Question

The monad instanc of Data.Stream is defined that way:

instance Monad Stream where
  return = repeat
  xs >>= f = join (fmap f xs)
    where
      join :: Stream (Stream a) -> Stream a
      join ~(Cons xs xss) = Cons (head xs) (join (map tail xss))

That means join takes the first element of the first stream, the second element of the second stream etc, so the resulting stream can be seen as “main diagonal”, discarding all other elements.

Now there is a way to go through an infinite two-dimensional table, discovered by Georg Cantor for his proof that there are as many rational numbers as natural numbers: http://www.jcu.edu/math/vignettes/infinity.htm

Now my question is if a join using a path along all secondary diagonals (visiting every element of every stream) would be a valid implementation as well. Or would this violate one of the monad laws?

Answer 1

It would violate

return x >>= f === f x

Consider

f k = Cons k (f (k+1))

Now fmap f (return 1) is repeat (f 1) and if join went through all elements, in the resulting Stream, elements would repeat.

As a two-dimensional table, fmap f (return 1) looks like

1 2 3 4 ...
1 2 3 4 ...
1 2 3 4 ...

and if you traverse that following the secondary diagonals, you get

1 1 2 1 2 3 1 2 3 4 ...

and not 1 2 3 4 5 ... as with f 1.