6.005 — Software Construction
Fall 2014

Reading 8: Abstract Data Types

Software in 6.005

Safe from bugsEasy to understandReady for change
Correct today and correct in the unknown future. Communicating clearly with future programmers, including future you. Designed to accommodate change without rewriting.

Objectives

Today’s class introduces several ideas:

In this reading, we look at a powerful idea, abstract data types, which enable us to separate how we use a data structure in a program from the particular form of the data structure itself.

Abstract data types address a particularly dangerous problem: clients making assumptions about the type’s internal representation. We’ll see why this is dangerous and how it can be avoided. We’ll also discuss the classification of operations, and some principles of good design for abstract data types.

Access Control in Java

You should already have read: Controlling Access to Members of a Class in the Java Tutorials.

MITx → Access control

What Abstraction Means

Abstract data types are an instance of a general principle in software engineering, which goes by many names with slightly different shades of meaning. Here are some of the names that are used for this idea:

As a software engineer, you should know these terms, because you will run into them frequently. The fundamental purpose of all of these ideas is to help achieve the three important properties that we care about in 6.005: safety from bugs, ease of understanding, and readiness for change.

User-Defined Types

In the early days of computing, a programming language came with built-in types (such as integers, booleans, strings, etc.) and built-in procedures, e.g., for input and output. Users could define their own procedures: that’s how large programs were built.

A major advance in software development was the idea of abstract types: that one could design a programming language to allow user-defined types, too. This idea came out of the work of many researchers, notably Dahl (the inventor of the Simula language), Hoare (who developed many of the techniques we now use to reason about abstract types), Parnas (who coined the term information hiding and first articulated the idea of organizing program modules around the secrets they encapsulated), and here at MIT, Barbara Liskov and John Guttag, who did seminal work in the specification of abstract types, and in programming language support for them – and developed the original 6.170, the predecessor to 6.005. Barbara Liskov earned the Turing Award, computer science’s equivalent of the Nobel Prize, for her work on abstract types.

The key idea of data abstraction is that a type is characterized by the operations you can perform on it. A number is something you can add and multiply; a string is something you can concatenate and take substrings of; a boolean is something you can negate, and so on. In a sense, users could already define their own types in early programming languages: you could create a record type date, for example, with integer fields for day, month, and year. But what made abstract types new and different was the focus on operations: the user of the type would not need to worry about how its values were actually stored, in the same way that a programmer can ignore how the compiler actually stores integers. All that matters is the operations.

In Java, as in many modern programming languages, the separation between built-in types and user-defined types is a bit blurry. The classes in java.lang, such as Integer and Boolean are built-in; whether you regard all the collections of java.util as built-in is less clear (and not very important anyway). Java complicates the issue by having primitive types that are not objects. The set of these types, such as int and boolean, cannot be extended by the user.

Classifying Types and Operations

Types, whether built-in or user-defined, can be classified as mutable or immutable. The objects of a mutable type can be changed: that is, they provide operations which when executed cause the results of other operations on the same object to give different results. So Date is mutable, because you can call setMonth and observe the change with the getMonth operation. But String is immutable, because its operations create new String objects rather than changing existing ones. Sometimes a type will be provided in two forms, a mutable and an immutable form. StringBuilder, for example, is a mutable version of String (although the two are certainly not the same Java type, and are not interchangeable).

The operations of an abstract type are classified as follows:

We can summarize these distinctions schematically like this (explanation to follow):

These show informally the shape of the signatures of operations in the various classes. Each T is the abstract type itself; each t is some other type. The + marker indicates that the type may occur one or more times in that part of the signature, and the * marker indicates that it occurs zero or more times. For example, a producer may take two values of the abstract type, like String.concat() does. The occurrences of t on the left may also be omitted, since some observers take no non-abstract arguments, and some take several.

Mutators are often signaled by a void return type. A method that returns void must be called for some kind of side-effect, since otherwise it doesn’t return anything. But not all mutators return void. For example, Set.add() returns a boolean that indicates whether the set was actually changed. In Java’s graphical user interface toolkit, Component.add() returns the object itself, so that multiple add() calls can be chained together.

