Artificial Intelligence

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Artificial Intelligence

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Title: Artificial Intelligence

1
Artificial Intelligence

Universitatea Politehnica Bucuresti
Adina Magda Florea

2
Lecture No. 5

Theorem proving through resolution refutation
Standard form
Expression unification
Resolution in propositional logic
Resolution in predicate logic
Resolution strategies
Answers through resolution

3
1. Standard form (clausal)

Clause
Horn Clause
Definite Horn Clause
Empty clause
Transformation into clausal form

4
2. Expression unification

Substitution ?
Unifier
Most general unifier ?
Expression
Unification algorithm

Algorithm Unify(E1,E2) Expression unification
1. if E1 and E2 are constants
then
1.1. if E1E2
then return
1.2. return FAIL
2. if E1 is a variable or E2 is a variable
then
2.1. Exchange E1 with E2 such as E1 to be a
variable
2.2. if E1E2
then return
2.3. if E1 occurs in E2
then return FAIL
2.4. return E1/E2

3. if E1simb(t11,,t1n) and E2simb(t21,t2n)
then
3.1 x ? t11, y ? t21
3.2 Rest1 ? t12,,t1n , Rest2 ? t22,t2n
3.3 ?1 ? Unify(x,y)
3.4 if ?1 FAIL
then return FAIL
3.5 G1 ? Rest1 / ?1, G2 ? Rest2 / ?1
3.6 ?2 ? Unify(Rest1, Rest2)
3.7 if ?2 FAIL
then return FAIL
else return (?1, ?2 )
4. return FAIL
end.

7
3. Resolution in propositional logic

Resolvent
Clauses that resolve
Proof principle
Proof tree

Algorithm Resolution refutation in propositional
logic.
1. Transform the set of axioms A in clausal form
and get S0
2. Negate the theorem, transform the negated
theorem in clausal form and add it to S0
3. repeat
3.1. Select two clauses C1 and C2 from S
3.2. Compute R Res(C1,C2)
3.3. if R ? ?
then add R to S
until R ? or there are no two other clauses
that resolve
4. if R ?
then the theoren is proven
5. else it is not a theorem
end.

9
Example

Mihai has money
The car is white
The car is beautiful
If the car is white or the car is beautiful and
Mihai has money then Mihai goes to the mountain
B
A
F
(A ? F) ? B ? C

10
4. Resolution in predicate logic

Resolvent
Clauses that resolve
Factor of a clause

Algorithm Resolution refutation in predicate
logic.
1. Transform the set of axioms A in clausal form
and get S0
2. Negate the theorem, transform the negated
theorem in clausal form and add it to S0
3. repeat
3.1. Select two clauses C1 and C2 from S
3.2. Compute R Res(C1,C2)
3.3. if ??R
then add R to S
until ??R or there are no two other clauses that
resolve or a predefined amount of effort has been
exhausted
4. if ??R
then the theoren is proven
5. else
if there are no two other clauses that resolve
then it is not a theorem
else there is not a definite conclusion
end.

12
Example

Horses are faster than dogs and there is a
greyhound that is faster than every rabbit. We
know that Harry is a horse and that Ralph is a
rabbit. Derive that Harry is faster than Ralph.
Horse(x) Greyhound(y)
Dog(y) Rabbit(z)
Faster(y,z))

?x ?y Horse(x) ? Dog(y) ? Faster(x,y)
?y Greyhound(y) ? (?z Rabbit(z) ? Faster(y,z))
Horse(Harry)
Rabbit(Ralph)
?y Greyhound(y) ? Dog(y)
?x ?y ?z Faster(x,y) ? Faster(y,z) ? Faster(x,z)
13