Parametric Polymorphism

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Parametric Polymorphism

• COS 441
• Princeton University
• Fall 2004

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Polymorphic Polymorphism

• Polymorphism has different meanings in different
  contexts.
• Cardelli and Wegner (1985) decompose it as
  follows

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Varieties of Polymorphism

• Universal polymorphism is obtained when a
  function works uniformly on a range of types
• Ad-hoc polymorphism is obtained when a function
  works, or appears to work, on several different
  types and may behave in unrelated ways for each
  type.
• From On Understanding Types, Data Abstraction,
  and Polymorphism, Luca Cardelli and Peter Wegner
  Computing Surveys, (December, 1985)

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Forms of Universal Polymorphism

• parametric polymorphism (generics)
• ML-style, based on type parameters
• Focus of this lecture
• inclusion polymorphism (subtyping)
• OO-style based on subtype inclusion
• Will study this Friday

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Forms of Parametric Polymorphism

• Harper distinguishes between first-class and
  second-class forms of parametric polymorphism
• ML supports only second-class parametric
  polymorphism because of limitations in
  type-inference
• Supporting first-class versions is easy if you
  dont care about type-inference

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First vs Second Class

• First-class
• ? int ?1 ! ?2 t 8t(?)
• Example 8t(t ! t) ! 8t(t ! t)
• Second-class
• ? ? 8t(?)
• ? int ?1 ! ?2 t
• Example 8t(8t((t ! t) ! (t ! t)))

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Second Class Polymorphism

• Second-class or prenex polymorphism the 8 are
  all at the front of the type
• ? ? 8t(?) (polytypes)
• ? int ?1 ! ?2 t (monotypes)
• Polytypes ?, 8t(?), 8 t(8 t(?)), int
• Monotypes t, int, t ! int,

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PolyMinML

• Supports second-class parametric polymorphism
• Only instantiate polytypes with monotypes

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Dynamic Semantics
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Static Semantics

• Must modify judgment forms to account for free
  type variables occurring in types during type
  checking

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Static Semantics

• Must modify judgment forms to account for free
  type variables occurring in types during type
  checking

Typo should be ?
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Well-formed Type
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Well-Formed Expressions

• Remaining rules maintain/assume the following
  invariant

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Example Polymorphic Compose
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Type Soundness

• Similar to Substitution Lemma
• Proofs are all similar to previous proofs just
  with more tedium!

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First-Class Polymorphism

• Easy to obtain just relax distinction between
  polytypes and monotypes
• Zero changes to the proof or operational
  semantics
• Not supported for many reasons
• Interacts badly with type-inference
• Does not allow type-erasure in opsem

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Effects/Erasure and Polymorphism

• The expressions Fun and inst similar to roll and
  unroll
• No real need on a executing machine
• Just to make type checking and type-soundness
  easier
• In second-order system with value restriction can
  be erased from program
• ML compilers use type-erasure during compilation

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Value Restriction

• Consider v ?
• v must be of the form
• Fun t1 in Fun t2 in Fun tn in e
• Value restriction only allows type-abstraction
  over values so with the value restriction we know
  that e is also a value
• If it is a value than it is safe to remove all
  the Fun

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Example Value Restriction

• Consider program below
• the program is sound because there is a unique
  reference created with each instantiation of r

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Example Value Restriction

• Consider erased version below
• the program is unsound because there is just one
  reference created with each instantiation of r.
• SML rejects the above

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Polymorphism and Abstraction

• Parametric polymorphism can be used to enforce
  data abstraction
• Parametricity theorem lets us infer information
  about a value just by inspecting its type
• Similar to a canonical forms lemmas
• Gives us a semantic description of a value based
  on its type

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Example 8t(t ! t)

• Consider function f the type 8t(t ! t)
• assuming we only care about interesting
  functions (those that terminate) how can
  functions of that type behave?
• All functions of that type must simply return
  their argument. i.e. they all behave as the
  identity function

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Semantics vs. Syntax

• Note parametricity theorem states any function
  must act like the identity function it may
  look different
• (fn x gt x)
• (fn x gt (((fn f gt f) (fn y gt y)) x))
• (fn x gt (12x))
• Compare to canonical forms lemmas which states
  values must be a specific piece of syntax

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Other Theorems

• 8t(t)
• No interesting values of this type
• 8t(t list ! t list)
• Interesting functions only examine the spine
  of the list

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A Simple Abstraction Theorem

• Consider N 8t(t ! (t ! t) ! t)
• Interesting functions of this form can only apply
  their first and second arguments.
• Fun t in fn zt is fn st! t is z
• Fun t in fn zt is fn st! t is s(z)
• Fun t in fn zt is fn st! t is s(s(z))

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Formal Definition of Parametricity

• Define a notion of equivalence of expressions
  indexed by type

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Definition of Parametricity (cont.)

• Define a notion of equivalence of closed values
  indexed by type

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Definition of Parametricity (cont.)

• Interesting case is definition of all

Almost arbitrary relation
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Parametricity Theorems
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Summary

• Many different forms of polymorphism
• Parametric polymorphism comes in both first-class
  and second-class forms
• Parametricity theorems basis for formal claims of
  data-abstraction