IF614 – Inteligensia Semu
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5.4 Resolution
5.4.1 Conversion to Clause Form
 Problem :
x:[Roman(x)know(x,Marcus)]
[hate(x,Caesar)
(y:z:hate(y,z)thinkcrazy(x,y)]
 Solution :
 Flatten
 Separate out quantifiers
 Conjunctive Normal Form :
Roman(x)know(x,Marcus)
hate(x,Caesar) hate(y,z)
thinkcrazy(x,z)
 Clause Form :
 Conjunctive Normal Form
 No instances of 
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Algorithm : Convert to Clause Form
1. Eliminate ,using : a  b= a  b.
2. Reduce the scope of each  to a single
term, using :
 (p) = p
de morgan’s laws : (ab) = a  b
(ab) = a  b
 x : P(x) =x : P(x)
 x : P(x) = x : P(x)
3. Standardize variables.
4. Move all quantifiers to the left of the
formula without changing their relative
order.
5. Eliminate existential quantifiers
inserting Skolem functions.
by
6. Drop the prefix.
7. Convert the matrix into a conjuction of
disjuncts,
using
associativity
and
distributivity.
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IF614 – Inteligensia Semu
8. Create a
conjuct.
separate
07 / 03 - 10
clause
for each
9. Standardize the variables in the set of
clauses generated in step 8, using the
fact that
(x ; P(x)Q(x)) = x :P(x)  x : Q(x)
Examples of Conversion to Clause
Form
Example :
x : [Roman(x)  know(x,Marcus)]
[hate(x,Caesar)(y :z:hate(y,z) 
thinkcrazy(x,z))]
1. Eliminate .
x :[Roman(x)know(x,Marcus)]
[hate(x,Caesar)(y : (z : hate(y,z))
 thinkcrazy(z,y))]
2. Reduce scope of .
x :Roman(x)  know(x,Marcus)]
[hate(x,Caesar)  (y : z : hate(y,z) 
thinkcrazy(x,y))]
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3. Standardize variables.
x : P(x)  x : Q(x)
will be convert to
x : P(x)  y : Q(y)
4. Move quantifiers.
x : y : z :[Roman(x)  know(x,Marcus)]
[hate(x,Caesar)  (hate(y,z)  thinkcrazy(x,y))]
5. Eliminate existential quantifiers.
y : President(y)
will be convert to
President(S1)
while
x : y : father-of(y,x)
will be converted to
x : father-of(S2(x),x)
6. Drop the prefix.
[Roman(x)  know(x,Marcus)]  [hate(x,Caesar)
 (hate(y,z)  thinkcrazy(x,y))]
7. Convert to a conjuction of disjuncts.
Roman(x)  know(x,Marcus)  hate(x,Caesar)
 hate(y,z)  thinkcrazy(x,y)
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IF614 – Inteligensia Semu
07 / 05 - 10
5.4.2 The Basis of Resolution
Given :
winter  summer
 winter  cold
We can conclude :
summer  cold
5.4.3 Resolution in Propositional
Logic
A Few Facts in Propositional Logic
Given Axioms
P
(P  Q)R
(S  T)Q
Clause Form
P
P  Q  R
S  Q
T  Q
T
T
(1)
(2)
(3)
(4)
(5)
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Resolution in Propositional Logic
P  Q  R
R
P  Q
T  Q
P
Q
T
T
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5.4.4 The Unification Algorithm
Unification
Q(x)
P(x)
FAIL
P(x)
P(y)
x/y
P(Marcus)
P(y)
Marcus/y
P(Marcus)
P(Julius)
FAIL
P(x,x)
P(y,z)
(y/x)
P(y,y)
(z/y)(y/x)
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Finding General Substitution
Given :
hate(x,y)
hate(Marcus,z)
We could produce :
(Marcus/x, z/y)
(Marcus/x, y/z)
(Marcus/x, Caesar/y, Caesar/z)
(Marcus/x, Polonius/y, Polonius/z)
5.4.5 Resolution in Predicate Logic
Example :
1. man(Marcus)
2. man(x1)  mortal(x1)
Yields the substitution :
Marcus/x1
So it does not yield the resolvent :
mortal(x1)
It does yield :
mortal(Marcus)
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Algorithm : Resolution
1. Convert all the proposition of F to clause form.
2. Negate P and convert the result to clause form.
Add it to the set of clause in step
3. Repeat until either a contradiction is found or
no progress can be made,or a predetermined
amount of effort has been expended.
(a) Select two clause. Call these the parent
clauses.
(b) Resolve them together. The resolvent will be
the disjunction of all of the literals of both
parent clauses with appropriate substitutions
performed and with the following exception: If
there is one pairs of literals T1 and T2 such
that one of the parent clauses contains T1
and the other contains  T2 and if T1 and T2
are unifiable,then neither T1 nor  T2 should
appear in the resolvent.If there is more than
one pair of complementary literals,only one
pair should be ommitted from the resolvent.
(c) If the resolvent is the empty clause, then a
contradiction has been found. If it is not, then
add it to the set of clause available to the
procedure.
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5.4.6 The Need to Try Several
Substitutions
5.4.7 Question Answering
5.5 Natural Deduction
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