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RESOLUTION: A COMPLETE INFERENCE PROCEDURE
I
Then we certainly want to be able to
conclude S(A); S(A) is true if S(A) or R(A) is
true, and one
of those must be true
because either P(A) is true or ¬P(A) is true.
Canonical forms for resolution
In the first version of the resolution rule, every sentence is a disjunction of literals.
All the disjunction in the KB are assumed to be joined in one big, implicit conjunction
(as in a normal KB>, so this form is called conjunctive normal form (or CNF), even
though each individual sentence is a disjunction (confusing, isn't it ?).
In the second version of the resolution rule, each sentence is an implication with a
conjunction of atoms on the left and a disjunction of atoms on the right. We call this
implicative normal form (or INF), although the name is not standard. We can
transform the sentences in (17.1) into either of the two forms, as we now show. (Notice
that we have standardized apart
the variable names in these sentences.)
T
It is important to recognize that resolution is a generalization of
modus ponens. Clearly, the implicative normal form is more
general than Horn form, because the right-hand side can be a
disjunction, not just a single atom. But at first glance it seems
that Modus Ponens has the ability to combine atoms with an
implication to infer a conclusion in a way that resolution cannot
do. This is just an illusion-once we realize that an atomic
sentence a in implicative normal form is written as True →a, we
can see that modus ponens is just a special case of resolution:
EXAMPLE PROOFS
Example proof
We will now show how to apply the
conversion procedure and the resolution
refutation procedure on a more complicated
example, which is stated in English as:
Jack owns a dog.
Every dog owner is an animal lover.
 No animal lover kills an animal.
 Either Jack or Curiosity killed the cat, who
is named Tuna.
 Did Curiosity kill the cat?
First, we express the original sentences (and
some background knowledge) in first-order
logic:
The
problem
is
now
to
show
that
Kills(Curiosity, Tuna) is true. We do
that
by
assuming
the
negation,
Kills(Curiosity, Tuna) →False, and
applying the resolution inference rule seven
times, as shown in Figure 17.3. We eventually
derive a contradiction, False, which means
that the assumption must be false, and
Kills(Curiosity, Tuna) is true after all. In English,
the proof could be paraphrased as follows:
Suppose Curiosity did not kill Tuna. We know that
either Jack or Curiosity did, thus Jack must have.
But Jack owns D, and D is a dog, so Jack is an
animal lover. Furthermore, Tuna is a cat, and cats
are animals, so Tuna is an animal. Animal lovers
don't kill animals, so Jack couldn't have killed Tuna.
But this is a contradiction, because we already
concluded that Jack must have killed Tuna. Hence,
the original supposition (that Curiosity did not kill
Tuna) must be wrong, and we have proved that
Curiosity did kill Tuna.