CS345A Midterm Examination
Monday,May 16, 1994, 2:15 { 3:30 pm
Directions
The exam is
open book
any written materials may be used. Answer all 6 questions on the
exam paper itself. The total number of points is 150. You maymake additional comments
at the bottoms of pages or at the end if you feel the need to clarify or justify your answers.
Do not forget to
sign the pledge
below.
Iacknowledge and accept the honor co de.
Printyour name here:
1 2 3
4 5 6
Problem 1:
(40 points) Consider the following rules:
p(X) :- int(X) & X >= 2 & not c(X)
c(X) :- int(X) & p(Y) & divides(X,Y) & X != Y
Think of
p
(
X
) as meaning \
X
is a prime" and
c
(
X
)as\
X
is composite." The EDB
predicate
int
(
X
)says that
X
is a positiveinteger, and in practice it will hold a nite set
of integers. The EDB predicate
divides
(
X Y
) means that
Y
evenly divides
X
.
Suppose that
int
=
f
1
2
3
4
g
, and
divides
is the expected relation on these four
integers that is,
divides
=
f
(1
1)
(2
1)
(3
1)
(4
1)
(2
2)
(4
2)
(3
3)
(4
4)
g
. If
we instantiate these rules in all possible ways, eliminate rules with a kno
wn false subgoal
and then eliminate known true subgoals from the remaining rules, we are left with the
following:
p(2) :- not c(2) c(2) :- p(1)
p(3) :- not c(3) c(3) :- p(1)
p(4) :- not c(4) c(4) :- p(1)
c(4) :- p(2)
a) In the table at the top of the next page, construct the well-founded model by the
denition (not the alternating xedpoint). Indicate what inferences are made about
the truth or falsehood of
p
(
i
) and
c
(
i
), for 1
i
4 at each round.
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