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The final examination for the cs540 introduction to artificial intelligence course offered in the fall of 2009. The exam covers topics such as first-order logic, resolution, perceptrons, back-propagation learning in neural networks, support vector machines, probability, bayesian networks, markov models, and hidden markov models.
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December 22, 2009
Problem Score Max Score
Total ___________ 100
1. [14] First-Order Logic
(a) [4] Which of the following are correct translations of “No two adjacent countries have the same color”?
(i) x , y Country x ( ) Country y ( ) Adjacent x y ( , ) ( Color x ( ) Color y ( ))
(ii) x , y ( Country x ( ) Country y ( ) Adjacent x y ( , )) ( Color x ( ) Color y ( ))
(iii) x , y Country x ( ) Country y ( ) Adjacent x y ( , ) ( Color x ( ) Color y ( ))
(iv) x , y ( Country x ( ) Country y ( ) Adjacent x y ( , )) Color x ( y )
(b) [3] Let C1 be the clause ŸRepublican(Mother( x )) ¤ Republican( x ) and let C2 be the clause ŸRepublican( y ) ¤ likes( y , Sarah) ¤ ŸResident( y , Alaska). What is the result of applying the resolution rule of inference^ to C1 and C2?
(c) [4] Which of the following are valid sentences?
(ii) x P x ( ) P x ( )
(d) [3] Are the following two expressions unifiable? If so, what is the most general unifier? If not, why not?
P ( x , g ( y , A, h ( y , B))) and P ( h (A,B), g (A, y , x ))
3. [12] Perceptrons
(a) [4] Can a Perceptron learn the SAME function of three binary inputs, defined to be 1 if all inputs are the same value and 0 otherwise? Either argue/show that this is impossible or construct a Perceptron that correctly represents this function.
(b) [4] Can a Perceptron learn to correctly classify the following data, where each consists of three binary input values and a binary classification value: (111,1), (110,1), (011,1), (010,0), (000,0)? Either show that this is impossible or construct such a Perceptron.
(c) [2] True or False: Training neural networks has the potential problem of over-fitting the training data.
(d) [2] True or False: The Perceptron Learning Rule is a sound and complete method for a Perceptron to learn to correctly classify any two-class problem.
4. [14] Back-Propagation Learning in Neural Networks
(a) [4] What is the search space and what is the search method used by the back- propagation algorithm for training neural networks?
(b) [3] Back-propagation minimizes what quantity?
(c) [3] Does the back-propagation algorithm, when run until a minimum is achieved, always find the same solution no matter what the initial set of weights are? Briefly explain why or why not.
(d) [4] Instead of using a sigmoid function as the activation function at each unit, which of the following are mathematically legitimate when using the back-propagation algorithm for training, assuming x = S wi ai where ai are the inputs to the unit. Explain briefly.
(i) g ( x ) = sin( x )
(ii) g ( x ) = +1 if x > 0; -1 otherwise
6. [14] Probability
(a) [8] Fill in the missing values in the following joint probability table assuming that A and B are independent. Show your work.
(b) [6] Consider two arbitrary Boolean random variables, C and D. Assuming P ( C =F, D =T) / P ( C =T, D =T) = 2, what is P ( C =T | D =T)?
7. [10] Bayesian Networks
Consider a (Naïve) Bayesian network B A C where the variables are all Boolean.
Complete the following CPTs assuming P ( A =T, B =T, C =T) = 1/18, and P ( A =F, B =F, C =F) = 1/24. Show your work.
P ( B =T | A =F) = 1/ P ( C =T | A =T) = 1/ P ( C =T | A =F) = 3/ P ( A =T) =? P ( B =T | A =T) =?
