Homework 1 Solutions
1a. (B (C D) is valid and C is unsatisable. To prove B D is valid:
Suppose bwoc B D is not valid.
a truth-value assignment, say 1 , such that 1 (B ) = T (1)
& 1 (D) = F [by defns of validity and ]. (2)
Since (B (C D) is
Due on Thursday, March 4, 2010
No Late Submissions
1. (25 points) Let B , C , and D be formulae and 1 , 2 be sets of formulae
Prove or disprove the following statements. To disprove a statement, provide appropriate
Homework 2 Solutions
1a. To show that res (A B ) (B C ) (C A) B ):
First, negate what were trying to prove (i.e., to show res consider ):
(A B ) (B C ) (C A) B )
Second, convert from statement-form into a set of clauses:
cfw_[(A B ) (B C ) (C
Due on Thursday, March 18, 2010
res (AB ) (B C )(C A)B ).
b. cfw_(A B ) (C D) res (A (C D) (B (C D)
c. cfw_(C B )B ), (C A), (B C )A) res (B C ).
2. You may use any of the Theorems/Lemmas/Propositions from the text book
Homework 3 Solutions
1a. (x)P (x) (x)Q(x) (x)(P (x) Q(x)
To satisfy: QM = DM
Example: D = cfw_0, 1, . . . (henceforth referred to as N),
P M = cfw_0, 2, 4, . . ., and
QM = D M .
M = P M = QM
To falsify: D
Example: D = N,
P M = cfw_0, 2, 4, .
Due on Thursday, April 22, 2010
1. (35 points) For each of the following wfs, give a structure that satises
the wf and another structure that falsies it.
a. (x)P (x) (x)Q(x) (x)(P (x) Q(x).
b. (x)(P (x) P (a).
c. (x)(y )[P (x, y )
Homework 4 Solutions
1a. To show z xP (x, z, f (x, z ) z xyP (x, z, y ) is valid using resolution.
Negate: (z xP (x, z, f (x, z ) z xyP (x, z, y )
Convert to prenex: z1 z2 x2 y x1 (P (x1 , z1 , f (x1 , z1 ) P (x2 , z2 , y )
Skolemize: z2 y x1
Due date Thursday May 4, 2010
NO LATE SUBMISSIONS
1. (20 points) Using resolution, prove the following are valid:
a. z xP (x, z, f (x, z ) z xyP (x, z, y )
b. x(P (x) Q(x) (xP (x) xQ(x).
2. (10 points) Using resolution, show that (
Unication and Resolution
Let L be a set of literals that we wish to unify.
Let sub [ ].
While | Lsub |= 1
Scan left to right till a disagreement is found
If neither term (at place of disagreement) is a variable then return(not uniable)