Reading 15: Equality

In the previous readings we’ve developed a rigorous notion of data abstraction by creating types that are characterized by their operations, not by their representation. For an abstract data type, the abstraction function explains how to interpret a concrete representation value as a value of the abstract type, and we saw how the choice of abstraction function determines how to write the code implementing each of the ADT’s operations.

In this reading we turn to how we define the notion of equality of values in a data type: the abstraction function will give us a way to cleanly define the equality operation on an ADT.

In the physical world, every object is distinct – at some level, even two identical snowflakes are different, even if the distinction is just the position they occupy in space. (This isn’t strictly true of all subatomic particles, but true enough of large objects like snowflakes and baseballs and people.) So two physical objects are never truly “equal” to each other; they only have degrees of similarity.

In the world of human language, however, and in the world of mathematical concepts, you can have multiple names for the same thing. So it’s natural to ask when two expressions represent the same thing: 1+2, √9, and 3 are alternative expressions for the same ideal mathematical value.

The first part of this reading focuses on defining equality for immutable types, so that we can understand the concepts without the complication of mutation. Then we’ll expand to consider mutable types as well.

Now that we have a mathematical foundation for all equality operators, let’s look at equality in the context of abstract data types, starting with immutable types.

Formally, we can define equality on immutable types in two ways.

Using the abstraction function. Recall that an abstraction function AF: R → A maps concrete instances of a data type to their corresponding abstract values. To use AF as a definition for equality, we would say that a equals b if and only if AF(a) = AF(b).

Using observation. We can say that two objects are equal when they cannot be distinguished by observation – every operation we can apply produces the same result for both objects. Consider the set expressions {1,2} and {2,1}. Using the observer operations available for sets, cardinality |…| and membership ∈, these expressions are indistinguishable:

• |{1,2}| = 2 and |{2,1}| = 2
• 1 ∈ {1,2} is true, and 1 ∈ {2,1} is true
• 2 ∈ {1,2} is true, and 2 ∈ {2,1} is true
• 3 ∈ {1,2} is false, and 3 ∈ {2,1} is false
• … and so on

In terms of abstract data types, “observation” means calling operations on the objects. So two objects are equal if and only if they cannot be distinguished by calling any operations of the abstract data type.

Note that “operations of the abstract data type” means the operations that are part of the spec of the ADT. Java has language features that allow a client to break the abstraction barrier and observe differences between objects that are irrelevant to their abstract values. For example, Java’s == can detect that two objects representing identical sets are actually stored at different places in memory. System.identityHashCode() computes a hash code based on an object’s memory address, so it could observe the same distinction. But neither of these operations would be part of the spec of the set ADT, so the distinctions they make are not considered when deciding if two set objects are equal by observation.

The two definitions of equality – by abstraction function, and by observation – should be consistent with each other, or something is wrong.

Example: Duration

public class Duration {
    private final int mins;
    private final int secs;
    // Rep invariant:
    //    mins >= 0, secs >= 0
    // Abstraction function:
    //    AF(min, secs) = the span of time of mins minutes and secs seconds

    /** Make a duration lasting for m minutes and s seconds. */
    public Duration(int m, int s) {
        mins = m; secs = s;
    }
    /** @return length of this duration in seconds */
    public long getLength() {
        return (long)mins*60 + secs;
    }
}

Duration d1 = new Duration (1, 2);
Duration d2 = new Duration (1, 3);
Duration d3 = new Duration (0, 62);
Duration d4 = new Duration (1, 2);

/** Immutable type representing a set of letters, ignoring case */
class LetterSet {
    private String s;

    // Abstraction function:
    //    AF(s) = the set of the letters that are found in s
    //            (ignoring non-letters and alphabetic case)
    // Rep invariant:
    //    true

    /**
     * Make a LetterSet consisting of the letters found in chars 
     * (ignoring alphabetic case and non-letters).
     */
    public LetterSet(String chars) {
        s = chars;
    }

    ... // observer and producer operations
}

/**
 * @return the size of this set
 */
public int size() { ... }

/**
 * @param letter must be a letter 'a'...'z' or 'A'...'Z'
 * @return true iff this set contains letter, ignoring alphabetic case
 */
public boolean contains(char letter) { ... }

/**
 * @return the length of the string that this set was constructed from
 */
public int length() { ... }

/**
 * @return the union of this and that
 */
public LetterSet union(LetterSet that) { ... }

/**
 * @return true if and only if all the letters in this set are lowercase
 */
public boolean isAllLowercase() { ... }

/**
 * @return the first letter in s */
public char first() { ... }

/** Immutable type representing lines in the plane that are neither horizontal nor vertical. */
public class MyLine {
    private double a;
    private double b;

    // Abstraction function:
    //    TODO
    // Rep invariant:
    //    TODO

    /**
     *  Make a MyLine that passes through all the points (p[2i], p[2i+1]).
     *  p must contain at least 2 points and all points in p must be collinear
     *  on a line that is neither horizontal nor vertical.
     *  For example, MyLine({ 0,0,  1,2 }) passes through both (0,0) and (1,2).
     */
    public MyLine(int[] p) { ... }

    /**
     * Get the slope of the line.
     */
    public double slope() { ... }
}

Like many languages, Java has two different operations for testing equality between objects, with different semantics.

