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Mon Aug 15 12:49:11 CEST 2011

Applicative?

So..  I have the following: series / parallel composition and pure
functions.

serSSM :: ((s1, b)  -> (s1, c)) ->
          ((s2, a)  -> (s2, b)) ->
          ((s1,s2), a) -> ((s1,s2), c)
serSSM f g = fg where
  fg ((s1,s2), a) = ((s1',s2'), c) where
    (s2', b) = g (s2, a)
    (s1', c) = f (s1, b)
    
parSSM :: ((s1, a)  -> (s1, c)) ->
          ((s2, b)  -> (s2, d)) ->
          ((s1,s2), (a,b)) -> ((s1,s2), (c,d))
parSSM f g = fg where          
  fg ((s1,s2), (a,b)) = ((s1',s2'), (c,d)) where
    (s1',c) = f (s1, a)
    (s2',d) = g (s2, b)
    

pureSSM :: (a -> b) -> ((), a) -> ((), b)
pureSSM f (_,a) = ((), f a)


I wonder if parallel composition is really necessary since it seems to
be mostly useful for doing things like:
   a -> b -> (a,b) -> (a+b)

I.e. replacing the tuple by an arithmetic reduction.

Maybe it's enough to just allow for input/output/state to contain
curried functions.  Then Applicative can be used.

This doesn't work because my abstraction is Arrow-like (explicit
input/output, kind * -> * -> *) and Applicative is Functor-like (* ->
*).  See also [1].  It seems Applicative needs a "generator" approach
instead of a "processor" approach.

I'm first trying for Functor:

instance Functor (SSM i) where
  fmap f a = (SSM () $ pureSSM f) . a

This works!
*Main> fmap (1+) int
[2,2,2,2,2,2,2,2,2,2]

The case for mapping over inputs should be completely symmetric,
though I don't know how to "flip" the arguments of the type
constructor SSM.  As mentioned here[2] it's possible to use a
Bifunctor instead.



So the trick seems to be to focus on the output.  Let's try
Applicative.  There it seems to go wrong.  Paraphrased: x is a value
while pureSSM expects a function.

instance Applicative (SSM i) where  
  pure x = SSM () $ pureSSM x


So.. is it really necessary?  It doesn't look like it.  Ser/par +
Functor seem to be really enough.



[1] http://cdsmith.wordpress.com/2011/07/30/arrow-category-applicative-part-i/
[2] http://stackoverflow.com/questions/2335967/manipulating-the-order-of-arguments-to-type-constructors
[3] http://hackage.haskell.org/packages/archive/category-extras/0.53.5/doc/html/Control-Functor.html



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