csci538 fall2007 midterm sols

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					Name: ___Solutions__________                                            SID: ______________________

                                 CSci 538: Artificial Intelligence
                                         Midterm Exam, Fall 2007
   For official grading use only.                                                    Total: ______(+1) / 100
     Q1              Q2             Q3         Q4            Q5               Q6         Q7               Q8
  ____ / 15     ____ / 15       ____ / 12   ____ / 12     ____ / 10      ____ / 15    ____ / 10     ____ / 10

1. Tree Search (15 points)
Consider the tree shown below. The numbers on the arcs are the step costs; the numbers inside states A, L, C,
B and R are the heuristic estimates; all other states have a heuristic estimate of 0 (not shown). Perform the
indicated searches using X as the starting state, and G as the goal state.


                                     h=1                          h=4
                                     A                            L

                          h=3                 h=2                        h=1
                          C          M         B              G           R

                                Y                 T                  O          Q
For search methods D use a priority based method, assume that a tie in terms of priority will result in the fall
back method of a child node being selected and expanded based on alphabetical order (e.g. first occurring
alphabetically will be taken from queue and expanded first). For search methods that do not use priority, simply
assume that expansion will take place from left to right on the page. These two rules should eliminate all
ambiguity in terms of the results of your tree search. What order would the states be expanded by each type of
search. Write only the sequence of states expanded by each search.

     Search Type                                             List of states
Breadth First Search          XALCMBG

Depth First Search            XACDYMBTLG

Iterative Deepening           X XALXAC MB LG

Csci 538, Midterm                                   1                                         Fall 2007
Name: ___Solutions__________                                               SID: ______________________

Uniform-Cost Search          X LA R C B G

A* search                    XALRG

Q: If we changed the tie-breaker method for priority-based search to select the node that had first been placed
on the queue rather then select alphabetically, would there have been any differences in any of the priority based
searches? If so which ones and how would it have differed?

Uniform-Cost Search would have differed. In the original specified heuristic, B & G have the same estimated
cost (8), therefore by the tie-breaker specified in the problem we choose B because it occurs before G
alphabetically. But G was placed on the UCS queue before B (on the expansion of node L), therefore as the
question states if we are choosing by first-on-queue in the case of a tie, we would have choosen G before B and
thus UCS would have had the expansion sequence: X L A R C G

Breadth First, Depth First and Iterative Deepening are not priority based searches (e.g. we do not use a priority
queue in the implementation of the search heuristic). Therefore, there is no concept of priority, nor ties based
on priority, in these searches.

2. The A* search for this tree also involves a tie of priorities (G & O have an
   evaluation of 8 in this case), however using either first-on-queue or
   alphabetical ordering to break the tie results in the same sequence of
   expansions in this case, thus there is no difference for the A* search.Graph
   Search (15 points)
Consider the graph shown below where the numbers on the links are step costs and the numbers next to the
states are heuristic estimates. Note that arcs are undirected (e.g. it is valid to follow the link in either direction).
Let Commerce be the start state and Paris be the goal state.               Paris

                                                                      h = 33

                                                  h = 40                                  h = 24
            h = 21
            Farm                                   Com                                    Sulph
Csci 538,   Midterm
            ersvill                                   2
                                                   merc                                     ur       Fall 2007
               e                                     e                                     Springs
Name: ___Solutions__________                                            SID: ______________________

Simulate A* search with a strict close list on this graph. At each step, show the path to the state of the node
that's being expanded (use the first letter of the city name for abbreviation), the length of that path, the total
estimated cost of the path (actual + heuristic), and the current value of the close list (as a list of states). You
should use the scratch paper provided to show your work on the simulated search for this problem, especially
indicating your queue and the addition/removal of partial solutions therein. Please transcribe (only) the
information requested into the table given below of your final results (all rows may not necessarily be filled).
Path to State Expanded       Length of Path              Total Estimated Cost         Close List
C                            0                           40                           {C}
F← C                         20                          41                           {CF}
S←C                          20                          44                           {C F S }
L← C                         19                          52                           {CFSL}
P← L← C                      46                          46                           { C FS LP}

3. Heuristics and A* Search (12 points)
    1.   Is the heuristic given in Problem 2 admissible? Explain?
         No. One clear example is the path between Lake (L) and Paris (P). The heuristic evaluation of
         cost to goal at L is 33, but the actual path cost is only 27, so the heuristic function
         OVERESTIMATES the cost to the goal. In order for a heuristic to be admissible it must always
         give an estimate equal to or less then (underestimate) the actual cost to goal of the state being
         evaluated. Greenville is another overestimated cost-to-solution node in the problem.