Abstract Data Type Examples

Here are some examples of abstract data types, along with some of their operations, grouped by kind.

int is Java’s primitive integer type. int is immutable, so it has no mutators.

List is Java’s list interface. List is mutable. List is also an interface, which means that other classes provide the actual implementation of the data type. These classes include ArrayList and LinkedList.

String is Java’s string type. String is immutable.

This classification gives some useful terminology, but it’s not perfect. In complicated data types, there may be an operation that is both a producer and a mutator, for example. Some people reserve the term producer only for operations that do no mutation.

MITx → Operations

Designing an Abstract Type

Designing an abstract type involves choosing good operations and determining how they should behave. Here are a few rules of thumb.

It’s better to have a few, simple operations that can be combined in powerful ways, rather than lots of complex operations.

Each operation should have a well-defined purpose, and should have a coherent behavior rather than a panoply of special cases. We probably shouldn’t add a sum operation to List, for example. It might help clients who work with lists of Integers, but what about lists of Strings? Or nested lists? All these special cases would make sum a hard operation to understand and use.

The set of operations should be adequate in the sense that there must be enough to do the kinds of computations clients are likely to want to do. A good test is to check that every property of an object of the type can be extracted. For example, if there were no get operation, we would not be able to find out what the elements of a list are. Basic information should not be inordinately difficult to obtain. For example, the size method is not strictly necessary for List, because we could apply get on increasing indices until we get a failure, but this is inefficient and inconvenient.

The type may be generic: a list or a set, or a graph, for example. Or it may be domain-specific: a street map, an employee database, a phone book, etc. But it should not mix generic and domain-specific features. A Deck type intended to represent a sequence of playing cards shouldn’t have a generic add method that accepts arbitrary objects like ints or Strings. Conversely, it wouldn’t make sense to put a domain-specific method like dealCards into the generic type List.

Representation Independence

Critically, a good abstract data type should be representation independent. This means that the use of an abstract type is independent of its representation (the actual data structure or data fields used to implement it), so that changes in representation have no effect on code outside the abstract type itself. For example, the operations offered by List are independent of whether the list is represented as a linked list or as an array.

You won’t be able to change the representation of an ADT at all unless its operations are fully specified with preconditions and postconditions, so that clients know what to depend on, and you know what you can safely change.

Example: Different Representations for Strings

Let’s look at a simple abstract data type to see what representation independence means and why it’s useful. The MyString type below has far fewer operations than the real Java String, and their specs are a little different, but it’s still illustrative. Here are the specs for the ADT:

/** String represents an immutable sequence of characters. */
public class MyString { 

    //////////////////// Example of a creator operation ///////////////
    /** @param b a boolean value
      * @return string representation of b, either "true" or "false" */
    public static MyString valueOf(boolean b) { ... }

    //////////////////// Examples of observer operations ///////////////
    /** @return number of characters in this string */
    public int length() { ... }

    /** @param i character position (requires 0 <= i < string length)
      * @return character at position i
      */
    public char charAt(int i) { ... }

    //////////////////// Example of a producer operation ///////////////    
    /** Get the substring between start (inclusive) and end (exclusive).
     * @param start starting index
     * @param end ending index.  Requires 0 <= start <= end <= string length.
     * @return string consisting of charAt(start)...charAt(end-1)
     */
    public MyString substring (int start, int end) { ... }
}

These public operations and their specifications are the only information that a client of this data type is allowed to know. Following the test-first programming paradigm, in fact, the first client we should create is a test suite that exercises these operations according to their specs. At the moment, however, writing test cases that use assertEquals directly on MyString objects wouldn’t work, because we don’t have an equality operation defined on these MyStrings. We’ll talk about how to implement equality carefully in the next reading. For now, the only operations we can perform with MyStrings are the ones we’ve defined above: valueOf, length, charAt, and substring. Our tests have to limit themselves to those operations. For example, here’s one test for the valueOf operation:

MyString s = MyString.valueOf(true);
assertEquals(4, s.length());
assertEquals('t', s.charAt(0));
assertEquals('r', s.charAt(1));
assertEquals('u', s.charAt(2));
assertEquals('e', s.charAt(3));

We’ll come back to the question of testing ADTs in a later section of this reading.

