Intersection Types in Java

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook

Martin Plumicke¨

University of Cooperative Education Stuttgart/Horb

September 11, 2008

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Overview

Motivation/Tooldemonstration

Method calls with intersection types Semantics First approach The algorithm

Principal Typing

Conclusion and Outlook

Martin Plumicke¨ Intersection Types in Java Purpuse: Byte-code generation for methods with intersection types

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook

Purpuse: Byte-code generation for methods with intersection types

Martin Plumicke¨ Intersection Types in Java ... m ( ... ) { ... receiver.method(t1, ..., tn); ... }

0 0 0 I method: ty1 × ... × tyn → ty0 & ... & ty1 × ... × tyn → ty0 has an intersection type

I t1 , ... , tn have unambiguous types during execution

I by the argument types and the result type the typing of method is determined

I method is executed with the determined typing.

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Semantics of type–inferred Java programs

I control-structures have the same semantics as in standard Java

I method calls diﬀer, as there are intersection types

Martin Plumicke¨ Intersection Types in Java I t1 , ... , tn have unambiguous types during execution

I by the argument types and the result type the typing of method is determined

I method is executed with the determined typing.

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Semantics of type–inferred Java programs

I control-structures have the same semantics as in standard Java

I method calls diﬀer, as there are intersection types

... m ( ... ) { ... receiver.method(t1, ..., tn); ... }

0 0 0 I method: ty1 × ... × tyn → ty0 & ... & ty1 × ... × tyn → ty0 has an intersection type

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Semantics of type–inferred Java programs

I control-structures have the same semantics as in standard Java

I method calls diﬀer, as there are intersection types

... m ( ... ) { ... receiver.method(t1, ..., tn); ... }

0 0 0 I method: ty1 × ... × tyn → ty0 & ... & ty1 × ... × tyn → ty0 has an intersection type

I t1 , ... , tn have unambiguous types during execution

I by the argument types and the result type the typing of method is determined

I method is executed with the determined typing.

Martin Plumicke¨ Intersection Types in Java main:Integer → Integer is determined

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Semantics example

class OL { Integer m(x) { return x + x; } //Integer → Integer Boolean m(x) { return x || x; } //Boolean → Boolean }

class Main { main(x) { // Integer → Integer & Boolean → Boolean ol; ol = new OL(); return ol.m(x); } } ... Main rec = new Main(); Integer r = rec.main(x);

class OL { Integer m(x) { return x + x; } //Integer → Integer Boolean m(x) { return x || x; } //Boolean → Boolean }

class Main { main(x) { // Integer → Integer & Boolean → Boolean ol; ol = new OL(); return ol.m(x); } } ... Main rec = new Main(); Integer r = rec.main(x); main:Integer → Integer is determined

Martin Plumicke¨ Intersection Types in Java Result for Main: class Main { Integer main(Integer x) { OL ol; ol = new OL(); return ol.m(x); }

Boolean main(Boolean x) { OL ol; ol = new OL(); return ol.m(x); }}

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Code generation for method with intersection types

I Byte-code allows no intersection types I First approach: generate for each element of the intersection type an own method

I Byte-code allows no intersection types I First approach: generate for each element of the intersection type an own method Result for Main: class Main { Integer main(Integer x) { OL ol; ol = new OL(); return ol.m(x); }

Boolean main(Boolean x) { OL ol; ol = new OL(); return ol.m(x); }} Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Example: Multiplication of matrices I

class Matrix extends Vector> { mul(m) { ret = new Matrix(); int i = 0; while(i (); int j = 0; while(j < v1.size()) { int erg = 0; int k = 0; while(k < v1.size()) { erg = erg + v1.elementAt(k) * m.elementAt(k).elementAt(j); k++; } v2.addElement(new Integer(erg)); j++; } ret.addElement(v2); i++; } return ret; }}

Martin Plumicke¨ Intersection Types in Java class Matrix extends Vector> { Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } ... Vector> mul(Vector> m) { ... } ... Vector> mul(Matrix m) { ... } Not a correct Java program

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Example: Multiplication of matrices II

mul: &β,α(β→α), where β ≤∗ Vector>, Matrix ≤∗ α

Martin Plumicke¨ Intersection Types in Java Not a correct Java program

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Example: Multiplication of matrices II

mul: &β,α(β→α), where β ≤∗ Vector>, Matrix ≤∗ α

class Matrix extends Vector> { Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } ... Vector> mul(Vector> m) { ... } ... Vector> mul(Matrix m) { ... }

mul: &β,α(β→α), where β ≤∗ Vector>, Matrix ≤∗ α

class Matrix extends Vector> { Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } Matrix mul(Vector> m) { ... } ... Vector> mul(Vector> m) { ... } ... Vector> mul(Matrix m) { ... } Not a correct Java program

