CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #5

David Mix Barrington

Assignment posted Thu 20 Nov 2003

Due Wed 3 Dec 2003, 2:00 p.m. by fax to 413-545-1249, attention Prof. Barrington.

Make sure you have read the statement on academic honesty on the homework assignment index page.

• Question 1 (15): Prove that n n-bit numbers can be sorted by a logspace algorithm. (The numbers are given as read-only input, the sorted numbers are written to write-only output in order, and only O(log n) bits of read-write memory are used.

• Question 2 (15): Suppose that a, b and m are numbers each of O(log n) bits. Show that the number a^b mod m can be calculated in O(log n) space.

• Question 3 (15): Write the HAM-CIRCUIT problem as a second-order existential boolean query, where the input predicate is the edge relation of a directed graph.

• Question 4 (20): Give poly-time reductions from the HAM-CIRCUIT problem to the HAM-PATH problem and vice versa. Make sure that you follow the definition of a poly-time reduction carefully.

• Question 5 (15): Prove that the 2-SAT problem is NL-complete. (If you use a reference, indicate what you used and explain the argument clearly in your own words.)

• Question 6 (20): Let N be an NFA with k states and let w be a string of length n. We can express membership of w in L(N) as a REACH problem on a particular graph, where the vertices are pairs (q,i) where q is a state of N and i is a number from 0 through n, and there is an edge ((q,i),(r,i+1)) whenever (q,r) is a possible transition of N on the input character w_i+1.

  Consider testing w for membership in L(N) by applying middle-first search to this path problem. Explain how this can be carried out in O(log n) space rather than the O(log² n) space required by middle-first search on an arbitrary graph of n vertices.

Last modified 20 November 2003