Reading 7: Designing Specifications

In this reading we’ll look at different specs for similar behaviors, and talk about the tradeoffs between them. We’ll look at three dimensions for comparing specs:

• How deterministic it is. Does the spec define only a single possible output for a given input, or allow the implementor to choose from a set of legal outputs?

• How declarative it is. Does the spec just characterize what the output should be, or does it explicitly say how to compute the output?

• How strong it is. Does the spec have a small set of legal implementations, or a large set?

Not all specifications we might choose for a module are equally useful, and we’ll explore what makes some specifications better than others.

Recall the two example implementations of find we began with in the previous reading:

static int findFirst(int[] arr, int val) {
    for (int i = 0; i < arr.length; i++) {
        if (arr[i] == val) return i;
    }
    return arr.length;
}

static int findLast(int[] arr, int val) {
    for (int i = arr.length - 1 ; i >= 0; i--) {
        if (arr[i] == val) return i;
    }
    return -1;
}

The subscripts _First and _Last are not actual Java syntax. We’re using them here to distinguish the two implementations for the sake of discussion. In the actual code, both implementations should be Java methods called find.

Here is one possible specification of find:

static int findExactlyOne(int[] arr, int val)
  requires: val occurs exactly once in arr
  effects:  returns index i such that arr[i] = val

This specification is deterministic: when presented with a state satisfying the precondition, the outcome is completely determined. Only one return value and one final state is possible. There are no valid inputs for which there is more than one valid output.

Both findFirst and findLast satisfy the specification, so if this is the specification on which the clients relied, the two implementations are equivalent and substitutable for one another.

Here is a slightly different specification:

static int findOneOrMore,AnyIndex(int[] arr, int val)
  requires: val occurs in arr
  effects:  returns index i such that arr[i] = val

This specification is not deterministic. It doesn’t say which index is returned if val occurs more than once. It simply says that if you look up the entry at the index given by the returned value, you’ll find val. This specification allows multiple valid outputs for the same input.

Note that this is different from nondeterministic in the usual sense of that word. Nondeterministic code sometimes behaves one way and sometimes another, even if called in the same program with the same inputs. This can happen, for example, when the code’s behavior depends on a random number, or when it depends on the timing of concurrent processes. But a specification which is not deterministic doesn’t have to have a nondeterministic implementation. It can be satisfied by a fully deterministic implementation.

To avoid the confusion, we’ll refer to specifications that are not deterministic as underdetermined.

This underdetermined find spec is again satisfied by both findFirst and findLast, each resolving the underdeterminedness in its own (fully deterministic) way. A client of findOneOrMore,AnyIndex spec can’t rely on which index will be returned if val appears more than once. The spec would be satisfied by a nondeterministic implementation, too — for example, one that tosses a coin to decide whether to start searching from the beginning or the end of the array. But in almost all cases we’ll encounter, underdeterminism in specifications offers a choice that is made by the implementor at implementation time. An underdetermined spec is typically implemented by a fully-deterministic implementation.

static int findFirst(int[] arr, int val) {
    for (int i = 0; i < arr.length; i++) {
        if (arr[i] == val) return i;
    }
    return arr.length;
}

static int findLast(int[] arr, int val) {
    for (int i = arr.length - 1 ; i >= 0; i--) {
        if (arr[i] == val) return i;
    }
    return -1;
}

static int find(int[] arr, int val)
  effects: returns largest index i such that
             arr[i] = val, or -1 if no such i

static int findOneOrMore,AnyIndex(int[] arr, int val)
  requires: val occurs in arr
  effects:  returns index i such that arr[i] = val

static int find(int[] arr, int val)
  requires: val occurs exactly once in arr
  effects:  returns index i such that arr[i] = val

static int find(int[] arr, int val)
  effects: returns largest index i such that
             arr[i] = val, or -1 if no such i

static int find(int[] arr, int val)
  requires: val occurs in arr
  effects: returns largest index i such that arr[i] = val

Roughly speaking, there are two kinds of specifications. Operational specifications give a series of steps that the method performs; pseudocode descriptions are operational. Declarative specifications don’t give details of intermediate steps. Instead, they just give properties of the final outcome, and how it’s related to the initial state.

