[email protected] 1 a model for java with wildcards nicholas cameron sophia drossopoulou erik...

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A Model for Java with Wildcards

Nicholas CameronSophia Drossopoulou

Erik Ernst

[email protected] 2

• Background• Wildcards• Previous attempts• Existential types• Subtyping• Typing

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Background

• Java 5.0• 2004

• Adding Wildcards to the Java LanguageTorgersen, Ernst, Plesner Hansen, von der Ahé, Bracha, Gafter. SAC 04.

• Existential types to model wildcards

[email protected] 4

• Wild FJTorgersen, Ernst, Plesner Hansen. FOOL 05.

• ‘On the fly’ existential types• No soundness proof

• JCameron, Ernst, Drossopoulou. FtfJP 07.

• Explicit existential types• Explicit open/close expressions• Partial model

• Variant Parametric TypesIgarashi, Viroli. ECOOP 02, TOPLAS 06.

• Partial model• No capture conversion

[email protected] 5

• On Decidability of Nominal Subtyping with Variance Kennedy, Pierce. FOOL 07.

• Variance and Generalized Constraints for C# Generics

Emir, Kennedy, Russo, Yu. ECOOP 06

• Typeless Programming in Java 5.0Plümicke, Bäuerle. PPPJ 06

• A Flow-Based Approach for Variant Parametric Types

Chin, Craciun, Khoo, Popeea. OOPSLA 06.

• Subtyping Existential TypesWeir, Theimann. FTfJP 08.

[email protected] 6

[email protected]

Wildcards

A very brief description

Java Type

[email protected] 8

Java Generics Type

Cup<Tea>[email protected] 9

Generics - Invariant Subtyping

Cup<Tea> <: Cup<Object>

[email protected] 10

/

Java Wildcards Type

Cup<?>[email protected] 11

Bounds

Cup<? extends Drink>


Generics - Invariant Subtyping



/

Wildcards - Variant Subtyping


Cup<Tea> <: Cup<?>

Cup<Tea> <:Cup<? extends Drink>

Cup<? extends Tea> <:Cup<? extends Drink>


/

Wildcard Capture

void <X> test(Cup<X> cx) {...}

void m(Cup<?> c) {

this.test(c);

}


Wildcard Capture

void <X> test(Cup<X> cx) {...}

void m(Cup<?> c) {

this.test(c);

//this.<Z>test(c);

//Z fresh

}


Our Previous Attempts

Naïve Approach

Cup<?>


Naïve Approach

Cup<?>


Expressible but not

denotable types

'On the fly' Existential Types

capture(Cup<?>) = X.Cup<X>


'On the fly' Existential Types

capture(Cup<?>) = X.Cup<X>

Too complicated


Explicit Existential Types

open... close...


Explicit Existential Types

open... close...

Lower Bounds

Expressivity


Scope Violationvoid m(X.Cup<X> x) { this.<X>m2(x);

}


Scope Violationvoid m(X.Cup<X> x) { this.<X>m2(x);

}

Alpha Renaming


...


[email protected]

Modelling Concepts

Tame FJ

• Explicit existential typesImplicit packing (in subtyping (XS-ENV))

Implicit unpacking (in type rules)

• Inferred type parametersthis.<>m(x);

• Separation of subtypingSubclassing, extended subclassing, subtyping

• Guarding environment... e : T |

Tracks unpacked type variables



Existential Types

Cup<?>

X.Cup<X>


Cup<? extends Drink>

XDrink.Cup<X>


But Why?

But Why?

Capture conversion

Expressible but not denotable types

<X> Pair<X,X> make(List<X> x) {...}

<X> void test (Pair<X, X> x) {...}

void m(List<?> l, Pair<?, ?> p) {

test(p);

}


X.Pair<X,X>

Type is expressible but not denotable




void m(X.List<X> l, X,Y.Pair<X, Y> p) {

test(p);

test(make(l));

X.Pair<X, X> q = make(l);}





test(p);

test(make(l));




