-- | Dataflow analysis to be applied once basic block analysis is complete. {-# LANGUAGE FlexibleContexts, PatternGuards, ScopedTypeVariables, TupleSections #-} module Language.Fortran.Analysis.DataFlow ( dominators, iDominators, DomMap, IDomMap , postOrder, revPostOrder, preOrder, revPreOrder, OrderF , dataFlowSolver, showDataFlow, InOut, InOutMap, InF, OutF , liveVariableAnalysis, reachingDefinitions , genUDMap, genDUMap, duMapToUdMap, UDMap, DUMap , genFlowsToGraph, FlowsGraph , genVarFlowsToMap, VarFlowsMap , genBlockMap, genDefMap, BlockMap, DefMap , genCallMap, CallMap , loopNodes, genBackEdgeMap, sccWith, BackEdgeMap , genLoopNodeMap, LoopNodeMap , genInductionVarMap, InductionVarMap , genInductionVarMapByASTBlock, InductionVarMapByASTBlock , noPredNodes ) where import Data.Generics.Uniplate.Data import Data.Generics.Uniplate.Operations import Data.Data import Data.Function import Control.Monad.State.Lazy import Control.Monad.Writer import Text.PrettyPrint.GenericPretty (pretty, Out) import Language.Fortran.Analysis import Language.Fortran.Analysis.BBlocks import Language.Fortran.AST import qualified Data.Map as M import qualified Data.IntMap as IM import qualified Data.Set as S import qualified Data.IntSet as IS import Data.Graph.Inductive hiding (trc) import Data.Graph.Inductive.PatriciaTree (Gr) import Data.Graph.Inductive.Query.BFS (bfen) import Data.Maybe import Data.List (foldl', (\\), union, delete, nub, intersect) -------------------------------------------------- -- | DomMap : node -> dominators of node type DomMap = IM.IntMap IS.IntSet -- | Compute dominators of each bblock in the graph. Node A dominates -- node B when all paths from the start node (0) must pass through -- node A in order to reach node B. That will be represented as the -- relation (B, [A, ...]) in the DomMap. dominators :: BBGr a -> DomMap dominators = IM.fromList . map (fmap IS.fromList) . flip dom 0 -- | IDomMap : node -> immediate dominator of node type IDomMap = IM.IntMap Int -- | Compute the immediate dominator of each bblock in the graph. The -- immediate dominator is, in a sense, the 'closest' dominator of a -- node. Given nodes A and B, you can say that node A is immediately -- dominated by node B if there does not exist any node C such that: -- node A dominates node C and node C dominates node B. iDominators :: BBGr a -> IDomMap iDominators gr = IM.unions [ IM.fromList . flip iDom n $ gr | n <- noPredNodes gr ] -- | An OrderF is a function from graph to a specific ordering of nodes. type OrderF a = BBGr a -> [Node] -- | The postordering of a graph outputs the label after traversal of children. postOrder :: OrderF a postOrder gr = concatMap postorder . dff (noPredNodes gr) $ gr -- | Reversed postordering. revPostOrder :: OrderF a revPostOrder = reverse . postOrder -- | The preordering of a graph outputs the label before traversal of children. preOrder :: OrderF a preOrder gr = concatMap preorder . dff (noPredNodes gr) $ gr -- | Reversed preordering. revPreOrder :: OrderF a revPreOrder = reverse . preOrder -- | Compute the set of nodes with no predecessors. noPredNodes :: Graph g => g a b -> [Node] -- noPredNodes = flip ufold [] $ \ ctx ns -> if null (pre' ctx) then node' ctx : ns else ns -- doesn't work, though it should noPredNodes gr = filter (null . pre gr) (nodes gr) -------------------------------------------------- -- | InOut : (dataflow into the bblock, dataflow out of the bblock) type InOut t = (t, t) -- | InOutMap : node -> (dataflow into node, dataflow out of node) type InOutMap t = IM.IntMap (InOut t) -- | InF, a function that returns the in-dataflow for a given node type InF t = Node -> t -- | OutF, a function that returns the out-dataflow for a given node type OutF t = Node -> t -- | Apply the iterative dataflow analysis