For now, let’s look at a simple representation for MyString: just an array of characters, exactly the length of the string (no extra room at the end). Here’s how that internal representation would be declared, as an instance variable within the class:

private char[] a;

With that choice of representation, the operations would be implemented in a straightforward way:

public static MyString valueOf(boolean b) {
    MyString s = new MyString();
    s.a = b ? new char[] { 't', 'r', 'u', 'e' } 
            : new char[] { 'f', 'a', 'l', 's', 'e' };
    return s;
}

public int length() {
    return a.length;
}

public char charAt(int i) {
    return a[i];
}

public MyString substring (int start, int end) {
    MyString that = new MyString();
    that.a = new char[end - start];
    System.arraycopy(this.a, start, that.a, 0, end - start);
    return that;
}

Question to ponder: Why don’t charAt and substring have to check whether their parameters are within the valid range? What do you think will happen if the client calls these implementations with illegal inputs?

One problem with this implementation is that it’s passing up an opportunity for performance improvement. Because this data type is immutable, the substring operation doesn’t really have to copy characters out into a fresh array. It could just point to the original MyString’s character array and keep track of the start and end that the new substring object represents. The String implementation in some versions of Java do this.

To implement this optimization, we could change the internal representation of this class to:

private char[] a;
private int start;
private int end;

With this new representation, the operations are now implemented like this:

    public static MyString valueOf(boolean b) {
        MyString s = new MyString();
        s.a = b ? new char[] { 't', 'r', 'u', 'e' } 
                : new char[] { 'f', 'a', 'l', 's', 'e' };
        s.start = 0;
        s.end = s.a.length;
        return s;
    }

    public int length() {
        return end - start;
    }

    public char charAt(int i) {
      return a[start + i];
    }

    public MyString substring (int start, int end) {
        MyString that = new MyString();
        that.a = this.a;
        that.start = this.start + start;
        that.end = this.start + end;
        return that;
    }

Because MyString’s existing clients depend only on the specs of its public methods, not on its private fields, we can make this change without having to inspect and change all that client code. That’s the power of representation independence.

MITx → Representations

Interfaces

In the Java tutorial, read these pages:

Java’s interface is a useful language mechanism for expressing an abstract data type. An interface in Java is a list of method signatures, but no method bodies. A class implements an interface if it declares the interface in its implements clause, and provides method bodies for all of the interface’s methods. So one way to define an abstract data type in Java is as an interface, with its implementation as a class implementing that interface.

One advantage of this approach is that the interface specifies the contract for the client and nothing more. The interface is all a client programmer needs to read to understand the ADT. The client can’t create inadvertent dependencies on the ADT’s rep, because instance variables can’t be put in an interface at all. The implementation is kept well and truly separated, in a different class altogether.

Another advantage is that multiple different representations of the abstract data type can co-exist in the same program, as different classes implementing the interface. When an abstract data type is represented just as a single class, without an interface, it’s harder to have multiple representations. We saw that in the MyString example above, which was a single class. We couldn’t have both representations for MyString in the same program.

Java’s static type checking allows the compiler to catch many mistakes in implementing an ADT’s contract. For instance, it is a compile-time error to omit one of the required methods, or to give it the wrong return type. Unfortunately, the compiler doesn’t check for us that the code adheres to the specs of those methods that are written in documentation comments.

Example: Set

Java’s collection classes provide a good example of the idea of separating interface and implementation. Let’s consider as an example one of the ADTs from the Java collections library, Set. Set is the ADT of finite sets of elements of some other type E. Here is a simplified version of the Set interface:

public interface Set<E> {

We can match Java interfaces with our classification of ADT operations, starting with a creator:

    // example of creator method

    /** Make an empty set.
     * @return a new set instance, initially empty
     */
    public static Set<E> make() { ... }

Unfortunately, Java interfaces are not allowed to have constructors, but (as of Java 8) they are allowed to contain static methods. So we can implement creator operations as static methods. This design pattern, using a static method as a creator instead of a constructor, is called a factory method. The MyString.valueOf method we saw earlier is also a factory method.

