CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2014

Homework Assignment #6

Posted Thursday 17 April 2014

Due on paper in class or to the main office by 4:00 p.m., Wednesday 30 April 2014

There are fourteen questions for 100 total points plus 10 extra credit. All but three are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second edition). Problem numbers for the third edition are given in square brackets after the original number. 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 purple made 29 April 2014.

• Exercise 8.2 (5):

• Exercise 8.4 (5):

• Problem 8.10 (5):

• Problem 8.14 [8.15] (10) Note that for some reason Sipser exchanged the numbers of Problems 8.14 and 8.15 between the second and third edition -- I want the problem about two-person PUZZLE, not Cat-and-Mouse.

• Problem 8.17 (5):

• Problem 8.19 (10):

• Problem 8.20 (10): (Hint: Use the result that the composition of two logspace computable functions, from strings to strings, is itself logspace computable. We will prove this in class with reference to NL-completeness.)

• Problem 8.22 (10)

• Problem 8.24 (10XC):

• Problem 8.26 (10): Note an error in the second edition that is noted on Sipser's Errata page and corrected in the third edition -- the reduction should be from the complement of the language BIPARTITE, not from BIPARTITE itself.

• Problem 9.24 [9.23] (10): I accidently assigned the second edition's 9.23, not realizing that the numbering differed again. We did the parity problem in lecture, so please do the addition problem. 9.23

• Problem F-1 (5): Show that the NIM language of Problem 8.19 is in the class NC¹.

• Problem F-2 (5): The language IBMM (for "iterated boolean matrix multiplication") is the set of tuples <M₁,..., M_n, i, j> such that each M_i is an n by n matrix of booleans, and the (i, j) entry of the product of the M_i's, in the given order, is a 1. Place this language as low in the NC/AC hierarchy as you can. Boolean matrix multiplication is defined in terms of "multiplication" (AND) and "addition" (OR), or equivalently by adding "1 + 1 = 1" to the usual rules of arithmetic on 0 and 1 values..

• Problem F-3 (10): Repeat Problem F-2 for the similar language IZ₂MM where now the matrix multiplication is defined in terms of the multiplication and addition operations in the field Z₂, so that now 1 + 1 = 0.

Last modified 29 April 2014