`wise` after the event (was Re: Data.List.join)
Conor McBride
ctm at Cs.Nott.AC.UK
Mon Oct 23 06:27:58 EDT 2006
Hi all,
Bit of a sideways leap from joinWith versus intercalate (which is what
you must do if you need to take time out after you matriculate but
before you graduate), but I was inspired/provoked/reminded by this
remark of John's...
John Meacham wrote:
> Although, now that we lost the Monoid instance for functions (which is
> very dismaying), it is less useful, as the monoid functions were very
> useful to build things up efficiently with (String -> String) as a type.
...(with which I rather disagree) to finish a note about programming
with structure-indicating newtypes, like Endo.
I thought I'd also remind you of a little of what we bought by wrapping
the endofunction monoid, then making the function instance do pointwise
lifting instead.
I append it, for what it's worth.
All the best
Conor
-----------------------------------------------------------------------
> module Wise where
Perhaps it's quite foolish, but here's an exercise in modifying one's
manners. I would not presume to use the title of this module as an
adjective, but I rather like it as a /suffix/.
> import Applicative
> import Traversable
> import Monoid
Firstly, we shall need a useful little type class to capture the pattern
of using a special isomorph of a type to indicate not merely its raw data,
but also its relevant structure.
> class Unpack p u | p -> u where
> unpack :: p -> u
> (~~) :: (u -> p) -> p -> u
> (~~) _ = unpack
The idea is that the packed type |p| is a special-purpose relabelling
of some duller unpacked type |u|. We thus expect |p| to determine |u|,
but a given |u| may have many such |p|'s, eg |Sum Int| and |Product
Int|, from |Data.Monoid|, wrapping Int to indicate an intended
|Monoid| structure.
Let's have an example:
> newtype Id x = Id {getId :: x}
> instance Unpack (Id x) x where
> unpack (Id x) = x
The |(~~)| method is provided as a uniform way to make an explicit
inverse for the constructor. We can replace |getId| by |(Id ~~)|, and
we never have to remember its name again, or even waste namespace on
it in the first place. It's sometimes ambiguous and often unreadable
to use unpack directly, when a specific instance is intended.
Here, the structure I want to make explicit via |Id| is the notion of
pure computation, normally left implicit. But pure computation iss as
good a notion of computation as any other and quite often better.
> instance Applicative Id where
> pure = Id
> Id f <*> Id s = Id (f s)
We use |Id| to get ordinary |fmap| from the more general |traverse|,
explaining the manner of the traversal.
> myFMap :: Traversable c => (s -> t) -> c s -> c t
> myFMap f cs = Id ~~ traverse (Id . f) cs
Recall that |Applicative (Const m)| if |Monoid m|. Let us have
> instance Unpack (Const m x) m where
> unpack (Const c) = c
> myCrush :: (Traversable c, Monoid m) => (s -> m) -> c s -> m
> myCrush m cs = Const ~~ traverse (Const . m) cs
Spotting the pattern? One more.
> instance Unpack (Endo x) (x -> x) where
> unpack (Endo f) = f
> myFold :: Traversable c => (s -> t -> t) -> c s -> t -> t
> myFold f cs = Endo ~~ myCrush (Endo . f) cs
What are we doing here? We're specifying the manner in which we use a
structure-exploiting operator. We transform a function in a particular
way by wrapping the range of that function in the type which indicates
the structure to be exploited by the operator. We can capture this
pattern by writing a transformer-transformer, composing the original
function with a given packer, then composing the transformed function
with the determined unpacker.
> wise :: Unpack dp du =>
> (bu -> bp) -> ((a -> bp) -> c -> dp) -> (a -> bu) -> c -> du
> wise way changes how = unpack . changes (way . how)
Third-order programming: it's a whole other order.
So, now we have hints to the appropriate structure as parenthetical
remarks:
> travMap :: Traversable f => (s -> t) -> f s -> f t
> travMap = (Id `wise`) traverse
> crush :: (Monoid m, Traversable f) => (s -> m) -> f s -> m
> crush = (Const `wise`) traverse
> travFold :: (Traversable f) => (a -> b -> b) -> f a -> b -> b
> travFold = (Endo `wise`) crush
And, to boot,
> instance Unpack Any Bool where
> unpack (Any b) = b
> travExists :: (Traversable f) => (a -> Bool) -> f a -> Bool
> travExists = (Any `wise`) crush
> instance Unpack All Bool where
> unpack (All b) = b
> travAll :: (Traversable f) => (a -> Bool) -> f a -> Bool
> travAll = (All `wise`) crush
> instance Unpack (Sum x) x where
> unpack (Sum x) = x
> travSum :: (Num a, Traversable f) => f a -> a
> travSum = (Sum `wise`) crush id
I'm in the mood for some pointwise lifting (in pointfree style).
> instance Unpack tp tu => Unpack (s -> tp) (s -> tu) where
> unpack = (unpack .)
Now we can exploit the pointwise lifting of |Unpack| together with the
pointwise lifting of |Monoid|, yielding |travExists2|, a function which
checks out whether any pair of elements drawn respectively from two
traversable structures satisfies a relation. For example,
|travExists2 (==)| will check if the two structures have an element in
common.
> travExists2 :: (Traversable f, Traversable g) =>
> (a -> b -> Bool) -> f a -> g b -> Bool
> travExists2 = ((crush . (Any .)) `wise`) crush
How does it work? Well, we lift |r :: a -> b -> Bool| to
|(Any .) . r :: a -> b -> Any| to |crush . (Any .) . r :: a -> g b -> Any|.
Now, |g b -> Any| is a |Monoid| because Any is, so the outer crush lifts
us to an |f a -> g b -> Any|. Then the unpacker for |g b -> Any| takes the
returned function back to a |g b -> Bool|.
All of this goes to show that Haskell has taken us far beyond the
standard, unimaginative view of types that you see all over the
place. I mean the view 'We already know what the programs are and how
to run them; types just discriminate between good programs and bad
programs.' That's an appropriate view for a typed /core/ language, but
it misses so many great opportunities for a typed /programming/ language.
We program by exposing structure.
More to the point, locally, if we accept that some notion of type-level
function is here to stay, is it worth adopting Unpack as a uniform way
to work with those newtypes whose purpose is to indicate structure (as
opposed to those which are intended to encapsulate invariants)? I
don't know how wise 'wise' is, but I had fun playing with it. Unpack,
though, is something I makes heavy use of.
Thought I'd punt it out there...
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