Subtyping

Wildcards Subtyping

Cup<Tea> <: Cup<?>




Wildcards Subtyping

Cup<Tea> <: Cup<?>

.Cup<Tea> <: X.Cup<X>


.Cup<Tea> <: XDrink.Cup<X>


YTea.Cup<Y> <: XDrink.Cup<X>


,’ T <: B ’.[T/X]N <:XB.N

(S-ENV)


┴

┴

Cup<Tea> <: Cup<?>

.Cup<Tea> <: X.Cup<X>

Tea <: Object

.[Tea/X]Cup<X> <:X.Cup<X>(S-ENV)


┴

┴


.Cup<Tea> <: XDrink.Cup<X>

Tea <: Drink

.[Tea/X]Cup<X> <:XDrink.Cup<X>(S-ENV)


┴

┴


YTea.Cup<Y> <: XDrink.Cup<X>

YTea Y <:Tea <: Drink

YTea.[Y/X]Cup<X> <:XDrink.Cup<X>(S-ENV)


┴

┴

3 flavours of subtyping?

Java 4 Subtyping


Object

CoffeeTea

DrinkList

Vector

Java 5 Subtyping


Object

CoffeeTea

Drink

ZDrink.List<Z>

Z.List<Z>

List<Tea>

ZDrink.Vector<Z>

Z.Vector<Z>

Vector<Tea>

X

In class C<X extends Tea> ...

ZX.List<Z>

ZX.Vector<Z>

Tame FJ Subclassing


Object

CoffeeTea

Drink List<Z>

List<Tea>

Vector<Z>

Vector<Tea>

X


Tame FJ Extended Subtyping


ZDrink.List<Z>

Z.List<Z>

List<Tea>

ZDrink.Vector<Z>

Z.Vector<Z>

Vector<Tea>


ZX.List<Z> ZX.Vector<Z>

Tame FJ Subtyping

Tea

X


TameFJ Subtyping


Object

CoffeeTea

Drink

ZDrink.List<Z>

Z.List<Z>

List<Tea>

ZDrink.Vector<Z>

Z.Vector<Z>

Vector<Tea>

X


ZX.List<Z>

ZX.Vector<Z>

But Why?

• Subtyping is too flexible and complicated to use directly in the proofs

• SubclassingProperties of the class hierarchy – e.g., field/method lookup

• Extended SubclassingIf we need wildcard subtyping, but can't deal with lower bounds

E.g., preservation of existential types

• SubtypingStill required – e.g., result of inversion lemmas, due to T-SUBS



Typing

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}


class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}

X;this:Cup<X> c:Z.Cup<Z> | fType(f, Cup<Z>) = Mug<Z>


┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}


X;this:Cup<X> c.f:Mug<Z> | Z


┴┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}




┴┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}

X;this:Cup<X> c:Z.Cup<Z> |

fType(f, Cup<Z>) = Mug<Z>


[email protected] 69┴

┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}


fType(f, Cup<Z>) = Mug<Z> X;Z Mug<Z> <:Z.Mug<Z>



┴

┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}



X;this:Cup<X> c.f:Mug<Z> | Z X Z.Mug<Z> OK


┴

┴

┴

class Cup<X> {

Mug<X> f;

void m(Cup<?> c) {

c.f;

}

}




X;this:Cup<X> c.f:Z.Mug<Z> |


┴

┴

┴

┴

[email protected]

• Full model

• Proven sound

• Future Work• ‘More full’ - features, translation• More expressive?• Mechanical proof checking



Thank You!

http://www.doc.ic.ac.uk/~ncameron/papers/cameron_ecoop08_full.pdf

class Cup<X> {

Cup<X> f;

Cup<X> m2(String x) {...}

void m(Cup<?> c) {

c.f = c.f;

c.f = c;

c.f = c.m2(“a string”);

}

}


Variance in C# and Scala

• C# is invariant, but variance proposed

• Declaration site annotationsLess complex, but less flexible

Vs. use site declaration in Java

class CovariantCup<+X> ...

• Formalisation and soundnessC# - Emir, Kennedy, Russo, Yu. ECOOP 06

Scala - Cremet. PhD Thesis, 06 (Virtual types)


[email protected] 1 a model for java with wildcards nicholas cameron sophia drossopoulou erik...

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