Wildcards, Variance and Virtual Classes

Wildcards, Variance and Virtual Classes Transfer Talk Nicholas Cameron ncameron@doc.ic.ac.uk http://www.nicholascameron.co.uk

Halfway Already… • Parametric Polymorphism and Subtype Variance • Java Wildcards • Generics and wildcards • Programming with wildcards • Formalising wildcards • Virtual Classes • Virtual Types • Virtual Classes • Tribe • Parametric types Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Parametric Polymorphism * * Disclaimer: I don’t really write my shopping list in Excel. Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Parametric Polymorphism Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Generics class List<X> { X get(int i)... void add(X x)... } List<Dog> l = new List<Dog>(); l.add(new Dog()); Dog d = l.get(0); Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Virtual Types class List { virtual type X; X get(int i)... void add(X x)... } class DogList extends List { X = Dog; } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Invariance <: Dog <: Animal Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Invariance List<Dog> <: List<Animal> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcards Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

? • ? May be used as an actual type argument • A list of some type (but we don’t know (or care) which type) void m(List<?> aList) ... Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Bounds • Wildcards can have upper and lower bounds • ub is a list of some type that is a subtype of Animal • lb is a list of some type that is a supertype of Dog void m(List<? extends Animal> ub) ... void m(List<? super Dog> lb) ... Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Subtype Variance List<Dog> <: List<?> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Subtype Variance • Covariance • Contravariance List<Dog> <: List<? extends Animal> List<Animal> <: List<? super Dog> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

The trade-off void m1(List<? extends Dog> l) { l.add(new Dog()); //type error Dog d = l.get(0); //ok } void m2(List<? super Dog> l) { l.add(new Dog()); //ok Dog d = l.get(0); //type error } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcard Capture <X>void m1(List<X> x) {…} void m2 (List<?> y) { this.m1(y); } • Note: List<?>List<X> • BUT, we can pass it here • Capture conversion Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Formalisation J Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

J • Existential types List<?> = X.List<X> List<? super Dog> = Y[Dog Object].List<Y> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

J • Capture = open <X>void m1(List<X> x) {…} void m2 (Y.List<Y> y) { open y as z:List<Y> in this.<Y>m1(z); } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

J • Subtyping List<Dog> <: X.List<X> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Back to wildcard capture Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcard Capture • ...is not subtyping: <X>void m1(List<X> x1, List<X> x2) {…} void m2 (List<?> x1, List<?> x2) { m1(x1, x2); //Type error! } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcard Capture • …requires existential types: <X>void m1(Pair<X, X> x) {…} <X>Pair<X, X> m2(List<X> x) {…} void m3 (Pair<?, ?> xx, List<?> x) { m1(xx); //Type error m1(m2(x)); //OK! } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcard Capture <X>void m1(Pair<X, X> x) {…} <X>Pair<X, X> m2(List<X> x) {…} void m3 (Pair<?, ?> xx, List<?> x) { m1(xx); //Type error m1(m2(x)); //OK! } • xx:Y,Z.Pair<Y, Z> • m2(x):Y.Pair<Y, Y> Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcard Capture • m2(x):Y.Pair<Y, Y> <X>void m1(Pair<X, X> x) {…} <X>Pair<X, X> m2(List<X> x) {…} void m3 (Y.List<Y> x) { open x as y:List<Y> in this.<Y>m1(this.<Y>m2(y)); } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Type System • Expressible but not denotable types • Caused by wildcard capture • Solved by using existential types • Pack/close vs. subtyping • Modelling capture • Soundness requirement Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Soundness Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Soundness • Challenges caused by: • Type parameters in reduction of open expression • Lower bounds • Complexity of formalism Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Lemmas • uBound*(lBound(X)) <:uBound*(X) • C<P0…Pn> <: D<Q0…Qn> subclassing • T <: R T = R’ • Well formed type environments Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcards – further work • Finish proofs • Nearly done! • Formalise translation • Imperative version Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Questions? Thank you! Report: http://www.nicholascameron.co.uk/papers/transfer.pdf Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe • Nested virtual classes class Graph { class Node { } class Edge { Node n1; Node n2; } } class ColourGraph extends Graph { class Node { Colour c; } } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe • Path dependent types this.out.Node n1.out.Node this.out.n1.out.Edge .Graph.Node class Graph { class Node { } class Edge { Node n1; Node n2; } } Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe • Flexible, powerful, elegant • Undecidable subtyping? • Complete subtyping? Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe – further work • Parametric polymorphism and variance • Virtual types? • Extend Tribe with virtual types • Generics? • Orthogonal? • Type variables in paths • Genericity over families Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Tribe – further work • Decidable fragment – T9 • Tribal Ownership • Programming with virtual classes Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Wildcards, Variance and Virtual Classes