RESOLUTION
WHAT IS RESOLUTION ?
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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.
2.
3.
4.
5.
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
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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
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(⌐dog(X) U animal(X)) ∩ (⌐animal(Y) U die(Y)) ∩ (dog(fido))
1. Producing the clause form
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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.
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⌐ (⌐a) ≡ a
⌐ (X) a(X) ≡ (X) ⌐a(X)
⌐ (X) b(X) ≡ (X) ⌐b(X)
⌐ (a ∩ b) ≡ ⌐a U ⌐b
⌐ (a U b) ≡ ⌐a ∩ ⌐b
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3. Standardize by renaming all variables so that variables
bound by different quantifiers have unique names.
If we have a statement
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((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.
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(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))
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6. Drop all universal quantifiers.
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7. Convert the expression to the conjunct of disjuncts
form using the following equivalences.
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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)
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8. Call each conjunct a separate clause.
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Separate each conjunct as
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For eg.
(a U b) ∩ (a U c)
a U b and
aUc
9. Standardize the variables apart again.
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(X) (a(X) ∩ b(X)) ≡ (X) a(X) ∩ (Y) b(Y)
Example
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Consider the following expression
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Convert this expression to clause form.
Step 1. Eliminate the →.
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step 2:
Reduce the scope of negation.
The resolution proof procedure
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Suppose we are given the following axioms.
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1. b U c → a
2. b
3. d ∩ e → c
4. e U f
5. d ∩ ⌐f
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We want to prove “a‟ from these axioms.
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First convert the above predicates to clause
form.
1.
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b∩c→a
⌐ (b ∩ c) U a
⌐bU⌐cUa
a U ⌐b U ⌐c
2.
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d∩e→c
c U ⌐d U ⌐e
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We get the following clauses
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1.
2.
3.
4.
5.
6.
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
eUf
d
⌐f
The goal to be proved, a, is negated and
added to the clause set.
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Now we have
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a U ⌐b U ⌐c
b
c U ⌐d U ⌐e
eUf
d
⌐f
⌐a
Example 2
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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
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⌐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)
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Into these clauses is entered, in clause form, the negation
of the conclusion.
⌐happy (john)