method. dataFlowSolver :: Ord t => BBGr a -- ^ basic block graph -> (Node -> InOut t) -- ^ initialisation for in and out dataflows -> OrderF a -- ^ ordering function -> (OutF t -> InF t) -- ^ compute the in-flow given an out-flow function -> (InF t -> OutF t) -- ^ compute the out-flow given an in-flow function -> InOutMap t -- ^ final dataflow for each node dataFlowSolver gr initF order inF outF = converge (==) $ iterate step initM where ordNodes = order gr initM = IM.fromList [ (n, initF n) | n <- ordNodes ] step m = IM.fromList [ (n, (inF (snd . get m) n, outF (fst . get m) n)) | n <- ordNodes ] get m n = fromJustMsg "dataFlowSolver" $ IM.lookup n m -------------------------------------------------- -- | BlockMap : AST-block label -> AST-block -- Each AST-block has been given a unique number label during analysis -- of basic blocks. The purpose of this map is to provide the ability -- to lookup AST-blocks by label. type BlockMap a = IM.IntMap (Block (Analysis a)) -- | Build a BlockMap from the AST. This can only be performed after -- analyseBasicBlocks has operated, created basic blocks, and labeled -- all of the AST-blocks with unique numbers. genBlockMap :: Data a => ProgramFile (Analysis a) -> BlockMap a genBlockMap pf = IM.fromList [ (i, b) | gr <- uni pf , (_, bs) <- labNodes gr , b <- bs , let Just i = insLabel (getAnnotation b) ] where uni :: Data a => ProgramFile (Analysis a) -> [BBGr (Analysis a)] uni = universeBi -- | DefMap : variable name -> { AST-block label } type DefMap = M.Map Name IS.IntSet -- | Build a DefMap from the BlockMap. This allows us to quickly look -- up the AST-block labels that wrote into the given variable. genDefMap :: Data a => BlockMap a -> DefMap genDefMap bm = M.fromListWith IS.union [ (y, IS.singleton i) | (i, b) <- IM.toList bm, y <- allLhsVars b ] -------------------------------------------------- -- | Dataflow analysis for live variables given basic block graph. -- Muchnick, p. 445: A variable is "live" at a particular program -- point if there is a path to the exit along which its value may be -- used before it is redefined. It is "dead" if there is no such path. liveVariableAnalysis :: Data a => BBGr (Analysis a) -> InOutMap (S.Set Name) liveVariableAnalysis gr = dataFlowSolver gr (const (S.empty, S.empty)) revPreOrder inn out where inn outF b = (outF b S.\\ kill b) `S.union` gen b out innF b = S.unions [ innF s | s <- suc gr b ] kill b = bblockKill (fromJustMsg "liveVariableAnalysis kill" $ lab gr b) gen b = bblockGen (fromJustMsg "liveVariableAnalysis gen" $ lab gr b) -- | Iterate "KILL" set through a single basic block. bblockKill :: Data a => [Block (Analysis a)] -> S.Set Name bblockKill = S.fromList . concatMap blockKill -- | Iterate "GEN" set through a single basic block. bblockGen :: Data a => [Block (Analysis a)] -> S.Set Name bblockGen bs = S.fromList . fst . foldl' f ([], []) $ zip (map blockGen bs) (map blockKill bs) where f (bbgen, bbkill) (gen, kill) = ((gen \\ bbkill) `union` bbgen, kill `union` bbkill) -- | "KILL" set for a single AST-block. blockKill :: Data a => Block (Analysis a) -> [Name] blockKill = blockVarDefs -- | "GEN" set for a single AST-block. blockGen :: Data a => Block (Analysis a) -> [Name] blockGen = blockVarUses -------------------------------------------------- -- Reaching Definitions -- forward flow analysis (revPostOrder) -- GEN b@( definition of anything ) = {b} -- KILL b@( definition of y ) = DEFS y -- technically, except b, but it won't matter -- DEFS y = { all definitions of y } -- Within a basic block -- GEN [] = KILL [] = {} -- GEN [b_1 .. b_{n+1}] = GEN b_{n+1} `union` (GEN [b_1 .. b_n] `difference` KILL b_{n+1}) -- KILL [b_1 .. b_{n+1}] = KILL b_{n+1} `union` (KILL [b_1 .. b_n] `difference` GEN b_{n+1}) -- Between basic blocks -- REACHin bb = unions [ REACHout bb | bb <- pred bb ] -- REACHout bb = GEN bb `union` (REACHin bb `difference` KILL bb) -- | Reaching definitions dataflow analysis. Reaching definitions are -- the set of variable-defining AST-block labels that may reach a -- program point. Suppose AST-block with label A defines a variable -- named v. Label A may reach another program point labeled P if there -- is at least one program path from label A to label P that does not -- redefine variable v. reachingDefinitions :: Data a => DefMap -> BBGr (Analysis a) -> InOutMap IS.IntSet reachingDefinitions dm gr = dataFlowSolver gr (const (IS.empty, IS.empty)) revPostOrder inn out where inn outF b = IS.unions [ outF s | s <- pre gr b ] out innF b = gen `IS.union` (innF b IS.\\ kill) where (gen, kill) = rdBblockGenKill dm (fromJustMsg "reachingDefinitions" $ lab gr b) -- Compute the "GEN" and "KILL" sets for a given basic block. rdBblockGenKill :: Data a => DefMap -> [Block (Analysis a)] -> (IS.IntSet, IS.IntSet) rdBblockGenKill dm bs = foldl' f (IS.empty, IS.empty) $ zip (map gen bs) (map kill bs) where gen b | null (allLhsVars b) = IS.empty | otherwise = IS.singleton . fromJustMsg "rdBblockGenKill" . insLabel . getAnnotation $ b kill = rdDefs dm f (bbgen, bbkill) (gen, kill) = ((bbgen IS.\\ kill) `IS.union` gen, (bbkill IS.\\ gen) `IS.union` kill) -- Set of all AST-block labels that also define variables defined by AST-block b rdDefs :: Data a => DefMap -> Block (Analysis a) -> IS.IntSet rdDefs dm b = IS.unions [ IS.empty `fromMaybe` M.lookup y dm | y <- allLhsVars b ] -------------------------------------------------- -- | DUMap : definition -> { use } type DUMap = IM.IntMap IS.IntSet -- | def-use map: map AST-block labels of defining AST-blocks to the -- AST-blocks that may use the definition. genDUMap :: Data a => BlockMap a -> DefMap -> BBGr (Analysis a) -> InOutMap IS.IntSet -> DUMap genDUMap bm dm gr rdefs = IM.unionsWith IS.union duMaps where -- duMaps for each bblock duMaps = [ fst (foldl' inBBlock (IM.empty, is) bs) | (n, (is, _)) <- IM.toList rdefs, let Just bs = lab gr n ] -- internal analysis within bblock; fold over list of AST-blocks inBBlock (duMap, inSet) b = (duMap', inSet') where Just i = insLabel (getAnnotation b) bduMap = IM.fromListWith IS.union [ (i', IS.singleton i) | i' <- IS.toList inSet, overlap i' ] -- asks: does AST-block at label i' define anything used by AST-block b? overlap i' = not . null . intersect uses $ blockVarDefs b' where Just b' = IM.lookup i' bm uses = blockVarUses b duMap' = IM.unionWith IS.union duMap bduMap gen b | null (allLhsVars b) = IS.empty | otherwise = IS.singleton . fromJustMsg "genDUMap" . insLabel . getAnnotation $ b kill = rdDefs dm inSet' = (inSet IS.\\ (kill b)) `IS.union` (gen b) -- | UDMap : use -> { definition } type UDMap = IM.IntMap IS.IntSet -- | Invert the DUMap into a UDMap duMapToUdMap :: DUMap -> UDMap duMapToUdMap duMap = IM.fromListWith IS.union [ (use, IS.singleton def) | (def, uses) <- IM.toList duMap, use <- IS.toList uses ] -- | use-def map: map AST-block labels of variable-using AST-blocks to -- the AST-blocks that define those variables. genUDMap :: Data a => BlockMap a -> DefMap -> BBGr (Analysis a) -> InOutMap IS.IntSet -> UDMap genUDMap bm dm gr = duMapToUdMap . genDUMap bm dm gr -------------------------------------------------- -- | Convert a UD or DU Map into a graph. mapToGraph :: DynGraph gr => IM.IntMap a -> IM.IntMap IS.IntSet -> gr a () mapToGraph bm m = buildGr $ [ ([], i, l, jAdj) | (i, js) <- IM.toList m , let Just l = IM.lookup i bm , let jAdj = map ((),) $ IS.toList js ] ++ [ (iAdj, j, l, []) | (i, js) <- IM.toList m , j <- IS.toList js , let Just l = IM.lookup j bm , let iAdj = [((), i)] ] -- | FlowsGraph : nodes as AST-block (numbered by label), edges -- showing which definitions contribute