    2. Is the heuristic given in Problem 2 consistent? Explain?
         No. One example is that the heuristic cost for Greenville to Paris is 72, but the path cost to
         Commerce (15) plus the heuristics cost of Commerce to Paris (40) is only 65. There are many
         examples of inconsistency in the given graph.

    3. Did the A* algorithm with close list find the optimal path in the previous problem? Is it guaranteed to
       find the optimal solution? Explain why or why not and how you would fix any problems you detected.

4. The A* algorithm will happen to find the optimal path in this case. However, it
   is not guaranteed to find the optimal solution since the heuristic is neither

Csci 538, Midterm                                   3                                            Fall 2007
Name: ___Solutions__________                                          SID: ______________________

   admissible nor consistent. In order to guarantee optimality, both conditions
   of admissibility and consistency must be met by the heuristics function.
   Environment Properties (12 points)
A robot soccer player has to be able to track a soccer ball, keep track of other players (both opponents and
teammates), perform actions such as kicking the ball towards the goal, passing the ball to teammates, track its
own location on the field, move to locations on the playing field, etc.

For the general problem described above, where we want to field a team of robot soccer players to play a team
of real live humans at a game of soccer, determine the properties of the task environment. Choose the property
you think the environment has, and give a sentence of explanation. Some of the environment properties below
can be argued either way, so your explanation of how the robot players are engaging with the environment will
determine if your choice is correct.
Some of the environment properties can be argued either way depending on your formulation of the
agent or assumptions about the task. The most commonly assumed environmental properties for this
problem as stated might be:
     Fully Observable vs. Partially Observable:
        Both limitations of the agents sensors, as well as the apparent impossibility of observing the other
        agent's intentions and goals, would be strong indicators that the problem as stated will only be
        partially observable.
     Deterministic vs. Stochastic:
        Without an assumption of perfect information about initial conditions, even a simple prediction of,
        for example, the velocity, trajectory and position of the soccer ball for more than a few seconds
        into the future will become impossible. Therefore an assumption of randomness in the evolution of
        the physical state of the environment would probably be necessary.
     Episodic vs. Sequential:
        Though it might be possible to divide the agents experience into episodes, given real-humans and a
        need to act in continuous real-time, the environment is most usefully viewed as sequential.
     Static vs. Dynamic:
        It is useful to assume that there will be time pressures in this environment. If the agent stands still,
        the rest of the agents as well as the ball will continue to evolve and move.
     Discrete vs. Continuous:
        Similar to an argument of a dynamic environment, with real human opponents it will be difficult
        to discretize the time dimension of this problem, thus a real, continuous time formulation seems
     Single Agent vs. Multi Agent:
5. Treating the other players, both your teammates as well as your opponents,
   as    autonomous     agents    is   the   simplest  formulation   of  this
   environment/problem.Search Problem Formulation (10 points)
The general case of the robot soccer player, described in problem 4 would not easily be formulated as a standard
search problem. Explain why not:
Standard search problems require static states with discrete steps between successor states. Some
properties such as multiple-agents and stochasticity can be handled by extending standard search
formulations (using min-max and ? respectively for example to allow a form of search). However the
dynamic and especially fundamentally continuous nature of designing an agent to act in the real-world
tend to rule out simple search formulations.

Csci 538, Midterm                                  4                                         Fall 2007
Name: ___Solutions__________                                             SID: ______________________

You have to color a planar map using only four colors, in such a way that no two adjacent regions have the same
color. Formulate this problem by describing the state space, giving the initial state, goal test, successor function
and cost function. Choose a formulation that is precise enough to be implemented.
 State Space = ri (the color of region i, where i ranges from 1 to the number of regions, and the domain of
r is the four colors to be used, for example red, green, blue and yellow).
Initial State = {} (the empty, initially start with no regions colored)
Goal Test = All ri assigned a color, no adjacent regions assigned the same color (if ri adjacent to rj: ri != rj)
Successor Function = For all unassigned variables of the initial state, create four new successor states
   where we assign the unassigned variable one of the 4 color values and add it to the end of the original
Cost Function = Use a constant step cost function of 1 for each variable assignment. Path cost is not
    relevant to this problem as all solutions appear at the same level of the search tree where all variables
    have been assigned.