Martin Plumicke¨ Intersection Types in Java Code–execution: Callgraph of the method declarations

CG( cl.m : τ ) Callgraph of the method m in the class cl with the typing τ. Supertype of function types: Subtyping ordering ∗ 0 ∗ 0 θi ≤ θi , θ ≤ θ ⇒ 0 ∗ 0 0 θ1 × ... × θn → θ ≤ θ1 × ... × θn → θ

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Group elements of the intersection type

Idea: 1. Group all elements which I executes the same code I have a common supertype 2. Generate new methods only for the groups

Martin Plumicke¨ Intersection Types in Java Supertype of function types: Subtyping ordering ∗ 0 ∗ 0 θi ≤ θi , θ ≤ θ ⇒ 0 ∗ 0 0 θ1 × ... × θn → θ ≤ θ1 × ... × θn → θ

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Group elements of the intersection type

Idea: 1. Group all elements which I executes the same code I have a common supertype 2. Generate new methods only for the groups

Code–execution: Callgraph of the method declarations

CG( cl.m : τ ) Callgraph of the method m in the class cl with the typing τ.

Idea: 1. Group all elements which I executes the same code I have a common supertype 2. Generate new methods only for the groups

Code–execution: Callgraph of the method declarations

Martin Plumicke¨ Intersection Types in Java Supertype Integer → Integer Boolean → Boolean

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Example class OL I

Callgraph CG( Main.main : Integer → Integer ) CG( Main.main : Boolean → Boolean ) = = Main.main: Integer−>Integer Main.main: Integer−>Integer & Boolean −> Boolean & Boolean −> Boolean

OL.m: Integer −> Integer Ol.m: Boolean −> Boolean

Callgraph CG( Main.main : Integer → Integer ) CG( Main.main : Boolean → Boolean ) = = Main.main: Integer−>Integer Main.main: Integer−>Integer & Boolean −> Boolean & Boolean −> Boolean

OL.m: Integer −> Integer Ol.m: Boolean −> Boolean

Supertype Integer → Integer Boolean → Boolean

Code generation

class Main { Integer main(Integer x) { OL ol; ol = new OL(); return ol.m(x); }

Boolean main(Boolean x) { OL ol; ol = new OL(); return ol.m(x); }}

Code is unchanged in comparison to the ﬁrst approach

Martin Plumicke¨ Intersection Types in Java Supertype:

Vector> → Matrix

Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook Example class Matrix I

Callgraph CG( Matrix.mul : τ ) for all τ

Matrix.mul: Vector< Vector>? −> Matrix & Vector< Vector>? −> Matrix & & ... &

Matrix −> Vector< Vector>

Vector.size:−> int Vector.elementAt: int −> T Vector.addElement: T −> void

Callgraph CG( Matrix.mul : τ ) for all τ

Matrix.mul: Vector< Vector>? −> Matrix & Vector< Vector>? −> Matrix & & ... &

Matrix −> Vector< Vector>

Vector.size:−> int Vector.elementAt: int −> T Vector.addElement: T −> void

Supertype:

Vector> → Matrix

Code generation (only one method!) Matrix mul(Vector> m) { Matrix ret = new Matrix(); int i = 0; while(i v1 = this.elementAt(i); Vector v2 = new Vector(); int j = 0; while(j < v1.size()) { int erg = 0; int k = 0; while(k < v1.size()) { erg = erg + ...; k++; } v2.addElement(new Integer(erg)); j++; } ret.addElement(v2); i++; } return ret; }} Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Semantics Method calls with intersection types First approach Principal Typing The algorithm Conclusion and Outlook The Algorithm

Input: A Java program p with inferred (intersection) types. Output: A Java program p0, where the methods have standard Java types. The semantics of p and p0 are equal. 1. Step: For every class cl in p consider for each method m the intersection type tym:

I Build the callgraph CG( cl.m : τ ) for each function type τ of the intersection type tym. I Group all elements τ of tym, where CG( cl.m : τ ) is the same graph and there is a supertype. 2. Step: Determine the supertype of the respective group. 3. Step: Generate for each group of function types the corresponding Java code with the supertype as standard typing in p0.