to which uses. type FlowsGraph a = Gr (Block (Analysis a)) () -- | "Flows-To" analysis. Represent def-use map as a graph. genFlowsToGraph :: Data a => BlockMap a -> DefMap -> BBGr (Analysis a) -> InOutMap IS.IntSet -- ^ result of reaching definitions -> FlowsGraph a genFlowsToGraph bm dm gr = mapToGraph bm . genDUMap bm dm gr -- | Represent "flows" between variables type VarFlowsMap = M.Map Name (S.Set Name) -- | Create a map (A -> Bs) where A "flows" or contributes towards the variables Bs. genVarFlowsToMap :: Data a => DefMap -> FlowsGraph a -> VarFlowsMap genVarFlowsToMap dm fg = M.fromListWith S.union [ (conv u, sconv v) | (u, v) <- edges fg ] where sconv i | Just v <- IM.lookup i revDM = S.singleton v | otherwise = S.empty conv i | Just v <- IM.lookup i revDM = v | otherwise = error $ "genVarFlowsToMap: convert failed, i=" ++ show i -- planning to make revDM a surjection, after I flatten-out Fortran functions revDM = IM.fromListWith (curry fst) [ (i, v) | (v, is) <- M.toList dm, i <- IS.toList is ] {-| Finds the transitive closure of a directed graph. Given a graph G=(V,E), its transitive closure is the graph: G* = (V,E*) where E*={(i,j): i,j in V and there is a path from i to j in G} -} tc :: (DynGraph gr) => gr a b -> gr a () tc g = newEdges `insEdges` insNodes ln empty where ln = labNodes g newEdges = [ toLEdge (u, v) () | (u, _) <- ln, (_, v) <- bfen (outU g u) g ] outU gr = map toEdge . out gr -------------------------------------------------- -- | BackEdgeMap : node -> node type BackEdgeMap = IM.IntMap Node -- | Find the edges that 'loop back' in the graph; ones where the -- target node dominates the source node. If the backedges are viewed -- as (m -> n) then n is considered the 'loop-header' genBackEdgeMap :: Graph gr => DomMap -> gr a b -> BackEdgeMap genBackEdgeMap domMap = IM.filterWithKey isBackEdge . IM.fromList . edges where isBackEdge s t = t `IS.member` (fromJustMsg "genBackEdgeMap" $ s `IM.lookup` domMap) -- | For each loop in the program, find out which bblock nodes are -- part of the loop by looking through the backedges (m, n) where n is -- considered the 'loop-header', delete n from the map, and then do a -- reverse-depth-first traversal starting from m to find all the nodes -- of interest. Intersect this with the strongly-connected component -- containing m, in case of 'improper' graphs with weird control -- transfers. loopNodes :: Graph gr => BackEdgeMap -> gr a b -> [IS.IntSet] loopNodes bedges gr = [ IS.fromList (n:intersect (sccWith n gr) (rdfs [m] (delNode n gr))) | (m, n) <- IM.toList bedges ] -- | LoopNodeMap : node -> { node } type LoopNodeMap = IM.IntMap IS.IntSet -- | Similar to loopNodes except it creates a map from loop-header to -- the set of loop nodes, for each loop-header. genLoopNodeMap :: Graph gr => BackEdgeMap -> gr a b -> LoopNodeMap genLoopNodeMap bedges gr = IM.fromList [ (n, IS.fromList (n:intersect (sccWith n gr) (rdfs [m] (delNode n gr)))) | (m, n) <- IM.toList bedges ] -- | The strongly connected component containing a given node. sccWith :: (Graph gr) => Node -> gr a b -> [Node] sccWith n g = case filter (n `elem`) $ scc g of [] -> [] c:_ -> c -- | Map of loop header nodes to the induction variables within that loop. type InductionVarMap = IM.IntMap (S.Set Name) -- | Basic induction variables are induction variables that are the -- most easily derived from the syntactic structure of the program: -- for example, directly appearing in a Do-statement. basicInductionVars :: Data a => BackEdgeMap -> BBGr (Analysis a) -> InductionVarMap basicInductionVars bedges gr = IM.fromListWith S.union [ (n, S.singleton v) | (_, n) <- IM.toList bedges , let Just bs = lab gr n , b@(BlDo {}) <- bs , v <- blockVarDefs b ] -- | For each loop in the program, figure out the names of the -- induction