Intensional Polymorphism in Type-Erasure Semantics

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Intensional Polymorphism in Type-Erasure Semantics

• Karl Crary, Stephanie Weirich, Greg Morrisett

Presentation by Nate Waisbrot
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Authors' background
Part of Cornell University's TAL (Typed Assembly
Language) group Creating intermediate languages
to compile into TAL
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Three styles of polymorphism

• Get a value and inspect its type
• Easy and efficient, but not powerful
• Pass types as first-class objects
• Powerful, but slow and complicated for the
  compiler
• Pass values which represent types
• Best of both worlds

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Passing values

• Recall the paper "Dynamically Typing in a
  Statically Typed Language" (Abadi, Cardelli,
  Pierce, and Plotkin)
• Types are tied to values -- you can't have one
  without the other

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Dynamic types
Dynamic(5 ? int) dynamic value f dynamic ?
dynamic dynamic function ?xdynamic. typecase x
of int ? ??? ? ? ? ? ?
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Passing types

• Values are described by types
• Types are described by kinds
• We could keep going (System F?)

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Overview of System F
??.?x?.x The polymorphic identity
function ??.??? The type of the identity
function iden int 5 Application of the identity
function
??.??.?x?.?y?. (x ? y) a pair function ??.??.
(? ? ?) its type pair int string 0
zero application
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Overview of ?iML
Start with System F, add the ability to work with
types alone ??.??. ? ? ? the pair-type
function Type?Type?Type the kind of the pair-type
function pairT int string this gives us the
type (int ? string) Now add the typecase
construct typecase ?.int ? substitute ? for ?
and return an int int ? 0
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?iML syntax
? Type ??? c ? int c?c c ? c
???.c c c Typerec c(cint, c?, c?) ? c
??? ? ? ? ???.? e x i ?x?.e fix
f?.v e e lte,egt ? e ???.v ec
typecase ?.? v i ?x?.e fix f?.v
ltv,vgt ???.v
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Typing hierarchy
Type?Type
Type
kinds
int
int?int
??.?
types
?xint.x
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values
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What's wrong with type passing?

• Types must be kept separate from values
• Doubles the type-checker's work
• Compiling it down to TAL is hard
• Language semantics require types to always be
  passed

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Solution type representations

• Make new type, "representation"
• Get back all the simplicity of normal
  value-passing
• As a bonus, gain some abstraction

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Overview of type representations
R?(Rint,Rint) The representation of int?int R(
R(int)?R(int) ) Its type
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Syntax of ?R
? Type ??? c ? int c?c c ? c
???.c c c R(c) Typerec c(cint, c?, c?,
cR) ? c ??? ? ? ? ???.? ???.? e
x i ?x?.e fix f?.v e e lte,egt ?
e ???.v ec pack e as ??.? hiding ?
unpack lt?,xgt e in e Rint R?(e,e)
R?(e,e) RR(e) typecase ?.c v i
?x?.e fix f?.v ltv,vgt ???.v pack v as
??.? hiding ? Rint R?(v,v) R?(v,v) RR(v)
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TypeToString dynamic types
typeToString dynamic?string ?xdynamic. typeca
se x of int ? "int" string ? "string" ??? ?
"function" ?? ? "lt" typeToString Dynamic(? ?
?1x) "," typeToString Dynamic(? ? ?2x)
"gt"
?
?
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TypeToString type-passing
fix typeToString ??Type.string
. ??Type. typecase ?.string ? of int ?
"int" string ? "string" ??? ? typeToString
? "?" typeToString ? ?? ? "lt"
typeToString ? "," typeToString ?
"gt"
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TypeToString type representations
fix typeToString ??Type.R(?)?string
. ??Type.?xR(?). typecase ?.string x
of Rint ? "int" Rstring ? "string" R?(x,y)
as ??? ? typeToString ? x "?"
typeToString ? y R(x,y) as ?? ? "lt"
typeToString ? x "," typeToString ?
y "gt"
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Type erasure

• A well-typed program in typed ?-calculus has an
  equivalent in untyped ?-calculus
• Typed ?x?.x
• Untyped ?x.x

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TypeToString with types erased
fix typeToString ??Type.R(?)?string
. ??Type.?xR(?). typecase ?.string x
of Rint ? "int" Rstring ? "string" R?(x,y)
as ??? ? typeToString ? x "?"
typeToString ? y R(x,y) as ?? ? "lt"
typeToString ? x "," typeToString ?
y "gt"
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Type refinement

• Do dead-code elimination based on the type of the
  representation
• Propagate information about the type of an
  argument back through the function

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Monomorphic closure
?x?.y A function with a free variable ??? Its
type (if y has type ?) We want to eliminate
free variables f (?(x ? e)? ? env.?ye) ?
ltygt The closed function ?env.((? ? env) ? ?) ?
env Its type
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Polymorphic closure
The same closed function, with type
passing ??Kind. ?envType. ???. ????. ((?
? env) ? ?) ? env
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Closure with representations
f ????x(? ? R(?)). ?2 x f (? ? R(?)) ?
? f pack (f ? R?) as ?env.((? ? env) ? ?) ?
env hiding R(?) f ?env.((? ? env) ? ?) ? env
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Summary

• Create a value to represent a type
• Pass the value, not the type
• Passing values is easy
• Knowing types at runtime is useful