# CMPSCI 401: Theory of Computation

### David Mix Barrington

### Spring, 2011

# Homework Assignment #6

#### Posted Wednesday 20 April 2011

#### Due on paper to me or to CMPSCI main office by 4:00 p.m., Tuesday 3 May 2011

There are twelve questions for 110 total points. (I didn't pick
out any particular problem as extra credit, but I will scale this as
if it were a 100-point assignment with 10 points extra credit.)
All are from
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 8.6 (5) (Use ≤
_{p} reductions.)
- Problem 8.10 (10)
- Problem 8.15 (10)
- Problem 8.19 (10)
- Problem 8.21 (10)
- Problem 8.23 (10)
- Problem 8.25 (10)
- Problem 8.29 (10)
- Problem 9.12 (5) (Read the statement of the Time Hierarchy
Theorem, and assume that it is true, which it is.)
- Problem 9.23 (10)
- Exercise 10.3 (10)
- Problem 10.8 (10) (Hint: If δ from Q × Σ to
Q is the transition function of a DFA, we can think of δ
instead as a function from Σ to (Q → Q), the set of
functions from Q to Q. Each letter gives rise to one of these
functions, the
*behavior* of the DFA on seeing that letter.
And the behavior of the DFA when reading a string of length n is the composition
of the h individual behavior functions. Can you design a circuit,
with the given size and depth, to compose n functions from Q to Q,
where Q is a fixed set? Then can you use this circuit to decide
whether your input is in the language of the DFA?)

Last modified 20 April 2011