CMPSCI 401: Theory of Computation

First Midterm Exam, Spring 2010

David Mix Barrington

22 February 2010

Directions:

• Answer the problems on the exam pages.
• There are seven problems 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 four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification of your answer -- a proof, counterexample, quotation from the book or from lecture, etc. -- note that there is no reason not to guess if you don't know.

  Q1: 10 points
  Q2: 10 points
  Q3: 10 points
  Q4: 10 points
  Q5: 35+10 points
  Q6: 20 points (note the exam paper said "30" here)
  Q7: 30 points

  Total: 125+10 points

Several questions refer to the following PDA called P. P has state set {q₀, q, f}, start state q₀, final state set {f}, input alphabet {a, b}, and stack alphabet {$, c}. It has six transitions: (q₀, ε, ε; q, $), (q, a, ε; q, c), (q, b, ε; q, c), (q, a, c; q, ε), (q, b, c; q, ε), and (q, ε, $; f, ε). (Remember that a transition (p, a, b; q, c) means that the machine may go from state p to state q reading a from the input, popping b, and then pushing c.)

The grammar G is defined to have rules S --> SS, S --> aSa, S --> aSb, S --> bSa, S --> bSb, and S --> ε. It was on the test paper but I only noticed in February 2011 that is was not defined here. (DAMB)

• Question 1 (10): True or false with justification: There exists a string of odd length (with an odd number of letters) in L(P).

• Question 2 (10): True or false with justification: Every string of even length is in L(P).

• Question 3 (10): True or false with justification: Let L be a regular language, the language of some DFA with k states. Then if w is any string in L with length at least k, there exist strings x, y, and z such that w = xyz and such that every string in L is of the form xyⁱ for some number i. (Clarification during test: The last phrase means ∀s:(s ∈ L) → ∃i: s = xyⁱz rather than ∃i:∀s: (s ∈ L) → s = xyⁱz.)

• Question 4 (10): True or false with justification: The languages L(P) and L(G) are equal.

• Question 5 (35+10): The grammar G is defined before Question 1 above.

  • (a,5) Give a parse tree showing that the string aabbab is in L(G).

  • (b,5) Show that for any non-negative integer n, the string aⁿbⁿaⁿbⁿ is in L(G).

  • (c,10) Is the language {aⁿbⁿaⁿbⁿ: n ≥ 0} a context-free language? Prove your answer.

  • (d,10) Describe either a top-down parser or a bottom-up parser for G (and indicate which you are doing).

  • (e,5) Is L(G) a regular language? Prove your answer.

  • (f,10XC) Let G' be the grammar obtained from G by deleting the two rules S → aSb and S → bSa. Is L(G') a regular language? Prove your answer.

• Question 6 (20): Let N₁ and N₂ be any two NFA's with k states each. Remember not to make any assumptions other than that they are NFA's with k states.

  • (a,10) Describe an NFA N₃ such that L(N₃) = L(N₁) ∪ L(N₂). How many states does your N₃ have?

  • (b,10) Describe an NFA N₄ such that L(N₄) = L(N₁) ∩ L(N₂). How many states does your N₄ have?

• Question 7 (30): Let M be an NFA with state set {1, 2, 3, 4}, start state 1, final state set {4}, and the transitions (1,b,2), (1,a,3), (1,a,4), (2,b,3), (2,b,4), and (3,b,4). (Remember that the transition (p,a,q), for example, means that the NFA can go from state p to state q while reading an a.) In Sipser's notation for NFA's, we have that δ(1,a) = {3,4}, δ(1,b) = {2}, δ(2,b) = {3,4}, δ(3,b) = {4}, and for all other states p and letters c, δ(p,c) = ∅.

  • (a,10) Give a regular expression denoting the language L(M). If you use the GNFA construction, note that M already satisfies the conditions to be a GNFA.

  • (b,10) Using the subset construction or otherwise, find a DFA D such that L(D) = L(M).

  • (c,10) Show that your DFA D is minimal, or otherwise find a DFA for L(M) that has the minimum number of states needed to decide that language.

Last modified 13 February 2011