Thanks for the reply. Here's the decomposition I had in mind. Start with
type List a = Maybe (a, List a)
Rewrite a bit
type List a = Maybe (Id a, List a)
Then make the type *constructor* pairing explicit
type List a = Maybe ((Id :*: List) a)
where
newtype (f :*: g) a = Prod { unProd :: (f a, g a) }
Then make the type-constructor composition explicit
type List = Maybe :. (Id :*: List)
(which isn't legal Haskell, due to the type synonym cycle). From there use
the Functor and Applicative instances for composition and pairing of type
constructors and for Id. I think the result is equivalent to ZipList.
To clarify my "cross products" question, I mean fs <*> xs = [f x | f <- fs,
x <- xs], as with lists.
Cheers, - Conal
On Mon, Mar 24, 2008 at 8:36 AM, apfelmus
(Sorry for the late reply)
Is there a known deconstruction of the list/backtracking applicative functor (AF)? If I decompose the list type into pieces (Maybe, product, composition), I think I can see where the ZipList AF comes from, but not
Conal Elliott wrote: the
list/backtracking AF.
So, you mean that the strange thing about the list monad is that the "natural" applicative structure for [a] is derived from the "composition"
[a] ~ Maybe (a, Maybe (a, ...)) ~ Maybe `O` (a,) `O` Maybe `O` (a,) `O` ...
? Well, this is not quite true since the applicativity you're seeking is in the extra argument a , not in the argument of the composition. In fact, this infinite composition doesn't have an argument (that's the whole point of taking the fixed point). In other words, every chain like
Maybe `O` (a,) `O` Maybe `O` (a,) Maybe `O` (a,) `O` Maybe `O` (a,) `O` Maybe `O` (a,)
etc. is an applicative functor in its argument, but not necessarily in a . So, there is more to the "natural" ZipList AF than Maybe, product and composition.
Is there some construction simpler than lists (non-recursive) that introduces cross products?
What do you mean with "cross products" here? Something with
sequence :: Applicative f => [f a] -> f [a]
being the cartesian product for the list monad? Or simpler
pure (,) :: Applicative f => (f a, f b) -> f (a,b)
somehow "crossing" the "elements" of f a and f b ?
Regards, apfelmus
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