free-5.2: 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 :: Type -> Type) 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 :: forall (f :: Type -> Type) m a. (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 :: forall (m :: Type -> Type) a. Monad m => Iter a -> IterT m a
- cutoff :: forall (m :: Type -> Type) a. Monad m => Integer -> IterT m a -> IterT m (Maybe a)
- never :: forall (f :: Type -> Type) m a. (Monad f, MonadFree f m) => m a
- untilJust :: Monad m => m (Maybe a) -> IterT m a
- interleave :: forall (m :: Type -> Type) a. Monad m => [IterT m a] -> IterT m [a]
- interleave_ :: forall (m :: Type -> Type) a. 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 :: Type -> Type) (m :: Type -> Type) | 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 :: Type -> Type) 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

Instances details

MonadTrans IterT # |
Instance details Defined in Control.Monad.Trans.Iter Methods lift :: Monad m => m a -> IterT m a #
Monad m => MonadFree Identity (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods wrap :: Identity (IterT m a) -> IterT m a #
MonadError e m => MonadError e (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods throwError :: e -> IterT m a # catchError :: IterT m a -> (e -> IterT m a) -> IterT m a #
MonadReader e m => MonadReader e (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods ask :: IterT m e # local :: (e -> e) -> IterT m a -> IterT m a # reader :: (e -> a) -> IterT m a #
MonadState s m => MonadState s (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods get :: IterT m s # put :: s -> IterT m () # state :: (s -> (a, s)) -> IterT m a #
MonadWriter w m => MonadWriter w (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods writer :: (a, w) -> IterT m a # tell :: w -> IterT m () # listen :: IterT m a -> IterT m (a, w) # pass :: IterT m (a, w -> w) -> IterT m a #
MonadIO m => MonadIO (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods liftIO :: IO a -> IterT m a #
Foldable1 m => Foldable1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods fold1 :: Semigroup m0 => IterT m m0 -> m0 # foldMap1 :: Semigroup m0 => (a -> m0) -> IterT m a -> m0 # foldMap1' :: Semigroup m0 => (a -> m0) -> IterT m a -> m0 # toNonEmpty :: IterT m a -> NonEmpty a # maximum :: Ord a => IterT m a -> a # minimum :: Ord a => IterT m a -> a # head :: IterT m a -> a # last :: IterT m a -> a # foldrMap1 :: (a -> b) -> (a -> b -> b) -> IterT m a -> b # foldlMap1' :: (a -> b) -> (b -> a -> b) -> IterT m a -> b # foldlMap1 :: (a -> b) -> (b -> a -> b) -> IterT m a -> b # foldrMap1' :: (a -> b) -> (a -> b -> b) -> IterT m a -> b #
Eq1 m => Eq1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods liftEq :: (a -> b -> Bool) -> IterT m a -> IterT m b -> Bool #
Ord1 m => Ord1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods liftCompare :: (a -> b -> Ordering) -> IterT m a -> IterT m b -> Ordering #
Read1 m => Read1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (IterT m a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [IterT m a] # liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (IterT m a) # liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [IterT m a] #
Show1 m => Show1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> IterT m a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [IterT m a] -> ShowS #
MonadCatch m => MonadCatch (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods catch :: (HasCallStack, Exception e) => IterT m a -> (e -> IterT m a) -> IterT m a #
MonadThrow m => MonadThrow (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods throwM :: (HasCallStack, Exception e) => e -> IterT m a #
Monad m => Alternative (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods empty :: IterT m a # (<|>) :: IterT m a -> IterT m a -> IterT m a # some :: IterT m a -> IterT m [a] # many :: IterT m a -> IterT m [a] #
Monad m => Applicative (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods pure :: a -> IterT m a # (<*>) :: IterT m (a -> b) -> IterT m a -> IterT m b # liftA2 :: (a -> b -> c) -> IterT m a -> IterT m b -> IterT m c # (*>) :: IterT m a -> IterT m b -> IterT m b # (<*) :: IterT m a -> IterT m b -> IterT m a #
Monad m => Functor (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods fmap :: (a -> b) -> IterT m a -> IterT m b # (<$) :: a -> IterT m b -> IterT m a #
Monad m => Monad (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods (>>=) :: IterT m a -> (a -> IterT m b) -> IterT m b # (>>) :: IterT m a -> IterT m b -> IterT m b # return :: a -> IterT m a #
Monad m => MonadPlus (IterT m) # | Capretta's `race` combinator. Satisfies left catch.
Instance details Defined in Control.Monad.Trans.Iter Methods mzero :: IterT m a # mplus :: IterT m a -> IterT m a -> IterT m a #
