Slide trí tuệ nhân tạo chapter7 1 using logic
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Using Logic
Chapter 7
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Using Propositional Logic
Representing simple facts
It is raining
RAINING
It is sunny
SUNNY
It is windy
WINDY
If it is raining, then it is not sunny
RAINING SUNNY
Cao Hoang Tru
CSE Faculty - HCMUT
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Propositional Logic Syntax
• Logical constants: true, false
• Propositional symbols: P, Q, …
• Logical connectives: , , , ,
• Sentences (formulas):
–
–
–
–
Logical constants
Proposition symbols
If is a sentence, then so are and ()
If and are sentences, then so are , , ,
and
Cao Hoang Tru
CSE Faculty - HCMUT
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Propositional Logic Semantics
• Interpretation: propositional symbol true/false
• The truth value of a sentence is defined by the truth
table
P
Q
P
PQ
PQ
PQ
PQ
false
false
true
false
false
true
true
false
true
true
false
true
true
false
true
false
false
false
true
false
false
true
true
false
true
true
true
true
Cao Hoang Tru
CSE Faculty - HCMUT
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Propositional Logic Semantics
• Satisfiable: true under an interpretation
• Valid: true under all interpretations
P
Q
P P
(P Q) Q
((P Q) Q) P
false
false
false
false
true
false
true
false
false
true
true
false
false
true
true
true
true
false
false
true
satisfiable
valid
unsatisfiable
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CSE Faculty - HCMUT
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Propositional Logic Semantics
• Model: an interpretation under which the sentence is
true
PQ
PQ
PQ
P
PQ
P
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Q
PQ
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CSE Faculty - HCMUT
Q
PQ
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Propositional Logic Semantics
• Entailment: KB = iff every model of KB is a model of
is a logical consequence of KB
P
Q
PQ
P Q, P
false
false
true
false
false
true
true
false
true
false
false
false
true
true
true
true
Cao Hoang Tru
CSE Faculty - HCMUT
CuuDuongThanCong.com
{P Q, P} = Q
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Propositional Logic Semantics
• Equivalence: iff = and =
P
Q
PQ
P Q
false
false
true
true
false
true
true
true
true
false
false
false
true
true
true
true
Cao Hoang Tru
CSE Faculty - HCMUT
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P Q P Q
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Propositional Logic Semantics
• Theorems:
– = iff is valid
KB = can be proved by validity of KB
– = iff is unsatisfiable
KB = can be proved by refutation of KB
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CSE Faculty - HCMUT
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Using Propositional Logic
• Theorem proving is decidable
• Cannot represent objects and quantification
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CSE Faculty - HCMUT
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Predicate Logic Syntax
• Constant symbols: a, b, c, John, …
to represent primitive objects
• Variable symbols: x, y, z, …
to represent unknown objects
• Predicate symbols: safe, married, love, …
to represent relations
married(John)
love(John, Mary)
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CSE Faculty - HCMUT
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Predicate Logic Syntax
• Function symbols: square, father, …
to represent simple objects
safe(square(1, 2))
love(father(John), mother(John))
• Terms:
to represent complex objects
– Constant symbols
– If f is a function symbol, and t1, t2, …, tn are terms,
then so is f(t1, t2, …, tn)
love(mother(father(John)), John)
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CSE Faculty - HCMUT
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Predicate Logic Syntax
• Logical connectives: , , , ,
• Universal quantifier: "x: p(x)
"x: love(father(x), mother(x))
• Existential quantifier: $x: p(x) "x: p(x)
$x: married(x)
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CSE Faculty - HCMUT
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Predicate Logic Syntax
• Sentences:
– Atomic sentences: p(t1, t2, …, tn)
– If is a sentence, then so are and ()
– If and are sentences, then so are , , ,
and
– If is a sentence, then so are " and $
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CSE Faculty - HCMUT
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Using Predicate Logic
• Can represent objects and quantification
• Theorem proving is semi-decidable
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CSE Faculty - HCMUT
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Using Predicate Logic
1.
Marcus was a man.
2.
Marcus was a Pompeian.
3.
All Pompeians were Romans.
4.
Caesar was a ruler.
5.
All Pompeians were either loyal to Caesar or hated him.
6.
Every one is loyal to someone.
7.
People only try to assassinate rulers they are not loyal to.
8.
Marcus tried to assassinate Caesar.
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Using Predicate Logic
1.
Marcus was a man.
man(Marcus)
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CSE Faculty - HCMUT
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Using Predicate Logic
2.
Marcus was a Pompeian.
Pompeian(Marcus)
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Using Predicate Logic
3.
All Pompeians were Romans.
"x: Pompeian(x) Roman(x)
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CSE Faculty - HCMUT
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Using Predicate Logic
4.
Caesar was a ruler.
ruler(Caesar)
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CSE Faculty - HCMUT
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Using Predicate Logic
5.
All Pompeians were either loyal to Caesar or hated him.
inclusive-or
"x: Roman(x) loyalto(x, Caesar) hate(x, Caesar)
exclusive-or
"x: Roman(x) (loyalto(x, Caesar) hate(x, Caesar))
(loyalto(x, Caesar) hate(x, Caesar))
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CSE Faculty - HCMUT
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Using Predicate Logic
6.
Every one is loyal to someone.
"x: $y: loyalto(x, y)
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CSE Faculty - HCMUT
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$y: "x: loyalto(x, y)
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Using Predicate Logic
7.
People only try to assassinate rulers they are not loyal to.
"x: "y: person(x) ruler(y) tryassassinate(x, y)
loyalto(x, y)
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CSE Faculty - HCMUT
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Using Predicate Logic
7.
People only try to assassinate rulers they are not loyal to.
"x: "y: person(x) ruler(y) tryassassinate(x, y)
loyalto(x, y)
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Using Predicate Logic
8.
Marcus tried to assassinate Caesar.
tryassassinate(Marcus, Caesar)
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