Martin Plumicke¨ Intersection Types in Java Generalization to the Java type system [Plumicke¨ 2007, PPPJ07]

“An intersection type-scheme for a declaration is a principal type-scheme, if any (non–intersection) type-scheme for the declaration is a subtype of a generic instance of one element of the intersection type-scheme.”

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Principal Typing

Deﬁnition [Damas, Milner 1982]:

“A type-scheme for a declaration is a principal type-scheme, if any other type-scheme for the declaration is a generic instance of it.”

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Principal Typing

Deﬁnition [Damas, Milner 1982]:

“A type-scheme for a declaration is a principal type-scheme, if any other type-scheme for the declaration is a generic instance of it.”

Generalization to the Java type system [Plumicke¨ 2007, PPPJ07]

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example principal typing I

import java.util.Vector; import java.util.Stack;

class Put { putElement(T ele, Vector v) { v.addElement(ele); }

putElement(T ele, Stack s) { s.push(ele); }

main(ele, x) { putElement(ele, x); } } Martin Plumicke¨ Intersection Types in Java But there is another principal type:

main : T × Vector → void,

as Stack ≤∗ Vector.

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example principal typing II

The inferred intersection type:

main : T × Vector → void & T × Stack → void.

is a principal type.

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example principal typing II

The inferred intersection type:

main : T × Vector → void & T × Stack → void.

is a principal type.

But there is another principal type:

main : T × Vector → void,

as Stack ≤∗ Vector.

Martin Plumicke¨ Intersection Types in Java The principal type main : T × Vector → void is correct but not meaningful!!

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example principal typing III

class Put { void putElement(T ele, Vector v) { ... }

void putElement(T ele, Stack s) { ... }

void main(T ele, Vector x) { x.putElement(ele, x); }

void main(T ele, Stack x) { x.putElement(ele, x); }}

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example principal typing III

class Put { void putElement(T ele, Vector v) { ... }

void putElement(T ele, Stack s) { ... }

void main(T ele, Vector x) { x.putElement(ele, x); }

void main(T ele, Stack x) { x.putElement(ele, x); }}

The principal type main : T × Vector → void is correct but not meaningful!!

Martin Plumicke¨ Intersection Types in Java This reﬁned deﬁnition guarantees, that for each method, which is generated by the resolving algorithm, at least one typing is contained in the principal type.

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Reﬁned deﬁnition of Principal typing

“An intersection type-scheme for a declaration is a principal type-scheme, if any (non–intersection) type-scheme θ for the declaration is a subtype of a generic instance of one element of the intersection type-scheme τ and θ and τ have the same callgraph.”

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Reﬁned deﬁnition of Principal typing

This reﬁned deﬁnition guarantees, that for each method, which is generated by the resolving algorithm, at least one typing is contained in the principal type.

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Example Put (cont.)

CG( Put.main : T × Vector → void ) 6= CG( Put.main : Integer × Stack → void ) = = Put.main: T x Vector −> void Put.main: T x Vector −> void & T x Stack −> void & T x Stack −> void

Put.putElement: T x Vector −> void Put.putElement: T x Stack −> void

main : T × Vector → void is no principal type, but main : T × Vector → void & T × Stack → void is a principal type.

Martin Plumicke¨ Intersection Types in Java Outlook At the moment: Type inference algorithm infers typings, which are later erased as subtypes by the resolving algorithm. Purpuse: Type inference algorithm infers only supertypes, such that no typings are erased in the resolving algorithm.

Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Conclusion and Outlook

Conclusion

I Semantics for Java methods with intersection types

I Resolving algorithm of intersection types

I Code generation for methods with intersection types possible ⇒ type selection is not longer necessary in the Eclipse plugin.

I (Redeﬁned) Principal type property

Martin Plumicke¨ Intersection Types in Java Motivation/Tooldemonstration Method calls with intersection types Principal Typing Conclusion and Outlook Conclusion and Outlook

Conclusion

I Semantics for Java methods with intersection types

I Resolving algorithm of intersection types

I Code generation for methods with intersection types possible ⇒ type selection is not longer necessary in the Eclipse plugin.

I (Redeﬁned) Principal type property

Outlook At the moment: Type inference algorithm infers typings, which are later erased as subtypes by the resolving algorithm. Purpuse: Type inference algorithm infers only supertypes, such that no typings are erased in the resolving algorithm.

Martin Plumicke¨ Intersection Types in Java