import Control.Monad
import Control.Monad.Trans
+import Control.Monad.State
import Data.Either
import Data.Maybe
import Data.Monoid
-import Data.List
+import Data.List
+import Data.Functor
+import Data.Tuple
import StdString
+import StdTuple
import StdList
import StdMisc
import StdEnum
-import RWST
import GenEq
from Text import class Text(concat), instance Text String
import AST
-:: Scheme = Forall [String] Type
-:: Gamma :== 'Map'.Map String Scheme
+:: Scheme = Forall [TVar] Type
+:: Gamma :== 'Map'.Map String Scheme //map from Variables! to types
+:: Typing a :== StateT (Gamma, [TVar]) (Either SemError) a
+:: Substitution :== 'Map'.Map TVar Type
:: Constraints :== [(Type, Type)]
-:: Infer a :== RWST Gamma Constraints [String] (Either SemError) a
:: SemError
= ParseError Pos String
| UnifyError Pos Type Type
+ | InfiniteTypeError Pos Type
| FieldSelectorError Pos Type FieldSelector
| OperatorError Pos Op2 Type
| UndeclaredVariableError Pos String
| SanityError Pos String
| Error String
-variableStream :: [String]
+instance zero Gamma where
+ zero = 'Map'.newMap
+
+variableStream :: [TVar]
variableStream = map toString [1..]
-sem :: AST -> SemOutput
-sem a=:(AST fd) = case foldM (const $ hasNoDups fd) () fd
- >>| foldM (const isNiceMain) () fd
- >>| hasMain fd of
+defaultGamma :: Gamma //includes all default functions
+defaultGamma = extend "print" (Forall ["a"] ((IdType "a") ->> VoidType))
+ $ extend "isEmpty" (Forall ["a"] ((ListType (IdType "a")) ->> BoolType))
+ $ extend "read" (Forall [] CharType)
+ zero
+
+sem :: AST -> Either [SemError] AST
+sem (AST fd) = case foldM (const $ hasNoDups fd) () fd
+ >>| foldM (const isNiceMain) () fd
+ >>| hasMain fd
+ >>| evalStateT (type fd) (defaultGamma, variableStream) of
Left e = Left [e]
- _ = pure (a, 'Map'.newMap)
+ Right (_,fds) = Right (AST fds)
where
hasNoDups :: [FunDecl] FunDecl -> Either SemError ()
hasNoDups fds (FunDecl p n _ _ _ _)
_ = Left $ SanityError p "main has to return Void")
isNiceMain _ = pure ()
-instance toString Scheme where
- toString (Forall x t) =
- concat ["Forall ": map ((+++) "\n") x] +++ toString t
+class Typeable a where
+ ftv :: a -> [TVar]
+ subst :: Substitution a -> a
-instance toString Gamma where
- toString mp =
- concat [concat [k, ": ", toString v, "\n"]\\(k, v)<-'Map'.toList mp]
+instance Typeable Scheme where
+ ftv (Forall bound t) = difference (ftv t) bound
+ subst s (Forall bound t) = Forall bound $ subst s_ t
+ where s_ = 'Map'.filterWithKey (\k _ -> not (elem k bound)) s
-instance toString SemError where
- toString (SanityError p e) = concat [toString p,
- "SemError: SanityError: ", e]
- toString se = "SemError: "
+instance Typeable [a] | Typeable a where
+ ftv types = foldr (\t ts-> ftv t ++ ts) [] types
+ subst s ts = map (\t->subst s t) ts
+
+instance Typeable Type where
+ ftv (TupleType (t1, t2)) = ftv t1 ++ ftv t2
+ ftv (ListType t) = ftv t
+ ftv (IdType tvar) = [tvar]
+ ftv (t1 ->> t2) = ftv t1 ++ ftv t2