variables: the variables that are used to represent the -- current iteration of the loop. genInductionVarMap :: Data a => BackEdgeMap -> BBGr (Analysis a) -> InductionVarMap genInductionVarMap = basicInductionVars -- | InductionVarMapByASTBlock : AST-block label -> { name } type InductionVarMapByASTBlock = IM.IntMap (S.Set Name) -- | Generate an induction variable map that is indexed by the labels -- on AST-blocks within those loops. genInductionVarMapByASTBlock :: forall a. Data a => BackEdgeMap -> BBGr (Analysis a) -> InductionVarMapByASTBlock genInductionVarMapByASTBlock bedges gr = loopsToLabs . genInductionVarMap bedges $ gr where lnMap = genLoopNodeMap bedges gr get = fromMaybe (error "missing loop-header node") . flip IM.lookup lnMap astLabels n = [ i | b <- (universeBi :: Maybe [Block (Analysis a)] -> [Block (Analysis a)]) (lab gr n) , let Just i = insLabel (getAnnotation b) ] loopsToLabs = IM.fromListWith S.union . concatMap loopToLabs . IM.toList loopToLabs (n, ivs) = (map (,ivs) . astLabels) =<< IS.toList (get n) -------------------------------------------------- -- | Show some information about dataflow analyses. showDataFlow :: (Data a, Out a, Show a) => ProgramFile (Analysis a) -> String showDataFlow pf = perPU =<< uni pf where uni = (universeBi :: Data a => ProgramFile (Analysis a) -> [ProgramUnit (Analysis a)]) perPU pu | Analysis { bBlocks = Just gr } <- getAnnotation pu = dashes ++ "\n" ++ p ++ "\n" ++ dashes ++ "\n" ++ dfStr gr ++ "\n\n" where p = "| Program Unit " ++ show (puName pu) ++ " |" dashes = replicate (length p) '-' dfStr gr = (\ (l, x) -> '\n':l ++ ": " ++ x) =<< [ ("callMap", show cm) , ("postOrder", show (postOrder gr)) , ("revPostOrder", show (revPostOrder gr)) , ("revPreOrder", show (revPreOrder gr)) , ("dominators", show (dominators gr)) , ("iDominators", show (iDominators gr)) , ("defMap", show dm) , ("lva", show (IM.toList $ lva gr)) , ("rd", show (IM.toList $ rd gr)) , ("backEdges", show bedges) , ("topsort", show (topsort gr)) , ("scc ", show (scc gr)) , ("loopNodes", show (loopNodes bedges gr)) , ("duMap", show (genDUMap bm dm gr (rd gr))) , ("udMap", show (genUDMap bm dm gr (rd gr))) , ("flowsTo", show (edges $ genFlowsToGraph bm dm gr (rd gr))) , ("varFlowsTo", show (genVarFlowsToMap dm (genFlowsToGraph bm dm gr (rd gr)))) , ("ivMap", show (genInductionVarMap bedges gr)) , ("ivMapByAST", show (genInductionVarMapByASTBlock bedges gr)) , ("noPredNodes", show (noPredNodes gr)) ] where bedges = genBackEdgeMap (dominators gr) gr perPU _ = "" lva = liveVariableAnalysis bm = genBlockMap pf dm = genDefMap bm rd = reachingDefinitions dm cm = genCallMap pf -------------------------------------------------- -- | CallMap : program unit name -> { name of function or subroutine } type CallMap = M.Map ProgramUnitName (S.Set Name) -- | Create a call map showing the structure of the program. genCallMap :: Data a => ProgramFile (Analysis a) -> CallMap genCallMap pf = flip execState M.empty $ do let (ProgramFile cm_pus _) = pf let uP = (universeBi :: Data a => ProgramFile a -> [ProgramUnit a]) forM_ (uP pf) $ \ pu -> do let n = puName pu let uS :: Data a => ProgramUnit a -> [Statement a] uS = universeBi let uE :: Data a => ProgramUnit a -> [Expression a] uE = universeBi m <- get let ns = [ varName v | StCall _ _ v@(ExpValue _ _ (ValVariable _ )) _ <- uS pu ] ++ [ varName v | ExpFunctionCall _ _ v@(ExpValue _ _ (ValVariable _)) _ <- uE pu ] put $ M.insert n (S.fromList ns) m -------------------------------------------------- -- helper: iterate until predicate is satisfied. converge :: (a -> a -> Bool) -> [a] -> a converge p (x:ys@(y:_)) | p x y = y | otherwise = converge p ys fromJustMsg _ (Just x) = x fromJustMsg msg _ = error msg -- Local variables: -- mode: haskell -- haskell-program-name: "cabal repl" -- End: