# [Haskell-cafe] Type constraints for class instances

Krzysztof Skrzętnicki gtener at gmail.com
Sun Mar 23 08:39:40 EDT 2008

```I fixed the code, see below. In fact, it works now for any listst of
type (YOrd a) => [a]. It works for things like
> ysort [[1..],[1..],[2..],[1..]]
Unfortunately, the performance of ysort is rather low. I belive that
it is impossible to create any sorting algorithm that uses ycmp
instead of compare, that is faster than O(n^2). In fact, ysort is
Theta(n^2), and it appears to be optimal. Why?
Well, consider the bubble sort algorithm. Then ycmp will be simply
sort of swap used there:

ycmp x y = case x `compare` y of
LT -> (x,y)
EQ -> (x,y)
GT -> (y,x)

And because it is the only possible operation here, it can't be
faster. (Though I may be wrong.)

Best regards,

Christopher Skrzętnicki.

---

--- http://hpaste.org/6536#a1

{-# OPTIONS_GHC -O2 #-}

module Data.YOrd (ysort, YOrd(..)) where

-- Well defined where Eq means equality, not only equivalence

class YOrd a where
ycmp :: a -> a -> (a,a)

instance (Ord a) => YOrd [a] where
ycmp = ycmpWith compare
where
ycmpWith _ xs [] = ([],xs)
ycmpWith _ [] xs = ([],xs)
ycmpWith cmp (xs'@(x:xs)) (ys'@(y:ys)) = case x `cmp` y of
LT -> (xs',ys')
GT -> (ys',xs')
EQ -> let (sm,gt)
= xs `ycmp` ys in
(x:sm,x:gt)
-- assumes that cmp is equality not equivalence relation here!

ycmpWrap cmp x y = case x `cmp` y of
LT -> (x,y)
EQ -> (x,y)
GT -> (y,x)

instance YOrd Integer where
ycmp = ycmpWrap compare
instance YOrd Char where
ycmp = ycmpWrap compare
instance YOrd Int where
ycmp = ycmpWrap compare

-- ysort : sorting in O(n^2)

ysort :: (YOrd a) => [a] -> [a]

ysort = head . mergeAll . wrap

wrap xs = map (:[]) xs

mergeAll [] = []
mergeAll [x] = [x]
mergeAll (a:b:rest) = mergeAll ((merge a b) : (mergeAll rest))

merge xs [] = xs
merge [] xs = xs
merge (x:xs) (y:ys) = let (sm,gt) = x `ycmp` y in
sm : (merge [gt] \$ merge xs ys)

2008/3/21 Stephen Marsh <freeyourmind at gmail.com>:
> Actually, infinite trees wouldn't work, for a similar reason to above. You
> can't decide sort order on the infinite left branches, so you could never
> choose the correct right branch.
>
> Stephen
>
>  2008/3/21 Stephen Marsh <freeyourmind at gmail.com>:
>
>
> > There is a bug in the code:
> >
> > *Main> ycmp [5,2] [2,5] :: ([Int], [Int])
> > ([2,2],[5,5])
> >
> > I think it is impossible to define a working (YOrd a) => YOrd [a]
> instance. Consider:
> >
> > let (a, b) = ycmp [[1..], [2..]] [[1..],[1..]]
> >
> > head (b !! 1) -- would be nice if it was 2, but it is in fact _|_
> >
> > We take forever to decide if [1..] is greater or less than [1..], so can
> never decide if [1..] or [2..] comes next.
> >
> > However Ord a => YOrd [a] can be made to work, and that is absolutely
> awesome, esp. once you start thinking about things like Ord a => YOrd
> (InfiniteTree a). This really is very cool, Krzysztof.
> >
> > Stephen
> >
> >
> > 2008/3/20 Krzysztof Skrzętnicki <gtener at gmail.com>:
> >
> > >
> > >
> > >
> > > 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 Skrzętnicki
> > >
> > > _______________________________________________
> > > Haskell-Cafe mailing list