Sebastian Fischer wrote:
Your `ListTo` type achieves space efficiency for Applicative composition of list functions by executing them in lock-step. Because of the additional laziness provided by the `Fmap` constructor, compositions like
interpret a . interpret b
can also be executed in constant space. However, we cannot use the space efficient Applicative combinators again to form parallel compositions of sequential ones because we are already in the meaning type.
We could implement composition for the `ListTo` type as follows
(<.) :: ListTo b c -> ListTo a [b] -> ListTo a c a <. b = interpret a <$> b
But if we use a result of this function as argument of <*>, then the advantage of using `ListTo` is lost. While
interpret ((,) <$> andL <*> andL)
runs in constant space,
interpret ((,) <$> (andL <. idL) <*> (andL <. idL))
does not.
The ListTransformer type supports composition in lock-step via a category instance. The meaning of `ListTransformer a b` is `[a] -> [b]` with the additional restriction that all functions `f` in the image of the interpretation function are incremental:
xs `isPrefixOf` ys ==> f xs `isPrefixOf` f ys
[..]
The Applicative instance for `ListTransformer` is different from the Applicative instance for `ListTo` (or `ListConsumer`). While
interpret ((,) <$> idL <*> idL)
is of type `[a] -> ([a],[a])`
transformList ((,) <$> idL <*> idL)
is of type `[a] -> [(a,a)]`. [..]
Ah, so ListTransformer is actually quite different from ListTo because the applicative instance yields a different type. Then again, the former can be obtained form the latter via unzip .
I have a gut feeling that the laziness provided by the `Fmap` constructor is too implicit to be useful for the kind of lock-step composition provided by ListTransformer. So I don't have high hopes that we can unify `ListConsumer` and `ListTransformer` into a single type.
Do you have an idea?
Well, the simple solution would be to restrict the type of (<.) to (<.) :: ListTo b c -> ListTransformer a b -> ListTo a c so that the second argument is guaranteed to be incremental. Of course, this is rather unsatisfactory. Fortunately, there is a nicer solution that keeps everything in the ListTo type. The main problem with Fmap is that it can be far from incremental, because we can plug in any function we like: example :: ListTo a [a] example = Fmap reverse Without an explicit guarantee that the function is incremental, we can't do anything here. But we can just add another constructor to that effect if we turn ListTo into a GADT: data ListTo a b where CaseOf :: b -> (a -> ListTo a b) -> ListTo a b Fmap :: (b -> c) -> ListTo a b -> ListTo a c FmapCons :: b -> ListTo a [b] -> ListTo a [b] The interpretation for this case is given by the morphism interpret (FmapCons x g) = fmap (x:) $ interpret g and sequential composition reads -- sequential composition -- interpret (a <. b) = interpret $ interpret a <$> b (<.) :: ListTo b c -> ListTo a [b] -> ListTo a c (CaseOf _ cons) <. (FmapCons y b) = cons y <. b (Fmap f a) <. (FmapCons y b) = Fmap f $ a <. (FmapCons y b) (FmapCons x a) <. (FmapCons y b) = FmapCons x $ a <. (FmapCons y b) a <. (CaseOf nil cons) = CaseOf (interpret a nil) ((a <.) . cons) a <. (Fmap f b) = fmap (interpret a . f) b Of course, the identity has to be redefined to make use of the new guarantee idL :: ListTo a [a] idL = caseOf [] $ \x -> FmapCons x idL I'm going to omit the new definition for the applicative instance, the full code can be found here: https://gist.github.com/1550676 With all these combinators in place, even examples like liftA2 (,) (andL <. takeL 3) (andL <. idL) should work as expected. While nice, the above solution is not perfect. One thing we can do with ListTransformer type is to perform an apply first and then do a sequential composition. a <. (b <*> c) This only works because the result of <*> is already zipped. And there is an even more worrisome observation: all this work would have been superfluous if we had *partial evaluation*, i.e. if it were possible to evaluate expressions like \xs -> f (4:xs) beneath the lambda. Then we could dispense with all the constructor yoga above and simply use a plain type ListTo a b = [a] -> b with the applicative instance instance Applicative (ListTo a) where pure b = const b (f <*> x) cs = case cs of [] -> f [] $ x [] (c:cs) -> let f' = f . (c:); x; = x . (c:) in f' `partialseq` x' `partialseq` (f' <*> x') to obtain space efficient parallel and sequential composition. In fact, by using constructors, we are essentially doing partial evaluation by hand. Best regards, Heinrich Apfelmus -- http://apfelmus.nfshost.com