CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2014

Homework Assignment #1

Posted Monday 27 January 2013

Due on paper in class, Monday 10 February 2013

There are sixteen questions for 100 total points plus fifteen extra credit. All but three 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 3 February 2014.

Corrections in orange added 7 February 2014.

• Problem A-1 (5): A graph is called a cycle if for any vertex x, there is a path from x to itself that visits all other vertices exactly once, and contains all the edges in the graph. Informally describe an algorithm that inputs a graph and determines whether it is a cycle. (This is different from whether it includes a cycle.)

• Problem A-2 (10): The Fibonacci numbers are defined inductively by the rules F(0) = 0, F(1) = 1, and for n ≥ 2, F(n) = F(n-1) + F(n-2). Let X be the language of all strings over {a, b} that do not begin with b and never have two b's in a row. Prove, by induction on all non-negative integers n, that the number of strings of length n in X is exactly F(n+1). Question corrected 3 February 2014.)

• Problem A-3 (10): Let G be a graph with n vertices, e edges, and c connected components, and assume that G does not include a cycle. Prove that n = c + e. (Hint: Let n be arbitrary and use induction on e. If an edge does not create a cycle, what is its effect on the number of connected components?)

• Exercise 1.4 part c only (5).

• Exercise 1.6 parts j, k, and l only (5).

• Exercise 1.7 parts g and h only (5).

• Exercise 1.13 (10).

• Exercise 1.17 (5). You may use any valid method.

• Exercise 1.19 (5 XC) (Added to the assignment 7 February since it was accidently assigned before.)

• Exercise 1.21 part b only (5).

• Exercise 1.29, part (b) only (5). You may use either the book's method or mine.

• Problem 1.35 (10). You may either use the Pumping Lemma or the Myhill-Nerode Theorem.

• Problem 1.39 (5). This is easy using the Myhill-Nerode Theorem.

• Problem 1.42 (10XC). (Hint: Assume you have DFA's for A and B and build an NFA for the shuffle language.)

• Problem 1.43 (10).

• Problem 1.48 (10). Clearly you need an equivalent description of the language.

Last modified 7 February 2014