{-| Code for manipulation of 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'. 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, domain, equiv, equivalent, equate, equateAll ) where import Control.Concurrent.MVar import Control.Monad import Data.Array.Diff import Data.IORef import Data.List import Data.Maybe 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 ] {- Convenience method for building "transparent" references. These must have the property that updating them makes no semantic change in their value; otherwise, references really need to be created in an IO block. -} ref :: a -> IORef a ref x = unsafePerformIO (newIORef x) {-| 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 = Equivalence (arrayFrom is (const 0)) (ref (arrayFrom is id)) {-| Gets the domain of an equivalence relation, as the ordered pair of index bounds. -} domain :: Ix i => Equivalence i -> (i, i) domain (Equivalence rs _) = bounds rs reprHelper :: Ix i => DiffArray i i -> i -> (Maybe (DiffArray i i), i) reprHelper ps i | pi == i = (Nothing, i) | otherwise = let (ps', r) = reprHelper ps pi in (Just (fromMaybe ps 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 class and 'Equivalence' value. This is not exported, because clients that use this are doing something wrong. Note that: * The representative chosen depends on the order in which the equivalence relation was built, and is not always the same for values that represent the same relation. * The representative is not particularly stable. Uses of 'equate' are highly likely to change it. * If all you need is some representative of the equivalence class, you have to provide one as input to the function anyway, so you may as well use that. The only guarantee provided is that repr always returns the same value for the exact same 'Equivalence'. -} repr :: Ix i => Equivalence i -> i -> i repr (Equivalence rs vps) i = unsafePerformIO $ do ps <- readIORef vps let (ps', r) = reprHelper ps (ps ! i) maybe (return ()) (writeIORef vps) ps' return r {-| 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 ps <- readIORef vps let (ps', px) = reprHelper ps x (ps'', py) = reprHelper (fromMaybe ps ps') y psFinal <- case ps' of Nothing -> do maybe (return ()) (writeIORef vps) ps'' return (fromMaybe ps ps'') Just t -> do writeIORef vps (fromMaybe t ps'') return (fromMaybe t ps'') return (go px py psFinal) where go px py ps | px == py = Equivalence rs vps | rx > ry = let ps' = ps // [(py, px)] in Equivalence rs (ref ps') | rx < ry = let ps' = ps // [(px, py)] in Equivalence rs (ref ps') | otherwise = let ps' = ps // [(py, px)] rs' = rs // [(px, (rx + 1))] in Equivalence rs' (ref 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