+ ftv _ = []
+ subst s (TupleType (t1, t2))= TupleType (subst s t1, subst s t2)
+ subst s (ListType t1) = ListType (subst s t1)
+ subst s (t1 ->> t2) = (subst s t1) ->> (subst s t2)
+ subst s t1=:(IdType tvar) = 'Map'.findWithDefault t1 tvar s
+ subst s t = t
+
+instance Typeable Gamma where
+ ftv gamma = concatMap id $ map ftv ('Map'.elems gamma)
+ subst s gamma = Mapmap (subst s) gamma
+
+extend :: String Scheme Gamma -> Gamma
+extend k t g = 'Map'.put k t g
+
+//// ------------------------
+//// algorithm U, Unification
+//// ------------------------
+instance zero Substitution where zero = 'Map'.newMap
+
+compose :: Substitution Substitution -> Substitution
+compose s1 s2 = 'Map'.union (Mapmap (subst s1) s2) s1
+//Note: just like function compositon compose does snd first
+
+occurs :: TVar a -> Bool | Typeable a
+occurs tvar a = elem tvar (ftv a)
+
+unify :: Type Type -> Either SemError Substitution
+unify t1 t2=:(IdType tv) | t1 == (IdType tv) = Right zero
+ | occurs tv t1 = Left $ InfiniteTypeError zero t1
+ | otherwise = Right $ 'Map'.singleton tv t1
+unify t1=:(IdType tv) t2 = unify t2 t1
+unify (ta1->>ta2) (tb1->>tb2) = unify ta1 tb1 >>= \s1->
+ unify ta2 tb2 >>= \s2->
+ Right $ compose s1 s2
+unify (TupleType (ta1,ta2)) (TupleType (tb1,tb2)) = unify ta1 tb1 >>= \s1->
+ unify ta2 tb2 >>= \s2->
+ Right $ compose s1 s2
+unify (ListType t1) (ListType t2) = unify t1 t2
+unify t1 t2 | t1 == t2 = Right zero
+ | otherwise = Left $ UnifyError zero t1 t2
+
+//// ------------------------
+//// Algorithm M, Inference and Solving
+//// ------------------------
+gamma :: Typing Gamma
+gamma = gets fst
+putGamma :: Gamma -> Typing ()
+putGamma g = modify (appFst $ const g) >>| pure ()
+changeGamma :: (Gamma -> Gamma) -> Typing Gamma
+changeGamma f = modify (appFst f) >>| gamma
+withGamma :: (Gamma -> a) -> Typing a
+withGamma f = f <$> gamma
+fresh :: Typing Type
+fresh = gets snd >>= \vars->
+ modify (appSnd $ const $ tail vars) >>|
+ pure (IdType (head vars))
+
+lift :: (Either SemError a) -> Typing a
+lift (Left e) = liftT $ Left e
+lift (Right v) = pure v
+
+//instantiate maps a schemes type variables to variables with fresh names
+//and drops the quantification: i.e. forall a,b.a->[b] becomes c->[d]
+instantiate :: Scheme -> Typing Type
+instantiate (Forall bound t) =
+ mapM (const fresh) bound >>= \newVars->
+ let s = 'Map'.fromList (zip (bound,newVars)) in
+ pure (subst s t)
+
+//generalize quentifies all free type variables in a type which are not
+//in the gamma
+generalize :: Type -> Typing Scheme
+generalize t = gamma >>= \g-> pure $ Forall (difference (ftv t) (ftv g)) t
+
+lookup :: String -> Typing Type
+lookup k = gamma >>= \g-> case 'Map'.member k g of
+ False = liftT (Left $ UndeclaredVariableError zero k)
+ True = instantiate $ 'Map'.find k g
+
+//The inference class
+//When tying it all together we will treat the program is a big
+//let x=e1 in let y=e2 in ....
