## Sample Exam Problems

### Problem 1:

Consider the following collection of sentences in the propositional logic.
1. P => (Q <=> R).

2. Q <=> ~(R^W)

3. Q => (P^W).

A. Convert this set to CNF. (You need not show the intermediate steps.)
B. Show how the Davis-Putnam assignment finds a satisfying assumption.
(Assume that, when a branch point is reached, the algorithm chooses the
first atom alphabetically and tries TRUE before FALSE.)

### Problem 2:

What is the result of doing alpha-beta pruning in the game tree shown
below?

### Problem 3:

Consider a domain where the individuals are people and languages.
Let L be the first-order language with the following primitives:
s(X,L) --- Person X speaks language L.
c(X,Y) --- Persons X and Y can communicate.
i(W,X,Y) --- Person W can serve as an interpreter between persons X and Y.
j,p,e,f --- Constants: Joe, Pierre, English, and French respectively.

A. Express the following statements in L:
- i. Joe speaks English.
- ii Pierre speaks French.
- iii. If X and Y both speak L, then X and Y can communicate.
- iv. If W can communicate both with X and with Y, then
W can serve as an interpreter between X and Y.
- v. For any two languages L and M, there is someone who speaks both
L and M.
- vi. There is someone who can interpret between Joe and Pierre.

B. Show how (vi) can be proven from (i)---(v) using backward-chaining
resolution. You must show the Skolemized form of each statement, and
every resolution that is used in the final proof. You need not show the
intermediate stages of Skolemization, or show resolutions that are
not used in the final proof.
### Problem 4:

Let A, B, C be Boolean random variables. Assume that
Prob(A=T) = 0.8
Prob(B=T | A=T) = 0.5.
Prob(B=T | A=F) = 1.
Prob(C=T | B=T) = 0.1
Prob(C=T | B=F) = 0.5
A and C are conditionally independent given B.

Evaluate the following terms. (If you wish, you can give your answer as an
arithmetic expression such as "0.8*0.5 / (0.8*1 + 0.5*0.1)")
- i. Prob(B=T).
- ii. Prob(A=T | B=T)
- iii. Prob(C=T | A=F).

### Problem 5:

A. Give an example of a decision tree with two internal nodes (including
the root), and explain how it classifies an example.

B. Describe the ID3 algorithm to construct decision
trees from training data.

C. What is the entropy of a classification C in table T? What is
the expected entropy of classification C if table T is split on
predictive attribute A?

D. What kinds of techniques can be used to counter the problem of over-fitting
in decision trees?
### Problem 6:

Consider the following data set with three Boolean predictive attributes,
W,X,Y and Boolean classification C.
W X Y C
----------------
T T T T
T F T F
T F F F
F T T F
F F F T

We now encounter a new example: W=F, X=T, Y=F. If we apply the Naive
Bayes method, what probability is assigned to the two values of C?
### Problem 7:

"Local minima can cause difficulties for a feed-forward, back-propagation
neural network." Explain. Local minima of what function of what arguments?
Why do they create difficulties?
### Problem 8:

Which of the following describes the process of task execution
(classifying input signal) in a feed-forward, back-propagation neural network?
Which describe the process of learning? (One answer is correct for each.)
- a. Activation levels are propagated from the inputs through the hidden
layers to the outputs.
- b. Activation levels are propagated from the outputs through the hidden
layers to the inputs.
- c. Weights on the links are modified based on messages propagated
from input to output.
- d. Weights on the links are modified based on messages propagated
from output to input.
- e. Connections in the network are modified, gradually shortening
the path from input to output.
- f. Weights at the input level are compared to the weights at the
output level, and modified to reduce the discrepancy.

### Problem 9

Explain briefly (2 or 3 sentences) the use of a training
set and a test set in evaluating learning programs.