Church.hs

{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}

{- | Provides 'NonDetC', a carrier for 'NonDet' effects providing choice and failure.

Under the hood, it uses a Church-encoded structure and a binary tree to prevent the problems associated with a naïve list-based implementation.

@since 1.0.0.0
-}

module Control.Carrier.NonDet.Church
( -- * NonDet carrier
  runNonDet
, runNonDetA
, runNonDetM
, NonDetC(..)
  -- * NonDet effects
, module Control.Effect.NonDet
) where

import Control.Algebra
import Control.Applicative
import Control.Effect.NonDet
import Control.Monad.Fail as Fail
import Control.Monad.Fix
import Control.Monad.IO.Class
import Control.Monad.Trans.Class
import Data.Coerce (coerce)
import Data.Functor.Identity

-- | Run a 'NonDet' effect, using the provided functions to interpret choice, leaf results, and failure.
--
-- @
-- runNonDet fork leaf nil ('pure' a '<|>' 'empty') = leaf a \`fork\` nil
-- @
--
-- @since 1.0.0.0
runNonDet
  :: (m b -> m b -> m b) -- ^ Handles choice ('<|>')
  -> (a -> m b)          -- ^ Handles embedding results ('pure')
  -> m b                 -- ^ Handles failure ('empty')
  -> NonDetC m a         -- ^ A nondeterministic computation to execute
  -> m b
runNonDet fork leaf nil (NonDetC m) = m fork leaf nil
{-# INLINE runNonDet #-}

-- | Run a 'NonDet' effect, collecting all branches’ results into an 'Alternative' functor.
--
-- Using @[]@ as the 'Alternative' functor will produce all results, while 'Maybe' will return only the first. However, unless used with 'Control.Effect.Cull.cull', this will still enumerate the entire search space before returning, meaning that it will diverge for infinite search spaces, even when using 'Maybe'.
--
-- @
-- 'runNonDetA' ('pure' a) = 'pure' [a]
-- @
-- @
-- 'runNonDetA' ('pure' a) = 'pure' ('Just' a)
-- @
--
-- @since 1.0.0.0
runNonDetA :: (Alternative f, Applicative m) => NonDetC m a -> m (f a)
runNonDetA = runNonDet (liftA2 (<|>)) (pure . pure) (pure empty)
{-# INLINE runNonDetA #-}

-- | Run a 'NonDet' effect, mapping results into a 'Monoid'.
--
-- @since 1.0.0.0
runNonDetM :: (Applicative m, Monoid b) => (a -> b) -> NonDetC m a -> m b
runNonDetM leaf = runNonDet (liftA2 mappend) (pure . leaf) (pure mempty)
{-# INLINE runNonDetM #-}

-- | A carrier for 'NonDet' effects based on Ralf Hinze’s design described in [Deriving Backtracking Monad Transformers](https://www.cs.ox.ac.uk/ralf.hinze/publications/#P12).
--
-- @since 1.0.0.0
newtype NonDetC m a = NonDetC (forall b . (m b -> m b -> m b) -> (a -> m b) -> m b -> m b)
  deriving (Functor)

instance Applicative (NonDetC m) where
  pure a = NonDetC (\ _ leaf _ -> leaf a)
  {-# INLINE pure #-}

  NonDetC f <*> NonDetC a = NonDetC $ \ fork leaf nil ->
    f fork (\ f' -> a fork (leaf . f') nil) nil
  {-# INLINE (<*>) #-}

instance Alternative (NonDetC m) where
  empty = NonDetC (\ _ _ nil -> nil)
  {-# INLINE empty #-}

  NonDetC l <|> NonDetC r = NonDetC $ \ fork leaf nil ->
    l fork leaf nil `fork` r fork leaf nil
  {-# INLINE (<|>) #-}

instance Monad (NonDetC m) where
  NonDetC a >>= f = NonDetC $ \ fork leaf nil ->
    a fork (runNonDet fork leaf nil . f) nil
  {-# INLINE (>>=) #-}

instance Fail.MonadFail m => Fail.MonadFail (NonDetC m) where
  fail s = lift (Fail.fail s)
  {-# INLINE fail #-}

-- | Separate fixpoints are computed for each branch.
instance MonadFix m => MonadFix (NonDetC m) where
  mfix f = NonDetC $ \ fork leaf nil ->
    mfix (runNonDetA . f . head)
    >>= runNonDet fork leaf nil . foldr
      (\ a _ -> pure a <|> mfix (liftAll . fmap tail . runNonDetA . f))
      empty where
    liftAll m = NonDetC $ \ fork leaf nil -> m >>= foldr (fork . leaf) nil
  {-# INLINE mfix #-}

instance MonadIO m => MonadIO (NonDetC m) where
  liftIO io = lift (liftIO io)
  {-# INLINE liftIO #-}

instance MonadPlus (NonDetC m)

instance MonadTrans NonDetC where
  lift m = NonDetC (\ _ leaf _ -> m >>= leaf)
  {-# INLINE lift #-}

instance Algebra sig m => Algebra (NonDet :+: sig) (NonDetC m) where
  alg hdl sig ctx = NonDetC $ \ fork leaf nil -> case sig of
    L (L Empty)  -> nil
    L (R Choose) -> leaf (True <$ ctx) `fork` leaf (False <$ ctx)
    R other      -> thread (dst ~<~ hdl) other (pure ctx) >>= run . runNonDet (coerce fork) (coerce leaf) (coerce nil)
    where
    dst :: Applicative m => NonDetC Identity (NonDetC m a) -> m (NonDetC Identity a)
    dst = run . runNonDet (liftA2 (liftA2 (<|>))) (pure . runNonDetA) (pure (pure empty))
  {-# INLINE alg #-}