Monad m => MonadFail (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods fail :: String -> IterT m a #
MonadFix m => MonadFix (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods mfix :: (a -> IterT m a) -> IterT m a #
Foldable m => Foldable (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods fold :: Monoid m0 => IterT m m0 -> m0 # foldMap :: Monoid m0 => (a -> m0) -> IterT m a -> m0 # foldMap' :: Monoid m0 => (a -> m0) -> IterT m a -> m0 # foldr :: (a -> b -> b) -> b -> IterT m a -> b # foldr' :: (a -> b -> b) -> b -> IterT m a -> b # foldl :: (b -> a -> b) -> b -> IterT m a -> b # foldl' :: (b -> a -> b) -> b -> IterT m a -> b # foldr1 :: (a -> a -> a) -> IterT m a -> a # foldl1 :: (a -> a -> a) -> IterT m a -> a # toList :: IterT m a -> [a] # null :: IterT m a -> Bool # length :: IterT m a -> Int # elem :: Eq a => a -> IterT m a -> Bool # maximum :: Ord a => IterT m a -> a # minimum :: Ord a => IterT m a -> a # sum :: Num a => IterT m a -> a # product :: Num a => IterT m a -> a #
(Monad m, Traversable m) => Traversable (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods traverse :: Applicative f => (a -> f b) -> IterT m a -> f (IterT m b) # sequenceA :: Applicative f => IterT m (f a) -> f (IterT m a) # mapM :: Monad m0 => (a -> m0 b) -> IterT m a -> m0 (IterT m b) # sequence :: Monad m0 => IterT m (m0 a) -> m0 (IterT m a) #
MonadCont m => MonadCont (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods callCC :: ((a -> IterT m b) -> IterT m a) -> IterT m a #
Monad m => Apply (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods (<.>) :: IterT m (a -> b) -> IterT m a -> IterT m b # (.>) :: IterT m a -> IterT m b -> IterT m b # (<.) :: IterT m a -> IterT m b -> IterT m a # liftF2 :: (a -> b -> c) -> IterT m a -> IterT m b -> IterT m c #
Monad m => Bind (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods (>>-) :: IterT m a -> (a -> IterT m b) -> IterT m b # join :: IterT m (IterT m a) -> IterT m a #
(Monad m, Traversable1 m) => Traversable1 (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods traverse1 :: Apply f => (a -> f b) -> IterT m a -> f (IterT m b) # sequence1 :: Apply f => IterT m (f b) -> f (IterT m b) #
(Monad m, Semigroup a, Monoid a) => Monoid (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods mempty :: IterT m a # mappend :: IterT m a -> IterT m a -> IterT m a # mconcat :: [IterT m a] -> IterT m a #
(Monad m, Semigroup a) => Semigroup (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods (<>) :: IterT m a -> IterT m a -> IterT m a # sconcat :: NonEmpty (IterT m a) -> IterT m a # stimes :: Integral b => b -> IterT m a -> IterT m a #
(Typeable m, Data (m (Either a (IterT m a))), Data a) => Data (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> IterT m a -> c (IterT m a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (IterT m a) # toConstr :: IterT m a -> Constr # dataTypeOf :: IterT m a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (IterT m a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (IterT m a)) # gmapT :: (forall b. Data b => b -> b) -> IterT m a -> IterT m a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> IterT m a -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> IterT m a -> r # gmapQ :: (forall d. Data d => d -> u) -> IterT m a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> IterT m a -> u # gmapM :: Monad m0 => (forall d. Data d => d -> m0 d) -> IterT m a -> m0 (IterT m a) # gmapMp :: MonadPlus m0 => (forall d. Data d => d -> m0 d) -> IterT m a -> m0 (IterT m a) # gmapMo :: MonadPlus m0 => (forall d. Data d => d -> m0 d) -> IterT m a -> m0 (IterT m a) #
(Read1 m, Read a) => Read (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods readsPrec :: Int -> ReadS (IterT m a) # readList :: ReadS [IterT m a] # readPrec :: ReadPrec (IterT m a) # readListPrec :: ReadPrec [IterT m a] #
(Show1 m, Show a) => Show (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods showsPrec :: Int -> IterT m a -> ShowS # show :: IterT m a -> String # showList :: [IterT m a] -> ShowS #
(Eq1 m, Eq a) => Eq (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods (==) :: IterT m a -> IterT m a -> Bool # (/=) :: IterT m a -> IterT m a -> Bool #
(Ord1 m, Ord a) => Ord (IterT m a) # |
Instance details Defined in Control.Monad.Trans.Iter Methods compare :: IterT m a -> IterT m a -> Ordering # (<) :: IterT m a -> IterT m a -> Bool # (<=) :: IterT m a -> IterT m a -> Bool # (>) :: IterT m a -> IterT m a -> Bool # (>=) :: IterT m a -> IterT m a -> Bool # max :: IterT m a -> IterT m a -> IterT m a # min :: IterT m a -> IterT m a -> 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 :: forall (f :: Type -> Type) m a. (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 :: forall (m :: Type -> Type) a. Monad m => Iter a -> IterT m a #

