-------------------------------------------------------------------------
-- |
-- Module      : Control.Monad.Logic.Class
-- Copyright   : (c) 2007-2014 Dan Doel,
--               (c) 2011-2013 Edward Kmett,
--               (c) 2014      Roman Cheplyaka,
--               (c) 2020-2021 Andrew Lelechenko,
--               (c) 2020-2021 Kevin Quick
-- License     : BSD3
-- Maintainer  : Andrew Lelechenko <andrew.lelechenko@gmail.com>
--
-- Adapted from the paper
-- <http://okmij.org/ftp/papers/LogicT.pdf Backtracking, Interleaving, and Terminating Monad Transformers>
-- by Oleg Kiselyov, Chung-chieh Shan, Daniel P. Friedman, Amr Sabry.
-- Note that the paper uses 'MonadPlus' vocabulary
-- ('mzero' and 'mplus'),
-- while examples below prefer 'empty' and '<|>'
-- from 'Alternative'.
-------------------------------------------------------------------------

{-# LANGUAGE CPP #-}

#if __GLASGOW_HASKELL__ >= 704
{-# LANGUAGE Safe #-}
#endif

module Control.Monad.Logic.Class (MonadLogic(..), reflect) where

import Control.Applicative
import Control.Monad
import Control.Monad.Reader (ReaderT(..))
import Control.Monad.Trans (MonadTrans(..))
import qualified Control.Monad.State.Lazy as LazyST
import qualified Control.Monad.State.Strict as StrictST

-- | A backtracking, logic programming monad.
class (Monad m, Alternative m) => MonadLogic m where
    -- | Attempts to __split__ the computation, giving access to the first
    --   result. Satisfies the following laws:
    --
    --   > msplit empty          == pure Nothing
    --   > msplit (pure a <|> m) == pure (Just (a, m))
    msplit     :: m a -> m (Maybe (a, m a))

    -- | __Fair disjunction.__ It is possible for a logical computation
    --   to have an infinite number of potential results, for instance:
    --
    --   > odds = pure 1 <|> fmap (+ 2) odds
    --
    --   Such computations can cause problems in some circumstances. Consider:
    --
    --   > two = do x <- odds <|> pure 2
    --   >          if even x then pure x else empty
    --
    --   >>> observe two
    --   ...never completes...
    --
    --   Such a computation may never consider 'pure' @2@, and
    --   therefore even 'Control.Monad.Logic.observe' @two@ will
    --   never return any results. By
    --   contrast, using 'interleave' in place of
    --   'Control.Applicative.<|>' ensures fair consideration of both
    --   branches of a disjunction.
    --
    --   > fairTwo = do x <- odds `interleave` pure 2
    --   >              if even x then pure x else empty
    --
    --   >>> observe fairTwo
    --   2
    --
    --   Note that even with 'interleave' this computation will never
    --   terminate after returning 2: only the first value can be
    --   safely observed, after which each odd value becomes 'Control.Applicative.empty'
    --   (equivalent to
    --   <http://lpn.swi-prolog.org/lpnpage.php?pagetype=html&pageid=lpn-htmlse45 Prolog's fail>)
    --   which does not stop the evaluation but indicates there is no
    --   value to return yet.
    --
    --   Unlike '<|>', 'interleave' is not associative:
    --
    --   >>> let x = [1,2,3]; y = [4,5,6]; z = [7,8,9] :: [Int]
    --   >>> x `interleave` y
    --   [1,4,2,5,3,6]
    --   >>> (x `interleave` y) `interleave` z
    --   [1,7,4,8,2,9,5,3,6]
    --   >>> y `interleave` z
    --   [4,7,5,8,6,9]
    --   >>> x `interleave` (y `interleave` z)
    --   [1,4,2,7,3,5,8,6,9]
    --
    interleave :: m a -> m a -> m a

