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Existential Types

Existential Types. Polymorphic Type Inference and Abstract Data Types by Martin Odersky. Table Of Content. Existential type Syntax of ExML Formal syntax of ExML Type inference rules Type inference algorithm Semantic function. Existential Type. [Mit88] and [Cardelli85]

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Existential Types

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  1. Existential Types Polymorphic Type Inference and Abstract Data Typesby Martin Odersky

  2. Table Of Content • Existential type • Syntax of ExML • Formal syntax of ExML • Type inference rules • Type inference algorithm • Semantic function

  3. Existential Type • [Mit88] and [Cardelli85] • abstype complex with creat: real →real → complex, plus: complex → complex, re: complex → real, im: complex → realispack real ∧ realλx: real. λy: real. <x, y>λz: real ∧ real. λw: real ∧ real. <fst(z) + fst(w), snd(z) + snd(w)>λz: real ∧ real. fst(z)λz: real ∧ real. snd(z)to ∃t. [(real → real → t) ∧ (t → t) ∧(t → real) ∧ (t → real)]

  4. Syntax of ExML • type [args] T = K1ofτ1 | … | Kkofτk • type KEY = Key of‘a * (‘a -> int) • Key(3, fun x -> 5) • Key([1, 2, 3], list_length) • Key(3, list_length) (* error *) • let (Key(v,f)) = x in f v • let (Key(v,f)) = x in v (* error *)

  5. Syntax of ExML (cont.) • type Key = {x : ‘a; f : ‘a -> int} • let z = {x = 3, f = fun x -> x + 2} in z.f z.x • let z = {x = 3, f = fun x -> x + 2} in z.f (* error *) • let z = {x = 3, f = fun x -> x + 2} inlet y = z in z.f y.x (* error *)

  6. Syntax of ExML (cont.) • type‘a STACK = {Self : ‘b; Push : ‘a -> ‘b -> ‘b; Pop : ‘b -> ‘b; Top : ‘b -> ‘a; Null : ‘b -> bool} • let push x s = s with {Self = s.Push x s.Self} • let top s = s.Top s.Self

  7. Formal Syntax of ExML

  8. Original Type Inference Rule

  9. Type Inference Rule Extension

  10. Original Type Inference Algorithm

  11. Type Inference Algorithm Extension

  12. Semantic Function for Expression

  13. Semantic Function for Types

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