Control.Monad.Trans.Iter

free-4.12.4: Monads for free

Copyright | (C) 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.Iter

Contents

- The iterative monad transformer
- Capretta's iterative monad
- Combinators
- Consuming iterative monads
- IterT ~ FreeT Identity
- Examples

Description

Based on Capretta's Iterative Monad Transformer

Unlike `Free`, this is a true monad transformer.

Synopsis

- newtype IterT m a = IterT {
  - runIterT :: m (Either a (IterT m a))
  }
- type Iter = IterT Identity
- iter :: Either a (Iter a) -> Iter a
- runIter :: Iter a -> Either a (Iter a)
- delay :: (Monad f, MonadFree f m) => m a -> m a
- hoistIterT :: Monad n => (forall a. m a -> n a) -> IterT m b -> IterT n b
- liftIter :: Monad m => Iter a -> IterT m a
- cutoff :: Monad m => Integer -> IterT m a -> IterT m (Maybe a)
- never :: (Monad f, MonadFree f m) => m a
- untilJust :: Monad m => m (Maybe a) -> IterT m a
- interleave :: Monad m => [IterT m a] -> IterT m [a]
- interleave_ :: Monad m => [IterT m a] -> IterT m ()
- retract :: Monad m => IterT m a -> m a
- fold :: Monad m => (m a -> a) -> IterT m a -> a
- foldM :: (Monad m, Monad n) => (m (n a) -> n a) -> IterT m a -> n a
- class Monad m => MonadFree f m | m -> f where
  - wrap :: f (m a) -> m a

# Documentation

Functions in Haskell are meant to be pure. For example, if an expression has type Int, there should exist a value of the type such that the expression can be replaced by that value in any context without changing the meaning of the program.

Some computations may perform side effects (`unsafePerformIO`), throw an exception (using `error`); or not terminate (`let infinity = 1 + infinity in infinity`).

While the `IO` monad encapsulates side-effects, and the `Either` monad encapsulates errors, the `Iter` monad encapsulates non-termination. The `IterT` transformer generalizes non-termination to any monadic computation.

Computations in `IterT` (or `Iter`) can be composed in two ways:

- Sequential: Using the `Monad` instance, the result of a computation can be fed into the next.
- Parallel: Using the `MonadPlus` instance, several computations can be executed concurrently, and the first to finish will prevail. See also the cabbage example.

# The iterative monad transformer

newtype IterT m a

The monad supporting iteration based over a base monad `m`.

    IterT ~ FreeT Identity

Constructors

IterT |
Fields runIterT :: m (Either a (IterT m a))

Instances

MonadTrans IterT |
MonadError e m => MonadError e (IterT m) |
MonadReader e m => MonadReader e (IterT m) |
MonadState s m => MonadState s (IterT m) |
MonadWriter w m => MonadWriter w (IterT m) |
Monad m => MonadFree Identity (IterT m) |
Monad m => Monad (IterT m) |
Monad m => Functor (IterT m) |
MonadFix m => MonadFix (IterT m) |
Monad m => Applicative (IterT m) |
Foldable m => Foldable (IterT m) |
(Monad m, Traversable m) => Traversable (IterT m) |
Monad m => Alternative (IterT m) |
Monad m => MonadPlus (IterT m) | Capretta's `race` combinator. Satisfies left catch.
MonadThrow m => MonadThrow (IterT m) |
MonadCatch m => MonadCatch (IterT m) |
MonadIO m => MonadIO (IterT m) |
MonadCont m => MonadCont (IterT m) |
(Functor m, Eq1 m) => Eq1 (IterT m) |
(Functor m, Ord1 m) => Ord1 (IterT m) |
(Functor m, Show1 m) => Show1 (IterT m) |
(Functor m, Read1 m) => Read1 (IterT m) |
(Monad m, Traversable1 m) => Traversable1 (IterT m) |
Monad m => Apply (IterT m) |
Monad m => Bind (IterT m) |
Foldable1 m => Foldable1 (IterT m) |
Eq (m (Either a (IterT m a))) => Eq (IterT m a) |
(Typeable (* -> *) m, Typeable * a, Data (m (Either a (IterT m a))), Data a) => Data (IterT m a) |
Ord (m (Either a (IterT m a))) => Ord (IterT m a) |
Read (m (Either a (IterT m a))) => Read (IterT m a) |
Show (m (Either a (IterT m a))) => Show (IterT m a) |
(Monad m, Monoid a) => Monoid (IterT m a) |

# Capretta's iterative monad

type Iter = IterT Identity

Plain iterative computations.

iter :: Either a (Iter a) -> Iter a

Builds an iterative computation from one first step.

    runIter . iter == id

runIter :: Iter a -> Either a (Iter a)

Executes the first step of an iterative computation

    iter . runIter == id

# Combinators

delay :: (Monad f, MonadFree f m) => m a -> m a

Adds an extra layer to a free monad value.

In particular, for the iterative monad `Iter`, this makes the computation require one more step, without changing its final result.

    runIter (delay ma) == Right ma

hoistIterT :: Monad n => (forall a. m a -> n a) -> IterT m b -> IterT n b

Lift a monad homomorphism from `m` to `n` into a Monad homomorphism from ``IterT` m` to ``IterT` n`.

liftIter :: Monad m => Iter a -> IterT m a

Lifts a plain, non-terminating computation into a richer environment. `liftIter` is a `Monad` homomorphism.

cutoff :: Monad m => Integer -> IterT m a -> IterT m (Maybe a)

Cuts off an iterative computation after a given number of steps. If the number of steps is 0 or less, no computation nor monadic effects will take place.

The step where the final value is produced also counts towards the limit.

Some examples (`n ≥ 0`):

    cutoff 0     _        ≡ return Nothing
    cutoff (n+1) . return ≡ return . Just
    cutoff (n+1) . lift   ≡ lift . liftM Just
    cutoff (n+1) . delay  ≡ delay . cutoff n
    cutoff n     never    ≡ iterate delay (return Nothing) !! n

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

never :: (Monad f, MonadFree f m) => m a

A computation that never terminates

untilJust :: Monad m => m (Maybe a) -> IterT m a

Repeatedly run a computation until it produces a `Just` value. This can be useful when paired with a monad that has side effects.

For example, we may have `genId :: IO (Maybe Id)` that uses a random number generator to allocate ids, but fails if it finds a collision. We can repeatedly run this with

    retract (untilJust genId) :: IO Id

interleave :: Monad m => [IterT m a] -> IterT m [a]

Interleaves the steps of a finite list of iterative computations, and collects their results.

The resulting computation has as many steps as the longest computation in the list.

interleave_ :: Monad m => [IterT m a] -> IterT m ()

Interleaves the steps of a finite list of computations, and discards their results.

The resulting computation has as many steps as the longest computation in the list.

Equivalent to ``void` `.` `interleave``.

# Consuming iterative monads

retract :: Monad m => IterT m a -> m a

`retract` is the left inverse of `lift`

    retract . lift = id

fold :: Monad m => (m a -> a) -> IterT m a -> a

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

foldM :: (Monad m, Monad n) => (m (n a) -> n a) -> IterT m a -> n a

Like `fold` with monadic result.

# IterT ~ FreeT Identity

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) |

# Examples

- Rendering the Mandelbrot set
- The wolf, the sheep and the cabbage