CMPSCI 401: Theory of Computation

First Midterm Exam, Spring 2013

David Mix Barrington

20 February 2013

Directions:

• Answer the problems on the exam pages.
• There are nine problems (some with multiple parts) for 125 total points plus 10 extra credit. Actual scale was A = 115, C = 75.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first five questions are statements -- in each case say whether the statement is true or false and give a convincing justification of your answer -- a proof, counterexample, quotation from the book or from lecture, etc. You get five points for the correct boolean answer (so there is no reason not to guess if you don't know) and up to five for the justification.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 10 points
  Q6: 10 points
  Q7: 10 points
  Q8: 15 points
  Q8: 40+10 points
 Total: 125+10 points

The language ABC over the alphabet Σ = {a, b, c} is defined as the set of all strings that contain at least one a, at least one b, and at least one c.

If X and Y are any two languages over the same alphabet, the symmetric difference X Δ Y is defined to be the set of strings that are in either X or Y, but not in both.

• Question 1 (10): True or false with justification: Let L be a language over some nonempty alphabet Σ such that there exists a positive integer k such that for every string w in L, w has at most k letters. (In symbols, ∃k: ∀w:(w ∈ L) → (|w| ≤ k).) Then L must be a regular language.

• Question 2 (10): True or false with justification: Let L be a language over some nonempty alphabet Σ such that for every string w in L, there exists a positive integer k such that w has at most k letters. (In symbols, ∀w: (w ∈ L) → ( ∃k: |w| ≤ k).) Then L must be a regular language.

• Question 3 (10): True or false with justification: The language ABC (defined above) is the language of some DFA with exactly six states.

• Question 4 (10): True or false with justification: The language {w: ∃k: ∃y: (k ≥ 1) ∧ (w = 1^k0y) ∧ (y has at most k ones)} is context-free. (Here the alphabet is {0, 1}.)

• Question 5 (10): True or false with justification: If X and Y are regular languages, then X Δ Y is also regular. (The symmetric difference operator is defined above.)

• Question 6 (10): We define a queue machine to be analogous to a PDA. It is nondeterministic and has a finite state set, a start state, a final state set, an input alphabet, and a queue alphabet. But its transitions enqueue and dequeue single characters from a queue rather than push and pop characters from a stack. Describe (in English) a queue machine whose language is not context-free.

• Question 7 (10): Find a regular expression either for the language ABC defined above, or for its complement. (Your choice, no extra credit for both. If you attempt both, tell me which one you want me to grade.)

• Question 8 (15): Let N be the NFA with state set {1, 2, 3, 4}, start state 1, final state set {2, 3}, alphabet {a, b}, and the following six transitions (arrows): (1, a, 2), (1, ε, 3), (2, b, 2), (2, ε, 3), (3, b, 4), and (4, a, 2).
  • (a, 5) Circle exactly the strings on this list that are in the language L(N): ε, a, b, aa, ab, ba, bb, aaa, aab, aba, abb, baa, bab, bba, bbb.

  • (b, 10) Construct an ordinary DFA D such that L(D) = L(N). (Any correct DFA gets full credit, but showing your reasoning is important to get partial credit for a wrong answer.)

• Question 9 (40+10): Let G be the grammar with nonterminals S and T, terminals a, b, and c, start symbol S, and rules S → TaT, T → bS, and T → c.
  • (a, 10): Give a PDA M such that L(M) = L(G). (The simplest thing is probably to give the top-down parser, but any correct PDA gets full credit. Explaining your reasoning may help for partial credit.)

  • (b, 10): Give a PDA M' such that L(M') = L(G) and such that M' meets the three conditions for the proof that any PDA may be simulated by a context-free grammar.

  • (c, 10): Prove that for any positive integer k, there is a string in L(G) with exactly 3k letters.

  • (d, 10): Prove that every string in L(G) has a number of letters that is divisible by 3. (Hint: One way to do this is to simultaneously prove a fact about the lengths of strings that are derivable from the nonterminal T in the grammar G.)

  • (e, 10XC): Prove that L(G) is not a regular language. (Hint: Work with the intersection of L(G) and the regular language b^*(a ∪ c)^* -- if the intersection is not regular than L(G) is not regular.)

Last modified 15 May 2013