Type Reconstruction and Universal Types

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Type Reconstructionand Universal Types

• Olesya Peshko
• CAS 706
• McMaster University
• March 31, 2004

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Type Variables and Substitutions

• Type variable (TVar) uninterpreted base type,
  placeholder for some types we do not know
• Substituting types for TVars
• Type substitution describe a mapping from
  TVars to types
• Apply to type to obtain an instance
• Substitution a finite mapping from TVars to
  types
• Example
• set of TVars at left-hand sides
  of pairs in
• set of TVars at right hand
  sides

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Application of Substitution
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Constraint-Based Typing

• Constraint set is a set of equations
• equations between type expressions
• Substitution unifies an equation
• if and are identical
• unifies if it unifies every equation in
  
• To check if is typable under context
• Analyze the term and collect the constraints
• Look for substitutions that satisfy

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Constraint Typing Rules
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Unification

• Substitution is less specific (more general)
  than a substitution , written , if
  
• for some substitution
• A principal unifier (most general unifier) for a
  constraint set is a substitution that
  satisfies and such that for
  every substitution satisfying

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Unification II

• Theorem The algorithm always
  terminates, failing when given a non-unifiable
  constraint set as input and otherwise returning a
  principal unifier
• halts, either by failing or by
  returning a substitution, for all
• If , then is a
  unifier for
• If is a unifier for , then
  
• with

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Unification Algorithm
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Principal Types

• Principal solution for is a
  solution
• such that, whenever is
  also a solution for , we have
• If is a principal solution, is a
  principal type of under
• Theorem If has any solution,
  then it has a principal one
• Corollary It is decidable whether has
  a solution

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Universal Types

• Functions that are applicable to a different type
  of arguments but share the same behavior
• Abstract out a type from a term and later
  instantiating this abstract term with concrete
  type annotations

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Varieties of Polymorphism

• Parametric single piece of code is typed
  generically, variables used in place of types,
  particular types instantiated as needed
• Impredicative (first-class)
• ML-style (Let-polymorphism)
• Ad-hoc polymorphic value exhibits different
  behaviors when viewed at different types
• Overloading
• Intensional permits restricted computations
  over types at run time
• Sybtype gives a single term many types using
  the rule of subsumption (you can selectively
  forget information about terms behavior)

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System F

• System F is a straightforward extension of the
  simply typed lambda-calculus
• Introduce new forms of abstraction and
  application
• Type abstraction
• Type application (or instantiation)
• Reduction rule
• Example polymorphic identity function

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System F. Syntax
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System F. Evaluation
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System F. Typing
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Basic Properties

• Theorem
• If and , then
• Theorem
• if is a closed, well-typed term, then
  either
• is a value or else there is some with
• Property of normalization the fact that the
  evaluation of every well-typed program terminates

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Erasure

• Type erasure function maps System F terms to
  untyped lambda-terms

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Typability and Type Reconstruction

• Term in the untyped lambda-calculus is
  typable if there is some well-typed term
• such that
• Theorem It is undecidable whether, given a
  closed term of the untyped lambda-calculus,
  there is some well-typed term in System F
  such that
• Not only full type reconstruction but also
  various forms of partial type reconstruction are
  known to be undecidable for System F

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Erasure and Evaluation Order

• Many polymorphic languages adopt a type-erasure
  semantics after the typechecking phase all the
  types are erased and the resulting untyped terms
  are interpreted or compiled
• If the language includes side-effecting features
  such as mutable reference cells or exceptions,
  the type-erasure function is defined differently
• Example exception-raising primitive
• Term
  evaluates to 0
• ( is syntactic value, body is
  never evaluated)
• Its erasure raises
  an exception when evaluated

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Erasure and Evaluation Order II

• New form of erasure erase a type abstraction to
  a term-abstraction
• where is some arbitrary
  untyped value

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Fragments of System F

• Fragments with some possibilities of type
  reconstruction are used for language design
• Let-polymorphism of ML (prenex polymorphism)
  type variables range only over quantifier-free
  types, quantified types are not allowed to appear
  on the left-hand sides of arrows.
• Rank-2 polymorphism no pass from root of the
  type to a quantifier passes to the left of 2
  or more arrows, when the type is drawn as a tree
• Rank-2
• Not of rank-2

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Reference

• B.C.Pierce, Types and Programming Languages, The
  MIT Press, Cambridge (Massachusetts), London
  (England), 2002