1 / 25

RESOLUTION

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. .

deva
Download Presentation

RESOLUTION

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

  12. 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.

  13. 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

  14. 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

  15. 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

  16. 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?

  17. 1. The sentences to predicate form:

  18. .. 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)

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

More Related