The Universe of Discourse

Haskell type checker complaint 184 of 698

I want to build an adjacency matrix for the vertices of a cube; this is a matrix that has m[a][b] = 1 exactly when vertices a and b share an edge. We can enumerate the vertices arbitrarily but a convenient way to do it is to assign them the numbers 0 through 7 and then say that vertices !!a!! and !!b!! are adjacent if, regarded as binary numerals, they differ in exactly one bit, so:

   import Data.Bits
   a `adj` b = if (elem (xor a b) [1, 2, 4]) then 1 else 0         

This compiles and GHC infers the type

   adj :: (Bits a, Num a, Num t) => a -> a -> t 

Fine.

Now I want to build the adjacency matrix, which is completely straightforward:

    cube = [ [a `adj` b | b <- [0 .. 7] ] | a <- [0 .. 7] ]  where
      a `adj` b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

Ha ha, no it isn't; in Haskell nothing is straightforward. This produces 106 lines of type whining, followed by a failed compilation. Apparently this is because because 0 and 7 are overloaded, and could mean some weird values in some freakish instance of Num, and then 0 .. 7 might generate an infinite list of 1-graded torsion rings or something.

To fix this I have to say explicitly what I mean by 0. “Oh, yeah, by the way, that there zero is intended to denote the integer zero, and not the 1-graded torsion ring with no elements.”

        cube = [ [a `adj` b | b <- [0 :: Integer .. 7] ] | a <- [0 .. 7] ]  where
          a `adj` b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

Here's another way I could accomplish this:

        zero_i_really_mean_it = 0 :: Integer
        cube = [ [a `adj` b | b <- [zero_i_really_mean_it .. 7] ] | a <- [0 .. 7] ] where       
          a `adj` b = if (elem (xor a b) [1, 2, 4]) then 1 else 0

Or how about this?

        cube = [ [a `adj` b | b <- numbers_dammit [0 .. 7] ] | a <- [0 .. 7] ] where
          p `adj` q = if (elem (xor p q) [1, 2, 4]) then 1 else 0
          numbers_dammit = id :: [Integer] -> [Integer] 

I think there must be something really wrong with the language design here. I don't know exactly what it is, but I think someone must have made the wrong tradeoff at some point.

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