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CSED321 Programming Languages Polymorphism

CSED321 Programming Languages Polymorphism. Kyungmin Bae Department of Computer Science and Engineering POSTECH Spring 2019. Polymorphism. Parametric Polymorphism operates on all types of objects in a uniform way. C++ templates, Java generics, … Subtype Polymorphism

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CSED321 Programming Languages Polymorphism

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  1. CSED321 Programming LanguagesPolymorphism Kyungmin Bae Department of Computer Science and Engineering POSTECH Spring 2019

  2. Polymorphism • Parametric Polymorphism • operates on all types of objects in a uniform way. • C++ templates, Java generics, … • Subtype Polymorphism • operates on subtypes of objects • subtypes on classes • Ad hoc Polymorphism • different behavior depending on the type of objects it operates on • operator overloading

  3. Outline • System F • Programming in System F • Type reconstruction • Implicit polymorphism • Type inference algorithm

  4. System F • Untyped -calculus + Parametric polymorphism • First step toward the polymorphic type system of OCaml

  5. Type of • Bind variable to some type • Decide the type of the resulting expression • In the simply typed -calculus:

  6. Type of : Goal • Bind variable to any type • Assign a polymorphic type to

  7. Ambiguity

  8. Type Abstraction • Declares as a fresh any type, or a fresh type variable

  9. Example: Identity Function

  10. Type Application • Type application • Example

  11. System F: Syntax

  12. System F: Syntax

  13. Reduction Rules

  14. Reduction Rules

  15. Example

  16. Typing Context

  17. Typing Judgement

  18. Type Judgement

  19. Typing Rules

  20. Example

  21. Type Safety • Type preservation If and , then • Progress If for some type , then either is a value or there exists such that

  22. Substitution Lemmas

  23. Outline • System F • Programming in System F • Type reconstruction • Implicit polymorphism • Type inference algorithm

  24. Power of System F • Can naturally encode complex data types • C.f. simply typed lambda calculus

  25. Programming in System F: Boolean • Type

  26. Programming in System F: Numerals • Type • zero • succ

  27. Programming in System F: Product Types • Type • pair • fst • snd

  28. Programming in System F: Sum Types • Type • inl • inr • case

  29. Self-application (a.k.a, Impredicativity)

  30. Strong Normalization • Every expression in System F eventually reduces to a value.

  31. Outline • System F • Programming in System F • Type reconstruction • Implicit polymorphism • Type inference algorithm

  32. Which is the Most Readable? fun f -> fun g -> (fun x -> g (f x)) fun f : ('a -> ‘b) -> fun g : ('b -> ‘c) -> (fun x : 'a -> g (f x)) fun f : ('a -> ‘b) -> fun g : ('b -> ‘c) -> ((fun x : 'a -> (g (f x)) : 'c): 'a -> 'c)

  33. Type Reconstruction • From: untyped (typeable) program • To: typed program

  34. Erasure Function

  35. Requirement of Type Reconstruction

  36. Example

  37. Bad News • Not every untyped expression is typeable in System F • Type reconstruction of System F is undecidable

  38. Predicative Polymorphic -Calculus • Sublanguage of System F • Decidable type reconstruction • Supports polymorphism

  39. Types in System F

  40. Types in Predicative Polymorphic -Calculus • A polytype in prenex form

  41. Predicative Polymorphic -Calculus: Syntax

  42. Typing Rules

  43. Typing Rules

  44. Example

  45. Let-Polymorphism • System F • Powerful, but undecidable type reconstruction • Predicative polymorphic -Calculus • Decidable type reconstruction, but useless • Let-Polymorphism • Decidable type reconstruction, and useful

  46. Let-Binding • binds to a polymorphic expression • not a syntactic sugar in terms of types!

  47. Example

  48. Syntax

  49. Let-Binding: Typing Rules

  50. Let-Binding: Reduction Rules

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