# CMPSCI 601: Theory of Computation

### Due in class Wednesday 5 or beforehand to CMPSCI office drop box

#### (fax by Fri 14 May 2:00 p.m. EDT for off-campus students)

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, ΣiL 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. Πi is similar starting with a Black phase of control. The class LSH is the union for all constants i of Σi. 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 x1,...,xs. 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 NCi (except NC0), ACi, ThCi, in their non-uniform versions, are closed downward under this reducibility. That is, if B is in non-uniform ACi, and the stated (possibly non-uniform) circuit family with B-gates exists, then A is in non-uniform ACi. (And similarly for NCi and ThCi.)

• Question 6 (10): Using the result of Question 5, prove that if NC has a complete problem under ≤circ reductions, then NC = NCi for some number i. (That is, "the NC hierarchy collapses".)