# Homework Assignment #6

#### 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.

• 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?)