{-
Tock: a compiler for parallel languages
Copyright (C) 2007 University of Kent
This program is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation, either version 2 of the License, or (at your
option) any later version.
This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License for more details.
You should have received a copy of the GNU General Public License along
with this program. If not, see .
-}
module ArrayUsageCheck where
import Control.Monad.Error
import Control.Monad.State
import Data.Array.IArray
import Data.Generics
import Data.List
import qualified Data.Map as Map
import Data.Maybe
import qualified AST as A
import CompState
import Errors
import FlowGraph
import Metadata
import Pass
import Types
import Utils
-- TODO we should probably calculate this from the CFG
checkArrayUsage :: Data a => a -> PassM a
checkArrayUsage tree = (mapM_ checkPar $ listify (const True) tree) >> return tree
where
-- TODO this doesn't actually check that the uses are in parallel;
-- they might be in sequence within the parallel!
checkPar :: A.Process -> PassM ()
checkPar (A.Par m _ p) = mapM_ (checkIndexes m) $ Map.toList $ Map.fromListWith (++) $ mapMaybe groupArrayIndexes $ listify (const True) p
checkPar _ = return ()
groupArrayIndexes :: A.Variable -> Maybe (String,[A.Expression])
-- TODO this is quite hacky:
groupArrayIndexes (A.SubscriptedVariable _ (A.Subscript _ e) (A.Variable _ n))
= Just (A.nameName n, [e])
groupArrayIndexes _ = Nothing
checkIndexes :: Meta -> (String,[A.Expression]) -> PassM ()
checkIndexes m (arrName, indexes)
= -- liftIO (putStr $ "Checking: " ++ show (arrName, indexes)) >>
case makeEquations indexes (makeConstant emptyMeta 1000000) of
Left err -> die $ "Could not work with array indexes for array \"" ++ arrName ++ "\": " ++ err
Right (varMapping, problem) ->
case uncurry solveProblem problem of
-- No solutions; no worries!
Nothing -> return ()
Just vm -> do sol <- formatSolution varMapping (getCounterEqs vm)
arrName' <- getRealName arrName
dieP m $ "Overlapping indexes of array \"" ++ arrName' ++ "\" when: " ++ sol
formatSolution :: Map.Map String CoeffIndex -> Map.Map CoeffIndex Integer -> PassM String
formatSolution varToIndex indexToConst = do names <- mapM valOfVar $ Map.assocs varToIndex
return $ concat $ intersperse " , " $ catMaybes names
where
valOfVar (varName,k) = case Map.lookup k indexToConst of
Nothing -> return Nothing
Just val -> do varName' <- getRealName varName
return $ Just $ varName' ++ " = " ++ show val
-- TODO this is surely defined elsewhere already?
getRealName :: String -> PassM String
getRealName s = lookupName (A.Name undefined undefined s) >>* A.ndOrigName
-- | A type for inside makeEquations:
data FlattenedExp = Const Integer | Scale Integer A.Variable deriving (Eq,Show)
-- | Given a list of expressions, an expression representing the upper array bound, returns either an error
-- (because the expressions can't be handled, typically) or a set of equalities, inequalities and mapping from
-- (unique, munged) variable name to variable-index in the equations.
-- TODO probably want to take this into the PassM monad at some point, to use the Meta in the error message
makeEquations :: [A.Expression] -> A.Expression -> Either String (Map.Map String Int, (EqualityProblem, InequalityProblem))
makeEquations es high = makeEquations' >>* (\(s,v,lh) -> (s,squareEquations (pairEqs v, getIneqs lh v)))
where
-- | The body of makeEquations; returns the variable mapping, the list of (nx,ex) pairs and a pair
-- representing the upper and lower bounds of the array (inclusive).
makeEquations' :: Either String (Map.Map String Int, [(Integer,EqualityConstraintEquation)], (EqualityConstraintEquation, EqualityConstraintEquation))
makeEquations' = do ((v,h),s) <- (flip runStateT) Map.empty $
do flattened <- lift (mapM flatten es)
eqs <- mapM makeEquation flattened
(1,high') <- (lift $ flatten high) >>= makeEquation
return (eqs,high')
return (s,v,(amap (const 0) h, h))
-- Note that in all these functions, the divisor should always be positive!
-- Takes an expression, and transforms it into an expression like:
-- (e_0 + e_1 + e_2) / d
-- where d is a constant (non-zero!) integer, and each e_k
-- is either a const, a var, const * var, or (const * var) % const [TODO].
