{-# LANGUAGE BangPatterns #-} {-# LANGUAGE CPP #-} {-# LANGUAGE Rank2Types #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE UndecidableInstances #-} {-# LANGUAGE MultiParamTypeClasses #-} #include "free-common.h" ----------------------------------------------------------------------------- -- | -- Module : Control.Monad.Free.Church -- Copyright : (C) 2011-2015 Edward Kmett -- License : BSD-style (see the file LICENSE) -- -- Maintainer : Edward Kmett -- Stability : provisional -- Portability : non-portable (rank-2 polymorphism) -- -- \"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. -- -- prop> fmap f . fmap g == fmap (f . g) -- -- * All uses of `>>=` are right associated, so that every new subtree created -- is final. -- -- prop> (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: -- -- * -- -- * ---------------------------------------------------------------------------- module Control.Monad.Free.Church ( F(..) , improve , fromF , iter , iterM , toF , retract , hoistF , foldF , MonadFree(..) , liftF , cutoff ) where import Control.Applicative import Control.Monad as Monad import Control.Monad.Fix import Control.Monad.Free hiding (retract, iter, iterM, cutoff) import Control.Monad.Reader.Class import Control.Monad.Writer.Class import Control.Monad.Cont.Class import Control.Monad.Trans.Class import Control.Monad.State.Class import Data.Foldable import Data.Traversable import Data.Functor.Bind import Data.Semigroup.Foldable import Data.Semigroup.Traversable import Prelude hiding (foldr) -- | The Church-encoded free monad for a functor @f@. -- -- It is /asymptotically/ more efficient to use ('>>=') for 'F' than it is to ('>>=') with 'Free'. -- -- newtype F f a = F { runF :: forall r. (a -> r) -> (f r -> r) -> r } -- | Tear down a 'Free' 'Monad' using iteration. iter :: (f a -> a) -> F f a -> a iter phi xs = runF xs id phi -- | Like iter for monadic values. iterM :: Monad m => (f (m a) -> m a) -> F f a -> m a iterM phi xs = runF xs return phi instance Functor (F f) where fmap f (F g) = F (\kp -> g (kp . f)) instance Apply (F f) where (<.>) = (<*>) instance Applicative (F f) where pure a = F (\kp _ -> kp a) F f <*> F g = F (\kp kf -> f (\a -> g (kp . a) kf) kf) -- | This violates the Alternative laws, handle with care. instance Alternative f => Alternative (F f) where empty = F (\_ kf -> kf empty) F f <|> F g = F (\kp kf -> kf (pure (f kp kf) <|> pure (g kp kf))) instance Bind (F f) where (>>-) = (>>=) instance Monad (F f) where return = pure F m >>= f = F (\kp kf -> m (\a -> runF (f a) kp kf) kf) instance MonadFix (F f) where mfix f = a where a = f (impure a) impure (F x) = x id (error "MonadFix (F f): wrap") instance Foldable f => Foldable (F f) where foldMap f xs = runF xs f fold {-# INLINE foldMap #-} foldr f r xs = runF xs f (foldr (.) id) r {-# INLINE foldr #-} #if MIN_VERSION_base(4,6,0) foldl' f z xs = runF xs (\a !r -> f r a) (flip $ foldl' $ \r g -> g r) z {-# INLINE foldl' #-} #endif instance Traversable f => Traversable (F f) where traverse f m = runF m (fmap return . f) (fmap wrap . sequenceA) {-# INLINE traverse #-} instance Foldable1 f => Foldable1 (F f) where foldMap1 f m = runF m f fold1 instance Traversable1 f => Traversable1 (F f) where traverse1 f m = runF m (fmap return . f) (fmap wrap . sequence1) -- | This violates the MonadPlus laws, handle with care. instance MonadPlus f => MonadPlus (F f) where mzero = F (\_ kf -> kf mzero) F f `mplus` F g = F (\kp kf -> kf (return (f kp kf) `mplus` return (g kp kf))) instance MonadTrans F where lift f = F (\kp kf -> kf (liftM kp f)) instance Functor f => MonadFree f (F f) where wrap f = F (\kp kf -> kf (fmap (\ (F m) -> m kp kf) f)) instance MonadState s m => MonadState s (F m) where get = lift get put = lift . put instance MonadReader e m => MonadReader e (F m) where ask = lift ask local f = lift . local f . retract instance MonadWriter w m => MonadWriter w (F m) where tell = lift . tell pass = lift . pass . retract listen = lift . listen . retract instance MonadCont m => MonadCont (F m) where callCC f = lift $ callCC (retract . f . fmap lift) -- | -- 'retract' is the left inverse of 'lift' and 'liftF' -- -- @ -- 'retract' . 'lift' = 'id' -- 'retract' . 'liftF' = 'id' -- @ retract :: Monad m => F m a -> m a retract (F m) = m return Monad.join {-# INLINE retract #-} -- | Lift a natural transformation from @f@ to @g@ into a natural transformation from @F f@ to @F g@. hoistF :: (forall x. f x -> g x) -> F f a -> F g a hoistF t (F m) = F (\p f -> m p (f . t)) -- | The very definition of a free monad is that given a natural transformation you get a monad homomorphism. foldF :: Monad m => (forall x. f x -> m x) -> F f a -> m a foldF f (F m) = m return (Monad.join . f) -- | Convert to another free monad representation. fromF :: MonadFree f m => F f a -> m a fromF (F m) = m return wrap {-# INLINE fromF #-} -- | Generate a Church-encoded free monad from a 'Free' monad. toF :: Functor f => Free f a -> F f a toF xs = F (\kp kf -> go kp kf xs) where go kp _ (Pure a) = kp a go kp kf (Free fma) = kf (fmap (go kp kf) fma) -- | Improve the asymptotic performance of code that builds a free monad with only binds and returns by using 'F' behind the scenes. -- -- This is based on the \"Free Monads for Less\" series of articles by Edward Kmett: -- -- * -- -- * -- -- and by Janis Voightländer. improve :: Functor f => (forall m. MonadFree f m => m a) -> Free f a improve m = fromF m {-# INLINE improve #-} -- | 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@): -- -- prop> cutoff 0 _ == return Nothing -- prop> cutoff (n+1) . return == return . Just -- prop> cutoff (n+1) . lift == lift . liftM Just -- prop> 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. {-# INLINE cutoff #-} cutoff :: (Functor f) => Integer -> F f a -> F f (Maybe a) cutoff n m | n <= 0 = return Nothing | n <= toInteger (maxBound :: Int) = cutoffI (fromInteger n :: Int) m | otherwise = cutoffI n m {-# SPECIALIZE cutoffI :: (Functor f) => Int -> F f a -> F f (Maybe a) #-} {-# SPECIALIZE cutoffI :: (Functor f) => Integer -> F f a -> F f (Maybe a) #-} cutoffI :: (Functor f, Integral n) => n -> F f a -> F f (Maybe a) cutoffI n m = F m' where m' kp kf = runF m kpn kfn n where kpn a i | i <= 0 = kp Nothing | otherwise = kp (Just a) kfn fr i | i <= 0 = kp Nothing | otherwise = let i' = i - 1 in i' `seq` kf (fmap ($ i') fr)