (FreeT f m a))
  }
- type Free f = FreeT f Identity
- free :: FreeF f a (Free f a) -> Free f a
- runFree :: Free f a -> FreeF f a (Free f a)
- liftF :: (Functor f, MonadFree f m) => f a -> m a
- iterT :: (Functor f, Monad m) => (f (m a) -> m a) -> FreeT f m a -> m a
- iterTM :: (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FreeT f m a -> t m a
- hoistFreeT :: (Monad m, Functor f) => (forall a. m a -> n a) -> FreeT f m b -> FreeT f n b
- transFreeT :: (Monad m, Functor g) => (forall a. f a -> g a) -> FreeT f m b -> FreeT g m b
- joinFreeT :: (Monad m, Traversable f) => FreeT f m a -> m (Free f a)
- cutoff :: (Functor f, Monad m) => Integer -> FreeT f m a -> FreeT f m (Maybe a)
- partialIterT :: Monad m => Integer -> (forall a. f a -> m a) -> FreeT f m b -> FreeT f m b
- intersperseT :: (Monad m, Functor f) => f a -> FreeT f m b -> FreeT f m b
- intercalateT :: (Monad m, MonadTrans t, Monad (t m)) => t m a -> FreeT (t m) m b -> t m b
- retractT :: (MonadTrans t, Monad (t m), Monad m) => FreeT (t m) m a -> t m a
- retract :: Monad f => Free f a -> f a
- iter :: Functor f => (f a -> a) -> Free f a -> a
- iterM :: (Functor f, Monad m) => (f (m a) -> m a) -> Free f a -> m a
- class Monad m => MonadFree f m | m -> f where
  - wrap :: f (m a) -> m a

# The base functor

data FreeF f a b

The base functor for a free monad.

Constructors

Pure a |
Free (f b) |

Instances

Functor f => Bifunctor (FreeF f) |
Traversable f => Bitraversable (FreeF f) |
Foldable f => Bifoldable (FreeF f) |
Eq1 f => Eq2 (FreeF f) |
Ord1 f => Ord2 (FreeF f) |
Show1 f => Show2 (FreeF f) |
Read1 f => Read2 (FreeF f) |
Functor f => Functor (FreeF f a) |
Foldable f => Foldable (FreeF f a) |
Traversable f => Traversable (FreeF f a) |
(Eq1 f, Eq a) => Eq1 (FreeF f a) |
(Ord1 f, Ord a) => Ord1 (FreeF f a) |
(Show1 f, Show a) => Show1 (FreeF f a) |
(Read1 f, Read a) => Read1 (FreeF f a) |
(Eq a, Eq (f b)) => Eq (FreeF f a b) |
(Ord a, Ord (f b)) => Ord (FreeF f a b) |
(Read a, Read (f b)) => Read (FreeF f a b) |
(Show a, Show (f b)) => Show (FreeF f a b) |

# The free monad transformer

newtype FreeT f m a

The "free monad transformer" for a functor `f`

Constructors

FreeT |
Fields runFreeT :: m (FreeF f a (FreeT f m a))