+class infer a :: a -> Typing (Substitution, Type)
+
+////---- Inference for Expressions ----
+
+instance infer Expr where
+ infer e = case e of
+ VarExpr _ (VarDef k fs) = lookup k >>= \t ->
+ foldM foldFieldSelectors t fs >>= \finalT ->
+ pure (zero, finalT)
+
+ Op2Expr _ e1 op e2 =
+ infer e1 >>= \(s1, t1) ->
+ infer e2 >>= \(s2, t2) ->
+ fresh >>= \tv ->
+ let given = t1 ->> t2 ->> tv in
+ op2Type op >>= \expected ->
+ lift (unify expected given) >>= \s3 ->
+ pure ((compose s3 $ compose s2 s1), subst s3 tv)
+
+ Op1Expr _ op e1 =
+ infer e1 >>= \(s1, t1) ->
+ fresh >>= \tv ->
+ let given = t1 ->> tv in
+ op1Type op >>= \expected ->
+ lift (unify expected given) >>= \s2 ->
+ pure (compose s2 s1, subst s2 tv)
+
+ EmptyListExpr _ = (\tv->(zero,tv)) <$> fresh
-uni :: Type Type -> Infer ()
-uni t1 t2 = tell [(t1, t2)]
+ TupleExpr _ (e1, e2) =
+ infer e1 >>= \(s1, t1) ->
+ infer e2 >>= \(s2, t2) ->
+ pure (compose s2 s1, TupleType (t1,t2))
-inEnv :: (String, Scheme) (Infer a) -> (Infer a)
-inEnv (x, sc) m = local scope m
- where
- scope e = 'Map'.put x sc ('Map'.del x e )
+ FunExpr _ f args fs = //todo: fieldselectors
+ lookup f >>= \expected ->
+ let accST = (\(s,ts) e->infer e >>= \(s_,et)->pure (compose s_ s,ts++[et])) in
+ foldM accST (zero,[]) args >>= \(s1, argTs)->
+ fresh >>= \tv->
+ let given = foldr (->>) tv argTs in
+ lift (unify expected given) >>= \s2->
+ let fReturnType = subst s2 tv in
+ foldM foldFieldSelectors fReturnType fs >>= \returnType ->
+ pure (compose s2 s1, returnType)
-fresh :: Infer Type
-fresh = (gets id) >>= \vars-> (put $ tail vars) >>| (pure $ IdType $ head vars)
+ IntExpr _ _ = pure $ (zero, IntType)
+ BoolExpr _ _ = pure $ (zero, BoolType)
+ CharExpr _ _ = pure $ (zero, CharType)
-class infer a :: a -> Infer Type
+foldFieldSelectors :: Type FieldSelector -> Typing Type
+foldFieldSelectors (ListType t) (FieldHd) = pure t
+foldFieldSelectors t=:(ListType _) (FieldTl) = pure t
+foldFieldSelectors (TupleType (t1, _)) (FieldFst) = pure t1
+foldFieldSelectors (TupleType (_, t2)) (FieldSnd) = pure t2
+foldFieldSelectors t fs = liftT $ Left $ FieldSelectorError zero t fs
-op2Type :: Op2 -> Infer Type
+op2Type :: Op2 -> Typing Type
op2Type op
| elem op [BiPlus, BiMinus, BiTimes, BiDivide, BiMod]
- = pure (IntType ->> IntType ->> IntType)
+ = pure (IntType ->> IntType ->> IntType)
| elem op [BiEquals, BiUnEqual]
- = fresh >>= \t1-> fresh >>= \t2-> pure (t1 ->> t2 ->> BoolType)
+ = fresh >>= \t1-> fresh >>= \t2-> pure (t1 ->> t2 ->> BoolType)
| elem op [BiLesser, BiGreater, BiLesserEq, BiGreaterEq]
- = pure (IntType ->> IntType ->> BoolType)
+ = pure (IntType ->> IntType ->> BoolType)
| elem op [BiAnd, BiOr]
- = pure (BoolType ->> BoolType ->> BoolType)
+ = pure (BoolType ->> BoolType ->> BoolType)
| op == BiCons
- = fresh >>= \t1-> pure (t1 ->> ListType t1 ->> ListType t1)
+ = fresh >>= \t1-> pure (t1 ->> ListType t1 ->> ListType t1)
-op1Type :: Op1 -> Infer Type
+op1Type :: Op1 -> Typing Type
op1Type UnNegation = pure $ (BoolType ->> BoolType)
op1Type UnMinus = pure $ (IntType ->> IntType)
-//instantiate :: Scheme -> Infer Type
-//instantiate (Forall as t) = mapM (const fresh) as
-
-lookupEnv :: String -> Infer Type
-lookupEnv ident = asks ('Map'.get ident)
- >>= \m->case m of
- Nothing = liftT $ Left $ UndeclaredVariableError zero ident
- Just (Forall as t) = pure t //instantiate ???
-
-instance infer Expr where
- infer (VarExpr _ (VarDef ident fs)) = lookupEnv ident
- infer (Op2Expr _ e1 op e2) = case op of
- BiPlus = pure IntType
- BiMinus = pure IntType
- BiTimes = pure IntType
- BiDivide = pure IntType
- BiMod = pure IntType
- BiLesser = pure IntType
- BiGreater = pure IntType
- BiLesserEq = pure IntType
- BiGreaterEq = pure IntType
- BiAnd = pure BoolType
- BiOr = pure BoolType
- BiEquals = infer e1
- BiUnEqual = infer e1 // maybe check e2?