    -- | __Fair conjunction.__ Similarly to the previous function, consider
    --   the distributivity law, naturally expected from 'MonadPlus':
    --
    --   > (a <|> b) >>= k = (a >>= k) <|> (b >>= k)
    --
    --   If @a@ '>>=' @k@ can backtrack arbitrarily many times, @b@ '>>=' @k@
    --   may never be considered. In logic statements,
    --   "backtracking" is the process of discarding the current
    --   possible solution value and returning to a previous decision
    --   point where a new value can be obtained and tried.  For
    --   example:
    --
    --   >>> do { x <- pure 0 <|> pure 1 <|> pure 2; if even x then pure x else empty } :: [Int]
    --   [0,2]
    --
    --   Here, the @x@ value can be produced three times, where
    --   'Control.Applicative.<|>' represents the decision points of that
    --   production.  The subsequent @if@ statement specifies
    --   'Control.Applicative.empty' (fail)
    --   if @x@ is odd, causing it to be discarded and a return
    --   to an 'Control.Applicative.<|>' decision point to get the next @x@.
    --
    --   The statement "@a@ '>>=' @k@ can backtrack arbitrarily many
    --   times" means that the computation is resulting in 'Control.Applicative.empty' and
    --   that @a@ has an infinite number of 'Control.Applicative.<|>' applications to
    --   return to.  This is called a conjunctive computation because
    --   the logic for @a@ /and/ @k@ must both succeed (i.e. 'pure'
    --   a value instead of 'Control.Applicative.empty').
    --
    --   Similar to the way 'interleave' allows both branches of a
    --   disjunctive computation, the '>>-' operator takes care to
    --   consider both branches of a conjunctive computation.
    --
    --   Consider the operation:
    --
    --   > odds = pure 1 <|> fmap (2 +) odds
    --   >
    --   > oddsPlus n = odds >>= \a -> pure (a + n)
    --   >
    --   > g = do x <- (pure 0 <|> pure 1) >>= oddsPlus
    --   >        if even x then pure x else empty
    --
    --   >>> observeMany 3 g
    --   ...never completes...
    --
    --   This will never produce any value because all values produced
    --   by the @do@ program come from the 'pure' @1@ driven operation
    --   (adding one to the sequence of odd values, resulting in the
    --   even values that are allowed by the test in the second line),
    --   but the 'pure' @0@ input to @oddsPlus@ generates an infinite
    --   number of 'Control.Applicative.empty' failures so the even values generated by
    --   the 'pure' @1@ alternative are never seen.  Using
    --   'interleave' here instead of 'Control.Applicative.<|>' does not help due
    --   to the aforementioned distributivity law.
    --
    --   Also note that the @do@ notation desugars to '>>=' bind
    --   operations, so the following would also fail:
    --
    --   > do a <- pure 0 <|> pure 1
    --   >    x <- oddsPlus a
    --   >    if even x then pure x else empty
    --
    --   The solution is to use the '>>-' in place of the normal
    --   monadic bind operation '>>=' when fairness between
    --   alternative productions is needed in a conjunction of
    --   statements (rules):
    --
    --   > h = do x <- (pure 0 <|> pure 1) >>- oddsPlus
    --   >        if even x then pure x else empty
    --
    --   >>> observeMany 3 h
    --   [2,4,6]
    --
    --   However, a bit of care is needed when using '>>-' because,
    --   unlike '>>=', it is not associative.  For example:
    --
    --   >>> let m = [2,7] :: [Int]
    --   >>> let k x = [x, x + 1]
    --   >>> let h x = [x, x * 2]
    --   >>> m >>= (\x -> k x >>= h)
    --   [2,4,3,6,7,14,8,16]
    --   >>> (m >>= k) >>= h -- same as above
    --   [2,4,3,6,7,14,8,16]
    --   >>> m >>- (\x -> k x >>- h)
    --   [2,7,3,8,4,14,6,16]
    --   >>> (m >>- k) >>- h -- central elements are different
    --   [2,7,4,3,14,8,6,16]
    --
    --   This means that the following will be productive:
    --
    --   > (pure 0 <|> pure 1) >>-
    --   >   oddsPlus >>-
    --   >     \x -> if even x then pure x else empty
    --
    --   Which is equivalent to
    --
    --   > ((pure 0 <|> pure 1) >>- oddsPlus) >>-
    --   >   (\x -> if even x then pure x else empty)
    --
    --   But the following will /not/ be productive:
    --
    --   > (pure 0 <|> pure 1) >>-
    --   >   (\a -> (oddsPlus a >>- \x -> if even x then pure x else empty))
    --
    --   Since do notation desugaring results in the latter, the
    --   @RebindableSyntax@ language pragma cannot easily be used
    --   either.  Instead, it is recommended to carefully use explicit
    --   '>>-' only when needed.
    --
    (>>-)      :: m a -> (a -> m b) -> m b
    infixl 1 >>-

