CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2016

Homework Assignment #2

Posted Tuesday 2 February 2016

Due on paper in class, Tuesday 16 February 2016 (Monday Schedule)

There are thirteen questions for 100 total points plus 20 extra credit. All but four are from the textbook, Introduction to the Theory of Computation by Michael Sipser (third edition, but the second edition numbers are the same for all but one last problem). Some problems are adapted as indicated. 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 B-1 (10): Recall that the index of a language, as defined in Problem 1.52, is the number of Myhill-Nerode classes it has, or equivalently the number of states in its minimal DFA. Give an example of a language X whose index is different from that of its reversal language X^R.

• Problem B-2 (15+10): Let Σ = {a, b}. Let Y be the set of strings over Σ that contain all four possible substrings. That is, Y is the intersection of the four languages Σ^*aaΣ^*, Σ^*abΣ^*, Σ^*baΣ^*, and Σ^*bbΣ^*.
  • (a, 5) Prove by any method that Y is a regular language.

  • (b, 10) Give an explicit DFA whose language is Y.

  • (c, 10XC) Determine the exact index of Y, following the definition in Problem 1.52.

• Problem 1.38 (10).

• Problem 1.53 (5).

• Problem 1.62 (10).

• Exercise 2.1 (5)

• Exercise 2.2 (10)

• Exercise 2.9 (5)

• Exercise 2.10 (5)

• Exercise 2.15 (5)

• Problem 2.20 (10)

• Problem 2.24 (10 XC)

• Problem 2.47 (10). (This is not in the second edition so I will repeat the question here.) Let Σ = {0. 1} and let B be the collection of strings that contain at least one 1 in their second half. In other words, B = {uv: u ∈ Σ^*, v ∈ Σ^*1Σ^*, and |u| ≥ |v|}. (Recall that "|u|" is the length of the string u.)
  • (a) Give a PDA that recognizes B.

  • (b) Give a CFG thta generates B.

Last modified 2 February 2016