Download CPSC 420-500 HW #3: Theorem Proving, Uncertainty, Probabilistic Reasoning - Prof. Y. Choe and more Assignments Computer Science in PDF only on Docsity!
CPSC 420-500 Homework
Due 11/11/08 (Tuesday) in class
Instructor: Yoonsuck Choe
1 Theorem Proving in First-Order Logic
Question 1 (15 pts): Show that R(A) is a logical consequence of the following. Use resolution (w, x, y, z are variables, and A is a constant).
- ¬P (x) → [Q(x, y) ∨ R(y)]
- ¬P (A)
- ¬Q(w, z) ∨ R(w)
2 Uncertainty and Probabilistic Reasoning
Question 2 (15 pts): Given the belief network as shown below, calculate the joint probability P (¬JohnCalls∧ M aryCalls ∧ Alarm ∧ ¬Earthquake ∧ Burglary).
B T^ T F^ F
E T^ F T^ F
P(A) . .29.
.
P(B)^ P(E).
Alarm
Earthquake
JohnCalls MaryCalls
Burglary
A P(J) T^ F .90.05^ A T^ ^ P(M) F^ .70.
.
3 Learning
3.1 Decision Tree Learning
Consider the following set of examples where you are trying to make a decision whether to buy a robot or not given the specifications in terms of the sensors used (Sensor), number of legs (Legs), and energy source used (Energy).
Example# Sensor # Legs Energy Source Decision (Buy?) 1 Vision 8 Oil Y 2 Sonar 8 Solar N 3 Sonar 8 Food Y 4 Vision 2 Oil Y 5 Sonar 2 Oil N 6 Sonar 2 Sun Y 7 Thermal 2 Food N 8 Vision 2 Oil Y 9 Sonar 2 Food Y 10 Thermal 8 Sun N
Question 3 (12 pts): For each attribute (Sensor, Legs, Energy), draw a one-depth decision tree. See slide06.pdf, page 12 for an example.
Question 4 (12 pts): For each of the cases above, calculate the information gain.
Question 5 (5 pts): Based on the information gain, which attribute would you choose first?
Question 6 (12 pts): Can a single perceptron unit solve the following classification problem?: In other words, can the perceptron learning rule find a set of weights to correctly classify all examples? (1) Answer “yes” or “no” to the question, (2) draw a geometric illustration of the problem in 3D and (3) justify your reasoning.
Input x Input y Input z Class 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 (^1 1 1 0) x
y
z
0
1
1
1
Question 7 (15 pts): Explain how a backpropagation network with 2 hidden units and one output unit can correctly learn the XOR function. Hint: try illustrating below how two decision boundaries (implemented by the hidden layer neurons) can be combined to give the desired result.
I
I
XOR
Question 8 (14 pts): What are the two quantitative measures of performance for the Self-Organizing Map? Explain which one of those two relate to encoding/prototyping and which one relates to feature mapping.
