CS 611: Lecture 33

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CS 611 Lecture 33

• Parametric Polymorphism
• November 15, 1999
• Cornell University Computer Science Department
• Andrew Myers

2
ML type reconstruction

• Last time introduced idea of type
  reconstruction, started on Algorithm W
• ML-like languagee n t f x e0 e1
  fn x e let (x e0) e1 letrec ((x1 e1)(xn
  en)) eb if e0 e1 e2
• Typest int bool t0?t1 a
  (monomorphic)s ?a1,...,an.t
  t (polymorphic)
• Typing context extended to allow polymorphic
  types
• G Var?s
• But expressions have monomorphic type
• G ? e t

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Typing rules

• Correctness of algorithm W is judged with respect
  to typing rules
• W(e, G, S) ?t, S?
• Reconstructing type of e in typing context G
  with respect to substitution S yields type t,
  stronger substitution S
• Correctness SG ? e St
• Rules of interest

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Making types generic

• Generic(t, G) converts a type into a polymorphic
  form
• Generic(t, G) ?a1,...,an.twhere a1,...,an
  FTV(t) - FTE(G)
• FTE(G) is removed to avoiding capturing type
  variables FTE(G) ?t?range(G) FTV(t)
• FTV(a) a FTV(int) ?
• FTV(t0?t1) FTV(t0)?FTV(t1)

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Algorithm W

• W(e, G, S) ?t, S? SG ? e St
• Unify(t1t2, S) S
• W(n, G, S) ?int, S?
• W(t, G, S) ?bool, S?
• W(x, G, S) case Gx of t ? ?t, S?
• ?a1,...,an.t ? ?ta1f/a1,..., S?
• W(e0 e1, G, S) let ?t0, S0? W(e0, G, S) in
• let ?t1, S1? W(e0, G, S0) in
• ?af, Unify(t1?af t0, S1)?
• W( fn x e, G, S) let ?t, S? W(e, Gx ? af,
  S) in ?af?t, S?

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Remainder of algorithm

• W(if e0 e1 e2, G, S) let ?t0, S0? W(e0, G, S)
  in
• let ?t1, S1? W(e1, G, Unify(t0bool, S0)) in
• let ?t2, S2? W(e2, G, S1) in
• ?t1,Unify(t1t2, S2)?
• W(let (x e0) e1, G, S) let ?t0, S0? W(e0, G,
  S) in
• W(e1, Gx ? Generic(S0t, S0G), S0)
• W(letrec ((x1 e1)(xn en)) eb, G, S0)
• let G Gx1 ? a1f, in
• let ?t1, S1? W(e1, G, S) in let ?tn, Sn?
  W(en, G, Sn-1) in
• let S1 Unify(a1f t1, Sn) in...let Sn
  Unify(anf tn, Sn-1) in
• let G Gx1 ? Generic(Snt1, SnG), in
• W(eb, G, Sn)

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Parametric polymorphism

• Polymorphism expression has multiple types
• Parametric polymorphism (?a1,...,an.t)
• types appear as parameters
• expression is written the same way for all types
• polymorphic value is instantiated on some types
• Predicative polymorphism polymorphic values are
  not first class (can only be instantiated)
• ML (implicit), CLU, PolyJ (explicit)
• Bounded parametric polymorphism (?aP.t) -- can
  only be instantiated on types satisfying P. P may
  be a kind or a fancier bound (CLU, PolyJ)
• Templates (C, Modula-3) polymorphic
  expression is not type checked until
  instantiated a macro

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Ad-hoc polymorphism

• Same name can be used to denote values with
  different types
• e.g. refers to one operator on int, another
  on float, yet another on String
• Examples C, Java overloading
• Can be modeled by allowing overloading in the
  type context G
• Not true polymorphism no polymorphic values

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Dependent types

• Types polymorphic with respect to a value
• Example CLU, some varieties of Pascal
• procedure quicksort(a array1..n of
  integer, n int)
• A generalization of bounded parametric
  polymorphism
• ?n int, a type. t
• Have to be careful about defining type
  equivalence if n can change...

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Subtype polymorphism

• One type S is a subtype of another type T if all
  the values of S are also values of T
• A value is polymorphic because it is a member of
  all types that are supertypes of its type
• Object-oriented languages support subtype
  polymorphism -- we will discuss in more detail
  later

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First-class polymorphism

• Polymorphic values to be used as first-class
  values impredicative parametric polymorphism
• 2nd-order polymorphic lambda calculus
• Examples GJ, Haskell