# CMPSCI 601: Theory of Computation

### David Mix Barrington

### Spring 2010

### Homework Assignment #6

#### Posted Monday 29 March 2010

#### Due in class Monday 12 April 2010

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

Problem F-1 corrected, Problem F-3 added 4 April 2010.

- Problem F-1 (5) Let T be a binary tree with n nodes. Prove that there
is an edge that splits T into two pieces, one with k nodes and one with n-k,
such that n/4 ≤ k ≤ 3n/4. I originally had 1/3 and
2/3 here -- as you can see from a four-node example, this is not always true
(but see Problem F-3). The revised result is enough to use below. You can
also prove n/3 - 1 ≤ k ≤ 2n/3 + 1 in the presence of unary internal
nodes.
- Exercise 6.13 (10) The revised
Problem F-1 is a useful lemma.
- Exercise 6.15 (5)
- Problem F-2 (10) A
**one-look ATM** is an alternating Turing
machine with the property that it only references the input once, by writing
the index of an input location on a tape. White writes a prediction for the
input bit (which may be constrained by the previous play) and wins the game iff
this is correct. Show that the classes ATISP(log^{i} n, log n) and
A-ALT-SP (log^{i} n, log n) can be defined in terms of one-look ATM's
without changing the classes, for i ≥ 2 in the first case and i ≥ 1 in
the second.
- Exercise 7.4 (5)
- Exercise 7.6 (10)
- Exercise 7.10 (5)
- Problem F-3 (5 extra credit) Prove the original
version of Problem F-1 with the added condition that every
*internal*
node of the tree has exactly two children (though the tree need not be
balanced).

Last modified 4 April 2010