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 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^*1Σ^* and |x| = |z| ≥ |y|} and C₂ = {xyz: x, z ∈ Σ^* and y ∈ Σ^*1Σ^*1Σ^* 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)

Last modified 24 February 2015