RESOLUTION

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RESOLUTION - PowerPoint PPT Presentation

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. .

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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.
• 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

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 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. 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.
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
• a U ⌐b U ⌐c
• b
• c U ⌐d U ⌐e
• e U f
• d
• ⌐f
• 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?
.. 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)