CPE/CSC 480-F05 Artificial Intelligence Lab Exercise 5

This lab exercise will use another tool in the Computational Intelligence Lab at UBC in Vancouver ( http://www.cs.ubc.ca/labs/lci/CIspace). As with the previous exercise, you can use the online version, or download the code, and run it locally on your machine as a Java application.

The topic of this exercise is local search, and we will use the n-queens problem to investigate some local search methods.

Constraint Satisfaction

Instructions

Invoke the Constraint Satisfaction applet, and load the sample "Five Queens Problem". Look at the graph displayed, and determine the relationship between the conventional representation of the queens problem on a chess board, and this one. Then use the tool to solve the problem. You can do this either in "Auto-Solve" mode, or manually: Select values for a variable, and then let the tool calculate compatible values for the other variables. This is done with the "Auto Arc Consistency" button. If you made a choice that does not have any possible solution, you can backtrack, and select different values. You can speed up the process by selecting "Very Fast" for the Arc-Consistency Speed in the CSP Options.

Once you're comfortable with the tool, try your luck on the eight queens problem. This will take longer, but even on my relatively slow Mac iBook it was tolerable, and I could generate 30 solutions in a few minutes.

Questions

• How is the queens problem represented in the constraint satisfaction (CS) graph? What is the meaning of the components that make up the CS graph:

  • nodes

    Variables in the CSP, representing a column with a queen in it. Each queen is assumed to be in a different column, and the values for the variables indicate the position of the queen within the column.

  • links

    The links represent constraints. Here they indicate that for any pair of nodes, the two nodes must not have the same value. As a consequence, no two queens may appear in the same position in any pair of columns.

  • labels on the links

    Their main purpose is to distinguish the constraints. They are also used to set properties of the constraints. For this problem, the constraints indicate disallowed combinations of values for the two variables involved: The number in the constraint (e.g. 2 in "Queens2" means that a difference in value for the two variables is not allowed) encodes the requirement that the two queens do not threaten each other diagonally. The numbers also indicate how many queens are "between" the pair of nodes (e.g. "Queens3" means there are two queens between A, D.)

• How is it guaranteed that the queens do not threaten each other

  • horizontally,

    no two variables may have the same value

  • vertically, and

    only one queen per column

  • diagonally?

    The difference in the values for two queens in adjacent columns must be at least 2. For example, if A = 3, B must not be 2 or 4.

• Is this representation easy for humans to follow?

  No, not really since it deviates considerably from the visual representation we are used to. However, once the mapping from the conventional board representation to the CS graph is given, it is not too difficult to understand.

• Write down at least two of the solutions you found for the five queens and two for the eight queens problem.

• How many solutions did you find for the five queens problem?

  10

• How many for eight queens (you don't need to examine all of them)?

  92; twelve if symmetry is considered

• Does this method use an incremental or a complete-state approach? Explain your answer!

  It is incremental: The possible locations for one queent (by default the one in column A) are restricted first, and checked for conflicts with the other queens. The method can also be used for a complete-state approach by selecting one value for each node, and using backtracking to modify the values.





• Is this method complete? Is it optimal? If so, under which circumstances? If not, what causes it to be incomplete or non-optimal? What does optimal mean here?

  It is complete. Backtracking guarantees that a solution is found if there is one.





• Is this a good method to determine how many solutions there are for the n-queens problem?

  It is reasonable. Although it is not too fast, it is straightforward to generate the solutions systemaically: Start with the lowest value for queen A, pick the lowest compatible value for queen B, then for queen C, etc. If necessary, backtrack.

Hill Climbing

Instructions

Invoke the Stochastic Local Search applet (hill.jar), and load the sample "Five Queens Problem". Select the "Greedy Descent" method, initialize the graph, and press the "Autosolve" button. You can view additional information about the progress on the "Plot" and "Trace" panes. The "Batch" button will initiate multiple runs of the sample; it is initially set to 100 runs.

After experimenting with the regular greedy descent method, try out some of the modifications to see the effects they have. The most relevant for this type of problems is probably the "min-conflicts" heuristic.

Finally, try out the "Simulated Annealing" method. For this one, I recommend to observe the plot, and do some batch runs. You can also try to increase the number of steps.

Questions

• Do these method use an incremental or a complete-state approach?

  Here a complete-state approach is used: The initialization step at the very beginning selects random values for the positions of the queens within the columns. Existing conflicts are then resolved.





• Write down at least two of the solutions you found for the five queens and two for the eight queens problem.





• Is this a good way to determine how many solutions there are for the n-queens problem?

  No, because it won't let you generate the solutions systematically. With random restart and enough runs, all solutions will eventually be found, but multiple solutions must be weeded out explicitly. It may be possible to systematically try out all possible initial configurations, but since there are random elements in there, it is not guaranteed that all solutions will be found.





• Are these methods complete and optimal? If so, under which circumstances? If not, what causes them to be incomplete or non-optimal? What does optimal mean here?

  It is complete for random restart and enough runs ("stochastically complete"). Optimal is meaningless here since all solutions are equal. It is possible to define ordering criteria on the solutions, e.g. symmetry, "pleasant arrangement" (whatever that may mean), or number of steps to find the solution (this is questionable, since it depends on the initial configuration).





• Did you notice any significant differences between the various versions of the greedy descent method?





• How did the simulated annealing method perform?

  It performs very poorly, and seems to find a solution only by chance. It doesn't try to reduce the number of conflicts (min-conflicts strategy), so a change in the value of a variable actually may make the configuration worse than it was.