COSC 6368 (Fall 2009)
Assignment 2: Logical Reasoning
Deadlines: Problem 8+9 is due Monday, March 9, 11p; problem 10 is due on
Mo., March 21, 11p; all problems are individual problems!
8) Translating Natural Language into FOPL
a) Fred send an e-mail to each student enrolled in COSC 6342 excluding himself.
b) Politician can fool some people all the time, and they can fool all the people some of
the time, but they can’t fool all the people all the time.
c) Persons cannot have multiple social security numbers.
d) Fred has at least 2 sisters.
e) If block A is on the top of block B, and block B is on the top of block C, then block A
is on the top of block C.
f) For any two real numbers r1 and r2 with r1r2 there always exists a real number r
such that: r1rr2
9) Resolution Proofs
a) Textbooks mention that “resolution for first order predicate logic is semi-decidable”.
What does this mean? [3]
b) Show using Resolution (and not by using other methods!) [11]:
(1) VxVy]z ( (Q(x,y) P(x,x)) S(x,z) )
(2) VaVb (S(a,b) R(a,b))
(3) ]cVd P(c,d)
(4) VsVt (Q(s,t) P(s,s))
|-
(X) Ve]f]g (Q(e,f) R(e,g))
(1) ~Q($x,$y) v ~P($x,$x) v S($x,Z($x,$y))
(2) ~S($a,$b) v R($a,$b)
(3) P(C,$d)
(4) ~Q($s,$t) v P($s,$s)
(Xa) Q(E,$f)
(Xb) ~R(E,$g)
(5) P(E,E) using Xa and 4 with (($s E))
(6) ~S(E,$g) using Xb and 2 with (($a E)($g $b))
(7) ~Q(E,$y) v ~P(E,E) using 6 and 1 with (($x E) ($g ...))
(8) ~Q(E,$y) using 5 and 7
(9) using Xa and 8
First transform the FOPL formulas into clauses, and then the hunt for the empty clause
can begin!
10) More on Resolution
a) Prove using resolution (and not using other methods)
(A1) Every man loves at least one woman.
(A2) Every woman loves at least one man.
