1. Existential Quantification for Variant Ownership Nicholas Cameron Sophia Drossopoulou Imperial College London (Victoria University of Wellington)‏ ncameron@ecs.vuw.ac.nz

2. Ownership

3. Statically partitions the heap Organises objects into an hierarchical structure Improves reasoning about programs  Effects  Concurrency  Memory Management  Garbage Collection Many flavours Encapsulation properties

4. Ownership

6. Parametric ownership Each object is owned by a context  context  object  owner Classes and types are parameterised by contexts

7. Ownership Classes are parameterised by formal contexts class C {...}  o1 is the owner of objects of class C  o2 is another context

8. Ownership Types are parameterised by actual contexts class C {...} C  this C  World, formal contexts C  Final variables, final fields

9. Ownership Ownership hierarchy  o1:C  o2:C  o1 is inside o3  o2 is inside o4  If o4 is inside o3 then o2 is inside o3

10. Existential Types

11. Used to model abstraction  Ǝ X.(X → int)‏ Or variance in generic object-oriented languages  Ǝ X.List  List (Java wildcards)‏

12. Existential Types Bounds  Ǝ X→[Merino Animal].List  Merino <: X <: Animal

13. Existential Types Safe covariance:  Ǝ X→[  Sheep].List  List

14. Existential Types Packing pack( List ) = Ǝ X.List  A is hidden – it is the witness type close expression

15. Existential Types Unpacking unpack( Ǝ X.List ) = List  Z is a fresh type variable open expression

16. Jo Ǝ - Motivation

17. Motivation Desire for variance  +, -, * in Variant Ownership Types Lu, Potter, ECOOP '06  ? in MOJO Cameron, Drossopoulou, Noble, Smith, OOPSLA '07  any / readonly in Universes Müller, Dietl, et al., many papers  any in Effective Ownership Lu, Potter, POPL '06 ...

18. Motivation All tailor-made solutions  No connection to established type theory  Limited expressivity

19. Motivation E.g., ? in MOJO  Strict substitution If ? In....  Strict method and field lookup Permeates entire calculus  Additional subtype rules

20. Motivation

21. Jo Ǝ

22. Goals:  Expressivity  Uniformity

23. Jo Ǝ Existential quantification of contexts Parametrisation by types and contexts  This combination gives improved expressivity

24. Jo Ǝ Class List { List next; X datum; X get() {...} void add(X x) {...} } context parameters type parameters

25. Jo Ǝ Generics and quantification for expressivity:  List >  Ǝ o.List >  List >  Ǝ o.List >

26. Jo Ǝ Flexible ownership language  Parametric methods Type and context parameters  Actual contexts  Explicit unpacking  Upper and lower bounds on formal contexts

27. Jo Ǝ Upper and lower bounds on formal contexts Ǝ o→[o5 o2].Sheep o is inside o2 o5 is inside o

28. Jo Ǝ Uniform  Uses only the purest form of existential types  No special cases to support variance  No tailored annotations  Variance is dealt with locally  Direct relation to long history of research on existential types

29. Jo Ǝ Describes the topology of the heap  This topology is fixed, not variant Type sound  Static description of topology reflects heap at runtime Ownership hierarchy can 'grow', but not 'mutate' Useful  e.g. effects

30. Jo Ǝ Can encode other systems  Generic Universes Dietl, Drossoploulou, Müller, ECOOP '07  Existential owners for downcasting Wrigstad, Clarke, JOT, May '07  Variant Ownership Types? Lu, Potter, ECOOP '06  Single owner case of MOJO Cameron, Drossopoulou, Noble, Smith, OOPSLA '07

31. How it Works

32. Jo Ǝ

33. Three environments Variable environment

34. Jo Ǝ Owner environment Quantified contexts are unpacked here  Does not affect this rule

35. Jo Ǝ Type environment Owner bounds

36. Jo Ǝ Substitute receiver for this

37. Jo Ǝ Receiver cannot be arbitrary expression But no loss of expressivity Thanks to open expression

38. Jo Ǝ Receiver must have class type Forces unpacking Don't account for quantification here

39. Summary Parametric ownership + Existential quantification of contexts + Generics = Jo Ǝ Expressive and Uniform Can encode existing systems

40. Thank you! ncameron@ecs.vuw.ac.nz

41. Ownership Encapsulation  Owners-as-Dominators  Owners-as-Modifiers  Link Soundness ... Owners-as-Dominators  All paths through the heap to an object are dominated by that object's owner  Common for parametric ownership systems  Useful for garbage collection, memory management,...

42. Ownership Owners-as-Dominators

43. Ownership Owners-as-Modifiers

44. Encapsulation

45. Jo Ǝ deep Owners-as-Dominators Strong encapsulation property Very small modifications  Well-formed heap  Well-formed classes  Class type checking

46. Jo Ǝ deep Treat existentially quantified context variables as their lower bounds Owners-as-dominators involves showing referenced objects are outside their owners If the lower bound is outside, then the hidden context must be outside