# The Universe of Discourse

Fri, 09 Nov 2018

(Previously: [1] [2])

I'm doing more work on matrix functions. A matrix represents a relation, and I am representing a matrix as a [[Integer]]. Then matrix addition is simply liftA2 (liftA2 (+)). Except no, that's not right, and this is not a complaint, it's certainly my mistake. The overloading for liftA2 for lists does not do what I want, which is to apply the operation to each pair of correponding elements. I want liftA2 (+) [1,2,3] [10,20,30] to be [11,22,33] but it is not. Instead liftA2 lifts an operation to apply to each possible pair of elements, producing [11,21,31,12,22,32,13,23,33]. And the twice-lifted version is similarly not what I want:

$$\require{enclose} \begin{pmatrix}1&2\\3&4\end{pmatrix}\enclose{circle}{\oplus} \begin{pmatrix}10&20\\30&40\end{pmatrix}= \begin{pmatrix} 11 & 21 & 12 & 22 \\ 31 & 41 & 32 & 42 \\ 13 & 23 & 14 & 24 \\ 33 & 43 & 34 & 44 \end{pmatrix}$$

No problem, this is what ZipList is for. ZipLists are just regular lists that have a label on them that advises liftA2 to lift an operation to the element-by-element version I want instead of the each-one-by-every-other-one version that is the default. For instance

    liftA2 (+) (ZipList [1,2,3]) (ZipList [10,20,30])


gives ZipList [11,22,33], as desired. The getZipList function turns a ZipList back into a regular list.

But my matrices are nested lists, so I need to apply the ZipList marker twice, once to the outer list, and once to each of the inner lists, because I want the element-by-element behavior at both levels. That's easy enough:

    matrix :: [[a]] -> ZipList (ZipList a)
matrix m = ZipList (fmap ZipList m)


(The fmap here is actually being specialized to map, but that's okay.)

Now

    (liftA2 . liftA2) (+) (matrix [[1,2],[3,4]]) (matrix [[10,20],[30, 40]])


does indeed produce the result I want, except that the type markers are still in there: instead of

    [[11,22],[33,44]]


I get

    ZipList [ ZipList [11, 22], ZipList [33, 44] ]


No problem, I'll just use getZipList to turn them back again:

    unmatrix :: ZipList (ZipList a) -> [[a]]
unmatrix m = getZipList (fmap getZipList m)


And now matrix addition is finished:

    matrixplus :: [[a]] -> [[a]] -> [[a]]
matrixplus m n = unmatrix \$ (liftA2 . liftA2) (+) (matrix m) (matrix n)


This works perfectly.

But the matrix and unmatrix pair bugs me a little. This business of changing labels at both levels has happened twice already and I am likely to need it again. So I will turn the two functions into a single higher-order function by abstracting over ZipList. This turns this

    matrix m = ZipList (fmap ZipList m)


into this:

    twice zl m = zl (fmap zl m)


with the idea that I will now have matrix = twice ZipList and unmatrix = twice getZipList.

The first sign that something is going wrong is that twice does not have the type I wanted. It is:

    twice ::  Functor f             => (f a -> a)   -> f (f a) -> a


where I was hoping for something more like this:

    twice :: (Functor f, Functor g) => (f a -> g a) -> f (f a) -> g (g a)


which is not reasonable to expect: how can Haskell be expected to figure out I wanted two diferent functors in there when there is only one fmap? And indeed twice does not work; my desired matrix = twice ZipList does not even type-check:

    <interactive>:19:7: error:
• Occurs check: cannot construct the infinite type: a ~ ZipList a
Expected type: [ZipList a] -> ZipList a
Actual type: [a] -> ZipList a
• In the first argument of ‘twice’, namely ‘ZipList’
In the expression: twice ZipList
In an equation for ‘matrix’: matrix = twice ZipList
• Relevant bindings include
matrix :: [[ZipList a]] -> ZipList a (bound at <interactive>:20:5)


Telling GHC explicitly what type I want for twice doesn't work either, so I decide it's time to go to lunch. w I take paper with me, and while I am eating my roast pork hoagie with sharp provolone and spinach (a popular local delicacy) I work out the results of the type unification algorithm on paper for both cases to see what goes wrong.

I get the same answers that Haskell got, but I can't see where the difference was coming from.

So now, instead of defining matrix operations, I am looking into the type unification algorithm and trying to figure out why twice doesn't work.

And that is yet another reason why I never finish my Haskell programs. (“What do you mean, λ-abstraction didn't work?”)