    -- | __Pruning.__ Selects one result out of many. Useful for when multiple
    --   results of a computation will be equivalent, or should be treated as
    --   such.
    --
    --   As an example, here's a way to determine if a number is
    --   <https://wikipedia.org/wiki/Composite_number composite>
    --   (has non-trivial integer divisors, i.e. not a
    --   prime number):
    --
    --   > choose = foldr ((<|>) . pure) empty
    --   >
    --   > divisors n = do a <- choose [2..n-1]
    --   >                 b <- choose [2..n-1]
    --   >                 guard (a * b == n)
    --   >                 pure (a, b)
    --   >
    --   > composite_ v = do _ <- divisors v
    --   >                   pure "Composite"
    --
    --   While this works as intended, it actually does too much work:
    --
    --   >>> observeAll (composite_ 20)
    --   ["Composite", "Composite", "Composite", "Composite"]
    --
    --   Because there are multiple divisors of 20, and they can also
    --   occur in either order:
    --
    --   >>> observeAll (divisors 20)
    --   [(2,10), (4,5), (5,4), (10,2)]
    --
    --   Clearly one could just use 'Control.Monad.Logic.observe' here to get the first
    --   non-prime result, but if the call to @composite@ is in the
    --   middle of other logic code then use 'once' instead.
    --
    --   > composite v = do _ <- once (divisors v)
    --   >                  pure "Composite"
    --
    --   >>> observeAll (composite 20)
    --   ["Composite"]
    --
    once       :: m a -> m a

    -- | __Inverts__ a logic computation. If @m@ succeeds with at least one value,
    --   'lnot' @m@ fails. If @m@ fails, then 'lnot' @m@ succeeds with the value @()@.
    --
    --   For example, evaluating if a number is prime can be based on
    --   the failure to find divisors of a number:
    --
    --   > choose = foldr ((<|>) . pure) empty
    --   >
    --   > divisors n = do d <- choose [2..n-1]
    --   >                 guard (n `rem` d == 0)
    --   >                 pure d
    --   >
    --   > prime v = do _ <- lnot (divisors v)
    --   >              pure True
    --
    --   >>> observeAll (prime 20)
    --   []
    --   >>> observeAll (prime 19)
    --   [True]
    --
    --   Here if @divisors@ never succeeds, then the 'lnot' will
    --   succeed and the number will be declared as prime.
    lnot :: m a -> m ()

    -- | Logical __conditional.__ The equivalent of
    --   <http://lpn.swi-prolog.org/lpnpage.php?pagetype=html&pageid=lpn-htmlse44 Prolog's soft-cut>.
    --   If its first argument succeeds at all,
    --   then the results will be fed into the success
    --   branch. Otherwise, the failure branch is taken.  The failure
    --   branch is never considered if the first argument has any
    --   successes.  The 'ifte' function satisfies the following laws:
    --
    --   > ifte (pure a) th el       == th a
    --   > ifte empty th el          == el
    --   > ifte (pure a <|> m) th el == th a <|> (m >>= th)
    --
    --   For example, the previous @prime@ function returned nothing
    --   if the number was not prime, but if it should return 'False'
    --   instead, the following can be used:
    --
    --   > choose = foldr ((<|>) . pure) empty
    --   >
    --   > divisors n = do d <- choose [2..n-1]
    --   >                 guard (n `rem` d == 0)
    --   >                 pure d
    --   >
    --   > prime v = once (ifte (divisors v)
    --   >                   (const (pure False))
    --   >                   (pure True))
    --
    --   >>> observeAll (prime 20)
    --   [False]
    --   >>> observeAll (prime 19)
    --   [True]
    --
    --   Notice that this cannot be done with a simple @if-then-else@
    --   because @divisors@ either generates values or it does not, so
    --   there's no "false" condition to check with a simple @if@
    --   statement.
    ifte       :: m a -> (a -> m b) -> m b -> m b

