David Evans CS201j: Engineering Software University of Virginia Computer Science Lecture 14: Substitution Principle

David Evanshttp://www.cs.virginia.edu/evans

CS201j: Engineering SoftwareUniversity of VirginiaComputer Science

Lecture 14: SubstitutionPrinciple

How do we know if saying B is a subtype of A

is safe?

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Substitution Principle

• If B is a subtype of A, everywhere the code expects an A, a B can be used instead

For a function f (A), if f satisfies its specification when passed anobject whose actual type is type A, f also satisfies its specification when passed an object whose actual type is B.


What needs to be true?For a function f (a A), if f satisfies its specification when passed an object whose actual type is type A, f also satisfies its specification when passed an object whose actual type is B.

public int f (a A, x X) { return a.m (x);}


Subtype Condition 1: Signature RuleWe can use a subtype method where a supertype methods is expected:

– Subtype must implement all of the supertype methods

– Argument types must not be more restrictive

– Result type must be at least as restrictive

– Subtype method must not throw exceptions that are not subtypes of exceptions thrown by supertype


Signature Rule

class A { public RA m (PA p) ;}

class B extends A { public RB m (PB p) ;}

RB must be a subtype of RA: RB <= RAPB must be a supertype of PA: PB >= PAcovariant for results, contravariant for parameters


Signature Examples

public class Object { public boolean equals (Object o) ;}

public class String extends Object { }

public boolean equals (Object o) ;


Signature Examples

public class CellSet { // A set of Cell’s. public Cell choose () throws NoSuchElementException ;}

public class ConwayCellSet extends CellSet {// A set of ConwayCell (<= Cell) objects.

public ConwayCell choose () } Return type can be a subtype (according to substitution

principle, not according to Java compiler rules)


Java’s Rule

• Java compiler is stricter than this: doesn’t allow any variation in types (“novariant”):– Overriding method must have same return

and parameter types– Overriding method can throw fewer

exceptions.

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What needs to be true?For a function f (a A), if f satisfies its specification when passed an object whose actual type is type A, f also satisfies its specification when passed an object whose actual type is B <= A.

public int f (a A, x X) { // REQUIRES: a is initialized // EFFECTS: returns a.value * x.value return a.m (x);}

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• Precondition of the subtype method must be weaker than the precondition of the supertype method.

mA.pre mB.pre

• Postcondition of the subtype method must be stronger than the postcondition of the supertype method.

mB.post mA.post

Subtype Condition 2: Methods Rule

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Methods Rule Example

public int f (a A, x X) { // REQUIRES: a is initialized // EFFECTS: returns a.value * x.value return a.m (x);}public class A {

// An A may be initialized or uninitialized. // An initialized A has an associated int value. public int m (x X) { // REQUIRES: this is initialized} public class B extends A {

// A B may be initialized or uninitialized. // A B may be awake or asleep. // An initialized B has an associated int value. public int m (x X) { // REQUIRES: this is initialized and awake } Can’t make the precondition

stronger! The callsite mightnot satisfy it.

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// An A may be initialized or uninitialized. // An initialized A has an associated int value. public int m (x X) { // REQUIRES: this is initialized} public class B extends A {

// A B may be initialized or uninitialized. // A B may be awake or asleep. // An initialized B has an associated int value. public int m (x X) { // REQUIRES: nothing }

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// An A may be initialized or uninitialized. // An initialized A has an associated int value. public int m (x X) { // ENSURES: Returns this.value * x.value} public class B extends A {

// A B may be initialized or uninitialized. // A B may be awake or asleep. // An initialized B has an associated int value. public int m (x X) { // ENSURES: Returns this.value + x.value. } Postcondition must imply

the supertype’s methodpostcondition.

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// An A may be initialized or uninitialized. // An initialized A has an associated int value. public int m (x X) // MODIFIES: this // ENSURES: Returns this.value * x.value. // Makes this initialized.

public class B extends A { // A B may be initialized or uninitialized. // A B may be awake or asleep. // An initialized B has an associated int value. public int m (x X) // MODIFIES: this // ENSURES: Returns this.value * x.value. Makes // this initialized and awake.

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Subtypes must preserve all properties described in the overview specification of the supertype.

Subtype Condition 3: Properties

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Properties Example

• B <= A

• If A is an immutable datatype, can B be mutable?

• If A is a mutable datatype, can B be immutable?

