{-| Code for manipulation equivalence classes on index types. An 'Equivalence' is an equivalence relation. The empty equivalence relation is constructed over a ranges of values using 'emptyEquivalence'. Less discerning equivalence relations can be obtained with 'equate' and 'equateAll'. The relation can be tested with 'equiv' and 'equivalent', and canonical representatives can be chosen with 'repr'. An example follows: > import Data.Equivalence.Persistent > > rel = equateAll [1,3,5,7,9] > . equate 5 6 > . equate 2 4 > $ emptyEquivalence (1,10) > > test1 = equiv rel 3 5 -- This is True > test2 = equiv rel 1 6 -- This is True > test3 = equiv rel 4 6 -- This is False -} module Data.Equivalence.Persistent ( Equivalence, emptyEquivalence, repr, equiv, equivalent, equate, equateAll ) where import Control.Concurrent.MVar import Control.Monad import Data.Array.Diff import Data.IORef import Data.List import System.IO.Unsafe arrayFrom :: (IArray a e, Ix i) => (i,i) -> (i -> e) -> a i e arrayFrom rng f = array rng [ (x, f x) | x <- range rng ] {-| An 'Equivalence' is an equivalence relation on a range of values of some index type. -} data Equivalence i = Equivalence { ranks :: DiffArray i Int, parents :: IORef (DiffArray i i) } {-| 'emptyEquivalence' is an equivalence relation that equates two values only when they are equal to each other. It is the most discerning such relation possible. -} emptyEquivalence :: Ix i => (i, i) -> Equivalence i emptyEquivalence is = unsafePerformIO $ do v <- newIORef (arrayFrom is id) return $ Equivalence (arrayFrom is (const 0)) v reprHelper :: Ix i => DiffArray i i -> i -> (DiffArray i i, i) reprHelper ps i | pi == i = (ps, i) | otherwise = let (ps', r) = reprHelper ps pi in (ps' // [(i,r)], r) where pi = ps ! i {-| 'repr' gives a canonical representative of the equivalence class containing @x@. It is chosen arbitrarily, but is always the same for a given equivalence relation. This function is slightly unsafe. In particular, it's possible to build the same equivalence relation by equating values in two different orders, and the choice of canonical representatives will differ. You can either think of a value of type 'Equivalence' as an equivalence relation together with a choice of canonical representatives, or you can consider this not a pure function. Since 'Equivalence' is not an instance of @Eq@ and equality is not observable, both perspectives are valid. -} repr :: Ix i => Equivalence i -> i -> i repr (Equivalence rs vps) i = unsafePerformIO $ atomicModifyIORef vps f where f ps = reprHelper ps (ps ! i) {-| Determines if two values are equivalent under the given equivalence relation. -} equiv :: Ix i => Equivalence i -> i -> i -> Bool equiv eq x y = repr eq x == repr eq y {-| Determines if all of the given values are equivalent under the given equivalence relation. -} equivalent :: Ix i => Equivalence i -> [i] -> Bool equivalent eq [] = True equivalent eq (x:xs) = all (== repr eq x) (map (repr eq) xs) {-| Construct the equivalence relation obtained by equating the given two values. This combines equivalence classes. -} equate :: Ix i => i -> i -> Equivalence i -> Equivalence i equate x y (Equivalence rs vps) = unsafePerformIO $ do (px, py, ps) <- atomicModifyIORef vps $ \ ps -> let (ps', px) = reprHelper ps x (ps'', py) = reprHelper ps' y in (ps'', (px, py, ps'')) return (go px py ps) where go px py ps | px == py = Equivalence rs vps | rx > ry = let ps' = ps // [(py, px)] in Equivalence rs (unsafePerformIO (newIORef ps')) | rx < ry = let ps' = ps // [(px, py)] in Equivalence rs (unsafePerformIO (newIORef ps')) | otherwise = let ps' = ps // [(py, px)] rs' = rs // [(px, (rx + 1))] in Equivalence rs (unsafePerformIO (newIORef ps')) where rx = rs ! px ry = rs ! py {-| Construct the equivalence relation obtained by equating all of the given values. This combines equivalence classes. -} equateAll :: Ix i => [i] -> Equivalence i -> Equivalence i equateAll [] eq = eq equateAll (x:xs) eq = foldl' (flip (equate x)) eq xs