# Homework Assignment #1

#### Due on paper in class, Friday 6 March 2015

There are thirteen 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/third edition). though some are adapted as indicated. The number in parentheses following each problem is its individual point value.

• Problem 2-48 (10). This is not in the second edition so I will copy it here. "Let Σ = {0, 1}. Let C1 be the language of all strings that contain one or more 1's in their middle third. Let C2 be the language of all strings that contain two or more 1's in their middle third. So C1 = {xyz: x, z ∈ Σ* and y ∈ Σ** and |x| = |z| ≥ |y|} and C2 = {xyz: x, z ∈ Σ* and y ∈ Σ*** and |x| = |z| ≥ |y|}. Part (a) is to show that C1 is a CFL and part (b) is to show that C2 is not a CFL."

• Problem 2-49 (10XC). This is also not in the second edition so I will copy it. "We defined the rotational closure of language A to be RC(A) = {yx| xy ∈ A}. Show that the class of CFL's is closed under rotational closure." That is, show "if A is a CFL, then RC(A) is also a CFL".

• Exercise 3.2 part d only (5)

• Exercise 3.6 (5)

• Exercise 3.8 part c only (5)

• Problem 3.11 (10)

• Problem 3.15 parts a and c only (10)

• Problem 3.19 (10)

• Problem C-1 (10): In the style of Example 3.23, describe a TM that inputs an undirected graph and decides whether it has a Hamilton circuit (look the definition up if you don't remember it from 311).

• Exercise 4.2 (5)

• Problem 4.10 (second)/4.11 (third) (10)

• Problem 4.17 (second)/4.18 (third) (10)

• Problem 4.19 (second)/4.21 (third) (10)