-- If the expression cannot be transformed into such a format, an error is returned
flatten :: A.Expression -> Either String (Integer,[FlattenedExp])
flatten (A.Literal _ _ (A.IntLiteral _ n)) = return (1,[Const (read n)])
flatten (A.Dyadic m op lhs rhs) | op == A.Add = combine' (flatten lhs) (flatten rhs)
| op == A.Subtr = combine' (flatten lhs) (liftM (transformPair id (scale (-1))) $ flatten rhs)
| op == A.Mul = multiplyOut' (flatten lhs) (flatten rhs)
-- TODO Div (either constant on bottom, or common (variable) factor(s) with top)
| otherwise = throwError ("Unhandleable operator found in expression: " ++ show op)
flatten (A.ExprVariable _ v) = return (1,[Scale 1 v])
flatten other = throwError ("Unhandleable item found in expression: " ++ show other)
--TODO we need to handle lots more different expression types in future.
multiplyOut' :: Either String (Integer,[FlattenedExp]) -> Either String (Integer,[FlattenedExp]) -> Either String (Integer,[FlattenedExp])
multiplyOut' x y = do {x' <- x; y' <- y; multiplyOut x' y'}
multiplyOut :: (Integer,[FlattenedExp]) -> (Integer,[FlattenedExp]) -> Either String (Integer,[FlattenedExp])
multiplyOut (lx,lhs) (rx,rhs) = do exps <- mapM (uncurry mult) pairs
return (lx * rx, exps)
where
pairs = product2 (lhs,rhs)
mult :: FlattenedExp -> FlattenedExp -> Either String FlattenedExp
mult (Const x) (Const y) = return $ Const (x*y)
mult (Scale n v) (Const x) = return $ Scale (n*x) v
mult (Const x) (Scale n v) = return $ Scale (n*x) v
mult (Scale _ v) (Scale _ v')
= throwError $ "Cannot deal with non-linear equations; during flattening found: "
++ show v ++ " * " ++ show v'
-- | Scales a flattened expression by the given integer scaling.
scale :: Integer -> [FlattenedExp] -> [FlattenedExp]
scale sc = map scale'
where
scale' (Const n) = Const (n * sc)
scale' (Scale n v) = Scale (n * sc) v
-- | An easy way of applying combine to two monadic returns
combine' :: Either String (Integer,[FlattenedExp]) -> Either String (Integer,[FlattenedExp]) -> Either String (Integer,[FlattenedExp])
combine' = liftM2 combine
-- | Combines (adds) two flattened expressions with a divisor.
-- Given (nx,ex) and (ny,ey), representing ex/nx and ey/ny, this becomes
-- ((ny*ex)+(nx*ey)/nx*ny (i.e. standard mathematics!).
combine :: (Integer,[FlattenedExp]) -> (Integer,[FlattenedExp]) -> (Integer,[FlattenedExp])
combine (nx, ex) (ny, ey) = (nx * ny, scale ny ex ++ scale nx ey)
-- | Finds the index associated with a particular variable; either by finding an existing index
-- or allocating a new one.
varIndex :: A.Variable -> StateT (Map.Map String Int) (Either String) Int
varIndex (A.Variable _ (A.Name _ _ varName))
= do st <- get
let (st',ind) = case Map.lookup varName st of
Just val -> (st,val)
Nothing -> let newId = (1 + (maximum $ 0 : Map.elems st)) in
(Map.insert varName newId st, newId)
put st'
return ind
-- | Pairs all possible combinations of the list of divided equations. That is for all pairs
-- in the list ((nx,ex),(ny,ey)) (representing ex/nx and ey/ny), forms the equation ny*ex = nx*ey
pairEqs :: [(Integer,EqualityConstraintEquation)] -> [EqualityConstraintEquation]
pairEqs = filter (any (/= 0) . elems) . map (uncurry pairEqs') . allPairs
where
pairEqs' (nx,ex) (ny,ey) = arrayZipWith' 0 (-) (amap (* ny) ex) (amap (* nx) ey)
-- | Given a (low,high) bound (typically: array dimensions), and a list of equations (nx,ex) representing (ex/nx),
-- forms the possible inequalities:
-- * ex/nx >= low (=> ex >= low * nx)
-- * ex/nx <= high (=> ex <= high * nx)
getIneqs :: (EqualityConstraintEquation, EqualityConstraintEquation) -> [(Integer,EqualityConstraintEquation)] -> [InequalityConstraintEquation]
getIneqs (low, high) = concatMap getLH
where
-- eq / sc >= low => eq - (sc * low) >= 0
-- eq / sc <= high => (high * sc) - eq >= 0
getLH :: (Integer,EqualityConstraintEquation) -> [InequalityConstraintEquation]
getLH (sc, eq) = [eq `addEq` (scaleEq (-sc) low),(scaleEq sc high) `addEq` amap negate eq]
addEq = arrayZipWith' 0 (+)
-- | Given a pair (nx,ex) representing ex/nx, forms an equation (e) from the latter part, and returns (nx,e)
makeEquation :: (Integer,[FlattenedExp]) -> StateT (Map.Map String Int) (Either String) (Integer,EqualityConstraintEquation)
makeEquation (divisor, summedItems)
= do eqs <- foldM makeEquation' Map.empty summedItems
return (divisor, mapToArray eqs)
where
makeEquation' :: Map.Map Int Integer -> FlattenedExp -> StateT (Map.Map String Int) (Either String) (Map.Map Int Integer)
makeEquation' m (Const n) = return $ add (0,n) m
makeEquation' m (Scale n v) = varIndex v >>* (\ind -> add (ind, n) m)
add :: (Int,Integer) -> Map.Map Int Integer -> Map.Map Int Integer
add = uncurry (Map.insertWith (+))
-- | Converts a map to an array. Any missing elements in the middle of the bounds are given the value zero.
