RESOLUTION

WHAT IS RESOLUTION ? Resolution is a technique for proving theorems

in the propositional or predicate calculus. Resolution proves a theorem by negating the

statement to be proved and adding this negated goal to the set of axioms

Resolution involve the following steps.

1. Put the premises or axioms in to clause form. 2. Add the negation of what is to be proved, in

clause form, to the set of axioms. 3. Resolve these clauses together, producing

new clauses that logically follow from them. 4. Produce a contradiction by generating the

empty clause. 5. The substitutions used to produce the empty

clause

Resolution requires that the axioms and the negation of the goal be placed in a normal form called clause form

Clause form represents the logical database as a set of disjunctions of literals.

The form is referred to as conjunction of disjuncts.

The following is an example of a fact represented in clause form (⌐dog(X) U animal(X)) ∩ (⌐animal(Y) U die(Y)) ∩

(dog(fido))

1. Producing the clause form

1. First we eliminate the → by using the equivalent form. For example a→b ≡ ⌐a U b.

2. Next we reduce the scope of negation. ⌐ (⌐a) ≡ a ⌐ (X) a(X) ≡ (X) ⌐a(X) ⌐ (X) b(X) ≡ (X) ⌐b(X) ⌐ (a ∩ b) ≡ ⌐a U ⌐b ⌐ (a U b) ≡ ⌐a ∩ ⌐b

3. Standardize by renaming all variables so that variables bound by different quantifiers have unique names.

If we have a statement ((X) a(X) U X b(X) ) ≡ (X) a(X) U (Y) b(Y)

4. Move all quantifiers to the left without changing their order.

5. Eliminate all existential quantifiers by a process called skolemization. (X) (Y) (mother (X, Y)) is replaced by (X) mother (X, m(X)) (X) (Y) (Z) (W) (foo (X, Y, Z, W)) is replaced with (X) (Y) (W) (foo (X, Y, f(X, Y), W))

6. Drop all universal quantifiers.

7. Convert the expression to the conjunct of disjuncts form using the following equivalences. a U (b U c) ≡ (a U b) U c a ∩ (b ∩ c) ≡ (a ∩ b) ∩ c a ∩ (b U c) is already in clause form. a U (b ∩ c) ≡ (a U b) ∩ (a U c)

8. Call each conjunct a separate clause. For eg. (a U b) ∩ (a U c)

Separate each conjunct as a U b and a U c

9. Standardize the variables apart again. (X) (a(X) ∩ b(X)) ≡ (X) a(X) ∩ (Y) b(Y)

Example Consider the following expression

Convert this expression to clause form.

Step 1. Eliminate the →.

step 2: Reduce the scope of negation.

The resolution proof procedure Suppose we are given the following axioms.

1. b U c → a 2. b 3. d ∩ e → c 4. e U f 5. d ∩ ⌐f

We want to prove “a‟ from these axioms.

First convert the above predicates to clause form. 1.

b ∩ c → a ⌐ (b ∩ c) U a ⌐ b U ⌐ c U a a U ⌐b U ⌐c

2. d ∩ e → c c U ⌐d U ⌐e

We get the following clauses 1. b U c → a 2. b 3. d ∩ e → c 4. e U f 5. d ∩ ⌐f

1. a U ⌐b U ⌐c 2. b 3. c U ⌐d U ⌐e 4. e U f 5. d 6. ⌐f

The goal to be proved, a, is negated and added to the clause set. Now we have

a U ⌐b U ⌐c b c U ⌐d U ⌐e e U f d ⌐f ⌐a

Example 2

Anyone passing history exams and winning the lottery is happy.

But anyone who studies or is lucky can pass all his exams.

John did not study but he is lucky. Anyone who is lucky wins the lottery. Is john happy?

1. The sentences to predicate form:

.. We get

⌐pass (X, history) U ⌐win (X, lottery) U happy (X)

⌐study (Y) U pass (Y, Z) ⌐lucky (V) U pass (V, W) ⌐study (john) lucky (john) ⌐lucky (U) U win (U, lottery)

Into these clauses is entered, in clause form, the negation of the conclusion.

⌐happy (john)