# Add instance Monad ZipList

David Feuer david.feuer at gmail.com
Fri Jun 5 05:49:55 UTC 2020

```I'm not really sure what you're getting at here. Codensity will turn
anything into a Monad. How does that relate to the question of whether
there's a valid `Monad ZipList` instance?

On Fri, Jun 5, 2020 at 1:42 AM Gershom B <gershomb at gmail.com> wrote:
>
> 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 <roma at ro-che.info>, wrote:
>
> 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)
> ]
>
> Roman
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