Modular Type Classes

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Modular Type Classes

• Derek Dreyer
• Robert Harper
• Manuel M.T. Chakravarty
• POPL 2007
• Nice, France

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Specifications

• Haskell

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Specifications

• Haskell

ML
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Implementations

• Haskell

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Implementations

• Haskell

ML
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Generic Implementations

• Haskell

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Generic Implementations

• Haskell

ML
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Type Classes vs. Modules

• Haskell type classes (Wadler and Blott, Jones, et
  al.)
• Elegant account of ad-hoc polymorphism
• May want different instances of integer equality
  to be canonical in different scopes
• Not a substitute for a module system
• ML modules (MacQueen, Harper, Leroy, et al.)
• Explicit, fine-grained control over linking and
  abstraction
• No support for implicit program construction

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Our Proposal

• Start with ML modules
• Classes are just signatures of a special form
• Instances are just structures and functors of a
  special form
• Allow programmers to designate certain instance
  modules as canonical within a particular scope
• No tradeoff necessary Get the best of both
  worlds!
• Have access to full-featured module system by
  default
• Can hook into the type inference engine and make
  use of Haskell-style overloading, if/when you
  want to

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Classes and Instances in ML

• Great, but now how do we create the eq function?

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Creating an Overloaded Function

• We employ an overload mechanism
• This creates a polymorphic value eq,
  represented internally (in the semantics) as an
  implicit functor
• Analogous to Haskells qualified types

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Using eq

• If we apply eq to some arguments of type int
  int
• This initiates a demand for a canonical module of
  signature
• But none exists ) Type error!

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Making an Instance Canonical

• Designate EqInt and EqProd as canonical in a
  certain scope

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Making an Instance Canonical

• Now if we apply eq in that scope
• Then the above code typechecks and translates
  internally to
• Similar to evidence translation in Haskell
• Here we use modules as evidence

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Class Hierarchies

• Haskell subclass declaration

How to support class hierarchies using modules?
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Class Hierarchies

• Haskell subclass declaration

• How to support class hierarchies using modules?
• Use module hierarchies!

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Composite Class Signatures

• Instances of ORD are automatically computed
  frominstances of EQ and instances of LT

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Associated Types

• Proposed by Chakravarty et al. (2005) as an
  alternative tofunctional dependencies
• Basic idea Add type components to classes
• Falls out naturally from modular framework
• Associated types are just type components of
  class signatures other than the one called t
• In Haskell, requires the introduction of
  equality constraints
• In ML, these are just where type constraints

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What Else is in the Paper

• Other design points
• Coercions between implicit and explicit functors
• Coherence in the presence of explicitly-scoped
  instances
• Handling of overlapping instances
• Type-theoretic formalization (details in the tech
  report)
• Declarative elaboration semantics (a la
  Harper-Stone)
• Type inference algorithm
• Algorithm is sound, but not complete(due to
  problems inherited from both Haskell and ML)
• Related work

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Thank You!
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Extra Slides
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Associated Types(Chakravarty et al. 05)

• A function sumColl that sums the elements of a
  collectionwould have polymorphic type

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Haskell Approach

• Beef up type classes
• Multi-parameter classes (Jones, Peyton Jones, et
  al.)
• Constructor classes (Jones 95)
• Functional dependencies (Jones 00)
• Associated types (Chakravarty et al. 05)
• and more
• Each extension brings classes closer to
  modules,but still no explicit control over
  linking
• Only attempt at explicit control isnamed
  instances (Kahl and Scheffczyk 01)

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Instance of Ord at Int

• Haskell

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Instance of Ord at Int

• Haskell

ML
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Instance of Ord at

• Haskell

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Instance of Ord at

• Haskell

ML
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Instance of Ord at

• Haskell

ML
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Computing a Composite Instance

• Assuming LtInt and LtProd have been used, the
  canonical module of signature
• is

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Another Example