A Framework for Extended Algebraic Data Types
نویسندگان
چکیده
Data Types have Existential Types data Stack a = forall s. Stack s -self (a->s->s) -push (s->s) -pop (s->a) -top (s->Bool) -empty push :: a -> Stack a -> Stack a push x (Stack s push’ pop top empty) = Stack (push’ x s) push’ pop top empty pop :: Stack a -> Stack a pop (Stack s push pop’ top empty) = Stack (pop’ s) push pop’ top empty A Framework for Extended Algebraic Data Types – p.6 Abstract Data Types have Existential TypesData Types have Existential Types data Stack a = forall s. Stack s -self (a->s->s) -push (s->s) -pop (s->a) -top (s->Bool) -empty top :: Stack a -> a top (Stack s push pop top’ empty) = top’ s empty :: Stack a -> Bool empty (Stack s push pop top empty’) = empty’ s -examples mkListStack :: [a] -> Stack a mkListStack xs = Stack xs (:) tail head null s1 = mkListStack [1::Int] s2 = push 2 s1 A Framework for Extended Algebraic Data Types – p.7 Type Classes with Existential Types Type classes (Wadler/Blott [WB89], Kaes): class Eq a where (==) :: a->a->Bool instance Eq a => Eq [a] where (==) (x:xs) (y:ys) = (x==y) && (xs==ys) ... Type Classes with Existential Types (Läufer [Läu96]): class Key a where getKey::a->Int data KEY = forall a. Key a => Mk a g :: KEY -> Int g (Mk x) = getKey x A Framework for Extended Algebraic Data Types – p.8 Type Classes with Existential Types Type classes with functional dependencies (Jones [Jon00]) and existential types: class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) pop :: Stack a -> Stack a pop (Stack s) = Stack (popM s) A Framework for Extended Algebraic Data Types – p.9 Type Classes with Existential Types -examples instance StackM [a] a where pushM = (:) popM = tail topM = head emptyM = null mkListStack :: [a] -> Stack a mkListStack xs = Stack xs s1 = mkListStack [1::Int] s2 = push 2 s1 A Framework for Extended Algebraic Data Types – p.10 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s How to type check function push. push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) A Framework for Extended Algebraic Data Types – p.11 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) A Framework for Extended Algebraic Data Types – p.11 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) x: a Stack s : Stack a s : s StackM s a A Framework for Extended Algebraic Data Types – p.11 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) x: a Stack s : Stack a s : s StackM s a A Framework for Extended Algebraic Data Types – p.11 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) x: a Stack s : Stack a s : s StackM s a StackM s a A Framework for Extended Algebraic Data Types – p.11 Type Classes with Existential Types class StackM s a | s -> a where pushM :: a->s->s popM :: s->s topM :: s->a emptyM :: s->Bool data Stack a = forall s. StackM s a => Stack s push :: a -> Stack a -> Stack a push x (Stack s) = Stack (pushM x s) x: a Stack s : Stack a s : s StackM s a ⊃ StackM s a A Framework for Extended Algebraic Data Types – p.11 Generalized Algebraic Data Types Enforce data invariants via types: Leijen and Meijer [LM99] Okasaki [Oka99] Cheney and Hinze [CH02] Baars and Swierstra [BS02] and of course Kiselyov! “Generalized” algebraic data types: Index types (Zenger [Zen99], Xi) Guarded recursive data types (Xi, Chen and Chen [XCC03]) Generalized algebraic data types (Peyton-Jones et. al.) First-class phantom types [CH03]. A Framework for Extended Algebraic Data Types – p.12 Well-Typed Expressions Won’t Fail e1 :: Exp e1 = App (Fun (\x-> Plus Zero x)) (Succ Zero) Eval> eval e1 Succ Zero What about e2 :: Exp e2 = App Zero (Succ Zero) Eval> eval e2 Program error! Prevent construction of ill-typed expressions A Framework for Extended Algebraic Data Types – p.13 Construction of Well-Typed Terms Typing rules: (App) ⊢ e1 : a → b ⊢ e2 : a ⊢ e1 e2 : b (Abs) x : a ⊢ e : b ⊢ λx.e : a → b
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