# The Universe of Discourse

Mon, 03 Sep 2018

Whenever I try to program in Haskell, the same thing always goes wrong. Here is an example.

I am writing a module to operate on polynomials. The polynomial !!x^3 - 3x + 1!! is represented as

Poly [1, -3, 0, 1]

[ Addendum 20180904: This is not an error. The !!x^3!! term is last, not first. Much easier that way. Fun fact: two separate people on Reddit both commented that I was a dummy for not doing it the easy way, which is the way I did do it. Fuckin' Reddit, man. ]

I want to add two polynomials. To do this I just add the corresponding coefficients, so it's just

(Poly a) + (Poly b) = Poly $zipWith (+) a b Except no, that's wrong, because it stops too soon. When the lists are different lengths, zipWith discards the extra, so for example it says that !!(x^2 + x + 1) + (2x + 2) = 3x + 3!!, because it has discarded the extra !!x^2!! term. But I want it to keep the extra, as if the short list was extended with enough zeroes. This would be a correct implementation: (Poly a) + (Poly b) = Poly$ addup a b   where

and I can write this off the top of my head.

But do I? No, this is where things go off the rails. “I ought to be able to generalize this,” I say. “I can define a function like zipWith that is defined over any Monoid, it will combine the elements pairwise with mplus, and when one of the lists runs out, it will pretend that that one has some memptys stuck on the end.” Here I am thinking of something like ffff :: Monoid a => [a] -> [a] -> [a], and then the (+) above would just be

(Poly a) + (Poly b) = Poly (ffff a b)

as long as there is a suitable Monoid instance for the as and bs.

I could write ffff in two minutes, but instead I spend fifteen minutes looking around in Hoogle to see if there is already an ffff, and I find mzip, and waste time being confused by mzip, until I notice that I was only confused because mzip is for Monad, not for Monoid, and is not what I wanted at all.

So do I write ffff and get on with my life? No, I'm still not done. It gets worse. “I ought to be able to generalize this,” I say. “It makes sense not just for lists, but for any Traversable… Hmm, or does it?” Then I start thinking about trees and how it should decide when to recurse and when to give up and use mempty, and then I start thinking about the Maybe version of it.

Then I open a new file and start writing

mzip :: (Traversable f, Monoid a) => f a -> f a -> f a
mzip as bs = …

And I go father and farther down the rabbit hole and I never come back to what I was actually working on. Maybe the next step in this descent into madness is that I start thinking about how to perform unification of arbitrary algebraic data structures, I abandon mzip and open a new file for defining class Unifiable

Actually when I try to program in Haskell there a lot of things that go wrong and this is only one of them, but it seems like this one might be more amenable to a quick fix than some of the other things.

[ Addendum 20180904: A lobste.rs user points out that I don't need Monoid, but only Semigroup, since I don't need mempty. True that! I didn't know there was a Semigroup class. ]

[ Addendum 20181109: More articles in this series: [2] [3] ]