CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2014

Homework Assignment #3

Posted Monday 24 February 2014

Due on paper in class, Friday 7 March 2014

There are fourteen questions for 100 total points plus 10 extra credit. All but one are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second edition). The number in parentheses following each problem is its individual point value.

Students are responsible for understanding and following the academic honesty policies indicated on this page.

Problem 2.43 (10):
Problem C-1 (10): A two-way DFA has a state set Q, input alphabet Σ, start state q_o, and final state set F like an ordinary DFA, but its transition function is from (Q × Σ) to (Q × {L, R}). It starts looking at the leftmost letter of its input. If it moves right from the rightmost letter of the input, it accepts if it is then in a final state and rejects if it is in a nonfinal state. If it moves left from the leftmost letter, it hangs and the computation ends. It is also possible for a computation to loop without ever leaving the input.
Prove that for any 2WDFA M, L(M) = {w: M accepts w} is a regular language. Hint: For any string w, define f₀(w) to be the state (if any) on which M leaves w to the right when started in state q₀ at the left of w. For each state q, define f_q(w) to be the state (if any) on which M leaves w to the right when started in state q at the right of w. Show that if f₀(u) = f₀(v) and for every state q, f_q(u) = f_q(v), then u and v are L(M)-equivalent. (If M loops or hangs when started in one of these situations, let the output of the corresponding f-function be a special value "*".) Then finish with Myhill-Nerode.
Exercise 3.2 part (c) only (5):
Exercise 3.4 (5):
Exercise 3.7 (5):
Exercise 3.8 part (b) only (5):
Problem 3.9 (10):
Problem 3.13 (10):
Problem 3.18 (10):
Exercise 4.3 (5):
Problem 4.10 (5):
Problem 4.14 (10):
Problem 4.20 (10):
Problem 4.25 (10XC):

Last modified 24 February 2014