Say what search technique would be most appropriate for this problem, and why.
6. I had originally intended you to formulate this problem as a standard search
   problem.       Obviously this problem is best formulated as a constraint
   satisfaction problem for reasons we have discussed. The path to the solution
   is irrelevant in this case, we only care about finding a correct coloring
   assignment for the map. If there are N regions in the map, all solutions will
   occur when all N variables have been assigned, which in this case will happen
   at level N of the tree. Without the simplifying heuristics of a CSP formulation
   (only assign one variable at a time, prune inconsistent assignment subtrees
   as soon as they are detected) in general solving a 4-coloring problem using a
   standard search will be impractical for N above 10 or so. However, if forced to
   solve this problem using a standard search, you definitely do not want to do
   breadth first search (since all solutions occur at the same depth N of the
   search tree). Iterative Deepening search is also not useful, since we know the
   depth to solution will be N (and there are no infinite paths in this problem).
   Therefore the one and only standard search is a straight-forward Depth First
   search.Constraint Satisfaction Problem (15 points)
You are trying to schedule observations on the space telescope. We have m scientists who have each submitted a
list of n telescope observations they would like to make. An observation is specified by a target, a telescope
instrument, and a time slot. Each scientist is working on a different project so the targets in each scientist’s
observations are different from those of other scientists. There are k total time slots, and the telescope has three
instruments, but all must be aimed at the same target at the same time.
   The greedy scientists cannot all be satisfied, so we will try to find a schedule that satisfies the following
 C1. Exactly two observations from each scientist’s list will be made (the choice of the two will be part of the
 C2. At most one observation per instrument per time slot is scheduled.
 C3. The observations scheduled for a single time slot must have the same target.
Note that for some set of requested observations, there may not be any consistent schedule, but that’s fine.

Consider the following two formulations of the problem.

Csci 538, Midterm                                    5                                           Fall 2007
Name: ___Solutions__________                                           SID: ______________________

A. The variables are the 3k instrument/time slots.
B. The variables are the m scientists.

For each formulation, specify
1. The value domain for the variables.
2. The size of the domain for the variables (in terms of k, m, and n).
3. Which of the constraints are necessarily satisfied because of the formulation.
4. Whether the constraints can be specified as binary constraints in this formulation. If they can,
   explain how. If not, provide a counterexample.
Formulation A: The variables are the 3k instrument/time slots.
    1. Domain: for each instrument/time slot, the set of observations requesting that instrument and time
       slot and the value “empty”
    2. Size of domain: at most m*n+1 per variable
    3. Satisfied constraints: C2, since each variable (instrument/time) gets at most one value, an
    4. Binary constraints?:
            C1 is not a binary constraint in this formulation. It requires checking all the variable
               assignments at once to make sure that exactly two observations from each scientist’s list are
            C3 is a binary constraint in this formulation. Place a constraint between the 3 variables
               with the same time slot and require that the targets of the assigned observation be equal if
               they are both non-empty.

Formulation B: The variables are the m scientists.
   1. Domain: for each scientist, the set of all pairs of observations that scientist requested.
                         n                    n2
   2. Size of domain: 2 , approximately 2 .
   3. Satisfied constraints: C1, since we will guarantee that exactly two of the scientist’s observations
      are scheduled.
   4. Binary constraints?:
          C2 is a binary constraint in this formulation. Place a constraint between every pair of
              variables and require that the instrument/time slot requests don’t conflict.
          C3 is a binary constraint in this formulation. Place a constraint between every pair of
              variables and require that the targets for observations with the same time slot don’t

     Problem 6 Work Area
7.                                                       Game Search (10 points)
Consider the game tree shown below. Assume the top node is a max node. The labels on the arcs are the moves.
The numbers in the bottom layer are the values of the different outcomes of the game to the max player.
                                                     -                               Max
                                  -                                -               Min
                                  1                                7
Csci 538, Midterm                                    6                                     Fall 2007
                           1             -                     5             -     Max
                           0             1                                   7
Name: ___Solutions__________                                         SID: ______________________

   1. What is the value of the game to the max player? -7

   2. What first move should the max player make? R

   3. Assuming the max player makes that move, what is the best next move for the min player, assuming that
      this is the entire game tree? R

8. Alpha-Beta Pruning (10 points)
In the following game tree, are there any alpha-beta cutoffs?


                         -2    3     1         1       -5   -3   5    1
   1. Consider the nodes from left to right, which nodes are cutoff ? Circle the nodes that are not examined
      and label them with L. (The M5 and R1 nodes of right subtree)

   2. Consider the nodes from right to left, which nodes are cutoff ? Circle the nodes that are not examined
      and label them with R. (The L1 node of middle subtree)

Csci 538, Midterm                                  7                                     Fall 2007

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