-- Could probably be moved to Utils
mapToArray :: (IArray a v, Num v, Num k, Ord k, Ix k) => Map.Map k v -> a k v
mapToArray m = accumArray (+) 0 (0, highest') . Map.assocs $ m
where
highest' = maximum $ Map.keys m
-- | Given a pair of equation sets, makes all the equations in the lists be the length
-- of the longest equation. All missing elements are of course given value zero.
squareEquations :: ([Array CoeffIndex Integer],[Array CoeffIndex Integer]) -> ([Array CoeffIndex Integer],[Array CoeffIndex Integer])
squareEquations (eqs,ineqs) = uncurry transformPair (mkPair $ map $ makeSize (0,highest) 0) (eqs,ineqs)
where
makeSize :: (Show i, Show e,IArray a e, Ix i, Enum i) => (i,i) -> e -> a i e -> a i e
makeSize size def arr = array size [(i,arrayLookupWithDefault def arr i) | i <- [fst size .. snd size]]
highest = maximum $ concatMap indices $ eqs ++ ineqs
type CoeffIndex = Int
type EqualityConstraintEquation = Array CoeffIndex Integer
type EqualityProblem = [EqualityConstraintEquation]
-- Assumed to be >= 0
type InequalityConstraintEquation = Array CoeffIndex Integer
type InequalityProblem = [InequalityConstraintEquation]
-- | As we proceed with eliminating variables from equations (with the possible
-- addition of one new variable), we perform substitutions like:
-- x_k = a_k'.x_k' + sum (i = 0 .. n without k) of a_i . x_i
-- where a_k' can be zero (no new variable is introduced).
--
-- We want to keep a record of these substitutions because then
-- if we end up with no remaining inequalities, we know the exact results
-- assigned to each of our variables
--
-- We need to know the substitution for x_k; that is,
-- we can map from x_k to the RHS of its substitution (including the resolved value for x_k').
-- We keep a map from the original variables into the current variables.
-- This does not require fractional coefficients.
type VariableMapping = Map.Map CoeffIndex EqualityConstraintEquation
-- | Given a maximum variable, produces a default mapping
defaultMapping :: Int -> VariableMapping
defaultMapping n = Map.fromList $ [ (i,array (0,n) [(j,if i == j then 1 else 0) | j <- [0 .. n]]) | i <- [0 .. n]]
-- | Adds a new variable to a map. The first parameter is (k,value of old x_k)
addToMapping :: (CoeffIndex,EqualityConstraintEquation) -> VariableMapping -> VariableMapping
addToMapping (k, subst) = addOldToNew
where
-- We want to update all the existing entries to be scaled according to the new substitution.
-- Additionally, iff there was no previous entry for k, we should add the new substitution.
--
-- In terms of maths, we want to replace cur_a_k . x_k with a value in terms of x_k':
-- cur_a_k . x_k = cur_a_k . (a_k'.x_k' + sum (i = 0 .. n without k) of a_i . x_i)
--
-- So we just add the substitution for x_k, scaled by cur_a_k.
--
-- As a more readable example, you currently have:
--
-- y = sigma + 3tau
--
-- You have a new subsitution:
--
-- tau = -2sigma - 1
--
-- Therefore you must update your reference for y by adding 3*tau:
--
-- y = sigma + (-6sigma - 3) = -5sigma - 3
addOldToNew :: Map.Map CoeffIndex EqualityConstraintEquation -> Map.Map CoeffIndex EqualityConstraintEquation
addOldToNew = (Map.insertWith ignoreNewVal k subst) . (Map.map updateSub)
where
ignoreNewVal = flip const
updateSub eq = arrayZipWith (+) (eq // [(k,0)]) $ scaleEq eq_k subst
where
eq_k = eq ! k
-- | Returns a mapping from i to constant values of x_i for the solutions of the equations.
