1. Modular Type Classes Derek Dreyer Robert Harper Manuel M.T. Chakravarty POPL 2007 Nice, France

2. Specifications Haskell:

3. Specifications ML:

4. Implementations Haskell:

5. Implementations ML:

6. Generic Implementations Haskell:

7. Generic Implementations Haskell: ML:

8. 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

9. 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

10. Classes and Instances in ML Great, but now how do we create the eq function?

11. 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 Haskell’s qualified types:

12. 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!

13. Making an Instance Canonical Designate EqInt and EqProd as canonical in a certain scope:

14. 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

15. Class Hierarchies Haskell subclass declaration: How to support class hierarchies using modules?

16. Class Hierarchies Haskell subclass declaration: How to support class hierarchies using modules? Use module hierarchies!

17. Composite Class Signatures Instances of ORD are automatically computed from instances of EQ and instances of LT

18. Associated Types Proposed by Chakravarty et al. (2005) as an alternative to “functional 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

19. 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

20. Thank You!

21. Extra Slides

22. Associated Types (Chakravarty et al. 05) A function sumColl that sums the elements of a collection would have polymorphic type:

23. 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 is “named instances” (Kahl and Scheffczyk 01)

24. Instance of Ord at Int Haskell:

25. Instance of Ord at Int Haskell: ML:

26. Instance of Ord at £ Haskell:

27. Instance of Ord at £ Haskell: ML:

28. Instance of Ord at £ Haskell: ML:

29. Computing a Composite Instance Assuming LtInt and LtProd have been “used”, the canonical module of signature is

30. Another Example