# CMPSCI 601: Theory of Computation

### David Mix Barrington

### Spring, 2003

### Homework Assignment #7

### Due by Friday 2 May 2003 9:05 a.m.

#### (Fri 9 May for off-campus students)

**Question 1 (10 points):** Do Problem 13.4.3 in [P].
**Question 2 (15 points):** Do Problem 13.4.4 (a) in [P].
**Question 3 (15 points):** Do Problem 13.4.5 in [P].
**Question 4 (40 points):**
Remember that we define ATIME(log n) with random-access input. If the
input is a structure, we assume that we can access a value of an input
predicate by writing the arguments on an index tape. For example, to
decide whether E(x,y) is true we write x and y in the right place on
the input tape.
- (a,20) Suppose that A is a language defined by a fixed first-order
formula φ, that is, A is the set of structures
B such that B models φ. Prove that A is also in ATIME(log n).
(Statement corrected 29 April 2003, see Q&A 7.8.)
- (b,20) Let PARITY be the set of binary strings with an odd number
of ones. (It is known that PARITY is not defined by any first-order
formula, though we won't prove that in this course.) Prove that PARITY
is also in ATIME(log n). (Hint: You need to define an ATM game that
White can win iff the input string has an odd number of ones. Some of
the O(log n) bits written down during the game can be moved onto the
index tape to determine what bit of the input is to be looked up. It
may help to start with inputs that are 1, 2, 4, 8, etc. bits long.)

**Question 5 (20 points):**
A *one-look alternating Turing machine*
has the property that it only looks at its input
once, on the last step of its computation. This problem is to show
that any ASPACE(log n) machine can be simulated by a space O(log n)
one-look alternating machine.
(You can do this directly, by having one player
responsible for naming each input bit to be read and having the other
player either accept or challenge the claim. Or you can use the Alternation
Theorem and show how to simulate a deterministic poly-time machine with
a log-space one-look alternating ATM.)
(Clarifying words added 30 April 2003.)

Last modified 30 April 2003