Exam 2
                                    CS 451/551
                               Thursday, April 15, 1999



This is an open-book, open-notes examination.  There are five
problems.  Do not spend too much time on any problem. Read them 
all through first and solve them in the order that allows you to 
make the most progress.  


------------------------------------------------------------------------------
Problem 1:  

A.  Give the clauses for the following theorem:

There was a party.  Ann and Bill and Carl and Dave were invited.
Either Ann or Bill (or both) came.  If Ann came then Bill came.  
If Bill came then Ann came.  If Ann and Bill came then Carl came.  
If Bill and Carl came then Dave came.  Therfore, all four of them 
came to the party.

B. Give a proof of the theorem using the resolution inference
system, with the set of support strategy.

------------------------------------------------------------------------------
Problem 2:  

A.  Give the following theorem in first order logic:

For every number, there is a larger number.  The less-than
predicate is antisymmetric (i.e., if a number is smaller than
another number, then the second number is not smaller than the
first number).  Therefore, there is no largest number.

Use the following predicate:
Less(x,y) - x is less than y

B.  Give the clauses for the theorem.

C.  Give a proof of the theorem using the resolution inference system.

------------------------------------------------------------------------------
Problem 3:  

A.  Give the following theorem in first order logic:

Every pair of numbers has a product.  x and y divide the product
of x and y.  Therefore, every number divides something.

Use the following predicates:
Prod(x,y,z) - z is the product of x and y
Div(x,y) - x divides y

B.  Give the clauses for the theorem.

C.  Give a proof of the theorem using the resolution inference system.

------------------------------------------------------------------------------
Problem 4:

A.  Give the step by step results of the unification algorithm
    for P(x,g(z),z) and P(f(y),y,a).

B.  Give the step by step results of the unification algorithm
    for P(x,x) and P(y,f(y)).

If you don't know how the algorithm works, you can get partial 
credit for giving the result of the algorithm.
------------------------------------------------------------------------------
Problem 5:  

Resolution without factoring is not complete.  Show why,
using the following set of clauses:

P(x) | Q(y).
-P(x) | Q(y).
P(x) | -Q(y).
-P(x) | -Q(y).

------------------------------------------------------------------------------