**Assigned:** Wed 11-17

**Due:** Wed 12-01

- Please review the
grading policy
outlined in the course information page.
- On the
*first page*of your solution write-up, you*must*make explicit which problems are to be graded for "regular credit", which problems are to be graded for "extra credit", and which problems you did not attempt. Please use a table something like the followingProblem 01 02 03 04 05 06 07 08 09 ... Credit RC RC RC EC RC RC NA RC RC ... where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not applicable" (not attempted). Failure to do so will result in an arbitrary set of problems being graded for regular credit, no problems being graded for extra credit, and a five percent penalty assessment.

- You must also write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.

**Required:** 5 of the following 7 problems

**Points:** 20 pts per problem

- Exercise 7.1 and Exercise 7.2. All parts except c,d. Include a one-line justification of each of your answers.
- Exercise 7.3. Note: you should use the Euclidean algorithm seen in class.
- Consider the language L = {0
^{n}| n is a power of 2}.

Show that L is in TIME(n log n) (recall this refers to single-tape Turing machines).

OPTIONAL CHALLENGE: Show that L is in TIME(n) if we allow Turing machines to have two tapes. - Exercise 7.6, Exercise 7.7.
- Exercise 7.9. Note: A triangle is a set of three distinct nodes that are all connected to each other.
- Exercise 7.11.
- Problem 7.13.

viola@ccs.neu.edu