# Homework Assignment #6

#### 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.

• 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 = NCi. (That is, NC "collapses" to one of its levels.)