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 resolutionIn 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 conjunc tion 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 proofWe 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.