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CSE 331: Software Design & Implementation

CSE 331: Section 4 — Loop Invariants & Data Abstraction

This section's questions concern the following ADT:

 /**
  * Represents a **mutable** integer set, or a collection of distinct integers.
  */
  public class MutableIntSet {
    /**
     * Determines whether n is in the set.
     * @param n the number to look for in the set
     * @return true if n is in the set, false otherwise
     */
    public boolean contains(int n);

    /**
     * Adds n to the set if not already present.
     * @param n the number to add to the new set.
     * @modifies this
     * @effects this is unchanged if this_0 contains n
     *          otherwise, this contains all of this_0 and n
     */
    public void add(int n);

    /**
     * Removes the desired int from the set.
     * @param n The int to remove
     * .... To complete in part d
     */
    public boolean remove(int n);
  }

Task 1 — Everybody Loops

In this problem, we will show the correctness of a method containing a loop that finds the quotient of \(x\) divided by 10, i.e., the largest value \(y\) such that \(10y \le x\). To say that \(y\) is the largest such value means that any larger value would not satisfy the inequality, i.e., that \(10(y+1) \not\le x\).

We denote the initial value of the parameter \(x\) at the top of the method by \(x_0\). This is explicitly stated in the precondition as the fact “\(x = x_0\)” (note this is not strictly necessary - you always know \(x = x_0\) until you modify it!). The first two facts of Q are from the spec postcondition, which say that \(y\) is the quotient of \(x_0\) divided by 10. The third fact is specific to our implementation, and says that \(x\) is the remainder, i.e., the remaining amount not divisible by 10.

This method calculates the quotient without division. Instead, it just uses subtraction. It operates by increasing y and decreasing x each time around. The first part of the invariant says that the distance from \(x_0\) down to \(10y\) (i.e., \(x_0 - 10y\)) is the same as the distance from \(x\) down to 0 (i.e., \(x - 0 = x\)). The second part of the invariant says that \(x\) has not moved below 0 (i.e., \(x \ge 0\)).

// Computes the integer quotient of x divided by 10
// @param x The numerator
// @requires x >= 0
// @return The largest integer y such that 10 * y <= x_0
public static int divideByTen(int x) {
  {{ x = x_0 and x_0 >= 0 }}
  int y = 0;
  {{ P1: _________________________ }}
  {{ Inv: x_0 - 10y = x and x >= 0 }}
  while (x >= 10) {
    {{ _________________________ }}
    y = y + 1;
    {{ _________________________ }}
    x = x - 10;
    {{ P3: _________________________ }}
    {{ Q2: _________________________ }}
  }
  {{ P2: _________________________ }}
  {{ Q1: 10y <= x_0 and x_0 < 10(y+1) and x = x_0 - 10y }}
  return y;
}

  1. Fill in P1, then show that the invariant is true when we get to the top of the loop the first time.

  2. Fill in P2, then show that Q1 holds when we exit the loop.

  3. Fill in Q2 (Hint: what do we know at the end of a loop?). Then, forward reason to P3. Show that P3 implies Q2, proving that the body of the loop is correct.

Task 2 — Rally the Loops

In this problem, you will implement the following function.

/**
 * Writes over each copy of y in A with the value z.
 * @param A the array to replace values in
 * @param y the value to be replaced in A
 * @param z the value to replace y with in A
 * @modifies A
 * @effects A = A_0 with every instance of y replaced with a z
 */
public void replace(int[] A, int y, int z) { .. }

With each loop invariant below, fill in the missing parts of the code to make it correct with the given invariant.

  1. int i = ____________________

// Inv: A[0 .. i] = A_0[0 .. i] with every y replaced with a z
// and A[i .. A.length] = A_0[i .. A.length]
while (________________________________________) {







}

  2. int i = ____________________

// Inv: A[0 .. i] = A_0[0 .. i] and A[i .. A.length] = A_0[i .. A.length] with
// every y replaced with a z
while (________________________________________) {







}

Task 3 — RIch AF

In this problem, we will return to the original specification of MutableIntSet, whose abstract state is a set of elements. We will consider three different concrete representations for it.

Note the notation for this = this.elems[0 .. size] means that this is the set containing the first size elements from this.elems just like how an ArrayList would work!

    public class MutableIntSetImpl implements MutableIntSet {

(1)   // AF: this = this.elems[0 .. size]
      private int[] elems;
      private int size;

(2)   // AF: this = this.elems[0 .. size]
      // RI: this.elems contains no dups
      private int[] elems;
      private int size;

(3)   // AF: this = this.elems[0 .. size]
      // RI: this.elems is sorted
      private int[] elems;
      private int size;

      public MutableIntSetImpl() {
        this.elems = new int[10];
        this.size = 0;
      }

For each of the methods shown below, state the concrete representations (1--3) for which it would satisfy the specification of the method in MutableIntSet. In each case, briefly explain why.

  1.   public boolean contains(int n) {
    return Arrays.binarySearch(this.elems, n) >= 0;
  }

    Note: Binary Search is an algorithm that finds the position of a target value within a sorted array.

  2.   /**
   * Determines whether n is in the set.
   * @param n the number to look for in the set
   * @return true if n is in the set, false otherwise
   */
  public boolean contains(int n) {
    for (int i = 0; i < this.size; i++) {
      if (this.elems[i] == n)
        return true;
    }
    return false;
  }

  3.   /**
   * Adds n to the set if not already present.
   * @param n the number to add to the new set.
   * @modifies this
   * @effects this is unchanged if this_0 contains n
   *          otherwise, this contains all of this_0 and n
   */
  public void add(int n) {
    if (!this.contains(n)) {
      if (size >= this.elems.length) {
        int[] temp = new int[size * 2 + 1];
        for (int i = 0; i < this.elems.length; i++) {
          temp[i] = this.elems[i];
        }
        this.elems = temp;
      }
      this.elems[size] = n;
      size++;
    }
  }

  4.   /**
   * Removes the desired int from the set.
   * @param n The int to remove
   * .... Based on your Task 1 part d solution
   */
  public boolean remove(int n) {
    for (int i = 0; i < size; i++) {
      if (this.elems[i] == n) {
        size--;
        for (int j = i; j < size; j++) {
          this.elems[j] = this.elems[j + 1];
        }
        return true;
      }
    }
    return false;
  }

Task 4 — Good News and Add News

Answer the following questions about the specification of MutableIntSet. Assume that T is an instance of this class whose abstract state is {1, 2, 3}.

  1. Would T.add(3) actually change this? If not, why is that allowed when it says @modifies this?

  2. Consider the following static method.

    /**
 * Adds n to the set if not already present.
 * @param old the set to add to
 * @param n the number to add to the new set.
 * @requires old is not null
 * @return a set with n and all of the elements of old.
 * If old.contains(n), the new set has all the same elements as 'old'.
 */
public static MutableIntSet add(MutableIntSet old, int n);

    Now, consider a call T.add(4). Explain how the operation of MutableIntSet.add differs from that of a call to static add(T, 4) in terms of this.

  3. What is the abstract state of \(T\) after the following code (This is forward reasoning.):

         T.add(4);
     T.add(2);
     T.add(0);

  4. Write a specification for the method remove. You should have two cases - n is in the set, and n is not. Clearly explain how the abstract state changes after the method call and what is returned.