# CMPSCI 401: Theory of Computation

### David Mix Barrington

### Spring, 2012

# Homework Assignment #6

#### Posted Tuesday 17 April 2012

#### Due on paper to me or to CMPSCI main office by 4:00 p.m.,
Tuesday 1 May 2012

There are twelve questions for 100 total points plus 20 extra credit.
All but three are in
the textbook, *Introduction to the Theory of Computation*
by Michael Sipser (second edition).
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.

- Exercise 8.2 (5)
- Problem 8.14 (10)
- Problem 8.16 (10)
- Problem 8.17 (5)
- Problem 8.20 (10) (Hint: Use the transitivity of ≤
_{L}.)
- Problem 8.24 (10XC)
- Problem 8.26 (10)
- Problem 8.27 (10)
- Problem 8.28 (10)
- Problem F-1 (15) A circuit is defined to have
**width** w
if the gates are divided into levels of at most w nodes each, the
input gates are on level 0, and
the
inputs to a gate on level i come from gates on level i-1. Prove
that a language is in DLOGSPACE if and only if it is the language of
a L-uniform circuit family of polynomial size and O(log n) width.
(Hint:
Adapt the proof of the Cook-Levin Theorem.)
- Problem F-2 (15) Sipser defines a
**branching program** on
page
376. Prove that a language is in DLOGSPACE if and only if it is the
language of an L-uniform family of branching programs of polynomial size.
- Problem F-3 (10XC) Prove that if NC = P, then there exists
an i such that NC = NC
^{i}. (That is, NC "collapses" to one
of its levels.)

Last modified 17 April 2012