CS4811: Homework 8 --- Reasoning Under Uncertainty (Chapters 5 and 9)


Due: Wednesday, April 19, 2005, beginning of class.
(Assigned: Wednesday, April 12, 2005)

Reminder: This is an individual assignment. All the work should be the author's and in accordance with the university's academic integrity policies. You are allowed to use any written source in preparing your answers, but if you use any other source than the textbook and the class notes, you should specify it on your assignment.
Each problem is 25 points.

Problem 1: Consider the following sentences:
   1. All final exams are easy.
   2. Easy final exams make the students who take them happy.
   3. Jane is taking at least one final exam.
   4. Someone is happy.

Part a. Represent the above four statements in predicate logic using
FE (X) for "X is a final exam,"
EASY (X) for "X is easy,"
TAKES (X,Y) for "X takes exam Y,"
H (X) for "X is happy."
Note that the logic formula for the second sentence is given below. Do not forget to negate the whole premise in parantheses when you eliminate implication. ∀ Y ∀ Z ( FE(Y) ∧ EASY (Y) ∧ TAKES (Z, Y)) → H(Z)

Part b. Set up sentences so that the fourth can be proven using the first three employing resolution refutation. Then convert the sentences to clause form using the following steps:
1. Eliminate → (implication)
2. Reduce the scope of negation
3. Standardize variables apart
4. Move all quantifiers to the left without changing their order
5. Eliminate existential quantifiers (Skolemize)
6. Drop all universal quantifiers
7. Convert expressions into conjunct of disjuncts form
8. Make each conjunct a separate clause
9. Standardize the variables apart again

Part c. Prove the fourth statement using resolution.

Problem 2: Consider the following statements:

(a)Express the above statements in nonmonotonic logic using the modal operator M.

(b)Draw an LTMS and show the initial labels.

(c) Expand and label the LTMS with IN or OUT as the following information comes into the knowledge base:
If the snow on the ground is plowed the ground will freeze.

(d) Expand and label the LTMS again with IN or OUT as the following information comes into the knowledge base:
The snow on the ground was plowed.

(e) Expand and label the LTMS again with IN or OUT as the following information comes into the knowledge base:
The previous report was incorrect, only the driveway was plowed.


Problem 3: Imagine that affordable robot helpers have been manufactured and both you and your neighbor have one. Yours is named Robby and your neighbors' is named By-a-rob. One day, Robby looks out of the window and says ``By-a-rob is walking the dog.'' Robby is 100% reliable in detecting that a dog being walked. On the other hand, 70% of the time he mistakes a human for a robot (or vice versa). You know that your neighbor takes the dog out himself 90% of the time, and lets By-a-rob do it at other times.

(a) Write down all the prior and conditional probabilities that you can gather from the above description. Briefly describe the propositions you use.

(b) What is the probability that By-a-rob is walking the dog?

(c) What is the probability that your neighbor is walking the dog?

Problem 4: Consider the following BBN




Symbolize and compute the following probabilities:
(a) What is the probability that there is construction, an accident, bad traffic and flashing lights?

(b) What is the probability that there is no construction, no accident, no bad traffic and no flashing lights?

(c) What is the probability of an accident given bad traffic?

(d) What is the probability of an accident given flashing lights?

(e) What is the probability of an accident given construction and bad traffic?

You are welcome to verify your answers using JavaBayes (http://www-2.cs.cmu.edu/~javabayes/Home/) but you should do the computation yourself, and show all the steps of your computation.