    // examples of observer methods

    /** Get size of the set.
     * @return the number of elements in this set. */
    public int size();

    /** Test for membership.
     * @param e an element
     * @return true iff this set contains e. */
    public boolean contains(E e);

Next we have two observer methods. Notice how the specs are in terms of our abstract notion of a set; it would be malformed to mention the details of any particular implementation of sets with particular private fields. These specs should apply to any valid implementation of the set ADT.


    // examples of mutator methods

    /** Modifies this set by adding e to the set.
     * @param e element to add. */
    public void add(E e);

    /** Modifies this set by removing e, if found.
     * If e is not found in the set, has no effect.
     * @param e element to remove.*/
    public void remove(E e);

The story for these three mutator methods is basically the same as for the observers. We still write specs at the level of our abstract model of sets.

In the Java tutorial, read these pages:

Why Interfaces?

Interfaces are used pervasively in real Java code. Not every class is associated with an interface, but there are a few good reasons to bring an interface into the picture.

MITx → Interfaces

Testing an Abstract Data Type

We build a test suite for an abstract data type by creating tests for each of its operations. These tests inevitably interact with each other, since the only way to test creators, producers, and mutators is by calling observers on the objects that result.

Here’s how we might partition the input spaces of the four operations in our MyString type:

// testing strategy for each operation of MyString:
//
// valueOf():
//    true, false
// length(): 
//    string len = 0, 1, n
//    string = produced by valueOf(), produced by substring()
// charAt(): 
//    string len = 1, n
//    i = 0, middle, len-1
//    string = produced by valueOf(), produced by substring()
// substring():
//    string len = 0, 1, n
//    start = 0, middle, len
//    end = 0, middle, len
//    end-start = 0, n
//    string = produced by valueOf(), produced by substring()

Then a compact test suite that covers all these partitions might look like:

@Test public void testValueOfTrue() {
    MyString s = MyString.valueOf(true);
    assertEquals(4, s.length());
    assertEquals('t', s.charAt(0));
    assertEquals('r', s.charAt(1));
    assertEquals('u', s.charAt(2));
    assertEquals('e', s.charAt(3));
}

@Test public void testValueOfFalse() {
    MyString s = MyString.valueOf(false);
    assertEquals(5, s.length());
    assertEquals('f', s.charAt(0));
    assertEquals('a', s.charAt(1));
    assertEquals('l', s.charAt(2));
    assertEquals('s', s.charAt(3));
    assertEquals('e', s.charAt(4));
}

@Test public void testEndSubstring() {
    MyString s = MyString.valueOf(true).substring(2, 4);
    assertEquals(2, s.length());
    assertEquals('u', s.charAt(0));
    assertEquals('e', s.charAt(1));
}

@Test public void testMiddleSubstring() {
    MyString s = MyString.valueOf(false).substring(1, 2);
    assertEquals(1, s.length());
    assertEquals('a', s.charAt(0));
}

@Test public void testSubstringIsWholeString() {
    MyString s = MyString.valueOf(false).substring(0, 5);
    assertEquals(5, s.length());
    assertEquals('f', s.charAt(0));
    assertEquals('a', s.charAt(1));
    assertEquals('l', s.charAt(2));
    assertEquals('s', s.charAt(3));
    assertEquals('e', s.charAt(4));
}

@Test public void testSubstringOfEmptySubstring() {
    MyString s = MyString.valueOf(false).substring(1, 1).substring(0, 0);
    assertEquals(0, s.length());
}

Notice that each test case typically calls a few operations that make or modify objects of the type (creators, producers, mutators) and some operations that inspect objects of the type (observers). As a result, each test case covers parts of several operations.

