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
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