-- This function should only be called if the VariableMapping comes from a problem that
-- definitely has constant solutions after all equalities have been eliminated.
-- If variables remain in the inequalities, you will get invalid\/odd answers from this function.
getCounterEqs :: VariableMapping -> Map.Map CoeffIndex Integer
getCounterEqs origToLast = Map.delete 0 $ Map.map expressAsConst origToLast
where
expressAsConst rhs = rhs ! 0
scaleEq :: (IArray a e, Ix i, Num e) => e -> a i e -> a i e
scaleEq n = amap (* n)
-- | Solves all the constraints in the Equality Problem (taking them to be == 0),
-- and transforms the InequalityProblems appropriately. It also records
-- a variable mapping so that we can feed back the final answer to the user
solveConstraints :: VariableMapping -> EqualityProblem -> InequalityProblem -> Maybe (VariableMapping, InequalityProblem)
solveConstraints vm p ineq
= normaliseEq p >>= (\p' -> execStateT (solve p') (vm,ineq))
where
-- | Normalises an equation by dividing all coefficients by their greatest common divisor.
-- If the unit coefficient (a_0) doesn't divide by this GCD, Nothing will be returned
-- (the constraints do not have an integer solution)
normaliseEq :: EqualityProblem -> Maybe EqualityProblem
normaliseEq = mapM normaliseEq' --Note the mapM; if any calls to normalise' fail, so will normalise
where
normaliseEq' :: EqualityConstraintEquation -> Maybe EqualityConstraintEquation
normaliseEq' e | g == 0 = Just e
| ((e ! 0) `mod` g) /= 0 = Nothing
| otherwise = Just $ amap (\x -> x `div` g) e
where g = mygcdList (tail $ elems e) -- g is the GCD of a_1 .. a_n (not a_0)
-- | Solves all equality problems in the given list.
-- Will either succeed (Just () in the Error\/Maybe monad) or fail (Nothing)
solve :: EqualityProblem -> StateT (VariableMapping, InequalityProblem) Maybe ()
solve [] = return ()
solve p = (solveUnits p >>* removeRedundant) >>= liftF checkFalsifiable >>= solveNext >>= solve
-- | Checks if any of the coefficients in the equation have an absolute value of 1.
-- Returns either Just or Nothing (there are no such coefficients in the equation).
-- This function only looks at a_1 .. a_n. That is, a_0 is ignored.
checkForUnit :: EqualityConstraintEquation -> Maybe CoeffIndex
checkForUnit = listToMaybe . map fst . filter coeffAbsVal1 . tail . assocs
where
coeffAbsVal1 :: (a, Integer) -> Bool
coeffAbsVal1 (_,x) = (abs x) == 1
-- | Finds the first unit coefficient (|a_k| == 1) in a set of equality constraints.
-- Returns Nothing if there are no unit coefficients. Otherwise it returns
-- (Just (equation, indexOfUnitCoeff), otherEquations); that is, the specified equation is not
-- present in the list of equations.
findFirstUnit :: EqualityProblem -> (Maybe (EqualityConstraintEquation,CoeffIndex),EqualityProblem)
findFirstUnit [] = (Nothing,[])
findFirstUnit (e:es) = case checkForUnit e of
Just ci -> (Just (e,ci),es)
Nothing -> let (me,es') = findFirstUnit es in (me,e:es')
-- | Substitutes a value for x_k into an equation. Given k, the value for x_k in terms
-- of coefficients of other variables (let's call it x_k_val), it subsitutes this into
-- all the equations in the list by adding x_k_val (scaled by a_k) to each equation and
-- then zeroing out the a_k value. Note that the (x_k_val ! k) value will be ignored;
-- it should be zero, in any case (otherwise x_k would be defined in terms of itself!).
substIn :: CoeffIndex -> Array CoeffIndex Integer -> (VariableMapping, EqualityProblem) -> (VariableMapping, EqualityProblem)
substIn k x_k_val = transformPair (addToMapping (k,x_k_val)) (map substIn')
where
substIn' eq = (arrayZipWith (+) eq scaled_x_k_val) // [(k,0)]
where
scaled_x_k_val = amap (* (eq ! k)) x_k_val
-- | Solves (i.e. removes by substitution) all unit coefficients in the given list of equations.
