CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2014

Homework Assignment #2

Posted Monday 10 February 2014

Due on paper in class, Wednesday 19 February 2014

There are thirteen questions for 75 total points plus 10 extra credit. All 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.

Correction in green added 11 Feb 2014.

• Problem 1.45 (10): Hint: Assume you have a DFA for A and build an NFA or DFA for A / B. Read the definition of "A / B" carefully -- it is not the same, for example, as the set subtraction "A \ B". Partial credit (5 points) for proving this with the additional assumption that B is also a regular language, so you have a DFA for it. Another hint for the full problem -- although B might be a bizarre uncomputable language, there are only a finite number of facts about it that are relevant.

• Problem 1.64 parts (a) and (c) (10): In part (a), you can easily show that if A is non-empty, there must be a string in A of length at most k - 1.

• Problem 1.64 part (b) (10 XC): All you need is an example, for some k, of an NFA with k states such that the complement of its language is non-empty and has a shortest string that is longer than k. There is an example with a one-letter input alphabet.

• (Problem 1.64 part (d)): We'll give up to ten extra-extra-credit points for a good answer to this, but it's too hard a problem to actually assign. Remember that if you use an outside source you must cite it.

• Exercise 2.6 part (b) only (5)

• Exercise 2.9 (5)

• Exercise 2.10 (5)

• Exercise 2.12 (5)

• Exercise 2.13 (10)

• Exercise 2.14 (10)

• Problem 2.20 (5) Note the similarity to 1.45 above.

• Problem 2.30 part (a) only (5)

• Problem 2.45 (10)

Last modified 11 February 2014