[Haskell] Arrows and XML filters

[David Menendez]

I don't know if this has come up before. If it has, I'd be interested in
a link to the discussion.

Last week, I noticed that filters used in the Haskell XML Toolbox (and
elsewhere) are easily made into arrows. Specifically, for any filter

  f :: a -> [b]
  
we can write
  
  Kleisli f :: Kleisli [] a b


It then turns out that many of the filter combinators correspond exactly
to arrow combinators. For example:

  Kleisli (g `o` f) === Kleisli f >>> Kleisli g
  Kleisli (f .> g)  === Kleisli f >>> Kleisli g
  Kleisli (f +++ g) === Kleisli f <+> Kleisli g
  Kleisli this      === returnA
  Kleisli none      === zeroArrow

Similarly, for the functions 'cat' and 'seqF', we can write 

> asum :: ArrowPlus a => [a b c] -> a b c
> asum = foldr (<+>) zeroArrow
> 
> seqA :: Arrow a => [a b b] -> a b b
> seqA = foldr (>>>) returnA

And so forth.


The predicate combinators can't be made as general, but they still work
in an arrow framework.

> orElse :: Kleisli [] a b -> Kleisli [] a b -> Kleisli [] a b
> orElse (Kleisli f) (Kleisli g) =
>   Kleisli (\t -> let res = f t in if null res then g t else res)

The tree filters work similarly:

> type TFilterA node = Kleisli [] (NTree node) (NTree node)
> processChildren :: TFilterA node -> TFilterA node
> processChildren (Kleisli f) =
>   Kleisli (\(NTree n cs) -> [NTree n (concatMap f cs)])


I don't know if this has any practical value. (Perhaps for future XML
libraries?) If nothing else, it's given me some experience with and (I
hope) a better understanding of arrows.
-- 
David Menendez <zednenem at psualum.com> <http://www.eyrie.org/~zednenem/>