[Haskell-beginners] applicative instance

[Graham Gill]

On 28-Jan-2017 4:53 AM, sasa bogicevic wrote:
> Is there a way to do it without defining a separate function like 
> plusPlus ?

My guess is there isn't. I'm unsure what you mean by your question though.

Your List is a reimplementation of Haskell []. So, List with the 
"Cartesian product" Applicative instance will, like Haskell [], extend 
to a Monad instance. In the Monad instance for [], join = concat, and 
the work of concat is done using ++.

For List, we can implement join using concatList:

concatList :: List (List a) -> List a
concatList Nil = Nil
concatList (Cons xs xss) = xs `plusPlus` (concatList xss)

and then we can add a Monad instance for List:

instance Monad List where
     return = pure
     xs >>= f = concatList (pure f <*> xs)

or equivalently, bind isgiven by
     xs >>= f = concatList (fmap f xs)

To ask whether you can define the Cartesian product Applicative instance 
for List without plusPlus, is, I think, like asking whether there is a 
Cartesian product Applicative instance for List (or []) which doesn't 
extend to a Monad instance. Because if it does extend to a Monad (that 
obeys the Monad laws), then there will exist an implementation of join 
:: List (List a) -> List a, and join will need to collapse a List of 
Lists into a List. A function like plusPlus is used to accomplish the 
collapse.

That's "proof by hand waving."

The Ziplist Applicative instance for List on the other hand can't be 
extended to a Monad instance without additional restrictions on the 
lengths of the lists. Your question led me to some interesting reading 
with a google search on "list monad vs ziplist". Thanks.


On 28-Jan-2017 4:53 AM, sasa bogicevic wrote:
> Yep that worked, thanks.
> Is there a way to do it without defining a separate function like plusPlus ?
>
>
>
>
>
>> On Jan 28, 2017, at 10:43, Francesco Ariis<fa-ml at ariis.it>  wrote:
>>
>> On Sat, Jan 28, 2017 at 10:09:10AM +0100, sasa bogicevic wrote:
>>> Ok so how would the implementation look to get the correct result ?
>>> I can't seem to write something that will compile except ZipList version.
>> One way is by implementing your own (++):
>>
>>     data List a = Nil | Cons a (List a) deriving (Eq, Show)
>>
>>     plusPlus :: List a -> List a -> List a
>>     plusPlus Nil         bs = bs
>>     plusPlus (Cons a as) bs = Cons a (as `plusPlus` bs)
>>
>>     instance Functor List where
>>         fmap f Nil = Nil
>>         fmap f (Cons a b) = Cons (f a) (fmap f b)
>>
>>     instance Applicative List where
>>         pure x = Cons x Nil
>>         Nil <*> _ = Nil
>>         _ <*> Nil = Nil
>>         (Cons x xy) <*> ys = (fmap x ys) `plusPlus` (xy <*> ys)
>>
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