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 |

Based on Capretta's Iterative Monad Transformer

Unlike `Free`

, this is a true monad transformer.

- newtype IterT m a = IterT {}
- 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

# Capretta's iterative monad

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

Builds an iterative computation from one first step.

runIter . iter == id

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

Executes the first step of an iterative computation

iter . runIter == id

# Combinators

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

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 Source

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

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

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

`.`

`cutoff`

n

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

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] Source

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 () Source

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

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

Like `fold`

with monadic result.

# IterT ~ FreeT Identity

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

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

smashes the lists flat.`join`

[[a]]

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

instead of `Free`

Pair`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`

.

Nothing