Re: [Haskell-cafe] Gluing pipes

On Thu, Dec 4, 2008 at 11:19 AM, Dan Piponi wrote:

On Wed, Dec 3, 2008 at 10:17 AM, Matt Hellige wrote:

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From time to time, I've wanted to have a more pleasant way of writing point-free compositions of curried functions.

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I'd like to be able to write something like: \ x y -> f (g x) (h y)

This particular composition of f with g and h is an example of composition in what mathematicians call an operad. I think operads glue things just how you want. Unfortunately (1) I don't think mathematicians have great notation for it either and (2) it's hard to combine operads with currying because they rely on having a well defined notion of the number of arguments of a function.

Yes, of course I should have mentioned operads! I've thought of operads in connection with this puzzle before (and I seem to remember a recent post from you on the subject), but this time around, it slipped my mind... Thanks for the reminder! Something like operadic composition for curried functions is exactly what I want to define, and I agree that your (1) and (2) are exactly the sticking points. I'm relatively happy with what I've come up with so far, but I do still wonder if it's possible to do better. An alternative approach would be to define an Operad type class, and attempt to coax curried functions into and out of an instance as needed. I hesitate to propose that it's impossible, but I guess it would take type class hacking beyond my abilities in any case. We could imagine being able to write something like: f [g, h] and I think this is the route you took in your blog post? But of course your way only works for uncurried functions, right? And there's a bit of boilerplate? I also believe that this approach would leave at least some arity checking to runtime, which I would consider a shortcoming. Perhaps it would be possible to overcome this with length-indexed lists? Finally, there's a further disadvantage to both of these approaches: it seems to me that neither approach allows us to "cross pipes". For instance, we can't define flip using these techniques, or any "deep flip": \ x y z -> f x z y Is it true in general that operads cannot express pipe-crossing compositions? Or is it just a shortcoming of this particular implementation? Of course this is getting farther afield, and I don't believe that my approach can express this either. Anyway I'd still love to know the answers the questions in my last email, as well as the new questions posed in this one. ;) Thanks for taking the time! Matt