CMPSCI 250: Introduction to Computation

Second Midterm Exam

David Mix Barrington

4 November 2013

Directions:

• Answer the problems on the exam pages.
• There are five problems, some with multiple parts, for 100 total points plus 10 extra credit. Actual scale A = 93, C = 63, but may be adjusted.
• Some useful definitions precede the questions below.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "2¹⁷ - 4" need not be reduced to a single integer.

  Q1: 10 points
  Q2: 15 points
  Q3: 15+10 points
  Q4: 30 points
  Q5: 30 points
Total: 100+10 points

Definitions and Other Notes:

N is the set of naturals, {0, 1, 2, 3,...}.

Question 3 uses an inductively defined function from {a, b}^* to N. We define f(λ) = 0, and for any string w we define f(wa) = f(w) + 1 and f(wb) = 2f(w).

Question 4 uses the following inductive definition of "set expression":

1. An atomic set expression has a value that is a set of int values.
2. If E and F are set expressions, so are E ∪ F, E ∩ F, and E-bar, denoting the union of E and F, the intersection of E and F, and the complement of E respectively.
3. Nothing else is a set expression.

Question 4 assumes that we have defined a real-Java class IntSet whose objects are sets of int values. The only method of this class that you will need is public boolean isIn (int x), which returns true if and only if the int value x is in the calling set. Note that Question 4 will ask you to write a different method isIn, for the SetExpression class.

We also have the following real-Java class for set expressions:

public class SetExpression {
   public static final int UNION = 1;
   public static final int INTERSECTION = 2;
   public static final int COMPLEMENT = 3;
   boolean isLeaf;
   IntSet leafValue;
   SetExpression left, right;
   int op;

   // usual constructors, getters, and setters

Question 5 uses the following labeled undirected graph:

  [a] ---1--- [s] ---1--- [f]
   |           |           |
   3           6           9
   |           |           |
  [b] ---1--- [d] ---1--- [g]
   |           |           |
   3           6           9
   |           |           |
  [c] ---1--- [e] ---1--- [h]

• Question 1 (10): Identify each of the following five concepts, giving enough detail to make it clear that you are familiar with them (2 points each):
  • (a, 2) the Fibonacci numbers

  • (b, 2) the one's complement of a binary string

  • (c, 2) an undirected cycle in an undirected graph

  • (d, 2) a heuristic for an A search

  • (e, 2) a bipartite graph

• Question 2 (15): Let R be a binary relation on some set A and assume that R is transitive. By the definition of transitivity, if R(x, y) and R(y, z) are true, then R(x, z) must be true. Suppsoe that x₁, x₂,..., x_k are any k elements of A, and that we know that R(x_i, x_i+1) is true for every i with i ≤ i < k. Prove that R(x₁, x_k) is true. (Hint: Use ordinary induction, with a base case of k = 3.)

• Question 3 (15+10): This question uses the function f from {a, b}^* to N defined above.
  • (a, 5) Evaluate f(aabb), f(baba), and f(abba).

  • (b, 10) Prove by induction on all naturals n that there exists a string w such that f(w) = n.

  • (c, 10) Prove that for any natural n, there exists a string w such that (1) w never has two a's in a row, and (2) f(w) = n. (Hint: Use strong induction on all naturals n. If you know that f(w) = n, how can you get strings whose f values are 2n and 2n + 1?

• Question 4 (30): This question uses the definition of set expressions and the Java classes IntSet and SetExpression, all defined above.
  • (a, 5) If A denotes the set {2, 3, 7}, B denotes the set {3, 6, 7}, and C denotes the set {2, 7, 9}, what set is denoted by the set expression A ∩ (B ∪ C-bar)?

  • (b, 15) Write a real-Java method public boolean isIn (int x) for the SetExpression class. It should return true if and only if the int value x is in the set denoted by the calling SetExpression object.

  • (c, 10) Prove by induction on all set expressions E that your method returns the correct result when called on E. Note that to prove somethgin about your method, your answer has better refer to the code of your method!

• Question 5 (30): This question uses the undirected graph G pictured above. In every one of these searches, (1) when two or more nodes go into the open list at the same time, they come out in alphabetical order, and (2) a closed list is used so that the search can recognize nodes that have already been processed, according to the respective types of search. Each search has start node s and goal node g.
  • (a, 10) Trace the depth-first search from s and draw the DFS tree, stopping when g has been found. Ignore the edge labels.

  • (b, 10) Trace the breadth-first search from s and draw the BFS tree, stopping when G has been found. Ignore the edge labels.

  • (c, 10) Perform a uniform-cost search of the graph from s, using the edge labels, and stopping when the algorithm may conclude that it has found the best-path distance from s to g.

Last modified 11 November 2013