# CS3800 09F: Homework 09

Created: Wed 02 Dec 2009

Assigned: Wed 02 Dec 2009
Due: Wed 09 Dec 2009

## General Instructions

1. Please review the grading policy outlined in the course information page.

2. On the first page of each part 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 following

Problem01020304 0506070809...
CreditRCRCRCECRC RCNARCRC...

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.

3. 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.

## Specific Instructions

1. This assignment cannot be accepted late since solutions for this assignment will be handed out the day that it is due.

## Problems

Required: 4 of the following 7 problems
Points: 25 pts per problem

1. Problem 7.20 (b). Note: You have already shown that LPATH is in NP (HW08); you need not repeat that portion of the proof.

2. Problem 7.21. Hint: Reduce from SAT or 3SAT. You may do so by adding one extra clause to the given formula.

3. Problem 7.24

4. Problem 7.28. Hint: Reduce from NOT-EQUAL-3SAT, defined in Problem 7.24. (You need not solve Problem 7.24 to solve this problem.)

5. Problem 7.26. Hint: Reduce from 3SAT. Each card will correspond to a variable, and flipping the card will correspond to setting its truth value. Each row on a card will correspond to a clause, and the absence or presence of a hole will correspond to whether that clause is satisfied by the variable setting or not. You will need one additional specially designed card as well.

6. Problem 7.34

7. Problem 7.41. Hint: Your NFA will "guess" (via epsilon transitions) the clause that will not be satisified and then verify that the clause is not satisfied. There are c clauses, and to verify that any clause is unsatisfied will require roughly m states, thus yielding an O(cm) sized NFA. Finally, if the cnf-formula is unsatifiable, what will be true about the language accepted by the NFA you construct? What is the minimum equivalent such NFA? How could you use this to solve SAT? See also Problem 6 from HW07.

jaa@ccs.neu.edu