# CMPSCI 601: Theory of Computation

### Homework Assignment #6

#### 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(logi n, log n) and A-ALT-SP (logi 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).