# CMPSCI 501: Theory of Computation

### David Mix Barrington

### Spring, 2015

# Homework Assignment #1

#### Posted Tuesday 24 February 2015

#### 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.

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

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

Last modified 24 February 2015