CSE321 Programming Languages Letpolymorphism

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CSE-321 Programming LanguagesLet-polymorphism
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• POSTECH
• May 28, 2007

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System F
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Typing Rules
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Identity Function
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Pairing Function
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Types in P.P. ?-Calculus
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Predicative Polymorphic ?-Calculus
A variable is assigned a monotype.
A type application uses a monotype.
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Useless Polytypes

• Let's apply id to two different types bool and
  int.
• Polytypes can be used only monomorphically.

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Let-polymorphism

• System F
• powerful, but undecidable type reconstruction
• Predicative polymorphic ?-calculus
• decidable type reconstruction, but useless

System F
let-polymorphism
P.P. ?-calculus
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Outline

• System F V
• Predicative polymorphic ?-calculus V
• Let-polymorphism
• Implicit polymorphism
• Value restriction

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Let-binding

• let x e in e'
• binds x to a polymorphic expression e.
• allows multiple occurrences of x in e'.

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Let-polymorphism
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Reduction of let-bindings
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Why not?
System F
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Outline

• System F V
• Predicative polymorphic ?-calculus V
• Let-polymorphism V
• Implicit polymorphism
• Value restriction

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Implicit Polymorphism

• No type abstraction, no type application
• No type annotation in ?-abstraction
• An expression may have multiple types
• ?x. x
• bool ! bool, int ! int, ..., 8?.? ! ?

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Type System
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8I and 8E vs. Gen and Spec
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Gen Generalizing Types

• Used to assign polytypes to expressions

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Spec Specializing Types

• Typically used to specialize types of let-bound
  variables

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Outline

• System F V
• Predicative polymorphic ?-calculus V
• Let-polymorphism V
• Implicit polymorphism V
• Value restriction

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Prototypical Example
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Value Restriction

• Expression in the rule Gen must be a syntactic
  value.
• If not, it must be used monomorphically.

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SML Example

• - let val id (fn y gt y) (fn z gt z) in id true
  end
• val it true bool
• - let val id (fn y gt y) (fn z gt z) in (id
  true, id 1) end
• stdIn2.40-2.56 Error operator and operand don't
  agree literal
• operator domain bool
• operand int
• in expression
• id 1
• - let val id (fn y gt y) (fn z gt z) in id end
• stdIn1.1-1.46 Warning type vars not generalized
  because of
• value restriction are instantiated to dummy
  types (X1,X2,...)
• val it fn ?.X1 -gt ?.X1