Computer Science Canada

predicate logic proofs
Author:  Fonzie [ Sun Oct 05, 2008 3:40 pm ]
Post subject:  predicate logic proofs
I have a question on my assignment that I'm having a great deal of trouble with.

In this question we will
consider a simplified model of the ?People you may know? application in Facebook. The
basic idea is that if two people have a common friend, then they may know each other. We
will also take into account the fact in Facebook the relation ?friend? is symmetric, that is,
if X is a friend of Y , then Y is a friend of X.
Thus, as a problem description P, we have the following two predicate logic sentences:

8X8Y [(9Z(friend(X, Z) ^ friend(Z, Y ))) $ may know(X, Y )]
8X8Y [friend(X, Y ) ! friend(Y,X)]

Our goal is to show, using resolution refutation, that the following sentence g is implied by P:

8X8Y [may know(X, Y ) ! may know(Y,X)].

8 = universal quantifier
9 = existential quantifier
$ = biconditional
! = implication

I have derived the following clauses (this is for a logical programming class):

c1: may_know(X,Y) :- friend(X,T), friend(T,Y)
c2: friend(X,f(X,Y,T)) :- may_know(X,Y)
c3: friend(f(X,Y,T),Y) :- may_know(X,Y)
c4: friend(Y,X) :- friend(X,Y)
c5: may_know(d,e) :-
c6: :- may_know(e,d)

I've attempted linear refutation which won't work. The question goes on to advise that you prove g from p "using your normal, mathematical, reasoning. Then, try to re-create the steps of your proof using resolution on your clauses." This is where I'm stuck, I'm not sure how I can prove g from p using either my clauses or my ordinary mathematical reasoning. Any help with this would be greatly appreciated.
Author:  Clayton [ Sun Oct 05, 2008 3:46 pm ]
Post subject:  RE:predicate logic proofs
If I wasn't having so much trouble reading the math, I might be able to help you out a bit better, I think the LaTeX tags work...

Author:  Fonzie [ Sun Oct 05, 2008 4:07 pm ]
Post subject:  Re: predicate logic proofs
revised problem

p:

Posted Image, might have been reduced in size. Click Image to view fullscreen.XPosted Image, might have been reduced in size. Click Image to view fullscreen.Y [(Posted Image, might have been reduced in size. Click Image to view fullscreen.Z(friend(X, Z) ^ friend(Z, Y ))) Posted Image, might have been reduced in size. Click Image to view fullscreen. may know(X, Y )]
Posted Image, might have been reduced in size. Click Image to view fullscreen.XPosted Image, might have been reduced in size. Click Image to view fullscreen.Y [friend(X, Y ) Posted Image, might have been reduced in size. Click Image to view fullscreen. friend(Y,X)]

g:

Posted Image, might have been reduced in size. Click Image to view fullscreen.XPosted Image, might have been reduced in size. Click Image to view fullscreen.Y [may know(X, Y ) Posted Image, might have been reduced in size. Click Image to view fullscreen. may know(Y,X)]


sorry, took me a bit to figure out how to get the pictures in there.

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