• The == operator compares references. More precisely, it tests reference equality. Two references are == if they point to the same storage in memory. In terms of the snapshot diagrams we’ve been drawing, two references are == if their arrows point to the same object bubble.
• The equals() operation compares object contents – in other words, object equality, in the sense that we’ve been talking about in this reading. The equals operation has to be defined appropriately for every abstract data type.

For comparison, here are the equality operators in several languages:

Note that == unfortunately flips its meaning between Java and Python. Don’t let that confuse you: == in Java just tests reference identity, it doesn’t compare object contents.

Note also that this table is only about equality for object types, like we use to define a new ADT. Built-in primitive types, like int in Java or number in JavaScript, may follow different rules. For int in Java, for example, you can’t use equals(), the == operator does object equality, and there is no notion of reference equality at all.

As programmers in any of these languages, we can’t change the meaning of the reference equality operator. In Java, == always means reference equality. But when we define a new data type, it’s our responsibility to decide what object equality means for values of the data type, and implement the equals() operation appropriately.

The equals() method is defined by Object, and its default implementation looks like this:

public class Object {
    ...
    public boolean equals(Object that) {
        return this == that;
    }
}

In other words, the default meaning of equals() is the same as reference equality. For immutable data types, this is almost always wrong. So you have to override the equals() method, replacing it with your own implementation.

Implementing equals() correctly can be subtle. As a prelude, you may want to refresh your understanding of dynamic dispatch.

The wrong way to implement equals()

public class Duration {
    ...   
    // Problematic definition of equals()
    public boolean equals(Duration that) {
        return this.getLength() == that.getLength();        
    }
}

Duration d1 = new Duration (1, 2);
Duration d2 = new Duration (1, 2);
Object o2 = d2;
d1.equals(d2) → true
d1.equals(o2) → false

public class Duration extends Object {
    // explicit method that we declared:
    public boolean equals(Duration that) {
        return this.getLength() == that.getLength();
    }
    // implicit method inherited from Object:
    public boolean equals(Object that) {
        return this == that;
    }
}

A better way to implement equals()

@Override
public boolean equals(Object that) {
    return that instanceof Duration && this.sameValue((Duration)that);
}

// returns true iff this and that represent the same abstract value
private boolean sameValue(Duration that) {
    return this.getLength() == that.getLength();
}

Duration d1 = new Duration(1, 2);
Duration d2 = new Duration(1, 2);
Object o2 = d2;
d1.equals(d2) → true
d1.equals(o2) → true

instanceof

The specification of the Object class is so important that it is often referred to as the Object Contract. The contract can be found in the method specifications for the Object class. Here we will focus on the contract for equals. When you override the equals method, you must adhere to its general contract. It states that:

• equals must define an equivalence relation – that is, a relation that is reflexive, symmetric, and transitive;
• equals must be consistent: repeated calls to the method must yield the same result (as long as neither object has been mutated in a way that affects the equals comparison);
• for a non-null reference x, x.equals(null) should return false;
• hashCode must produce the same result for two objects that are deemed equal by the equals method.

Breaking the equivalence relation

@Override
public boolean equals(Object that) {
    return that instanceof Duration && this.sameValue((Duration)that);
}

private static final int CLOCK_SKEW = 5; // seconds

// returns true iff this and that represent the same abstract value within a clock-skew tolerance
private boolean sameValue(Duration that) {
    return Math.abs(this.getLength() - that.getLength()) <= CLOCK_SKEW;
}

public class Duration {
    private final int mins;
    private final int secs;
    // Rep invariant:
    //    mins >= 0, secs >= 0
    // Abstraction function:
    //    AF(min, secs) = the span of time of mins minutes and secs seconds

    /**
     * Make a duration lasting for m minutes and s seconds.
     */
    public Duration(int m, int s) {
        mins = m; secs = s;
    }
    /**
     * @return length of this duration in seconds
     */
    public long getLength() {
        return (long)mins*60 + secs;
    }