solveUnits :: EqualityProblem -> StateT (VariableMapping, InequalityProblem) Maybe EqualityProblem
solveUnits p
= case findFirstUnit p of
(Nothing,p') -> return p' -- p' should equal p anyway
(Just (eq,ind),p') -> modify change >> change' p' >>= liftF normaliseEq >>= solveUnits
where
change = substIn ind (arrayMapWithIndex (modifyOthersZeroSpecific ind) eq)
change' p = do (mp,ineq) <- get
let (_,p') = change (undefined,p)
put (mp,ineq)
return p'
origVal = eq ! ind
-- Zeroes a specific coefficient, modifies the others as follows:
-- If the coefficient of x_k is 1, we need to negate the other coefficients
-- to get its definition. However, if the coefficient is -1, we don't need to
-- do this. For example, consider 2 + 3x_1 + x_2 - 4x_3 = 0. In this case
-- x_2 = -2 - 3x_1 + 4x_3; the negation of the original equation (ignoring x_2).
-- If however, it was 2 + 3x_1 - x_2 - 4x_3 = 0 then x_2 = 2 + 3x_1 - 4x_3;
-- that is, identical to the original equation if we ignore x_2.
modifyOthersZeroSpecific :: CoeffIndex -> (CoeffIndex -> Integer -> Integer)
modifyOthersZeroSpecific match ind
| match == ind = const 0 -- The specific value to zero out
| origVal == 1 = negate -- Original coeff was 1; negate
| otherwise = id -- Original coeff was -1; don't do anything
-- | Finds the coefficient with the smallest absolute value of a_1 .. a_n (i.e. not a_0)
-- that is non-zero (i.e. zero coefficients are ignored).
findSmallestAbsCoeff :: EqualityConstraintEquation -> CoeffIndex
findSmallestAbsCoeff = fst . minimumBy cmpAbsSnd . filter ((/= 0) . snd) . tail . assocs
where
cmpAbsSnd :: (a,Integer) -> (a,Integer) -> Ordering
cmpAbsSnd (_,x) (_,y) = compare (abs x) (abs y)
-- | Solves the next equality and returns the new set of equalities.
solveNext :: EqualityProblem -> StateT (VariableMapping, InequalityProblem) Maybe EqualityProblem
solveNext [] = return []
solveNext (e:es) = -- We transform the kth variable into sigma, effectively
-- So once we have x_k = ... (in terms of sigma) we add a_k * RHS
-- to all other equations, AFTER zeroing the a_k coefficient (so
-- that the multiple of sigma is added on properly)
modify change >> change' (e:es) >>= liftF normaliseEq
where
change' p = do (mp,ineq) <- get
let (_,p') = change (undefined,p)
put (mp,ineq)
return p'
change = transformPair (addToMapping (k,x_k_eq)) (map alterEquation)
-- | Adds a scaled version of x_k_eq onto the current equation, after zeroing out
-- the a_k coefficient in the current equation.
alterEquation :: EqualityConstraintEquation -> EqualityConstraintEquation
alterEquation eq = arrayZipWith (+) (eq // [(k,0)]) (amap (\x -> x * (eq ! k)) x_k_eq)
k = findSmallestAbsCoeff e
a_k = e ! k
m = (abs a_k) + 1
sign_a_k = signum a_k
x_k_eq = amap (\a_i -> sign_a_k * (a_i `mymod` m)) e // [(k,(- sign_a_k) * m)]
-- I think this is probably equivalent to mod, but let's follow the maths:
mymod :: Integer -> Integer -> Integer
mymod x y = x - (y * (floordivplushalf x y))
-- This is floor (x/y + 1/2). Probably a way to do it without reverting to float arithmetic:
floordivplushalf :: Integer -> Integer -> Integer
floordivplushalf x y = floor ((fromInteger x / fromInteger y) + (0.5 :: Double))
-- Removes all equations where the coefficients are all zero
removeRedundant :: EqualityProblem -> EqualityProblem
removeRedundant = mapMaybe (boolToMaybe (not . isRedundant))
where
isRedundant :: EqualityConstraintEquation -> Bool
isRedundant = all (== 0) . elems
-- Searches for all equations where only the a_0 coefficient is non-zero; this means the equation cannot be satisfied
checkFalsifiable :: EqualityProblem -> Maybe EqualityProblem
checkFalsifiable = boolToMaybe (not . any checkFalsifiable')
where
-- | Returns True if the equation is definitely unsatisfiable
checkFalsifiable' :: EqualityConstraintEquation -> Bool
checkFalsifiable' e = (e ! 0 /= 0) && (all (== 0) . tail . elems) e
mygcd :: Integer -> Integer -> Integer
mygcd 0 0 = 0
mygcd x y = gcd x y
mygcdList :: [Integer] -> Integer
mygcdList [] = 0
mygcdList [x] = abs x
mygcdList (x:xs) = foldl mygcd x xs
-- | Prunes the inequalities. It does what is described in section 2.3 of Pugh's ACM paper;
-- it removes redundant inequalities, fails (evaluates to Nothing) if it finds a contradiction
-- and turns any 2x + y <= 4, 2x + y >= 4 pairs into equalities. The list of such equalities
-- (which may well be an empty list) and the remaining inequalities is returned.