Instances

(Functor f, MonadError e m) => MonadError e (FreeT f m) |
(Functor f, MonadReader r m) => MonadReader r (FreeT f m) |
(Functor f, MonadState s m) => MonadState s (FreeT f m) |
(Functor f, MonadWriter w m) => MonadWriter w (FreeT f m) |
(Functor f, Monad m) => MonadFree f (FreeT f m) |
MonadTrans (FreeT f) |
(Functor f, Monad m) => Monad (FreeT f m) |
(Functor f, Monad m) => Functor (FreeT f m) |
(Functor f, Monad m) => Applicative (FreeT f m) |
(Foldable m, Foldable f) => Foldable (FreeT f m) |
(Monad m, Traversable m, Traversable f) => Traversable (FreeT f m) |
(Functor f, MonadPlus m) => Alternative (FreeT f m) |
(Functor f, MonadPlus m) => MonadPlus (FreeT f m) |
(Functor f, MonadThrow m) => MonadThrow (FreeT f m) |
(Functor f, MonadCatch m) => MonadCatch (FreeT f m) |
(Functor f, MonadIO m) => MonadIO (FreeT f m) |
(Functor f, MonadCont m) => MonadCont (FreeT f m) |
(Functor f, Eq1 f, Functor m, Eq1 m) => Eq1 (FreeT f m) |
(Functor f, Ord1 f, Functor m, Ord1 m) => Ord1 (FreeT f m) |
(Functor f, Show1 f, Functor m, Show1 m) => Show1 (FreeT f m) |
(Functor f, Read1 f, Functor m, Read1 m) => Read1 (FreeT f m) |
(Functor f, Monad m) => Apply (FreeT f m) |
(Functor f, Monad m) => Bind (FreeT f m) |
Eq (m (FreeF f a (FreeT f m a))) => Eq (FreeT f m a) |
Ord (m (FreeF f a (FreeT f m a))) => Ord (FreeT f m a) |
Read (m (FreeF f a (FreeT f m a))) => Read (FreeT f m a) |
Show (m (FreeF f a (FreeT f m a))) => Show (FreeT f m a) |

# The free monad

type Free f = FreeT f Identity

The "free monad" for a functor `f`.

free :: FreeF f a (Free f a) -> Free f a

Pushes a layer into a free monad value.

runFree :: Free f a -> FreeF f a (Free f a)

Evaluates the first layer out of a free monad value.

# Operations

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

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

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

Tear down a free monad transformer using iteration.

iterTM :: (Functor f, Monad m, MonadTrans t, Monad (t m)) => (f (t m a) -> t m a) -> FreeT f m a -> t m a

Tear down a free monad transformer using iteration over a transformer.

hoistFreeT :: (Monad m, Functor f) => (forall a. m a -> n a) -> FreeT f m b -> FreeT f n b

Lift a monad homomorphism from `m` to `n` into a monad homomorphism from ``FreeT` f m` to ``FreeT` f n`

    hoistFreeT :: (Monad m, Functor f) => (m ~> n) -> FreeT f m ~> FreeT f n

transFreeT :: (Monad m, Functor g) => (forall a. f a -> g a) -> FreeT f m b -> FreeT g m b

Lift a natural transformation from `f` to `g` into a monad homomorphism from ``FreeT` f m` to ``FreeT` g m`

joinFreeT :: (Monad m, Traversable f) => FreeT f m a -> m (Free f a)

Pull out and join `m` layers of ``FreeT` f m a`.

cutoff :: (Functor f, Monad m) => Integer -> FreeT f m a -> FreeT f m (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.

partialIterT :: Monad m => Integer -> (forall a. f a -> m a) -> FreeT f m b -> FreeT f m b

`partialIterT n phi m` interprets first `n` layers of `m` using `phi`. This is sort of the opposite for ``cutoff``.

Some examples (`n ≥ 0`):

    partialIterT 0 _ m              ≡ m
    partialIterT (n+1) phi . return ≡ return
    partialIterT (n+1) phi . lift   ≡ lift
    partialIterT (n+1) phi . wrap   ≡ join . lift . phi

intersperseT :: (Monad m, Functor f) => f a -> FreeT f m b -> FreeT f m b

`intersperseT f m` inserts a layer `f` between every two layers in `m`.

    intersperseT f . return ≡ return
    intersperseT f . lift   ≡ lift
    intersperseT f . wrap   ≡ wrap . fmap (iterTM (wrap . (<$ f) . wrap))

intercalateT :: (Monad m, MonadTrans t, Monad (t m)) => t m a -> FreeT (t m) m b -> t m b

`intercalateT f m` inserts a layer `f` between every two layers in `m` and then retracts the result.

    intercalateT f ≡ retractT . intersperseT f

retractT :: (MonadTrans t, Monad (t m), Monad m) => FreeT (t m) m a -> t m a

Tear down a free monad transformer using Monad instance for `t m`.

# Operations of free monad

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

`retract` is the left inverse of `liftF`

    retract . liftF = id

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

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

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

Like `iter` for monadic values.

# Free Monads With Class

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) |