1. Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor John Altidor Taming the Wildcards: Combining Definition- and Use-Site Variance

2. 2 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Outline  Motivation for Variance.  Existing Approaches. Java only supports use-site variance. C# and Scala only support definition-site variance.  What Is New Here: Combine Definition-Site and Use-Site Variance. Both in a single language, each using the other Case Study – Inferring Def-Site for Java. Insights into Formal Reasoning.  Summary.

3. 3 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Software Reusability  Writing reusable software is hard; many factors: Generality Adaptability Modularity Simplicity of interface / Ease of use …  Programming language support Enable reusable code without introducing bugs. Focus on integrating two flavors of genericity: Generics – Parametric Polymorphism Subtyping – Inclusion Polymorphism

4. 4 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Subtyping – Inclusion Polymorphism  Example: Java inheritance. class Animal { void speak() { } } class Dog extends Animal { void speak() { print(“bark”); } } class Cat extends Animal { void speak() { print(“meow”); } }

5. 5 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Subtyping – Example Client void performSpeak(Animal animal) { animal.speak(); } Can be a Dog, Cat, or any other Animal.

6. 6 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Common Code Structure class ListOfDogs { void add(Dog num) { … } Dog get(int i) { … } int size() { … } }

7. 7 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Common Code Structure (cont.) class ListOfAnimals { void add(Animal num) { … } Animal get(int i) { … } int size() { … } }

8. 8 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Generics – Parametric Polymorphism class List { void add(X x) { … } X get(int i) { … } int size() { … } } type parameter write X read X no X List ≡ List of Animal s List ≡ List of Dog s Customized Lists

9. 9 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Generics and Subtyping  Dog <: Animal (Dog is an Animal).  Cat <: Animal (Cat is an Animal).  List No!  A List can add a Cat to itself. A List cannot.

10. 10 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Variance Introduction  When is C a subtype of C ? class RList { X get(int i) { … } int size() { … } } Can read from but not write to.  It is safe to assume RList.  Why?

11. 11 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Flavors of Variance - Covariance Generic Covariance <:  Assuming Dog <: Animal (Dog is an Animal).

12. 12 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Flavors of Variance - Contravariance Generic Contravariance <:  Assuming Dog <: Animal (Dog is an Animal).

13. 13 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Four Flavors of Variance  How do programmers specify variance? Covariance:T <: U ⇒ C. Contravariance:T <: U ⇒ C. Bivariance: C,for all T and U. Invariance: C,if T <: U and U <: T.

14. 14 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Definition-Site Variance (C#/ Scala)  Programmer specifies variance in definition as in Scala and C#.  Variance of a type position. Return types: covariant. Arguments types: contravariant. class RList { X get(int i) { … } int size() { … } // no method to add } class WList { void add(X x) { … } int size() { … } // no method to get }

15. 15 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site Variance (Java Wildcards) class List { void add(X x) { … } X get(int i) { … } int size() { … } }

16. 16 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site Variance (Java Wildcards) class List { void add(X x) { … } X get(int i) { … } int size() { … } } List

17. 17 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site Variance (Java Wildcards) class List { void add(X x) { … } X get(int i) { … } int size() { … } } List

18. 18 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site Variance (Java Wildcards) class List { void add(X x) { … } X get(int i) { … } int size() { … } } List

19. 19 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Definition-Site: Pros  Conceptual Simplicity Simpler Type Expressions – no use-site annotations. Subtype policy stated in class definition.  Burden on library designers; not on users. Classes declare variance once and for all uses. class RList {... } class WList {... } class List extends RList, WList {... } covariant contravariant invariant

20. 20 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Definition-Site: Cons class RList {... } class WList {... } class List extends RList, WList {... } covariant contravariant invariant  Redundant Types scala.collection.immutable.Map scala.collection.mutable.Map Generic with n parameters ⇒ 3 n interfaces (or 4 n if bivariance is allowed)

21. 21 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site: Pros  Flexibility: co-, contra-, bivariant versions on the fly.  Easier on library designers Design classes in natural way No need for fractured classes. class List { void add(X x) { … } X get(int i) { … } int size() { … } }

22. 22 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Use-Site Cons:  Burden shifts to users of generics. Library designers are often users as well.  Type signatures quickly become complicated.  Heavy variance annotation required for subtyping.  From Apache Commons-Collections Library: Iterator > createEntrySetIterator( Iterator >)

23. 23 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Wildcards Criticism Iterator > createEntrySetIterator( Iterator >)  “We simply cannot afford another wildcards” – Joshua Bloch.  “Simplifying Java Generics by Eliminating Wildcards” – Howard Lovatt.