-- As an additional step not specified in the paper, equations with no variables in them are checked
-- for consistency. That is, all equations c >= 0 (where c is constant) are checked to
-- ensure c is indeed >= 0, and those equations are removed.
pruneIneq :: InequalityProblem -> Maybe (EqualityProblem, InequalityProblem)
pruneIneq ineq = do let (opps,others) = splitEither $ groupOpposites $ map pruneGroup groupedIneq
(opps', eq) <- mapM checkOpposite opps >>* splitEither
checked <- mapM checkConstantEq (concat opps' ++ others) >>* catMaybes
return (eq, checked)
where
groupedIneq = groupBy (\x y -> EQ == coeffSort x y) $ sortBy coeffSort $ map normaliseIneq ineq
normaliseIneq :: InequalityConstraintEquation -> InequalityConstraintEquation
normaliseIneq ineq | g > 1 = arrayMapWithIndex norm ineq
| otherwise = ineq
where
norm ind val | ind == 0 = normaliseUnits val
| otherwise = val `div` g
g = mygcdList $ tail $ elems ineq
-- I think div would do here, because g will always be positive, but
-- I feel safer using the mathematical description:
normaliseUnits a_0 = floor $ (fromInteger a_0 :: Double) / (fromInteger g)
coeffSort :: InequalityConstraintEquation -> InequalityConstraintEquation -> Ordering
coeffSort x y = compare (tail $ elems x) (tail $ elems y)
-- | Takes in a group of inequalities with identical a_1 .. a_n coefficients
-- and returns the equation with the smallest unit coefficient. Consider the standard equation:
-- a_1.x_1 + a_2.x_2 .. a_n.x_n >= -a_0. We want one equation with the maximum value of -a_0
-- (this will be the strongest equation), which is therefore the minimum value of a_0.
-- This therefore automatically removes duplicate and redundant equations.
pruneGroup :: [InequalityConstraintEquation] -> InequalityConstraintEquation
pruneGroup = minimumBy (\x y -> compare (x ! 0) (y ! 0))
-- | Groups all equations with their opposites, if found. Returns either a pair
-- or a singleton. O(N^2), but there shouldn't be that many inequalities to process (<= 10, I expect).
-- Assumes equations have already been pruned, and that therefore for every unique a_1 .. a_n
-- set, there is only one equation.
groupOpposites :: InequalityProblem -> [Either (InequalityConstraintEquation,InequalityConstraintEquation) InequalityConstraintEquation]
groupOpposites [] = []
groupOpposites (e:es) = case findOpposite e es of
Just (opp,rest) -> (Left (e,opp)) : (groupOpposites rest)
Nothing -> (Right e) : (groupOpposites es)
findOpposite :: InequalityConstraintEquation -> [InequalityConstraintEquation] -> Maybe (InequalityConstraintEquation,[InequalityConstraintEquation])
findOpposite _ [] = Nothing
findOpposite target (e:es) | negTarget == (tail $ elems e) = Just (e,es)
| otherwise = case findOpposite target es of
Just (opp,rest) -> Just (opp,e:rest)
Nothing -> Nothing
where
negTarget = map negate $ tail $ elems target
-- Checks if two "opposite" constraints are inconsistent. If they are inconsistent, Nothing is returned.
-- If they could be consistent, either the resulting equality or the inequalities are returned
--
-- If the equations are opposite, then setting z = sum (1 .. n) of a_n . x_n, the two equations must be:
-- b + z >= 0
-- c - z >= 0
-- The choice of which equation is which is arbitrary.
--
-- It is easily seen that adding the two equations gives:
--
-- (b + c) >= 0
--
-- Therefore if (b + c) < 0, the equations are inconsistent.
