Hello everyone,
I'm working on a small module for comparing things incomparable with Ord.
More precisely I want to be able to compare equal infinite lists like [1..].
Obviously
(1) compare [1..] [1..] = _|_
It is perfectly reasonable for Ord to behave this way.
Hovever, it doesn't have to be just this way. Consider this class
class YOrd a where
ycmp :: a -> a -> (a,a)
In a way, it tells a limited version of ordering, since there is no
way to get `==` out of this.
Still it can be useful when Ord fails. Consider this code:
(2) sort [ [1..], [2..], [3..] ]
It is ok, because compare can decide between any elements in finite time.
However, this one
(3) sort [ [1..], [1..] ]
will fail because of (1). Compare is simply unable to tell that two
infinite list are equivalent.
I solved this by producing partial results while comparing lists. If
we compare lists
(1:xs)
(1:ys)
we may not be able to tell xs < ys, but we do can tell that 1 will be
the first element of both of smaller and greater one.
You can see this idea in the code below.
--- cut here ---
{-# OPTIONS_GHC -O2 #-}
module Data.YOrd where
-- Well defined where Eq means equality, not only equivalence
class YOrd a where
ycmp :: a -> a -> (a,a)
instance (YOrd a) => YOrd [a] where
ycmp [] [] = ([],[])
ycmp xs [] = ([],xs)
ycmp [] xs = ([],xs)
ycmp xs'@(x:xs) ys'@(y:ys) = let (sm,gt) = x `ycmp` y in
let (smS,gtS) = xs `ycmp` ys in
(sm:smS, gt:gtS)
ycmpWrap x y = case x `compare` y of
LT -> (x,y)
GT -> (y,x)
EQ -> (x,y) -- biased - but we have to make our minds!
-- ugly, see the problem below
instance YOrd Int where
ycmp = ycmpWrap
instance YOrd Char where
ycmp = ycmpWrap
instance YOrd Integer where
ycmp = ycmpWrap
-- ysort : sort of mergesort
ysort :: (YOrd a) => [a] -> [a]
ysort = head . mergeAll . wrap
wrap :: [a] -> [[a]]
wrap xs = map (:[]) xs
mergeAll :: (YOrd a) => [[a]] -> [[a]]
mergeAll [] = []
mergeAll [x] = [x]
mergeAll (a:b:rest) = mergeAll ((merge a b) : (mergeAll rest))
merge :: (YOrd a) => [a] -> [a] -> [a]
merge [] [] = []
merge xs [] = xs
merge [] xs = xs
merge (x:xs) (y:ys) = let (sm,gt) = x `ycmp` y in
sm : (merge [gt] $ merge xs ys)
--- cut here ---
I'd like to write the following code:
instance (Ord a) => YOrd a where
ycmp x y = case x `compare` y of
LT -> (x,y)
GT -> (y,x)
EQ -> (x,y)
But i get an error "Undecidable instances" for any type [a].
Does anyone know the way to solve this?
Best regards
Christopher Skrzêtnicki