24. 24 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Our Approach: Take Best of Both Worlds  Take advantages. Remove disadvantages. Simpler type expressions than in Java (burden off clients). Less redundant type definitions in C# and Scala.  Inferring definition-site variance from only use-site annotations. [1] Added notion of def-site variance without extending Java. VarLang Calculus: Denotational and language neutral approach. [1] PLDI 2011 (Altidor, Huang, Smaragdakis)

25. 25 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Fewer Wildcard Annotations Iterator > createEntrySetIterator( Iterator >) Iterator > createEntrySetIterator( Iterator >)

26. 26 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor VarLang Calculus  Language neutral approach.  Simplifies and generalizes all previous work.  Sample Applications : Infer definition-site variance for Java. Add use-site variance to C#.

27. 27 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Variance Lattice  Ordering details later

28. 28 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Java to VarLang Calculus class C { X foo(C arg1) {... } void bar(D arg2) {... } } class D { void baz(C arg3) {... } } module C { X+, C -, void+, D - } module D { void+, C - }

29. 29 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Infer Definition-Site Variance  Def-Site Variance of C = c = +. (C is covariant)  Def-Site Variance of D = –. (D is contravariant) module C { X+, C -, void+, D - } module D { void+, C - }

30. 30 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Case Study: Definition-Site Inference for Java  How much benefit if Java’s type system inferred definition-site variance?  Mapped Java classes to VarLang modules.  Applied inference to large, standard libraries e.g., Sun’s JDK 1.6  Example inferences java.util.Iterator is covariant. java.util.Comparator is contravariant.

31. 31 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Sample Results from Case Study  Analysis was conservative (e.g. ignored method bodies). “ foo(List arg) ” could have been “ foo(List arg) ”. majority

32. 32 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Insights into Formal Reasoning  Variance composition: v 1 ⊗ v 2 = v 3  Variance binary predicate: v( T; T’ )  Variance lattice: v 1 ≤ v 2  Variance of a type: var( X ; T )  Relating variance to subtyping: Subtype Lifting Lemma  Variance of a position

33. 33 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Standard Modeling: Variance Lattice  Ordered by subtype constraint binary predicate

34. 34 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Variance of a Type  When is C a subtype of C ?  What about existential types? ∃ X->[ ⊥ -String].Stack  We answer a more general question: When is [U/X]T <: [U’/X]T ?  Key: defined very general predicate: var( X ; T ) = variance of type T with respect to type variable X.

35. 35 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Subtype Lifting Lemma If: (a) v ≤ var( X ; T ) (b) v( U; U’ ) Then: [U/X]T <: [U’/X]T  var( X; Iterator ) = + and + (Dog; Animal) ≡ Dog  We generalize Emir et al.’s subtype lifting lemma.  Goal property of var.

36. 36 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Variance Composition  Variance of variable X in type A >> ?  In general, variance of variable X in type C ?  v 1 ⊗ v 2 = v 3. If: Variance of variable X in type expression E is v 2. The def-site variance of class C is v 1. Then: variance of X in C is v 3. Transform Operator

37. 37 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Deriving Transform Operator Example Case: + ⊗ – = –  Class C is covariant.  Type E is contravariant in X.  Need to show C is contravariant in X.  For any T 1, T 2 :  Hence, C is contravariant in X.

38. 38 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Summary of Transform  Invariance transforms everything into invariance.  Bivariance transforms everything into bivariance.  Covariance preserves a variance.  Contravariance reverses a variance.

39. 39 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Definition of var predicate  Definition-site variance annotations are type checked using var predicate.  For further details: [2] ECOOP 2012 (Altidor, Reichenbach, Smaragdakis)

40. 40 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor  Variance composition: v 1 ⊗ v 2 = v 3  Variance binary predicate: v( T; T’ )  Variance lattice: v 1 ≤ v 2  Variance of a type: var( X ; T )  Relating variance to subtyping: Subtype Lifting Lemma  Variance of a position: See [2] for derivation. General Theory – Template for Adding Variance A >> Subtype Policy Comparing Variances Computing Variances Variance Soundness Check Def-Site Variance

41. 41 Taming the Wildcards: Combining Definition- and Use-Site Variance – Altidor Summary of Contributions  Generics and subtyping coexist fruitfully. Subtyping between two different instantiations.  Combine def-site and use-site variance to reap their advantages and remove disadvantages.  Generalize all previous related work.  Resolve central questions in the design of any language involving parametric polymorphism and subtyping. Variance of a type. Variance of a position.

42.  To my collaborators: Smaragdakis, Reichenbach, Huang, Palsberg, Tate, Cameron, Kennedy, Urban  To CLC for hosting.