# CMPSCI 601: Theory of Computation

### David Mix Barrington

### Spring 2010

### Homework Assignment #7

#### Posted Tuesday 13 April 2010

#### Due in discussion class Friday 23 April 2010, or
to the CMPSCI main office by 4:00 p.m. that day (due date changed 15 April).

Most of these questions are from the textbook, *Computational
Complexity: A Modern Approach* by Arora and Barak. There are seven
questions for 50 points plus 10 extra credit.

Question 8.3 uses the definition of BP · NP and
hence the definition of ≤_{r} on page 138 of [AB]. The latter
definition (7.16) contains a serious typo -- the last probability should be
"Pr[C(M(x)) = B(x)]" rather than "Pr[B(M(x)) = C(x)]".

- Exercise 8.1 (20)
- Exercise 8.3 (10) (see green note above)
- Exercise 8.4 (5)
- Problem G-1 (5) Define a family H of hash functions (as in Definition
8.14 on page 152 of [AB]) to be
**triplewise independent** if for any
three distinct strings x, x', and x'' in {0,1}^{n} and any three strings
y, y', and y'' in {0,1}^{k}, and for h uniformly randomly chosen from
H, the probability that h(x) = y, h(x') = y', and h(x'') = y'' is the product
of the probabilities of these three individual events. Prove that the hash
function family of Theorem 8.15 is not triplewise independent. Define a hash
function family that *is* triplewise independent.
- Exercise 8.5 (10XC)
- Exercise 9.2 (5)
- Exercise 9.5 (5)

Last modified 15 April 2010