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

cutoff :: forall (m :: Type -> Type) a. 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 :: forall (f :: Type -> Type) m a. (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 :: forall (m :: Type -> Type) a. 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_ :: forall (m :: Type -> Type) a. 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 :: Type -> Type) (m :: Type -> Type) | 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

default wrap :: forall (t :: (Type -> Type) -> Type -> Type) (n :: Type -> Type) a. (m ~ t n, MonadTrans t, MonadFree f n, Functor f) => f (m a) -> m a #

#### Instances

Instances details

Monad m => MonadFree Identity (IterT m) # |
Instance details Defined in Control.Monad.Trans.Iter Methods wrap :: Identity (IterT m a) -> IterT m a #
Functor f => MonadFree f (Free f) # |
Instance details Defined in Control.Monad.Free Methods wrap :: f (Free f a) -> Free f a #
Applicative f => MonadFree f (Free f) # |
Instance details Defined in Control.Monad.Free.Ap Methods wrap :: f (Free f a) -> Free f a #
Functor f => MonadFree f (F f) # |
Instance details Defined in Control.Monad.Free.Church Methods wrap :: f (F f a) -> F f a #
(Functor f, MonadFree f m) => MonadFree f (MaybeT m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (MaybeT m a) -> MaybeT m a #
(Functor f, Monad m) => MonadFree f (FreeT f m) # |
Instance details Defined in Control.Monad.Trans.Free Methods wrap :: f (FreeT f m a) -> FreeT f m a #
(Applicative f, Monad m) => MonadFree f (FreeT f m) # |
Instance details Defined in Control.Monad.Trans.Free.Ap Methods wrap :: f (FreeT f m a) -> FreeT f m a #
MonadFree f (FT f m) # |
Instance details Defined in Control.Monad.Trans.Free.Church Methods wrap :: f (FT f m a) -> FT f m a #
(Functor f, MonadFree f m) => MonadFree f (ExceptT e m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ExceptT e m a) -> ExceptT e m a #
(Functor f, MonadFree f m) => MonadFree f (IdentityT m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (IdentityT m a) -> IdentityT m a #
(Functor f, MonadFree f m) => MonadFree f (ReaderT e m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ReaderT e m a) -> ReaderT e m a #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (StateT s m a) -> StateT s m a #
(Functor f, MonadFree f m) => MonadFree f (StateT s m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (StateT s m a) -> StateT s m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (WriterT w m a) -> WriterT w m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (WriterT w m a) -> WriterT w m a #
(Functor f, MonadFree f m) => MonadFree f (ContT r m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (ContT r m a) -> ContT r m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (RWST r w s m a) -> RWST r w s m a #
(Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) # |
Instance details Defined in Control.Monad.Free.Class Methods wrap :: f (RWST r w s m a) -> RWST r w s m a #

# Examples

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