Almost always, declarative specifications are preferable. They’re usually shorter, easier to understand, and most importantly, they don’t inadvertently expose implementation details that a client may rely on (and then find no longer hold when the implementation is changed). For example, if we want to allow either implementation of find, we would not want to say in the spec that the method “goes down the array until it finds val,” since aside from being rather vague, this spec suggests that the search proceeds from lower to higher indices and that the lowest will be returned, which perhaps the specifier did not intend.

One reason programmers sometimes lapse into operational specifications is because they’re using the spec comment to explain the implementation for a maintainer. Don’t do that. When it’s necessary, use comments within the body of the method, not in the spec comment.

For a given specification, there may be many ways to express it declaratively:

static boolean startsWith(String str, String prefix)
 effects: returns true if and only if there exists String suffix
            such that prefix + suffix == str

static boolean startsWith(String str, String prefix)
 effects: returns true if and only if there exists integer i
            such that str.substring(0, i) == prefix

static boolean startsWith(String str, String prefix)
 effects: returns true if the first prefix.length() characters of str
            are the characters of prefix, false otherwise

It’s up to us to choose the clearest specification for clients and maintainers of the code.

static String join(String delimiter, String[] elements)
  effects: append together the strings in elements, but at each step,
             if there are more elements left, insert delimiter

Suppose you want to change a method – either how its implementation behaves, or the specification itself. There are already clients that depend on the method’s current specification. How do you compare the behaviors of two specifications to decide whether it’s safe to replace the old spec with the new spec?

A specification S2 is stronger than or equal to a specification S1 if

• S2’s precondition is weaker than or equal to S1’s,
  and
• S2’s postcondition is stronger than or equal to S1’s, for the states that satisfy S1’s precondition.

If this is the case, then an implementation that satisfies S2 can be used to satisfy S1 as well, and it’s safe to replace S1 with S2 in your program.

These two rules embody several ideas. They tell you that you can always weaken the precondition; placing fewer demands on a client will never upset them. And you can always strengthen the post-condition, which means making more promises.

For example, this spec for find:

static int findExactlyOne(int[] a, int val)
  requires: val occurs exactly once in a
  effects:  returns index i such that a[i] = val

can be replaced with:

static int findOneOrMore,AnyIndex(int[] a, int val)
  requires: val occurs at least once in a
  effects:  returns index i such that a[i] = val

which has a weaker precondition. This in turn can be replaced with:

static int findOneOrMore,FirstIndex(int[] a, int val)
  requires: val occurs at least once in a
  effects:  returns lowest index i such that a[i] = val

which has a stronger postcondition.

What about this specification:

static int findCanBeMissing(int[] a, int val)
  requires: nothing
  effects:  returns index i such that a[i] = val,
              or -1 if no such i

We’ll come back to findCanBeMissing in the exercises.

Imagine (very abstractly) the space of all possible Java methods.

Each point in this space represents a method implementation.

First we’ll diagram findFirst and findLast defined above. Look back at the code and see that findFirst and findLast are not specs. They are implementations, with method bodies that implement their actual behavior. So we denote them as points in the space.

A specification defines a region in the space of all possible implementations. A given implementation either behaves according to the spec, satisfying the precondition-implies-postcondition contract (it is inside the region), or it does not (outside the region).

Both findFirst and findLast satisfy findOneOrMore,AnyIndex, so they are inside the region defined by that spec.

We can imagine clients looking in on this space: the specification acts as a firewall.

• Implementors have the freedom to move around inside the spec, changing their code without fear of upsetting a client. This is crucial in order for the implementor to be able to improve the performance of their algorithm, the clarity of their code, or to change their approach when they discover a bug, etc.

• Clients don’t know which implementation they will get. They must respect the spec, but also have the freedom to change how they’re using the implementation without fear that it will suddenly break.

How will similar specifications relate to one another? Suppose we start with specification S1 and use it to create a new specification S2.

