# CMPSCI 601: Theory of Computation

### Due by Friday 2 May 2003 9:05 a.m.

#### (Fri 9 May for off-campus students)

• Question 1 (10 points): Do Problem 13.4.3 in [P].

• Question 2 (15 points): Do Problem 13.4.4 (a) in [P].

• Question 3 (15 points): Do Problem 13.4.5 in [P].

• Question 4 (40 points): Remember that we define ATIME(log n) with random-access input. If the input is a structure, we assume that we can access a value of an input predicate by writing the arguments on an index tape. For example, to decide whether E(x,y) is true we write x and y in the right place on the input tape.

• (a,20) Suppose that A is a language defined by a fixed first-order formula φ, that is, A is the set of structures B such that B models φ. Prove that A is also in ATIME(log n). (Statement corrected 29 April 2003, see Q&A 7.8.)

• (b,20) Let PARITY be the set of binary strings with an odd number of ones. (It is known that PARITY is not defined by any first-order formula, though we won't prove that in this course.) Prove that PARITY is also in ATIME(log n). (Hint: You need to define an ATM game that White can win iff the input string has an odd number of ones. Some of the O(log n) bits written down during the game can be moved onto the index tape to determine what bit of the input is to be looked up. It may help to start with inputs that are 1, 2, 4, 8, etc. bits long.)

• Question 5 (20 points): A one-look alternating Turing machine has the property that it only looks at its input once, on the last step of its computation. This problem is to show that any ASPACE(log n) machine can be simulated by a space O(log n) one-look alternating machine. (You can do this directly, by having one player responsible for naming each input bit to be read and having the other player either accept or challenge the claim. Or you can use the Alternation Theorem and show how to simulate a deterministic poly-time machine with a log-space one-look alternating ATM.) (Clarifying words added 30 April 2003.)