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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. .

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what is 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
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
slide4

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. 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
slide6

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))
slide7

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)
slide8

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
Example
  • Consider the following expression
  • Convert this expression to clause form.
the resolution proof procedure
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
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
slide17

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
the goal to be proved a is negated and added to the clause set
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
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
.. 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)