CS 5541 Study Questions and Homework 3

Homework (15 points) due Monday, October 27, 2003.

Updates: Corrected typo on question 6d, 10/27/03, 10:30am

Homework assignment

For this Homework, turn in your answers for the following study questions:

Q1 [4 pts]
Q5 [4 pts]
Q6 [4 pts]
Q11 [3 pts]

Answers must be typewritten.

Study questions

1. Using the truth table (enumeration) method, prove that the modus ponens inference rule preserves truth (only generates entailed sentences) in a knowledge base.
2. Clearly distinguish between induction and deduction in the manner that has been considered in class. Give specific examples to illustrate each concept.
3. In logic, what is a model? What is a model of a sentence? Give specific and concrete definitions.
4. In propositional logic, modus tollens is defined as: Given that we have E1 => E2, and ~E2, then we have that ~E1 must be true. Prove that modus tollens is a truth preserving inference rule.
5. Given the following sentences:
   a) John likes all kinds of foods.
   b) Apples are food.
   c) Chicken is food.
   d) Anything anyone eats and remains alive is food.
   e) Bill eats peanuts and is alive.
   f) Sue eats everything that Bill eats.
   

   i) Give an ontology (your logic terms; e.g., functions, predicates, constants) for this set of sentences.
   ii) Translate these sentences in to first-order logic.
   iii) Convert the logic sentences into clausal form.
   iv) Prove, by resolution refutation (contradiction), that John likes peanuts.

6. Assuming upper case letters indicate constants (or functions or predicates), and lower case letters indicate variables, find the most general unifier (MGU), if it exists, for the following pairs of formulas. In this case, give the binding set. If a unifier does not exist, make a clear argument to explain why the unifier does not exist.
   a) P(A, B, B), P(x, y, z)
   b) Q(y, G(A, B)), Q(G(x, x), y)
   c) Older(Father(y), y), Older(Father(x), John)
   d) Knows(Father(y), y), Knows(x, x)
   
7. Show why unify(['P', '?x'], ['P', ['R', '?x']]) returns 'nix' or 'failure' in terms of the definition of Unification:

   SUBST(theta, form1) = SUBST(theta, form2))

8. Write Python code to define a RenameVariables function that renames all of the variables in a formula. Assume that you are given a function Substitute(theta, formula) that takes a binding set (theta) and a logic formula, and performs the substitutions indicated by the binding set on the formula, returning the (potentially changed) formula as the result of calling the function.
9. Give two specific reasons for incorporating learning techniques into artificially intelligent systems.
10. Clearly discriminate between inductive vs. deductive methods of learning. Which one (inductive or deductive) is empiricist-based? Which one is rationalist-based? Explain.
11. Define: (a) supervised learning, (b) unsupervised learning, (c) generalization.
12. What kind of learning does a decision tree provide?
13. Suppose you are representing a feature, S (size) with attribute values: huge, large, medium, small, tiny. (a) With no other assumptions, how many bits of information will it take for you to represent a single item's category? (b) Say that we know some frequency information about the size category. Suppose P(S=huge) = .2, P(S=large) = .3, P(S= medium) = .4, P(S =small) = .04, P(S=tiny)= 0.06. Calculate the average information required to represent an item's category, given this frequency information, and given that we are representing many items.
14. Generate a minimal height decision tree for the following examples. The question being asked is: Should tennis be played? Use the information theory approach to compute the next feature(s) to be included in the decision tree.

    Day	Outlook	Temperature	Humidity	Wind	PlayTennis?
    1	Sunny	Hot	High	Weak	no
    2	Sunny	Hot	High	Strong	no
    3	Overcast	Hot	High	Weak	Yes
    4	Rain	Mild	High	Weak	Yes
    5	Rain	Cool	Normal	Weak	Yes
    6	Rain	Cool	Normal	Strong	no
    7	Overcast	Cool	Normal	Strong	Yes
    8	Sunny	Mild	High	Weak	no
    9	Sunny	Cool	Normal	Weak	Yes
    10	Rain	Mild	Normal	Weak	Yes
    11	Sunny	Mild	Normal	Strong	Yes
    12	Overcast	Mild	High	Strong	Yes
    13	Overcast	Hot	Normal	Weak	Yes
    14	Rain	Mild	High	Strong	no

15. For the last question, express in English the rule formed by the decision tree. (Example due to T. M. Mitchell)