-instance toString SemError where
- toString (ParseError p e) = concat [
- toString p,"SemError: ParseError: ", e]
- toString (Error e) = "SemError: " +++ e
- toString (UnifyErrorStub t1 t2) = toString (UnifyError {line=0,col=0} t1 t2)
- toString (UnifyError p t1 t2) = concat [
- toString p,
- "SemError: Cannot unify types. Expected: ",
- toString t1, ". Given: ", toString t2]
-
-sem :: AST -> SemOutput
-sem (AST vd fd) = case runStateT m ('Map'.newMap, getRandomStream 0) of
- Left e = Left [e]
- Right ((vds, fds), gamma) = Right ((AST vds fds), gamma)
-where
- m :: Env (([VarDecl], [FunDecl]))
- m = (mapM semVarDecl vd) >>= \vds ->
- mapM semFunDecl fd >>= \fds ->
- pure (vds, fds)
-
-semFunDecl :: FunDecl -> Env FunDecl
-semFunDecl f = pure f
-
-semVarDecl :: VarDecl -> Env VarDecl
-semVarDecl (VarDecl pos type ident ex) = unify type ex
- >>= \t-> putIdent ident t >>| (pure $ VarDecl pos t ident ex)
-
-typeExpr :: Expr -> Env Type
-typeExpr (IntExpr _ _) = pure IntType
-typeExpr (CharExpr _ _) = pure CharType
-typeExpr (BoolExpr _ _) = pure BoolType
-typeExpr (Op1Expr p UnNegation expr) = unify BoolType expr
-typeExpr (Op1Expr p UnMinus expr) = unify IntType expr
-typeExpr (TupleExpr p (e1, e2)) = typeExpr e1
- >>= \t1-> typeExpr e2 >>= \t2-> pure $ TupleType (t1, t2)
-//Int
-typeExpr (Op2Expr p e1 BiPlus e2) = unify IntType e1 >>| unify IntType e2
-typeExpr (Op2Expr p e1 BiMinus e2) = unify IntType e1 >>| unify IntType e2
-typeExpr (Op2Expr p e1 BiTimes e2) = unify IntType e1 >>| unify IntType e2
-typeExpr (Op2Expr p e1 BiDivide e2) = unify IntType e1 >>| unify IntType e2
-typeExpr (Op2Expr p e1 BiMod e2) = unify IntType e1 >>| unify IntType e2
-//bool, char of int
-typeExpr (Op2Expr p e1 BiEquals e2) = typeExpr e1 >>= \t1 -> unify t1 e2
- >>| pure BoolType //todo, actually check t1 in Char,Bool,Int
-typeExpr (Op2Expr p e1 BiUnEqual e2) = typeExpr (Op2Expr p e1 BiEquals e2)
-//char of int
-typeExpr (Op2Expr p e1 BiLesser e2) = typeExpr e1 >>= \t1 -> unify t1 e2
- >>| pure BoolType //todo, actually check t1 in Char, Int
-typeExpr (Op2Expr p e1 BiGreater e2) = typeExpr (Op2Expr p e1 BiLesser e2)
-typeExpr (Op2Expr p e1 BiLesserEq e2) = typeExpr (Op2Expr p e1 BiLesser e2)
-typeExpr (Op2Expr p e1 BiGreaterEq e2) = typeExpr (Op2Expr p e1 BiLesser e2)
-//bool
-typeExpr (Op2Expr p e1 BiAnd e2) = unify BoolType e1 >>| unify BoolType e2
-typeExpr (Op2Expr p e1 BiOr e2) = unify BoolType e1 >>| unify BoolType e2
-//a
-typeExpr (Op2Expr p e1 BiCons e2) = typeExpr e1 >>= \t1-> typeExpr e2
- >>= \t2-> unify (ListType t1) t2
-//typeExpr (FunExpr p FunCall) = undef
-typeExpr (EmptyListExpr p) = freshIdent >>= \frsh-> let t = IdType frsh in
- putIdent frsh t >>| pure t
-//typeExpr (VarExpr Pos VarDef) = undef //when checking var-expr, be sure to
- //put the infered type in the context
-
-class unify a :: Type a -> Env Type
-
-instance unify Expr where
- unify (_ ->> _) e = liftT $ Left $ ParseError (extrPos e)
- "Expression cannot be a higher order function. Yet..."
