Using Roman’s smallcheck code (thanks!) here’s some evidence that codensity turns a bad diagonalizing ziplist instance into a good one, by fixing associativity. I’ve been pondering this for some time, and I’m glad this thread kicked me into making it work out. Also, as David noted, this works with or without the “take i” in the code, which enforces that minimum criteria I mentioned. So I suppose there’s a range of possibilities here.
If this works out, it looks like this also shows that a “purely algebraic” argument as to why ZipList can’t be a monad doesn't exist. I.e. there’s no conflict in the laws. It’s just that using a plain list as the underlying datastructure can’t force a uniform associativity.
To make a real “monadic ziplist” out of this, I think the codensity stuff would just need to be inlined under the ziplist constructor.
Cheers,
Gershom
import Data.List
import Data.Maybe
import Test.SmallCheck.Series
import Test.Tasty
import Test.Tasty.SmallCheck
import Control.Monad
import Control.Applicative
import System.Environment
data ZL a = ZL {unZL :: [a]} deriving (Eq, Show)
instance Functor ZL where
fmap f (ZL xs) = ZL (fmap f xs)
joinZL :: ZL (ZL a) -> ZL a
joinZL (ZL []) = ZL []
joinZL (ZL zs) = ZL (chop . diag (0,[]) $ map unZL zs)
where diag :: (Int,[a]) -> [[a]] -> (Int,[a])
diag (i,acc) [] = (i,acc)
diag (i,acc) (x:xs) = case drop i x of
[] -> (length x, acc)
(y:_) ->diag (i+1, (y : acc)) xs
chop (i,acc) = take i $ reverse acc
instance Applicative ZL where
pure = return
f <*> x = joinZL $ fmap (\g -> fmap g x) f
instance Monad ZL where
return x = ZL (repeat x)
x >>= f = joinZL $ fmap (f $) x
newtype Codensity m a = Codensity { runCodensity :: forall b. (a -> m b) -> m b }
instance Functor (Codensity k) where fmap f (Codensity m) = Codensity (\k -> m (\x -> k (f x)))
instance Applicative (Codensity f) where
pure x = Codensity (\k -> k x)
Codensity f <*> Codensity g = Codensity (\bfr -> f (\ab -> g (\x -> bfr (ab x))))
instance Monad (Codensity f) where
return = pure
m >>= k = Codensity (\c -> runCodensity m (\a -> runCodensity (k a) c))
lowerCodensity :: Monad m => Codensity m a -> m a
lowerCodensity a = runCodensity a return
lift m = Codensity (m >>=)
-- tests
instance Serial m a => Serial m (ZL a) where
series = ZL <$> series
instance Serial m a => Serial m (Codensity ZL a) where
series = lift <$> series
instance Show (Codensity ZL Int) where
show x = show (lowerCodensity x)
instance Show (Codensity ZL Bool) where
show x = show (lowerCodensity x)
main = do
setEnv "TASTY_SMALLCHECK_DEPTH" "4"
defaultMain $ testGroup "Monad laws"
[ testProperty "Right identity" $ \(z :: Codensity ZL Int) ->
lowerCodensity (z >>= return) == lowerCodensity z
, testProperty "Left identity" $ \(b :: Bool) (f :: Bool -> Codensity ZL Bool) ->
lowerCodensity (return b >>= f) == lowerCodensity (f b)
, testProperty "Associativity" $
\(f1 :: Bool -> Codensity ZL Bool)
(f2 :: Bool -> Codensity ZL Bool)
(z :: Codensity ZL Bool) ->
lowerCodensity (z >>= (\x -> f1 x >>= f2)) == lowerCodensity ((z >>= f1) >>= f2)
]
On Jun 4, 2020, 4:04 PM -0400, Roman Cheplyaka
On 04/06/2020 09.53, Dannyu NDos wrote:
instance Monad ZipList where ZipList [] >>= _ = ZipList [] ZipList (x:xs) >>= f = ZipList $ do let ZipList y' = f x guard (not (null y')) let ZipList ys = ZipList xs >>= ZipList . join . maybeToList . fmap snd . uncons . getZipList . f head y' : ys
instance MonadFail ZipList where fail _ = empty
instance MonadPlus ZipList
While others have commented on the general feasibility of the idea, the problem with this specific instance is that it appears to violate the associativity law:
% ./ziplist --smallcheck-depth=3 Monad laws Right identity: OK 21 tests completed Left identity: OK 98 tests completed Associativity: FAIL (0.04s) there exist {True->ZipList {getZipList = [True]};False->ZipList {getZipList = [False,True]}} {True->ZipList {getZipList = [True,True]};False->ZipList {getZipList = []}} ZipList {getZipList = [True,False]} such that condition is false
1 out of 3 tests failed (0.05s)
Here's the code I used for testing:
{-# LANGUAGE ScopedTypeVariables, FlexibleInstances, MultiParamTypeClasses #-} import Control.Applicative import Control.Monad import Data.List import Data.Maybe import Test.SmallCheck.Series import Test.Tasty import Test.Tasty.SmallCheck
instance Monad ZipList where ZipList [] >>= _ = ZipList [] ZipList (x:xs) >>= f = ZipList $ do let ZipList y' = f x guard (not (null y')) let ZipList ys = ZipList xs >>= ZipList . join . maybeToList . fmap snd . uncons . getZipList . f head y' : ys
instance Serial m a => Serial m (ZipList a) where series = ZipList <$> series
main = defaultMain $ testGroup "Monad laws" [ testProperty "Right identity" $ \(z :: ZipList Int) -> (z >>= return) == z , testProperty "Left identity" $ \(b :: Bool) (f :: Bool -> ZipList Bool) -> (return b >>= f) == f b , testProperty "Associativity" $ \(f1 :: Bool -> ZipList Bool) (f2 :: Bool -> ZipList Bool) (z :: ZipList Bool) -> (z >>= (\x -> f1 x >>= f2)) == ((z >>= f1) >>= f2) ]
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