If S2 is stronger than S1, how will these specs appear in our diagram?

• Let’s start by strengthening the postcondition. If S2’s postcondition is now stronger than S1’s postcondition, then S2 is the stronger specification.

  Think about what strengthening the postcondition means for implementors: it means they have less freedom, the requirements on their output are stronger. Perhaps they previously satisfied findOneOrMore,AnyIndex by returning any index i, but now the spec demands the lowest index i. So there are now implementations inside findOneOrMore,AnyIndex but outside findOneOrMore,FirstIndex.

  Could there be implementations inside findOneOrMore,FirstIndex but outside findOneOrMore,AnyIndex? No. All of those implementations satisfy a stronger postcondition than what findOneOrMore,AnyIndex demands.

• Think through what happens if we weaken the precondition, which will again make S2 a stronger specification. Implementations will have to handle new inputs that were previously excluded by the spec. If they behaved badly on those inputs before, we wouldn’t have noticed, but now their bad behavior is exposed.

We see that when S2 is stronger than S1, it defines a smaller region in this diagram; a weaker specification defines a larger region.

In our figure, since findLast iterates from the end of the array arr, it does not satisfy findOneOrMore,FirstIndex and is outside that region.

Another specification S3 that is neither stronger nor weaker than S1 might overlap (such that there exist implementations that satisfy only S1, only S3, and both S1 and S3) or might be disjoint. In both cases, S1 and S3 are incomparable.

static int findExactlyOne(int[] a, int val)
  requires: val occurs exactly once in a
  effects:  returns index i such that a[i] = val

 static int findOneOrMore,AnyIndex(int[] a, int val)
  requires: val occurs at least once in a
  effects:  returns index i such that a[i] = val

 static int findOneOrMore,FirstIndex(int[] a, int val)
  requires: val occurs at least once in a
  effects:  returns lowest index i such that a[i] = val

static int findCanBeMissing(int[] a, int val)
  requires: nothing
  effects:  returns index i such that a[i] = val,
              or -1 if no such i

What makes a good method? Designing a method means primarily writing a specification.

About the form of the specification: it should obviously be succinct, clear, and well-structured, so that it’s easy to read.

The content of the specification, however, is harder to prescribe. There are no infallible rules, but there are some useful guidelines.

The specification should be coherent

static int sumFind(int[] a, int[] b, int val)
  effects: returns the sum of all indices in arrays a and b at which
             val appears

public static int LONG_WORD_LENGTH = 5;
public static String longestWord;

/**
 * Update longestWord to be the longest element of words, and print
 * the number of elements with length > LONG_WORD_LENGTH to the console.
 * @param words list to search for long words
 */
public static void countLongWords(List<String> words)

The results of a call should be informative

static V put (Map<K,V> map, K key, V val)
  requires: val may be null, and map may contain null values
  effects:  inserts (key, val) into the mapping,
              overriding any existing mapping for key, and
              returns old value for key, unless none,
              in which case it returns null

The specification should be strong enough

static void addAll(List<T> list1, List<T> list2)
  effects: adds the elements of list2 to list1,
             unless it encounters a null element,
             at which point it throws a NullPointerException

The specification should also be weak enough

static File open(String filename)
  effects: opens a file named filename

The specification should use abstract types where possible

static ArrayList<T> reverse(ArrayList<T> list)
  effects: returns a new list which is the reversal of list, i.e.
             newList[i] == list[n-i-1]
             for all 0 <= i < n, where n = list.size()

Another design issue is whether to use a precondition, and if so, whether the method code should attempt to make sure the precondition has been met before proceeding. In fact, the most common use of preconditions is to demand a property precisely because it would be hard or expensive for the method to check it.

As mentioned above, a non-trivial precondition inconveniences clients, because they have to ensure that they don’t call the method in a bad state (that violates the precondition); if they do, there is no predictable way to recover from the error. So users of methods don’t like preconditions. That’s why the Java API classes, for example, tend to specify (as a postcondition) that they throw unchecked exceptions when arguments are inappropriate. This approach makes it easier to find the bug or incorrect assumption in the caller code that led to passing bad arguments. In general, it’s better to fail fast, as close as possible to the site of the bug, rather than let bad values propagate through a program far from their original cause.

