1. (15 points) (Problem 5.8) Use the AC-3 algorithm to show that arc consistency is able to detect the inconsistency of the partial assignment
{ WA = red, V = blue} for the problem shown in Figure 5.1.

2. (15 points) (Problem 5.9) What is the worst-case complexity of running the tree version of AC-3 on a tree-structured CSP?

3. (20 points) Create networks with the following properties. Use a problem different than map coloring:

• The original CSP is not arc consistent, the equivalent arc consistent network has empty domains.
• The original CSP is not arc consistent, the equivalent arc consistent network has no empty domains but has no solutions.
• The original CSP is not arc consistent, the equivalent arc consistent network has exactly one solution.
• The original CSP is not arc consistent, the equivalent arc consistent network has multiple solutions.

4. (15 points) (Problem 12.6.1) Prove that path-consistency for the interval algebra can be achieved in O(n³).

5. (15 points) (Problem 12.6.2) Define IA as a CSP where the variables are relationships between pairs of intervals, their values are the possible relations, and constraints are defined via the composition tables.

6. (20 points) Show that Example 12.7's scenario where John takes the car and Fred takes the carpool is consistent. Show that the alternate scenario, in which John used a bus and Fred used a carpool is not consistent (see the last sentence of Example 12.8 on page 350 of Dechter's book).