RESOLUTION

1. RESOLUTION

2. 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

3. Resolution involve the following steps. • Put the premises or axioms in to clause form. • Add the negation of what is to be proved, in clause form, to the set of axioms. • Resolve these clauses together, producing new clauses that logically follow from them. • Produce a contradiction by generating the empty clause. • The substitutions used to produce the empty clause

4. 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))

5. 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

6. 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))

7. 6. Drop all universalquantifiers. • 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. 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)

9. Example • Consider the following expression • Convert this expression to clause form.

10. Step 1. Eliminate the →.

11. step 2: Reduce the scope of negation.

15. 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.

16. 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

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

18. 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

20. 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?

21. 1. The sentences to predicate form:

22. .. 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)

23. Into these clauses is entered, in clause form, the negation of the conclusion. • ⌐happy (john)