Sometimes, it’s not feasible to check a condition without making a method unacceptably slow, and a precondition is often necessary in this case. If we wanted to implement the find method using binary search, we would have to require that the array be sorted. Forcing the method to actually check that the array is sorted would defeat the entire purpose of the binary search: to obtain a result in logarithmic and not linear time.

The decision of whether to use a precondition is an engineering judgment. The key factors are the cost of the check (in writing and executing code), and the scope of the method. If it’s only called locally in a class, the precondition can be discharged by carefully checking all the sites that call the method. But if the method is public, and used by other developers, it would be less wise to use a precondition. Instead, like the Java API classes, you should throw an exception.

We have been using public for almost all of our methods, without really thinking about it. The decision to make a method public or private is actually a decision about the contract of the class. Public methods are freely accessible to other parts of the program. Making a method public advertises it as a service that your class is willing to provide. If you make all your methods public — including helper methods that are really meant only for local use within the class — then other parts of the program may come to depend on them, which will make it harder for you to change the internal implementation of the class in the future. Your code won’t be as ready for change.

Making internal helper methods public will also add clutter to the visible interface your class offers. Keeping internal things private makes your class’s public interface smaller and more coherent (meaning that it does one thing and does it well). Your code will be easier to understand.

We will see even stronger reasons to use private in the next few classes, when we start to write classes with persistent internal state. Protecting this state will help keep the program safe from bugs.

We have also been using static for almost all of our methods, again without much discussion. Static methods are not associated with any particular instance of a class, while instance methods (declared without the static keyword) must be called on a particular object.

Specifications for instance methods are written just the same way as specifications for static methods, but they will often refer to properties of the instance on which they were called.

For example, by now we’re very familiar with this specification:

static int find(int[] arr, int val)
  requires: val occurs in arr
  effects:  returns index i such that arr[i] = val

Instead of using an int[], what if we had a class IntArray designed for storing arrays of integers? The IntArray class might provide an instance method with the specification:

int find(int val)
  requires: val occurs in this array
  effects:  returns index i such that the value at index i in this array
              is val

We’ll have much more to say about specifications for instance methods in future classes!

public static int secondToLastIndexOf(int[] arr, int val)
  requires: val appears in arr an odd number of times
  effects:  returns the 2nd-largest i such that arr[i] == val

public static int secondToLastIndexOf(int[] arr, int val)
  requires: val appears in arr an odd number of times
  effects:  returns the 2nd-largest i such that arr[i] == val

public static int secondToLastIndexOf(int[] arr, int val)
  requires: val appears in arr an odd number of times
  effects:  returns the 2nd-largest i such that arr[i] == val

A specification acts as a crucial firewall between implementor and client — both between people (or the same person at different times) and between code. As we saw last time, it makes separate development possible: the client is free to write code that uses a module without seeing its source code, and the implementor is free to write the implementation code without knowing how it will be used.

Declarative specifications are the most useful in practice. Preconditions (which weaken the specification) make life harder for the client, but applied judiciously they are a vital tool in the software designer’s repertoire, allowing the implementor to make necessary assumptions.

As always, our goal is to design specifications that make our software:

• Safe from bugs. Without specifications, even the tiniest change to any part of our program could be the tipped domino that knocks the whole thing over. Well-structured, coherent specifications minimize misunderstandings and maximize our ability to write correct code with the help of static checking, careful reasoning, testing, and code review.

• Easy to understand. A well-written declarative specification means the client doesn’t have to read or understand the code. You’ve probably never read the code for, say, Python dict.update, and doing so isn’t nearly as useful to the Python programmer as reading the declarative spec.

• Ready for change. An appropriately weak specification gives freedom to the implementor, and an appropriately strong specification gives freedom to the client. We can even change the specs themselves, without having to revisit every place they’re used, as long as we’re only strengthening them: weakening preconditions and strengthening postconditions.