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CS 540-1: Introduction to Artificial Intelligence
Midterm Exam: 7:15-9:15pm, March 22, 1999
CLOSED BOOK
(one page of notes and a calculator allowed)
Write your answers on these pages and show your work. If you feel that a question is not fully specified, state any assumptions you need to make in order to solve the problem. You may use the backs of these sheets for scratch work.
Write your name on this and all other pages of this exam. Make sure your exam contains six problems.
Name ________________________________________________
Student ID ________________________________________________
Problem Score Max Score
1 _____ 25
2 _____ 25
3 _____ 8
4 _____ 20
5 _____ 10
6 _____ 12
Total _____ 100
PROBLEM 1 - Decision Trees (25 points)
Assume you are given the following features with the possible values shown.
F1 ∈ {A, B, C} F2 ∈ {D, E} F3 ∈ {G, H, I} F4 ∈ {J, K}
a) Consider the following training examples.
ex1 F1 = A F2 = E F3 = H F4 = J category = + ex2 F1 = A F2 = E F3 = G F4 = K category = - ex3 F1 = A F2 = D F3 = G F4 = J category = + ex4 F1 = B F2 = D F3 = G F4 = J category = + ex5 F1 = B F2 = E F3 = H F4 = J category = + ex6 F1 = B F2 = E F3 = H F4 = K category = - ex7 F1 = C F2 = D F3 = H F4 = K category = - ex8 F1 = C F2 = D F3 = G F4 = J category = -
i. What score would the information gain formula assign to each of the features? Be sure to show all your work. ( Some calculations that might be of use: I(x,y)=I(y,x), I(1,0)=0, I(1/2,1/2)=1, I(1/3,2/3)=0.92, I(1/4,3/4)=0.81, I(1/5,4/5)=0.72, I(1/6,5/6)=0.65, I(1/7,6/7)=0.59, I(1/8,7/8)=0.54)
ii. Which feature would be chosen as the root? _________ (Break any ties in favor of F1 over F2 over F3 over F4 , and ’+’ over ’-’.)
PROBLEM 2 - Search Strategies (25 points)
a) Assume you have the following search graph, where S is the start node and G1 and G2 are goal nodes. Arcs are labeled with the cost of traversing them and the estimated cost to a goal is reported inside nodes.
A S
B
D
C
G
G
4
E
11
2
6
(^2 6 10) KEY
0
7
2
0
4
(^5 )
1 5
8
7
4
7 4
5
Estimated cost tonearest goal is X Y Cost of traversingthis arc is Y
X
For each of the search strategies listed below, indicate which goal state is reached (if any) and list, in order, all the states expanded. (Recall that a state is expanded when it is removed from the OPEN list.) When all else is equal, nodes should be expanded in alphabetical order.
Breadth-First
Goal state reached: _____ States expanded: ____________________________________
Iterative Deepening
Goal state reached: _____ States expanded: ____________________________________
Hill Climbing (using the h function only)
Goal state reached: _____ States expanded: ____________________________________
A*
Goal state reached: _____ States expanded: ____________________________________
b) Imagine that the simulated annealing algorithm is at node A and has randomly chosen node D is the candidate next state. Assuming the temperature equals 5, what is the probability that node D will be accepted as the next state?
Repeat the above, but this time assume node E is the candidate next state.
PROBLEM 3 - Game Playing (8 points)
a) Apply the minimax algorithm to the partial game tree below, where it is the minimizer’s turn to play. The values estimated by the static-board evaluator (SBE) are indicated in the leaf nodes. Write the estimated values of the intermediate nodes inside their circles, and indicate the proper move of the minimizer by circling one of the root’s outgoing arcs.
11 12
minimizer
maximizer
9 -7 4 -8 -9 -
b) Indicate, by crossing out, one (1) unnecessary call to the static-board evaluator. Explain why this call to the SBE is unnecessary.
c) Given the well-formed formulae (wff’s) below, show that (S ∨ R) logically follows (don’t do more than 10 deductive steps):
WFF Justification hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
1 P given
2 (Q ∧ W) given
3 (P ∧ Q) ⇒ (S ∨ Z) given
4 (A ∨ R ∨ ¬ Z) given
5 ¬ A given
PROBLEM 5 - Miscellaneous Short Answers (10 points)
Provide brief answers to the following questions.
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
A. Why do we use test sets in machine learning? What is wrong with using the same subset of data for one’s pruning set and test set in ID3?
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
B. If a solution of length N is known to exist, is beam-search with beam-width=2 N guaranteed to find it?
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
C. Express the following sentence in predicate calculus:
Every red ball is made of rubber.
Problem 6 - Information Theory and Expected Values (12 points)
Assume that in your morning email each day you get a message from Amazing.com. The last line of each message always says how many free dollars you have for use at their site that day. You’re told that 95% of the time, you’ll get $1 and 5% of the time you’ll get $10. (Be sure to show your work when answering the questions below.)
a) What is the expected dollar value of your daily message?
b) How many bits of information (about the free dollars awarded) do you expect to receive in each day’s email from Amazing.com?
c) You’ve discovered that whenever the phrase "big sale" appears in the email’s subject header, which happens half the time, you will always receive only the $1 award. How much information (in bits), about the award contained in the email, does this discovery provide you?
The End!
