Supervised by: Dr Sophia Drossopoulou, Dr Nobuko Yoshida [email protected] Wildcards, Variance...

[email protected]://

www.nicholascameron.co.uk

Supervised by: Dr Sophia Drossopoulou,Dr Nobuko Yoshida

Wildcards, Variance and Virtual Classes

Nicholas Cameron

Transfer Talk

Wildcards, Variance and Virtual ClassesNick Cameron, Imperial College London

Halfway Already…o Parametric Polymorphism and Subtype Variance

o Java Wildcardso Generics and wildcardso Programming with wildcardso Formalising wildcards

o Virtual Classeso Virtual Typeso Virtual Classeso Tribeo Parametric types


Parametric Polymorphism

*

* Disclaimer: I don’t really write my shopping list in Excel.


Parametric Polymorphism


Genericsclass 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);


Virtual Typesclass List {

virtual type X;

X get(int i)...

void add(X x)...

}

class DogList extends List {

X = Dog;

}


Invariance

<:

Dog <: Animal


Invariance

List<Dog> <: List<Animal>


Wildcards


?o ? May be used as an actual type

argument

o A list of some type (but we don’t know (or care) which type)

void m(List<?> aList) ...


Boundso Wildcards can have upper and lower

bounds

o ub is a list of some type that is a subtype of Animal

o 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) ...


Subtype Variance

List<Dog> <: List<?>


Subtype Varianceo Covariance

o Contravariance

List<Dog> <: List<? extends Animal>

List<Animal> <: List<? super Dog>


The trade-offvoid 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}


Wildcard Capture

o Note: List<?> List<X>o BUT, we can pass it here

o Capture conversion

<X>void m1(List<X> x) {…}

void m2 (List<?> y){ this.m1(y);}


Formalisation

J


Jo Existential types

List<?> = X.List<X>

List<? super Dog> =

Y[Dog Object].List<Y>


Jo 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);}


J

o Subtyping

List<Dog> <: X.List<X>


Back to wildcard capture


Wildcard Captureo ...is not subtyping:

<X>void m1(List<X> x1, List<X> x2) {…}

void m2 (List<?> x1, List<?> x2) { m1(x1, x2); //Type error!}


Wildcard Captureo …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!}


Wildcard Capture

o xx:Y,Z.Pair<Y, Z>o m2(x): Y.Pair<Y, Y>



void m3 (Pair<?, ?> xx, List<?> x) { m1(xx); //Type error m1(m2(x)); //OK!}


Wildcard Capture

o m2(x): Y.Pair<Y, Y>



void m3 (Y.List<Y> x) { open x as y:List<Y> in this.<Y>m1(this.<Y>m2(y));}


Type Systemo Expressible but not denotable types

o Caused by wildcard captureo Solved by using existential types

o Pack/close vs. subtyping

o Modelling capture

o Soundness requirement


Soundness


Soundnesso Challenges caused by:

o Type parameters in reduction of open expression

o Lower bounds

o Complexity of formalism


Lemmaso uBound*(lBound(X)) <: uBound*(X)

o C<P0…Pn> <: D<Q0…Qn> subclassing

o T <: R T = R’

o Well formed type environments


Wildcards – further worko Finish proofs

o Nearly done!

o Formalise translation

o Imperative version


Questions?

Thank you!

Report: http://www.nicholascameron.co.uk/papers/transfer.pdf


Tribe


Tribeo Nested virtual classes

class Graph{ class Node { } class Edge { Node n1; Node n2; }}

class ColourGraph extends Graph{ class Node { Colour c; }}


Tribeo 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; }}


Tribeo Flexible, powerful, elegant

o Undecidable subtyping?

o Complete subtyping?


Tribe – further worko Parametric polymorphism and variance

o Virtual types?o Extend Tribe with virtual types

o Generics?o Orthogonal?o Type variables in pathso Genericity over families


Tribe – further worko Decidable fragment – T9

o Tribal Ownership

o Programming with virtual classes

Supervised by: Dr Sophia Drossopoulou, Dr Nobuko Yoshida [email protected] Wildcards, Variance...

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