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Substitution Principle Summary• Signatures: subtype methods must be type correct

in supertype callsites: result is a subtype (covariant), parameters are supertypes (contravariant)

• Methods: subtype postconditions must be stronger than supertype postconditions (covariant); subtype preconditions must be weaker than supertype preconditions (contravariant);

• Properties: subtype must preserve all properties specified in supertype overview

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Substitution Principle… (in client code)MysteryType1 mt1;MysteryType2 mt2;MysteryType3 mt3;… (anything could be here)mt1 = mt2.m (mt3);

If the Java compiler is happy with this code, which of these are guaranteedto be true:a. The apparent type of mt2 is MysteryType2b. At the last statement, the actual type of mt2 is MysteryType2c. MysteryType2 has a method named md. The MysteryType2.m method takes a parameter of type MysteryType3e. The MysteryType2.m method returns a subtype of MysteryType1f. After the last statement, the actual type of mt1 is MysteryType1

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… (in client code)MysteryType1 mt1;MysteryType2 mt2;MysteryType3 mt3;… (anything could be here)mt1 = mt2.m (mt3);

If the Java compiler is happy with this code, which of these are guaranteedto be true:a. The apparent type of mt2 is MysteryType2

b. At the last statement, the actual type of mt2 is MysteryType2

c. MysteryType2 has a method named m

d. The MysteryType2.m method takes a parameter of type MysteryType3

e. The MysteryType2.m method returns a subtype of MysteryType1

f. After the last statement, the actual type of mt1 is MysteryType1

TRUE: the apparent type is obvious from the declaration.

FALSE: we only know the actual type <= MysteryType2

TRUE

FALSE: we only know it takes a parameter >= MysteryType3

TRUE: the assignment type checking depends on this

FALSE: we only know that the actual type <= MysteryType1

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Subtyping Rules

class A { public RA m (PA p) ;}

… (in client code)MysteryType1 mt1;MysteryType2 mt2;MysteryType3 mt3;…mt1 = mt2.m (mt3);

RA must be a subtype of MysteryType1:

RA <= MysteryType1MysteryType3 must be a subtype of PA:

PA >= MysteryType3

If A is MysteryType2, what do we know about RA and PA?

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Subtyping Rulesclass A { public RA m (PA p) ;}class B extends A { public RB m (PB a);}


RB must be a subtype of RA: RB <= RAPA must be a subtype of PB: PB >= PA

If B <= A, what do we know about RB and PB?

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Substitution Principleclass A { public RA m (PA p) ;}class B extends A { public RB m (PB a);}


Substitution Principle:Parameters PB >= PAPreconditions pre_A pre_B

Result RB <= RAPostconditions post_B post_A

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Substitution Principle

Is this the only way?

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Eiffel’s Rules

(Described in Bertrand Meyer paper out today)

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Eiffel Rules

Skierset_roommate (Skier)

Boy Girl

The types of the parameters in the subtype method may be subtypes of the supertype parameters.

How can Girl override set_roomate?set_roommate (Girl g)set_roommate (Boy b)

Opposite of substitution principle!

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Eiffel and I Can’t Get Up?

s: skier; g: girl; b: boy;s := g; ...s.set_roommate (b);

Skierset_roommate (Skier)

Boy Girlset_roomate (Girl)

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Meyer’s ExcuseStrangely enough, some workers in the field have been advocating a contravariant policy. Here it would mean that if we go for example to class RANKED_GIRL, where the result of roommate is naturally redefined to be of type RANKED_GIRL, we may for the argument of routine share use type GIRL, or rather scaringly, SKIER of the most general kind. One type that is never permitted in this case is RANKED_GIRL! Here is what, under various mathematical excuses, some professors have been promoting. No wonder teenage pregnancies are on the rise.

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Substitution Principle / Eiffelclass A { public RA m (PA p) ;}class B extends A { public RB m (PB a);}


Substitution Principle EiffelParameters PB >= PA PB <= PAPreconditions pre_A pre_B pre_B pre_A

Result RB <= RA RB <= RAPostconditions post_B post_A post_B post_A

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PS5• First part (due Thursday Oct 30):

– Questions about subtyping rules• Which rules make most sense?• Which rules are most useful for real programs?

– Exercises to learn about concurrency (Thursday)

– Design a distributed simulation of something interesting

• Second part (due Nov 11):– Implement your distributed simulation

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Charge

Must it be assumed that because we are engineers beauty is not our concern, and that while we make our constructions robust and durable we do not also strive to make them elegant?

Is it not true that the genuine conditions of strength always comply with the secret conditions of harmony?

Gustav Eiffel

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David Evans CS201j: Engineering Software University of Virginia Computer Science Lecture 14: Substitution Principle