## Joseph and the Amazing Technicolor Box

Consider the following data type in Haskell:

data Box a = B a

How many computable functions of type `Box a -> Box a` are there? If we strictly use denotational semantics, there are seven:

But if we furthermore distinguish the *source* of the bottom (a very operational notion), some functions with the same denotation have more implementations...

*Irrefutable pattern match:*`f ~(B x) = B x`. No extras.*Identity:*`f b = b`. No extras.*Strict:*`f (B !x) = B x`. No extras.*Constant boxed bottom:*Three possibilities:`f _ = B (error "1")`;`f b = B (case b of B _ -> error "2")`; and`f b = B (case b of B !x -> error "3")`.*Absent:*Two possibilities:`f (B _) = B (error "4")`; and`f (B x) = B (x `seq` error "5")`.*Strict constant boxed bottom:*`f (B !x) = B (error "6")`.*Bottom:*Three possibilities:`f _ = error "7"`;`f (B _) = error "8"`; and`f (B !x) = error "9"`.

List was ordered by colors of the rainbow. If this was hieroglyphics to you, may I interest you in this blog post?

*Postscript.* GHC can and will optimize `f b = B (case b of B !x -> error "3")`, `f (B x) = B (x `seq` error "5")` and `f (B !x) = error "9"` into alternative forms, because in general we don't say if `seq (error "1") (error "2")` is semantically equivalent `error "1"` or `error "2"`: any one is possible due to imprecise exceptions. But if you really care, you can use `pseq`. However, even with exception set semantics, there are more functions in this "refined" view of the normal denotational semantics.