CMPSCI 501: Theory of Computation

First Midterm Exam, Spring 2016

David Mix Barrington

17 February 2016

Directions:

• Answer the problems on the exam pages.
• There are twelve problems (some with multiple parts) for 130 total points. 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: 15 points
  Q8: 10 points
  Q9: 10 points
  Q10: 10 points
  Q11: 10 points
  Q12: 15 points
 Total: 130 points

The language X over the alphabet {a, b, c} is denoted by the regular expression (ab)^*c(ab)^*.

The language Y over the alphabet {a, b, c} is the set {wcw: w ∈ {a, b}^*}.

The grammar G has input alphabet {a, b, c}, nonterminals S and T, start symbol S, and rules S → aTb, T → bSa, and S → c.

• Question 1 (10): True or false with justification: If L is any language that is not context-free, and R is any regular language, then L ∩ R cannot be context-free.

• Question 2 (10): True or false with justification: The language L(G), where G is the grammar defined above, is regular.

• Question 3 (10): True or false with justification: For any non-negative integer n, let f(n) be the number of strings of length n in the language X defined above. Then f(n) = O(1), that is, there is some constant c such that for all sufficiently large n, f(n) ≤ c.

• Question 4 (10): True or false with justification: For any non-negative integer n, let g(n) be the number of strings of length n in the language Y defined above. Then g(n) is not polynomially bounded, that is, it is not the case there is a polynomial p such that, for sufficiently large n, we have g(n) ≤ p(n). (Note: The paraphrase of "polynomially bounded" was incorrect on the actual exam sheet.)

• Question 5 (10): True or false with justification: Let M and N by any two DFA's. Let Z be the language {u₁v₁u₂v₂...u_kv_k: k ≤ 0, u_i ∈ L(M), v_i L(N)}. Then Z must be a regular language.

• Question 6 (10): Prove that there does not exist any PDA M such that Y = L(M), wehre Y is defined above.

• Question 7 (15): Suppose that M' is a PDA in which the transition (p, ε, ε; q, ε) occurs. (Recall that ε is the empty string.)

  • (a, 5) If we transform M' into a PDA in the correct normal form for the PDA-to-CFG construction, what transitions and other entities will we add to replace the transition (p, ε, ε; q, ε)?

  • (b, 10) What rules of the resulting CFG will come from the transitions added in part (a)?

• Question 8 (10): Give an NFA N such that L(N) = X, using the regular expression for X given above. Remember that NFA's may include ε-moves.

• Question 9 (10): Build (by any valid method) a DFA D such that L(D) = X, where X is given above. Explain why you believe your DFA to be correct.

• Question 10 (10): Describe the equivalence classes for the relation of X-equivalence, in the sense of the Myhill-Nerode Theorem. (Again, X is the language given above.) You may use any valid method.

• Question 11 (10): Describe a PDA M_G such that L(M_G) = L(G), where G is the grammar given above. You may use any valid method.

• Question 12 (15): Let L₁, a subset of {a, b, c}^*, be the set of strings that contain exactly one c.
  • (a, 5) Build a DFA D₁ such that L(D₁) = L₁.

  • (b, 10) Use the construction given in lecture and in the text to create a regular expression for the language L₁ from your DFA.

Last modified 26 February 2016