- BiCons = infer e1 >>= \it1->pure $ ListType it1
- infer (Op1Expr _ op e) = case op of
- UnMinus = pure IntType
- UnNegation = pure BoolType
- infer (IntExpr _ _) = pure IntType
- infer (CharExpr _ _) = pure CharType
- infer (BoolExpr _ _) = pure BoolType
- infer (FunExpr _ _ _ _) = undef
- infer (EmptyListExpr _) = undef
- infer (TupleExpr _ (e1, e2)) =
- infer e1 >>= \et1->infer e2 >>= \et2->pure $ TupleType (et1, et2)
-
-//:: VarDef = VarDef String [FieldSelector]
-//:: FieldSelector = FieldHd | FieldTl | FieldFst | FieldSnd
-//:: Op1 = UnNegation | UnMinus
-//:: Op2 = BiPlus | BiMinus | BiTimes | BiDivide | BiMod | BiEquals | BiLesser |
-// BiGreater | BiLesserEq | BiGreaterEq | BiUnEqual | BiAnd | BiOr | BiCons
-//:: FunDecl = FunDecl Pos String [String] (Maybe Type) [VarDecl] [Stmt]
-//:: FunCall = FunCall String [Expr]
-//:: Stmt
-// = IfStmt Expr [Stmt] [Stmt]
-// | WhileStmt Expr [Stmt]
-// | AssStmt VarDef Expr
-// | FunStmt FunCall
-// | ReturnStmt (Maybe Expr)
-//:: Pos = {line :: Int, col :: Int}
-//:: AST = AST [VarDecl] [FunDecl]
-//:: VarDecl = VarDecl Pos Type String Expr
-//:: Type
-// = TupleType (Type, Type)
-// | ListType Type
-// | IdType String
-// | IntType
-// | BoolType
-// | CharType
-// | VarType
-// | VoidType
-// | (->>) infixl 7 Type Type
+////----- Inference for Statements -----
+applySubst :: Substitution -> Typing Gamma
+applySubst s = changeGamma (subst s)
+
+instance infer Stmt where
+ infer s = case s of
+ IfStmt e th el =
+ infer e >>= \(s1, et)->
+ lift (unify et BoolType) >>= \s2 ->
+ applySubst (compose s2 s1) >>|
+ infer th >>= \(s3, tht)->
+ applySubst s3 >>|
+ infer el >>= \(s4, elt)->
+ applySubst s4 >>|
+ lift (unify tht elt) >>= \s5->
+ pure (compose s5 $ compose s4 $ compose s3 $ compose s2 s1, subst s5 tht)
+
+ WhileStmt e wh =
+ infer e >>= \(s1, et)->
+ lift (unify et BoolType) >>= \s2 ->
+ applySubst (compose s2 s1) >>|
+ infer wh >>= \(s3, wht)->
+ pure (compose s3 $ compose s2 s1, subst s3 wht)
+
+ AssStmt (VarDef k fs) e =
+ lookup k >>= \expected ->
+ infer e >>= \(s1, given)->
+ foldM reverseFs given (reverse fs) >>= \varType->
+ lift (unify expected varType) >>= \s2->
+ let s = compose s2 s1 in
+ applySubst s >>|
+ changeGamma (extend k (Forall [] (subst s varType))) >>|
+ pure (s, VoidType)
+
+ FunStmt f es _ = pure (zero, VoidType)
+
+ ReturnStmt Nothing = pure (zero, VoidType)
+ ReturnStmt (Just e) = infer e
+
+reverseFs :: Type FieldSelector -> Typing Type
+reverseFs t FieldHd = pure $ ListType t
+reverseFs t FieldTl = pure $ ListType t
+reverseFs t FieldFst = fresh >>= \tv -> pure $ TupleType (t, tv)
+reverseFs t FieldSnd = fresh >>= \tv -> pure $ TupleType (tv, t)
+
+//The type of a list of statements is either an encountered
+//return, or VoidType
+instance infer [a] | infer a where
+ infer [] = pure (zero, VoidType)
+ infer [stmt:ss] =
+ infer stmt >>= \(s1, t1) ->
+ applySubst s1 >>|
+ infer ss >>= \(s2, t2) ->
+ applySubst s2 >>|
+ case t1 of
+ VoidType = pure (compose s2 s1, t2)
+ _ = case t2 of
+ VoidType = pure (compose s2 s1, t1)
+ _ = lift (unify t1 t2) >>= \s3 ->
+ pure (compose s3 $ compose s2 s1, t1)
+
+//the type class inferes the type of an AST element (VarDecl or FunDecl)
+//and adds it to the AST element
+class type a :: a -> Typing (Substitution, a)
+
+instance type VarDecl where