- unify VoidType e = liftT $ Left $ ParseError (extrPos e)
- "Expression cannot be a Void type."
- unify (IdType _) e = liftT $ Left $ ParseError (extrPos e)
- "Expression cannot be an polymorf type."
- unify VarType e = typeExpr e
- //we have to cheat to decorate the error, can be done nicer?
- unify t e = StateT $ \s0 -> let res = runStateT m s0 in case res of
- Left err = Left $ decErr e err
- Right t = Right t //note, t :: (Type, Gamma)
- where m = typeExpr e >>= \tex-> unify t tex
-
-instance unify Type where
- unify IntType IntType = pure IntType
- unify BoolType BoolType = pure BoolType
- unify CharType CharType = pure CharType
- unify (ListType t1) (ListType t2) = unify t1 t2
- unify t1 t2 = liftT $ Left $ UnifyError zero t1 t2
-
-instance zero Pos where
- zero = {line=0,col=0}
-
-decErr :: Expr SemError -> SemError
-decErr e (UnifyError _ t1 t2) = UnifyError (extrPos e) t1 t2
-decErr e (ParseError _ s) = ParseError (extrPos e) s
-decErr e err = err
-
-dc2 :: Expr (Either SemError a) -> Either SemError a
-dc2 e (Right t) = Right t
-dc2 e (Left err) = Left err
-
-extrPos :: Expr -> Pos
-extrPos (VarExpr p _) = p
-extrPos (Op2Expr p _ _ _) = p
-extrPos (Op1Expr p _ _) = p
-extrPos (IntExpr p _) = p
-extrPos (CharExpr p _) = p
-extrPos (BoolExpr p _) = p
-extrPos (FunExpr p _) = p
-extrPos (EmptyListExpr p) = p
-extrPos (TupleExpr p _) = p
+class Typeable a where
+ ftv :: a -> [TVar]
+ subst :: Substitution a -> a
+
+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 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 "isEmpty" = ListType <$> fresh
+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) = (\t->(zero,t)) <$> lookup k
+ //instantiate is key for the let polymorphism!
+ //TODO: field selectors
+
+ 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
+
+ TupleExpr _ (e1, e2) =
+ infer e1 >>= \(s1, t1) ->
+ infer e2 >>= \(s2, t2) ->
+ pure (compose s2 s1, TupleType (t1,t2))
+
+ 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->
+ pure (compose s2 s1, subst s2 tv)
+
+ IntExpr _ _ = pure $ (zero, IntType)
+ BoolExpr _ _ = pure $ (zero, BoolType)
+ CharExpr _ _ = pure $ (zero, CharType)
+
+
+op2Type :: Op2 -> Typing Type
+op2Type op
+| elem op [BiPlus, BiMinus, BiTimes, BiDivide, BiMod]
+ = pure (IntType ->> IntType ->> IntType)
+| elem op [BiEquals, BiUnEqual]
+ = fresh >>= \t1-> fresh >>= \t2-> pure (t1 ->> t2 ->> BoolType)
+| elem op [BiLesser, BiGreater, BiLesserEq, BiGreaterEq]
+ = pure (IntType ->> IntType ->> BoolType)
+| elem op [BiAnd, BiOr]
+ = pure (BoolType ->> BoolType ->> BoolType)
+| op == BiCons
+ = fresh >>= \t1-> pure (t1 ->> ListType t1 ->> ListType t1)
+
+op1Type :: Op1 -> Typing Type
+op1Type UnNegation = pure $ (BoolType ->> BoolType)
+op1Type UnMinus = pure $ (IntType ->> IntType)
+
+////----- 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)->
+ lift (unify expected given) >>= \s2->
+ let s = compose s2 s1 in
+ applySubst s >>|
+ changeGamma (extend k (Forall [] given)) >>| //todo: fieldselectors
+ pure (s, VoidType)
+
+ FunStmt f es = undef //what is this?
+
+ ReturnStmt Nothing = pure (zero, VoidType)
+ ReturnStmt (Just e) = infer e
+
+//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];