Multiple functions applied to a single value

[Graham Klyne]

At 21:03 27/11/03 -0500, Derek Elkins wrote:
>On Thu, 27 Nov 2003 14:56:03 +0000
>Graham Klyne <gk at ninebynine.org> wrote:
>
>(perhaps a more serious and to the point reply later)
>
> > But not all cases I encounter involve lists or monads.  A different
> > case might look like this:
>
>Are you sure this doesn't involve monads?

No, I'm not, and yours is very much the kind of response I was hoping to 
elicit...  but I think I may need a little more help to properly "get it".

I'm looking at:
[1] http://www.haskell.org/hawiki/MonadReader
[2] http://www.haskell.org/ghc/docs/latest/html/base/Control.Monad.Reader.html
[3] http://www.nomaware.com/monads/html/readermonad.html

You say of my examples "(these work fine with a Monad instance ((->) r) 
which is a Reader monad)".  If I get this correctly, (->) used here is a 
type constructor for a function type [ah yes... p42 of the Haskell report, 
but not in the index].

In [2] I see ((->) r) as an instance of MonadReader r, which you also 
say.  I think this means that a function from r to something is an instance 
MonadReader r.  So in my definition of eval:

   eval f g1 g2 a = f (g1 a) (g2 a)

g1 and g2 are instances of MonadReader a.  Which I can see means that eval 
is liftM2 as you say:  it takes a 2-argument function f and 'lifts' it to 
operate on the monads g1 and g2.

So far, so good, but what are the implications of g1 and g2 being monads?
 From [2], we have:
   class (Monad m) => MonadReader r m | m -> r where
   MonadReader r ((->) r)
So ((->) r) must be a Monad.
How are the standard monad operators implemented for ((->) r)?  Maybe:

instance Monad ((->) r) where
     return a = const a       -- is this right?  As I understand,
                              -- return binds some value into a monad.
     -- (>>=) :: m a -> (a -> m b) -> m b
     g1 >>= f = \e -> f (g1 e) e

<aside>
so, if f is \a -> g2, we get:
     g1 >>= f = \e -> (\a -> g2) (g1 e) e
              = \e -> g2 e
              = g2
</aside>

Hmmm... this seems plausible, but I'm not clear-sighted enough to see if I 
have the ((->) r) monad right.  [Later: though it seems to work as intended.]

Looking at [3], I get a little more insight.  It seems that ((->) r) is a 
function with a type of "Computations which read values from a shared 
environment", where r is the type of the shared environment.  Monadic 
sequencing (>>=) passes the result from one monad/function to the 
next.  The monad is used by applying it to an instance of the shared 
environment.

So, returning to my example, it would appear that the idiom I seek is:
     liftM2 f g1 g2
or:
     liftM3 f g1 g2 g3
etc.

Provided that ((->) r) is appropriately declared as an instance of 
Monad.  Does this work with the above declaration?

     liftM2 f g1 g2
         = do { g1' <- g1 ; g2' <- g2 ; return (f g1' g2') }          [from 
Monad]
         = g1 >>= \g1' -> g2 >>= \g2' -> return (f g1' 
g2')           [do-notation]
         = \e1 -> (\g1' -> g2 >>= \g2' -> return (f g1' g2')) (g1 e1) e1
                                               [above:  g1 >>= f = (\e -> f 
(g1 e) e)]
         = \e1 -> (\g1' -> \e2 -> (\g2' -> return (f g1' g2')) (g2 e2) e2) 
(g1 e1) e1
                                               [again]
         = \e1 -> (\e2 -> (return (f (g1 e1) (g2 e2))) e2) e1
                                               [apply fns: g1' = g1 e1,g2' 
= g2 e2]
         = \e1 -> (return (f (g1 e1) (g2 e1))) e1
                                               [apply fn:  e2 = e1]
         = \e1 -> (return (f (g1 e1) (g2 e1))) e1
                                               [apply fn:  e2 = e1]
         = \e1 -> (const (f (g1 e1) (g2 e1))) e1
                                               [above: return = const]
         = \e1 -> (f (g1 e1) (g2 e1)))
                                               [apply const]

Which is the desired result (!)

>(these work fine with a Monad instance ((->) r) which is a Reader monad)

Hmmm... is it true that ((->) r) *is* a reader monad?  It seems to me that 
it is a Monad which can be used to build a reader monad.

...

The more I do with Haskell the more impressed I am by the folks who figured 
out this Monad wizardry.

A question I find myself asking at the end:  why isn't ((->) r) declared as 
a Monad instance in the standard prelude?  If I'm following all this 
correctly, it seems like a natural to include there.

Thanks for pointing me in this direction.  I hope my ramblings are 
on-track, and not too tedious to wade through.

#g
--


> >  >  eval :: (b->c->d) -> (a->b) -> (a->c) -> (a->d)
> >  >  eval f g1 g2 a = f (g1 a) (g2 a)
>
>eval :: Monad m => (b -> c -> d) -> m b -> m c -> m d
>eval = liftM2
>
> > So, for example, a function to test of the two elements of a pair are
> > the same might be:
> >
> >  > pairSame = eval (==) fst snd
> >
> > giving:
> >
> >  > pairSame (1,2) -- false
> >  > pairSame (3,3) -- true
> >
> >
> > Or a function to subtract the second and subsequent elements of a list
> > from the first:
> >
> >  > firstDiffRest = eval (-) head (sum . tail)
> >
> >  > firstDiffRest [10,4,3,2,1] -- 0


------------
Graham Klyne
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