Clarifying remarks and corrections (in green) added on 4 May 2004.

**Question 1 (15):**(as promised) Define DTIME(log n) in terms of multitape TM's with an index tape as in Lecture 22. On a given step the machine's move depends on whether the input position named on the index tape contains a 0, contains a 1, or is larger than the input length. Explain how the function taking a binary string w to its length is in F(DTIME(log n)).**Question 2 (15):**Using the result of Question 1, prove that the following language is in DTIME(log n): the set of 4-tuples (a,b,c,y) such that a, b, and c are binary numbers, a + b = c, and c is less than the length of y. As explained in the Q&A, you may assume that the coding scheme assures you that the input is made up of exactly four strings.**Question 3 (20):**Following the definitions of PH and the logtime hierarchy in Lecture 23, define the logspace hierarchy as follows. For any i, Σ_{i}L is the set of languages of logspace alternating TM's that have at most i phases of control and start with a White phase of control. Π_{i}is similar starting with a Black phase of control. The class LSH is the union for all constants i of Σ_{i}. Prove that LSH equals NL. Note that LSH is not the same class as AL, because an AL machine might have a number of alternations that is not bounded by a constant as n increases.**Question 4 (20):**Let ODDREACH be the set of directed acyclic graphs (with marked nodes s and t) such that there are an odd number of different paths from s to t. Prove that ODDREACH is in the class ASPACE[log n]*without quoting the Alternation Theorem*. That is, I want you to describe the ATM whose language is ODDREACH, using either the accepting-configuration or the game semantics. You may assume G is given with node numbers and edges only going from lower gate numbers to higher, so that it is clear that G is acyclic. Hint: It may help to assume that G has out-degree at most two, then show that you can convert the original G in deterministic logspace to a graph with this assumption that has the same number of paths from s to t.**Question 5 (20):**Let A and B be languages. We say that A ≤_{circ}B, A*circuit-reduces*to B, if for every n there exists a circuit that decides membership in A for strings of length n, and this circuit has constant depth, polynomial (in n) size, and unbounded fan-in gates for AND, OR, and membership in B. A "gate for membership in B" has a certain number of boolean inputs x_{1},...,x_{s}. These inputs form a string x of length s, and the B-gate outputs 1 iff this string is in the language B. Prove that all the classes NC^{i}(except NC^{0}), AC^{i}, ThC^{i}, in their non-uniform versions, are closed downward under this reducibility. That is, if B is in non-uniform AC^{i}, and the stated (possibly non-uniform) circuit family with B-gates exists, then A is in non-uniform AC^{i}. (And similarly for NC^{i}and ThC^{i}.)**Question 6 (10):**Using the result of Question 5, prove that if NC has a complete problem under ≤_{circ}reductions, then NC = NC^{i}for some number i. (That is, "the NC hierarchy collapses".)

Last modified 4 May 2004