    @Override
    public boolean equals(Object that) {
        return that instanceof Duration && this.sameValue((Duration)that);
    }

    private static final int CLOCK_SKEW = 5; // seconds

    // returns true iff this and that represent the same abstract value within a clock-skew tolerance
    private boolean sameValue(Duration that) {
        return Math.abs(this.getLength() - that.getLength()) <= CLOCK_SKEW;
    }
}

Duration d_0_60 = new Duration(0, 60);
Duration d_1_00 = new Duration(1, 0);
Duration d_0_57 = new Duration(0, 57);
Duration d_1_03 = new Duration(1, 3);

1 @Override
2 public boolean equals(Object that) {
3     return that instanceof Duration 
4            && this.sameValue((Duration)that);
  }

  // returns true iff this and that represent the same abstract value
5 private boolean sameValue(Duration that) {
6     return this.getLength() == that.getLength();
  }

@Override
public boolean equals(Object that) {
    return this.toString().equals(that.toString());
}

@Override
public String toString() {
    return mins + ":" + secs;  // e.g. "2:25" for 2 minutes, 25 seconds
}

Breaking hash tables

public class Object {
  ...
  public boolean equals(Object that) { return this == that; }
  public int hashCode() { return /* the memory address of this */; }
}

Duration d1 = new Duration(1, 2);
Duration d2 = new Duration(1, 2);
d1.equals(d2) → true
d1.hashCode() → 2392
d2.hashCode() → 4823

@Override
public int hashCode() {
    return (int) getLength();
}

class Person {
  private String firstName;
  private String lastName;
  ...

  @Override
  public boolean equals(Object that) {
      return that instanceof Person && this.sameValue(that);
  }

  // returns true iff this and that represent the same abstract value
  private boolean sameValue(Person that) {
      return this.lastName.toUpperCase().equals(that.lastName.toUpperCase());
  }

  @Override
  public int hashCode() {
      // TODO
  }
}

We’ve been focusing on equality of immutable objects so far in this reading. What about mutable objects?

Equality must still be an equivalence relation, as required by the Object contract.

We also want equality to respect the abstraction function and respect operations. But with mutable objects, there is a new possibility: by calling a mutator on one of the objects before doing the observation, we may change its state and thus create an observable difference between the two objects.

So let’s refine our definition and allow for two notions of equality based on observation:

• observational equality means that two references cannot be distinguished now, in the current state of the program. A client can try to distinguish them only by calling operations that don’t change the state of either object (i.e. only observers and producers, not mutators) and comparing the results of those operations. This tests whether the two references “look” the same for the current state of the object.
• behavioral equality means that two references cannot be distinguished now or in the future, even if a mutator is called to change the state of one object but not the other. This tests whether the two references will “behave” the same, in this and all future states.

For immutable objects, observational and behavioral equality are identical, because there aren’t any mutator methods that can change the state of the objects.

For mutable objects, it can be very tempting to use observational equality as the design choice. Java uses observational equality for most of its mutable data types, in fact. If two distinct List objects contain the same sequence of elements, then equals() reports that they are equal.

List<Integer> listA = List.of(1, 2, 3);
List<Integer> listB = new ArrayList<>(listA);
List<Integer> listC = listB;

List<Integer> listA = List.of(1, 2, 3);
List<Integer> listB = new ArrayList<>(listA);
List<Integer> listC = listB;

int[] arrayA = new int[] { 1, 2, 3 };
int[] arrayB = arrayA;
int[] arrayC = new int[] { 1, 2, 3 };

Set<Integer> setA = Set.of(1, 2, 3);
Set<Integer> setB = Set.of(3, 2, 1);
Set<Integer> setC = new HashSet<>(setA);

Using observational equality for mutable types seems at first like a reasonable idea. It allows, for example, two Lists of the same integers in the same order to be equal().

But the presence of mutators unfortunately leads to subtle bugs, because it means that equality isn’t consistent over time. Two objects that are observationally equal at one moment in the program may stop being equal after a mutation.

This fact allows us to easily break the rep invariants of other collection data structures. Suppose we make a List, and then drop it into a HashSet:

List<String> list = new ArrayList<>();
list.add("a");

Set<List<String>> set = new HashSet<List<String>>();
set.add(list);

We can check that the set contains the list we put in it, and it does:

set.contains(list) → true

But now we mutate the list:

list.add("goodbye");

And it no longer appears in the set!

set.contains(list) → false!

It’s worse than that, in fact: when we iterate over the members of the set, we still find the list in there, but contains() says it’s not there!

for (List<String> l : set) { 
    set.contains(l) → false! 
}

If the set’s own iterator and its own contains() method disagree about whether an element is in the set, then the set clearly is broken. You can see this code in action.

What’s going on? List<String> is a mutable object. In the standard Java implementation of collection classes like List, mutations affect the result of equals() and hashCode(). When the list is first put into the HashSet, it is stored in the hash bucket corresponding to its hashCode() result at that time. When the list is subsequently mutated, its hashCode() changes, but HashSet doesn’t realize it should be moved to a different bucket. So it can never be found again.

When equals() and hashCode() can be affected by mutation, we can break the rep invariant of a hash table that uses that object as a key.

Here’s a telling quote from the specification of java.util.Set:

Note: Great care must be exercised if mutable objects are used as set elements. The behavior of a set is not specified if the value of an object is changed in a manner that affects equals comparisons while the object is an element in the set.

The Java library is unfortunately inconsistent about its interpretation of equals() for mutable classes. Collections like List, Set, and Map use observational equality, but other mutable classes like StringBuilder, and arrays, use behavioral equality.

The lesson we should draw from this example is that equals() should implement behavioral equality. Otherwise instances of the type cannot be safely stored in a data structure that depends on hashing, like HashMap and HashSet.

For immutable types:

• Behavioral equality is the same as observational equality.
• equals() must be overridden to compare abstract values.
• hashCode() must be overriden to map the abstract value to an integer.

For mutable types:

• Behavioral equality is different from observational equality.
• equals() should generally not be overriden, but inherit the implementation from Object that compares references, just like ==.
• hashCode() should likewise not be overridden, but inherit Object’s implementation that maps the reference into an integer.

For a mutable type that needs a notion of observational equality (whether two mutable objects “look” the same in the current state), it’s better to define a completely new operation, which might be called similar() or sameValue(). Its implementation would be similar to the private sameValue() helper method we have been writing for immutable types, but it would be a public operation available to clients. Unfortunately the Java library did not make that design decision.

/** make an empty bag */
public Bag<E>()

/** modify this bag by adding an occurrence of e, and return this bag */
public Bag<E> add(E e)

/** modify this bag by removing an occurrence of e (if any), and return this bag */
public Bag<E> remove(E e)

/** return number of times e occurs in this bag */
public int count(E e)

Bag<String> b1 = new Bag<>().add("a").add("b");
Bag<String> b2 = new Bag<>().add("a").add("b");
Bag<String> b3 = b1.remove("b");
Bag<String> b4 = new Bag<>().add("b").add("a"); // swap!

Autoboxing and equality

Integer x = new Integer(3);
Integer y = new Integer(3);
x.equals(y) → true

x == y // returns false

(int)x == (int)y // returns true

Map<String, Integer> a = new HashMap<>(), b = new HashMap<>();
String c = "c";
a.put(c, 130); // put ints into the maps
b.put(c, 130);
a.get(c) == b.get(c) → ?? // what do we get out of the maps? are they equal?

Map<String, Integer> a = new HashMap<>(), b = new HashMap<>();
String c = "c";
a.put(c, 130); // put ints into the maps
b.put(c, 130);

Map<String, Integer> a = new HashMap<>(), b = new HashMap<>();
String c = "c";
a.put(c, 130); // put ints into the maps
b.put(c, 130);

int i = a.get(c);
int j = b.get(c);
boolean isEqual = (i == j);

• Equality should be an equivalence relation (reflexive, symmetric, transitive).
• Equality and hash code must be consistent with each other, so that data structures that use hash tables (like HashSet and HashMap) work properly.
• The abstraction function is the basis for equality in immutable data types (which means immutable types must override equals(), and therefore hashCode()).
• Reference equality is the basis for equality in mutable data types; this is the only way to ensure consistency over time and avoid breaking rep invariants of hash tables.

Equality is one part of implementing an abstract data type, and we’ve already seen how important ADTs are to achieving our three primary objectives. Let’s look at equality in particular:

• Safe from bugs. Correct implementation of equality and hash codes is necessary for use with collection data types like sets and maps. It’s also highly desirable for writing tests. Since every object in Java inherits the Object implementations, immutable types must override them.

• Easy to understand. Clients and other programmers who read our specs will expect our types to implement an appropriate equality operation, and will be surprised and confused if we do not.

• Ready for change. Correctly-implemented equality for immutable types separates equality of reference from equality of abstract value, hiding from clients our decisions about whether values are shared. Choosing behavioral rather than observational equality for mutable types helps avoid unexpected aliasing bugs.

If you would like to get more practice with the concepts covered in this reading, you can visit the question bank. The questions in this bank were written in previous semesters by students and staff, and are provided for review purposes only – doing them will not affect your classwork grades.