CMPSCI 501: Theory of Computation

First Midterm Exam, Spring 2014

David Mix Barrington

19 February 2014

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 = 70.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first six 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: 40 points
  Q8: 25 points
  Q9: +10 points
 Total: 125+10 points

The language X is the set of all strings in {a, b}^* in which every b has an a immediately before it and an a immediately after it.

If w is any string, w^R is w written backwards. For example, (abaab)^R = baaba.

The function f from strings in {a, b}^* to strings in {a, b}^* replaces each a in the input with ab and each b in the input with ba. Thus, f(aba) = abbaab.

Language Y is the set of all strings of the form w(f(w))^R where w is any string in {a, b}^R. For example, taking w = aba, we see that ababaabba is in Y because it is the concatenation of w = aba and f(w)^R = baabba.

Language Z is the set of all strings of the form wf(w) where w is any string in {a, b}^*. For example, taking w = aba, we find that abaabbaab is in Z.

The NFA N has input alphabet {a}, state set {i, p, q, r, s, t}, final state set {i, p, r, s}, and transition function δ defined as follows: δ(i, a) = {p, r}, δ(p, a) = {q}, &delta(q, a) = {p}, δ(r, a) = {s}, δ(s, a) = {t}, δ(t, a) = {r}, and for all states u, δ(u, ε) = ∅. This last definition means that there are no ε-moves in N.

• Question 1 (10): True or false with justification: The language Y, defined above, is regular.

• Question 2 (10): True or false with justification: The language Z, defined above, is context-free.

• Question 3 (10): True or false with justification: The language {w: f(w) ∈ X} is regular. (The function f and the language X are defined above.)

• Question 4 (10): True or false with justification: The language {f(w): w ∈ X} is not regular. (Again, f and X are defined above.)

• Question 5 (10): True or false with justification: Let M be a pushdown automaton that follows the rules for our construction of an equivalent context-free grammar. (It has exactly one final state that is not its start state, it accepts only with empty stack, and every transition either pushes or pops a letter but not both.) Assume in addition that M reads exactly one input letter on each transition. Then every string in L(M) has even length.

• Question 6 (10): True or false with justification: There is an example of a string aⁱ that is in the language L(N) while the string aⁱ⁺⁶ is not. (The NFA N is defined above.)

• Question 7 (40): These questions deal with finite automaton and regular language constructions. The language X and the NFA N are defined above.
  • (a, 10) Build a DFA whose language is X. You may use any method, but be sure that your DFA is correct.

  • (b, 10) Find a set of four strings that are pairwise X-distinguishable, and show that they have this property. Explain (without proof) what the existence of this set implies about the set of correct DFA's for the language X. (Hint: You must choose two strings in X and two strings not in X.)

  • (c, 10) Find a regular expression for X, using your DFA from part (a).

  • (d, 10) Using the Subset Construction, find a DFA equivalent to the NFA N.

• Question 8 (25): Here are three questions about the languages Y and Z, which are defined above.
  • (a, 5) Give three examples of six-letter strings in Y and three examples of six-letter strings in Z. Explain (without proof) exactly which non-negative integers can be the lengths of strings in Y, and which can be the lengths of strings in Z.

  • (b, 10) Give a context-free grammar whose language is Y. You should justify your answer, but you need not prove that your grammar is correct.

  • (c, 10) Describe a pushdown automaton whose language is Y, either by construction from your answer to part (b) or otherwise.

• Question 9 (10 extra credit): The function f defined above is an example of a homomorphism, a function from strings to strings that satisfies the rule f(uv) = f(u)f(v) for all strings u and v. (A homomorphism is completely determined by its outputs for single-letter inputs.)

  The homomorphic image of a language L under a homomorphism f is the set {f(w): w ∈ L}. The inverse homomorphic image of L under f is the set {w: f(w) ∈ L}.

  If L is a context-free language and f is any homomorphism, is its homomorphic image guaranteed to be a context-free language? Justify your answer.

  If L is a CFL and f is a homomorphism, is its inverse homomorphic image guaranteed to be a context-free language? Justify your answer.

Last modified 22 February 2014