One problem with such monad implementations is efficiency. Let's define
step :: (MonadPlus m) => Int -> m Int
step i = choose [i, i + 1]
-- repeated application of step on 0:
stepN :: (Monad m) => Int -> m (S.Set Int)
stepN = runSet . f
where
f 0 = return 0
f n = f (n-1) >>= step
Then `stepN`'s time complexity is exponential in its argument. This is
because `ContT` reorders the chain of computations to right-associative,
which is correct, but changes the time complexity in this unfortunate way.
If we used Set directly, constructing a left-associative chain, it produces
the result immediately:
step' :: Int -> S.Set Int
step' i = S.fromList [i, i + 1]
stepN' :: Int -> S.Set Int
stepN' 0 = S.singleton 0
stepN' n = stepN' (n - 1) `setBind` step'
where
setBind k f = S.foldl' (\s -> S.union s . f) S.empty k
See also: Constructing efficient monad instances on `Set` (and other
containers with constraints) using the continuation monad <
http://stackoverflow.com/q/12183656/1333025>
Best regards,
Petr Pudlak
2013/4/11
The question of Set monad comes up quite regularly, most recently at http://www.ittc.ku.edu/csdlblog/?p=134
Indeed, we cannot make Data.Set.Set to be the instance of Monad type class -- not immediately, that it. That does not mean that there is no Set Monad, a non-determinism monad that returns the set of answers, rather than a list. I mean genuine *monad*, rather than a restricted, generalized, etc. monad.
It is surprising that the question of the Set monad still comes up given how simple it is to implement it. Footnote 4 in the ICFP 2009 paper on ``Purely Functional Lazy Non-deterministic Programming'' described the idea, which is probably folklore. Just in case, here is the elaboration, a Set monad transformer.
{-# LANGUAGE TypeSynonymInstances, FlexibleInstances #-}
module SetMonad where
import qualified Data.Set as S import Control.Monad.Cont
-- Since ContT is a bona fide monad transformer, so is SetMonadT r. type SetMonadT r = ContT (S.Set r)
-- These are the only two places the Ord constraint shows up
instance (Ord r, Monad m) => MonadPlus (SetMonadT r m) where mzero = ContT $ \k -> return S.empty mplus m1 m2 = ContT $ \k -> liftM2 S.union (runContT m1 k) (runContT m2 k)
runSet :: (Monad m, Ord r) => SetMonadT r m r -> m (S.Set r) runSet m = runContT m (return . S.singleton)
choose :: MonadPlus m => [a] -> m a choose = msum . map return
test1 = print =<< runSet (do n1 <- choose [1..5] n2 <- choose [1..5] let n = n1 + n2 guard $ n < 7 return n) -- fromList [2,3,4,5,6]
-- Values to choose from might be higher-order or actions test1' = print =<< runSet (do n1 <- choose . map return $ [1..5] n2 <- choose . map return $ [1..5] n <- liftM2 (+) n1 n2 guard $ n < 7 return n) -- fromList [2,3,4,5,6]
test2 = print =<< runSet (do i <- choose [1..10] j <- choose [1..10] k <- choose [1..10] guard $ i*i + j*j == k * k return (i,j,k)) -- fromList [(3,4,5),(4,3,5),(6,8,10),(8,6,10)]
test3 = print =<< runSet (do i <- choose [1..10] j <- choose [1..10] k <- choose [1..10] guard $ i*i + j*j == k * k return k) -- fromList [5,10]
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