    -- All the class functions besides msplit can be derived from msplit, if
    -- desired
    interleave m1 m2 = msplit m1 >>=
                        maybe m2 (\(a, m1') -> pure a <|> interleave m2 m1')

    m >>- f = msplit m >>= maybe empty
      (\(a, m') -> interleave (f a) (m' >>- f))

    ifte t th el = msplit t >>= maybe el (\(a,m) -> th a <|> (m >>= th))

    once m = msplit m >>= maybe empty (\(a, _) -> pure a)

    lnot m = msplit m >>= maybe (pure ()) (const empty)

-------------------------------------------------------------------------------
-- | The inverse of 'msplit'. Satisfies the following law:
--
-- > msplit m >>= reflect == m
reflect :: Alternative m => Maybe (a, m a) -> m a
reflect Nothing = empty
reflect (Just (a, m)) = pure a <|> m

-- An instance of MonadLogic for lists
instance MonadLogic [] where
    msplit []     = pure Nothing
    msplit (x:xs) = pure $ Just (x, xs)

-- | Note that splitting a transformer does
-- not allow you to provide different input
-- to the monadic object returned.
-- For instance, in:
--
-- > let Just (_, rm') = runReaderT (msplit rm) r in runReaderT rm' r'
--
-- @r'@ will be ignored, because @r@ was already threaded through the
-- computation.
instance MonadLogic m => MonadLogic (ReaderT e m) where
    msplit rm = ReaderT $ \e -> do r <- msplit $ runReaderT rm e
                                   case r of
                                     Nothing -> pure Nothing
                                     Just (a, m) -> pure (Just (a, lift m))

-- | See note on splitting above.
instance (MonadLogic m, MonadPlus m) => MonadLogic (StrictST.StateT s m) where
    msplit sm = StrictST.StateT $ \s ->
                    do r <- msplit (StrictST.runStateT sm s)
                       case r of
                            Nothing          -> pure (Nothing, s)
                            Just ((a,s'), m) ->
                                pure (Just (a, StrictST.StateT (const m)), s')

    interleave ma mb = StrictST.StateT $ \s ->
                        StrictST.runStateT ma s `interleave` StrictST.runStateT mb s

    ma >>- f = StrictST.StateT $ \s ->
                StrictST.runStateT ma s >>- \(a,s') -> StrictST.runStateT (f a) s'

    ifte t th el = StrictST.StateT $ \s -> ifte (StrictST.runStateT t s)
                                                (\(a,s') -> StrictST.runStateT (th a) s')
                                                (StrictST.runStateT el s)

    once ma = StrictST.StateT $ \s -> once (StrictST.runStateT ma s)

-- | See note on splitting above.
instance (MonadLogic m, MonadPlus m) => MonadLogic (LazyST.StateT s m) where
    msplit sm = LazyST.StateT $ \s ->
                    do r <- msplit (LazyST.runStateT sm s)
                       case r of
                            Nothing -> pure (Nothing, s)
                            Just ((a,s'), m) ->
                                pure (Just (a, LazyST.StateT (const m)), s')

    interleave ma mb = LazyST.StateT $ \s ->
                        LazyST.runStateT ma s `interleave` LazyST.runStateT mb s

    ma >>- f = LazyST.StateT $ \s ->
                LazyST.runStateT ma s >>- \(a,s') -> LazyST.runStateT (f a) s'

    ifte t th el = LazyST.StateT $ \s -> ifte (LazyST.runStateT t s)
                                              (\(a,s') -> LazyST.runStateT (th a) s')
                                              (LazyST.runStateT el s)

    once ma = LazyST.StateT $ \s -> once (LazyST.runStateT ma s)