-- If (b + c) = 0, we can substitute into the original equations b = -c:
-- -c + z >= 0
-- c - z >= 0
-- Rearranging both gives:
-- z >= c
-- z <= c
-- This implies c = z. Therefore we can take either of the original inequalities
-- and treat them directly as equality (c - z = 0, and b + z = 0)
-- If (b + c) > 0 then the equations are consistent but we cannot do anything new with them
checkOpposite :: (InequalityConstraintEquation,InequalityConstraintEquation) ->
Maybe (Either [InequalityConstraintEquation] EqualityConstraintEquation)
checkOpposite (x,y) | (x ! 0) + (y ! 0) < 0 = Nothing
| (x ! 0) + (y ! 0) == 0 = Just $ Right x
| otherwise = Just $ Left [x,y]
-- The type of this function is quite confusing. We want to use in the Maybe monad, so
-- the outer type indicates error; Nothing is an error. Just x indicates non-failure,
-- but x may either be Just y (keep the equation) or Nothing (remove it). So the three
-- possible returns are:
-- * Nothing: Equation inconsistent
-- * Just Nothing: Equation redundant
-- * Just (Just e) : Keep equation.
checkConstantEq :: InequalityConstraintEquation -> Maybe (Maybe InequalityConstraintEquation)
checkConstantEq eq | all (== 0) (tail $ elems eq) = if (eq ! 0) >= 0 then Just Nothing else Nothing
| otherwise = Just $ Just eq
-- | Returns the number of variables (of x_1 .. x_n; x_0 is not counted)
-- that have non-zero coefficients in the given inequality problems.
numVariables :: InequalityProblem -> Int
numVariables ineq = length (nub $ concatMap findVars ineq)
where
findVars = map fst . filter ((/= 0) . snd) . tail . assocs
-- | Eliminating the inequalities works as follows:
--
-- Rearrange (and normalise) equations for a particular variable x_k to eliminate so that
-- a_k is always positive and you have:
-- A: a_Ak . x_k <= sum (i is 0 to n, without k) a_Ai . x_i
-- B: a_Bk . x_k >= sum (i is 0 to n, without k) a_Bi . x_i
-- C: equations where a_k is zero.
--
-- Determine if there is an integer solution for x_k:
--
-- If it is an inexact projection, the function recurses into both the real and dark shadow.
-- If necessary, it does brute-forcing.
--
--
-- Real shadow:
--
-- Form lots of new equations:
-- Given a_Ak . x_k <= RHS(A)
-- a_Bk . x_k >= RHS(B)
-- We can get (since a_Ak and a_bk are positive):
-- a_Ak . A_Bk . x_k <= A_Bk . RHS(A)
-- a_Ak . A_Bk . x_k >= A_Ak . RHS(B)
-- For every combination of the RHS(A) and RHS(B) generate an inequality: a_Bk . RHS(A) - a_Ak . RHS(B) >=0
-- Add these new equations to the set C, and iterate
--
-- Dark shadow:
--
-- Form lots of new equations:
-- Given a_Ak . x_k <= RHS(A)
-- a_Bk . x_k >= RHS(B)
-- We need to form the equations:
-- a_Bk . RHS(A) - a_Ak . RHS(B) - (a_Ak - 1)(a_Bk - 1) >= 0
--
-- That is, the dark shadow is the same as the real shadow but with the constant subtracted
fmElimination :: VariableMapping -> InequalityProblem -> Maybe VariableMapping
fmElimination vm ineq = fmElimination' vm (presentItems ineq) ineq
where
-- Finds all variables that have at least one non-zero coefficient in the equation set.
-- a_0 is ignored; 0 will never be in the returned list
presentItems :: InequalityProblem -> [Int]
presentItems = nub . map fst . filter ((/= 0) . snd) . concatMap (tail . assocs)
-- The real body of the function:
fmElimination' :: VariableMapping -> [Int] -> InequalityProblem -> Maybe VariableMapping
fmElimination' vm [] ineqs = pruneAndHandleIneq (vm,ineqs) >>* fst
-- We have to prune the ineqs when they have no variables to
-- ensure none are inconsistent
fmElimination' vm indexes@(ix:ixs) ineqs
= do (vm',ineqsPruned) <- pruneAndHandleIneq (vm,ineqs)
case listToMaybe $ filter (flip isExactProjection ineqsPruned) indexes of
-- If there is an exact projection (real shadow = dark shadow), eliminate that
-- variable, and therefore just recurse to process this shadow:
Just ex -> fmElimination' vm' (indexes \\ [ex]) (getRealShadow ex ineqs)
Nothing ->
-- Otherwise, check the real shadow first:
case fmElimination' vm' ixs (getRealShadow ix ineqs) of
-- No solution to the real shadow means no solution to the problem:
Nothing -> Nothing
-- Check the dark shadow:
Just vm'' -> case fmElimination' vm'' ixs (getDarkShadow ix ineqs) of
-- Solution to the dark shadow means there is a solution to the problem:
Just vm''' -> return vm'''
-- Solution to real but not to dark; we must brute force the problem.