MITx → Testing

Invariants

Resuming our discussion of what makes a good abstract data type, the final, and perhaps most important, property of a good abstract data type is that it preserves its own invariants. An invariant is a property of a program that is always true, for every possible runtime state of the program. Immutability is one crucial invariant that we’ve already encountered: once created, an immutable object should always represent the same value, for its entire lifetime. Saying that the ADT preserves its own invariants means that the ADT is responsible for ensuring that its own invariants hold. It doesn’t depend on good behavior from its clients.

When an ADT preserves its own invariants, reasoning about the code becomes much easier. If you can count on the fact that Strings never change, you can rule out that possibility when you’re debugging code that uses Strings – or when you’re trying to establish an invariant for another ADT that uses Strings. Contrast that with a string type that guarantees that it will be immutable only if its clients promise not to change it. Then you’d have to check all the places in the code where the string might be used.

Immutability

We’ll see many interesting invariants. Let’s focus on immutability for now. Here’s a specific example:

/**
 * This immutable data type represents a tweet from Twitter.
 */
public class Tweet {

    public String author;
    public String text;
    public Date timestamp;

    /**
     * Make a Tweet.
     * @param author    Twitter user who wrote the tweet.
     * @param text      text of the tweet
     * @param timestamp date/time when the tweet was sent
     */
    public Tweet(String author, String text, Date timestamp) {
        this.author = author;
        this.text = text;
        this.timestamp = timestamp;
    }
}

How do we guarantee that these Tweet objects are immutable – that, once a tweet is created, its author, message, and date can never be changed?

The first threat to immutability comes from the fact that clients can — in fact must — directly access its fields. So nothing’s stopping us from writing code like this:

Tweet t = new Tweet("justinbieber", 
                    "Thanks to all those beliebers out there inspiring me every day", 
                    new Date());
t.author = "rbmllr";

This is a trivial example of representation exposure, meaning that code outside the class can modify the representation directly. Rep exposure like this threatens not only invariants, but also representation independence. We can’t change the implementation of Tweet without affecting all the clients who are directly accessing those fields.

Fortunately, Java gives us language mechanisms to deal with this kind of rep exposure:

public class Tweet {

    private final String author;
    private final String text;
    private final Date timestamp;

    public Tweet(String author, String text, Date timestamp) {
        this.author = author;
        this.text = text;
        this.timestamp = timestamp;
    }

    /** @return Twitter user who wrote the tweet */
    public String getAuthor() {
        return author;
    }

    /** @return text of the tweet */
    public String getText() {
        return text;
    }

    /** @return date/time when the tweet was sent */
    public Date getTimestamp() {
        return timestamp;
    }

}

The private and public keywords indicate which fields and methods are accessible only within the class and which can be accessed from outside the class. The final keyword also helps by guaranteeing that the fields of this immutable type won’t be reassigned after the object is constructed.

retweetLater breaking Tweet's immutability

But that’s not the end of the story: the rep is still exposed! Consider this perfectly reasonable client code that uses Tweet:

/** @return a tweet that retweets t, one hour later*/
public static Tweet retweetLater(Tweet t) {
    Date d = t.getTimestamp();
    d.setHours(d.getHours()+1);
    return new Tweet("rbmllr", t.getText(), d);
}

retweetLater takes a tweet and should return another tweet with the same message (called a retweet) but sent an hour later. The retweetLater method might be part of a system that automatically echoes funny things that Twitter celebrities say.

What’s the problem here? The getTimestamp call returns a reference to the same date object referenced by tweet t. t.timestamp and d are aliases to the same mutable object. So when that date object is mutated by d.setHours(), this affects the date in t as well, as shown in the snapshot diagram.

Tweet’s immutability invariant has been broken. The problem is that Tweet leaked out a reference to a mutable object that its immutability depended on. We exposed the rep, in such a way that Tweet can no longer guarantee that its objects are immutable. Perfectly reasonable client code created a subtle bug.

