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Modulefree-5.2Haskell2010

Control.Monad.Free.Church

"Free Monads for Less"

The most straightforward way of implementing free monads is as a recursive datatype that allows for arbitrarily deep nesting of the base functor. This is akin to a tree, with the leaves containing the values, and the nodes being a level of Functor over subtrees.

For each time that the fmap or >>= operations is used, the old tree is traversed up to the leaves, a new set of nodes is allocated, and the old ones are garbage collected. Even if the Haskell runtime optimizes some of the overhead through laziness and generational garbage collection, the asymptotic runtime is still quadratic.

On the other hand, if the Church encoding is used, the tree only needs to be constructed once, because:

  • All uses of fmap are collapsed into a single one, so that the values on the _leaves_ are transformed in one pass.

Property
fmap f . fmap g == fmap (f . g)

  • All uses of >>= are right associated, so that every new subtree created is final.

Property
(m >>= f) >>= g == m >>= (\x -> f x >>= g)

Asymptotically, the Church encoding supports the monadic operations more efficiently than the naïve Free.

This is based on the "Free Monads for Less" series of articles by Edward Kmett:

  • 1 type
  • 1 class
  • 10 values
  • Packagefree-5.2
  • Exports12
  • LanguageHaskell2010
  • LicenceBSD-3-Clause
  • SourceChurch.hs
newtypenewtype F (f :: Type -> Type) a
#

The Church-encoded free monad for a functor f.

It is asymptotically more efficient to use (>>=) for F than it is to (>>=) with Free.

https://ekmett.github.io/reader/2011/free-monads-for-less-2/

Constructors

  • F
    • runF :: forall r. (a -> r) -> (f r -> r) -> r
Instances18MonadTrans, MonadReader, MonadState, MonadWriter, MonadFree, Monad, …
valuefromF :: MonadFree f m => F f a -> m a
#

Convert to another free monad representation.

valueiter :: (f a -> a) -> F f a -> a
#

Tear down a Free Monad using iteration.

valueiterM :: Monad m => (f (m a) -> m a) -> F f a -> m a
#

Like iter for monadic values.

valuetoF :: Functor f => Free f a -> F f a
#

Generate a Church-encoded free monad from a Free monad.

valuehoistF :: (forall x. f x -> g x) -> F f a -> F g a
#

Lift a natural transformation from f to g into a natural transformation from F f to F g.

valuefoldF :: Monad m => (forall x. f x -> m x) -> F f a -> m a
#

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

classclass 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.

Methods

  • wrap :: f (m a) -> m a

    Add a layer.

    wrap (fmap f x) ≡ wrap (fmap return x) >>= f
Instances18MonadFree, …
valueliftF :: (Functor f, MonadFree f m) => f a -> m a
#

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

valuecutoff :: Functor f => Integer -> F f a -> F 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):

Property
cutoff 0     _        == return Nothing
Property
cutoff (n+1) . return == return . Just
Property
cutoff (n+1) . lift   == lift . liftM Just
Property
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.