+ type (VarDecl p expected k e) =
+ infer e >>= \(s1, given) ->
+ applySubst s1 >>|
+ case expected of
+ Nothing = pure zero
+ Just expected_ = lift (unify expected_ given)
+ >>= \s2->
+ applySubst s2 >>|
+ let vtype = subst (compose s2 s1) given in
+ generalize vtype >>= \t ->
+ changeGamma (extend k t) >>|
+ pure (compose s2 s1, VarDecl p (Just vtype) k e)
+
+instance type FunDecl where
+ type (FunDecl p f args expected vds stmts) =
+ gamma >>= \outerScope-> //functions are infered in their own scopde
+ introduce f >>|
+ mapM introduce args >>= \argTs->
+ type vds >>= \(s1, tVds)->
+ applySubst s1 >>|
+ infer stmts >>= \(s2, result)->
+ applySubst s1 >>|
+ let argTs_ = map (subst $ compose s2 s1) argTs in
+ //abort (concat $ intersperse "\n" $ map toString argTs_) >>|
+ let given = foldr (->>) result argTs_ in
+ (case expected of
+ Nothing = pure zero
+ Just expected_ = lift (unify expected_ given))
+ >>= \s3 ->
+ let ftype = subst (compose s3 $ compose s2 s1) given in
+ generalize ftype >>= \t->
+ putGamma outerScope >>|
+ changeGamma (extend f t) >>|
+ pure (compose s3 $ compose s2 s1, FunDecl p f args (Just ftype) tVds stmts)
+
+instance type [a] | type a where
+ type [] = pure (zero, [])
+ type [v:vs] =
+ type v >>= \(s1, v_)->
+ applySubst s1 >>|
+ type vs >>= \(s2, vs_)->
+ applySubst (compose s2 s1) >>|
+ pure (compose s2 s1, [v_:vs_])
+
+introduce :: String -> Typing Type
+introduce k =
+ fresh >>= \tv ->
+ changeGamma (extend k (Forall [] tv)) >>|
+ pure tv
+
+instance toString Scheme where
+ toString (Forall x t) =
+ concat ["Forall ": intersperse "," x] +++ concat [". ", toString t];
+
+instance toString Gamma where
+ toString mp =
+ concat [concat [k, ": ", toString v, "\n"]\\(k, v)<-'Map'.toList mp]
+
+instance toString Substitution where
+ toString subs =
+ concat [concat [k, ": ", toString t, "\n"]\\(k, t)<-'Map'.toList subs]
+
+instance toString SemError where
+ toString (SanityError p e) = concat [toString p,
+ "SemError: SanityError: ", e]
+ toString (ParseError p s) = concat [toString p,
+ "ParseError: ", s]
+ toString (UnifyError p t1 t2) = concat [toString p,
+ "Can not unify types, expected|given:\n", toString t1,
+ "\n", toString t2]
+ toString (InfiniteTypeError p t) = concat [toString p,
+ "Infinite type: ", toString t]
+ toString (FieldSelectorError p t fs) = concat [toString p,
+ "Can not run fieldselector '", toString fs, "' on type: ",
+ toString t]
+ toString (OperatorError p op t) = concat [toString p,
+ "Operator error, operator '", toString op, "' can not be",
+ "used on type: ", toString t]
+ toString (UndeclaredVariableError p k) = concat [toString p,
+ "Undeclared identifier: ", k]
+ toString (ArgumentMisMatchError p str) = concat [toString p,
+ "Argument mismatch: ", str]
+ toString (Error e) = concat ["Unknown error during semantical",
+ "analysis: ", e]
+
+instance toString (Maybe a) | toString a where
+ toString Nothing = "Nothing"
+ toString (Just e) = concat ["Just ", toString e]
+
+instance MonadTrans (StateT (Gamma, [TVar])) where
+ liftT m = StateT \s-> m >>= \a-> return (a, s)
+
+Mapmap :: (a->b) ('Map'.Map k a) -> ('Map'.Map k b)
+Mapmap _ 'Map'.Tip = 'Map'.Tip
+Mapmap f ('Map'.Bin sz k v ml mr) = 'Map'.Bin sz k (f v)
+ (Mapmap f ml)
+ (Mapmap f mr)
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