We can patch this kind of rep exposure by using defensive copying: making a copy of a mutable object to avoid leaking out references to the rep. Here’s the code:

public Date getTimestamp() {
    return new Date(Date.getTime());
}

Mutable types often have a copy constructor that allows you to make a new instance that duplicates the value of an existing instance. In this case, Date’s copy constructor uses the timestamp value, measured in seconds since January 1, 1970. As another example, StringBuilder’s copy constructor takes a String. Another way to copy a mutable object is clone(), which is supported by some types but not all. There are unfortunate problems with the way clone() works in Java. For more, see Josh Bloch, Effective Java, item 10.

tweetEveryHourToday breaking Tweet's immutability

So we’ve done some defensive copying in the return value of getTimestamp. But we’re not done yet! There’s still rep exposure. Consider this (again perfectly reasonable) client code:

/** @return a list of 24 inspiring tweets, one per hour today */
public static List<Tweet> tweetEveryHourToday () {
    List<Tweet> list = new ArrayList<Tweet>(); 
    Date date = new Date();
    for (int i=0; i < 24; i++) {
        date.setHours(i);
        list.add(new Tweet("rbmllr", "keep it up! you can do it", date));
    } 
    return list;
}

This code intends to advance a single Date object through the 24 hours of a day, creating a tweet for every hour. But notice that the constructor of Tweet saves the reference that was passed in, so all 24 Tweet objects end up with the same time, as shown in this snapshot diagram.

Again, the immutability of Tweet has been violated. We can fix this problem too by using judicious defensive copying, this time in the constructor:

public Tweet(String author, String text, Date timestamp) {
    this.author = author;
    this.text = text;
    this.timestamp = new Date(timestamp.getTime());
}

In general, you should carefully inspect the argument types and return types of all your ADT operations. If any of the types are mutable, make sure your implementation doesn’t return direct references to its representation. Doing that creates rep exposure.

You may object that this seems wasteful. Why make all these copies of dates? Why can’t we just solve this problem by a carefully written specification, like this?

/**
 * Make a Tweet.
 * @param author    Twitter user who wrote the tweet.
 * @param text      text of the tweet
 * @param timestamp date/time when the tweet was sent. Caller must never 
 *                   mutate this Date object again!
 */
public Tweet(String author, String text, Date timestamp) {

This approach is sometimes taken when there isn’t any other reasonable alternative – for example, when the mutable object is too large to copy efficiently. But the cost in your ability to reason about the program, and your ability to avoid bugs, is enormous. In the absence of compelling arguments to the contrary, it’s almost always worth it for an abstract data type to guarantee its own invariants, and preventing rep exposure is essential to that.

An even better solution is to prefer immutable types. If we had used an immutable date object, like java.time.ZonedDateTime, instead of the mutable java.util.Date, then we would have ended this section after talking about public and private. No further rep exposure would have been possible.

Immutable Wrappers Around Mutable Data Types

The Java collections classes offer an interesting compromise: immutable wrappers.

Collections.unmodifiableList() takes a (mutable) List and wraps it with an object that looks like a List, but whose mutators are disabled – set(), add(), remove() throw exceptions. So you can construct a list using mutators, then seal it up in an unmodifiable wrapper (and throw away your reference to the original mutable list), and get an immutable list.

The downside here is that you get immutability at runtime, but not at compile time. Java won’t warn you at compile time if you try to sort() this unmodifiable list. You’ll just get an exception at runtime. But that’s still better than nothing, so using unmodifiable lists, maps, and sets can be a very good way to reduce the risk of bugs.

How to Establish Invariants

An invariant is a property that is true for the entire program – which in the case of an invariant about an object, reduces to the entire lifetime of the object.

To make an invariant hold, we need to:

Translating this in terms of the types of ADT operations, this means:

The risk of rep exposure makes the situation more complicated. If the rep is exposed, then the object might be changed anywhere in the program, not just in the ADT’s operations, and we can’t guarantee that the invariant still holds after those arbitrary changes. So the full rule for proving invariants is:

Structural induction. If an invariant of an abstract data type is

  1. established by creators and producers;
  2. preserved by mutators, and observers; and
  3. no representation exposure occurs,

then the invariant is true of all instances of the abstract data type.

MITx → Invariants

Summary

These ideas connect to our three key properties of good software as follows: