CMPSCI 601: Theory of Computation

David Mix Barrington

Fall, 2004

Homework Assignment #3

Posted Fri 12 Nov 2004

Due Mon 24 Nov, 2:00 p.m. by fax to 413-545-1249

Covers Lectures 16-20

These problems all concern complexity measures and complexity classes.

• Question 1: Suppose that I have a procedure Augment that takes a set of sentences Γ in the language of number theory, finds the true sentence φ that is unprovable from Γ given by our proof of Godel's Theorem, and returns Γ&union;{φ}. Consider the program "while (true) Γ = Augment(Γ);". Explain why the set of strings eventually put into Γ by this program is r.e. but not recursive. Why does it not provide a counterexample to Godel's Theorem?

• Question 2: Suppose that there were an algorithm to multiply a sequence of n boolean matrices, each n by n, using O(log n) space. Show that REACH would then be computable in O(log n) space, and thus that L = NL.

• Question 3: Show that there is an algorithm to multiply a sequence of n boolean matrices, each n by n, in O(log²) space. Remember that you must use read-only input and write-only output. (Hint: Your worktape has room for a stack.)

• Question 4: Write a first-order formula in the language of directed graphs that expresses that the graph has a directed triangle. Following the discussion on slide 6 of Lecture 19, write a pseudo-Java method that tests this property and uses O(log n) space.

• Question 5: On slide 7 of Lecture 20 I outline a direct reduction from REACH to LEVELLED-REACH. Carefully define this reduction and prove that it is a correct log-space reduction.

Last modified 12 November 2004