-- If we find any solutions during the brute-forcing, we have our solution.
-- Otherwise there is no solution
Nothing -> listToMaybe $ mapMaybe (uncurry $ solveProblem' vm'') $ getBruteForceProblems ix ineqs
-- Prunes the inequalities. If any equalities arise, those are processed, so
-- that the return is only inequalities
pruneAndHandleIneq :: (VariableMapping, InequalityProblem) -> Maybe (VariableMapping, InequalityProblem)
pruneAndHandleIneq (vm,ineq)
= do (eq,ineq') <- pruneIneq ineq
if null eq then return (vm,ineq') else solveConstraints vm eq ineq'
-- We need to partition the related equations into sets A,B and C.
-- C is straightforward (a_k is zero).
-- In set B, a_k > 0, and "RHS(B)" (as described above) is the negation of the other
-- coefficients. Therefore "-RHS(B)" is the other coefficients as-is.
-- In set A, a_k < 0, and "RHS(A)" (as described above) is the other coefficients, untouched
-- So we simply zero out a_k, and return the rest, associated with the absolute value of a_k.
splitBounds :: Int -> InequalityProblem -> ([(Integer, InequalityConstraintEquation)], [(Integer,InequalityConstraintEquation)], [InequalityConstraintEquation])
splitBounds k = (\(x,y,z) -> (concat x, concat y, concat z)) . unzip3 . map partition'
where
partition' e | a_k == 0 = ([],[],[e])
| a_k < 0 = ([(abs a_k, e // [(k,0)])],[],[])
| a_k > 0 = ([],[(abs a_k, e // [(k,0)])],[])
where
a_k = e ! k
getRealShadow :: Int -> InequalityProblem -> InequalityProblem
getRealShadow k ineqs = eqC ++ map (uncurry pairIneqs) (product2 (eqA,eqB))
where
(eqA,eqB,eqC) = splitBounds k ineqs
pairIneqs :: (Integer, InequalityConstraintEquation) -> (Integer, InequalityConstraintEquation) -> InequalityConstraintEquation
pairIneqs (x,ex) (y,ey) = arrayZipWith (+) (amap (* y) ex) (amap (* x) ey)
getDarkShadow :: Int -> InequalityProblem -> InequalityProblem
getDarkShadow k ineqs = eqC ++ map (uncurry pairIneqsDark) (product2 (eqA,eqB))
where
(eqA,eqB,eqC) = splitBounds k ineqs
pairIneqsDark :: (Integer, InequalityConstraintEquation) -> (Integer, InequalityConstraintEquation) -> InequalityConstraintEquation
pairIneqsDark (x,ex) (y,ey) = addConstant (-1*(y-1)*(x-1)) (arrayZipWith (+) (amap (* y) ex) (amap (* x) ey))
-- Adds a constant value to an equation:
addConstant :: Integer -> InequalityConstraintEquation -> InequalityConstraintEquation
addConstant x e = e // [(0,(e ! 0) + x)]
-- Checks if eliminating the specified variable would yield an exact projection (real shadow = dark shadow):
isExactProjection :: Int -> InequalityProblem -> Bool
isExactProjection n ineqs = (all (== 1) $ pos n ineqs) || (all (== (-1)) $ neg n ineqs)
where
pos :: Int -> InequalityProblem -> [Integer]
pos n ineqs = filter (> 0) $ map (! n) ineqs
neg :: Int -> InequalityProblem -> [Integer]
neg n ineqs = filter (< 0) $ map (! n) ineqs
getBruteForceProblems :: Int -> InequalityProblem -> [(EqualityProblem,InequalityProblem)]
getBruteForceProblems k ineqs = concatMap setLowerBound eqB
where
(eqA,eqB,_) = splitBounds k ineqs
largestUpperA = maximum $ map fst eqA
setLowerBound (b,beta) = map (\i -> ([addConstant (-i) (beta // [(k,b)])],ineqs)) [0 .. max]
where
max = ((largestUpperA * b) - largestUpperA - b) `div` largestUpperA
solveProblem' :: VariableMapping -> EqualityProblem -> InequalityProblem -> Maybe VariableMapping
solveProblem' vm eq ineq = solveConstraints vm eq ineq >>= uncurry fmElimination
solveProblem :: EqualityProblem -> InequalityProblem -> Maybe VariableMapping
solveProblem eq ineq = solveProblem' (defaultMapping maxVar) eq ineq
where
maxVar = if null eq && null ineq then 0 else
if null eq then snd $ bounds $ head ineq else snd $ bounds $ head eq