CS 462: TEST #1
Deadline: E-mail to zlatareva@ccsu.edu by March 25, 2003

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Please remember that everybody must work independently.
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PROBLEM 1: Here is a function FACTORS that returns a list of prime
factors of a number.

   (defun factors (number)
     (factors-aux number 2))

   (defun factors-aux (number1 number2)
     (cond ((equal number1 1) nil)
           ((zerop (rem number1 number2)) 
               (cons number2 (factors-aux (/ number1 number2) number2)))
           (t (factors-aux number1 (+ number2 1)))))

Here the REM function is a remainder division number1 by number2.
FACTORS returns the list of prime factors of a number. For example:

* (factors 60)
(2 2 3 5) 

Number 60 has factors 2 2 3 and 5 (60 = 2 * 2 * 3 * 5). Here is a
factorization tree for 60:

        60
      /    \
     2     30
          /   \
         2    15
             /  \
            3    5

Write a similar function, FACTOR-TREE, that returns a factorization 
tree, i.e.

* (factor-tree 60)
(60 2 (30 2 (15 3 5)))


PROBLEM 2:  Suppose that we run a greedy search algorithm with
h(n) = -g(n). What sort of search will the greedy search emulate?
What if h(n) = g(n)? What sort of search does greedy search emulate
in this case? Explain your answer -- just stating this or that search 
is not enought.


Problem 3:   Consider the sentence "Heads I win; tails you lose."
We can represent this sentence and associated domain knowledge
by means of the following PL formulas:

   Heads => I-Win
   Tails => You-Lose
   not Heads => Tails
   You-Lose => I-win

   A.) Is it possible to prove I-win using only modus ponens and
       these four formulas? If yes, give the proof, if not -
       explain why.
   B.) Convert each of these formulas to a disjunction of literals,
       and for each of the resulting disjunctions, specify if it is 
       a Horn formula.
   C.) Is it possible to prove I-win using only the resolution rule
       and these four formulas? If yes, give the proof, if not -
       explain why.


Problem 4:  Consider a world in which there are only four propositions
A, B, C and D. How many models are there for the following sentences:

   A.) A and B
   B.) A or B
   C.) A and B and C


Problem 5:  Explain the limitations of PL as a knowledge representation 
language. Give examples to illustrate your answer.