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Knowledge Representation Using
Predicate Logic
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Representing Simple Facts in Logic
Representing Instance and Isa
Relationships
Computable Functions and
Predicates
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Resolution
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Natural Deduction
Chapter 5
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Representing Simple Facts in
Logic
1. Marcus was a man
man(Marcus)
2. Marcus was a Pompiean
Pompiean(Marcus)
3. All Pmpien are Romans
νx: Pompiean(x) -> Roman(x)
4. Caesar was a ruler
ruler(Caesar )
5. All Romans were either loyal to
Caesar or hated him
ν x: Roman(x) -> loyalto(x,Caesar) v hate(x,Caesar)
Chapter 5
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6. Everyone is loyal to someone
ν x:∋ y : loyal(x,y)
7. People only try to assassinate
rulers they are not loyal to
ν x: ν
y : person(x) Λ ruler(y) Λ tryassassinate(x,y) -> not
loyalto(x,y)
8. Marcus tried to assassinate Caesar
tryassassinate(Marcus,Caesar)
9. All men are people
ν x: man(x) -> person(x)
Chapter 5
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not loyalto(Marcus,Caesar)
(7, substitution)
person(Marcus)
ruler(Caesar)
tryassassinate(Marcus,Caesar)
(4)
person(Marcus)
tryassassinate(Marcus,Caesar)
(8)
person(Marcus)
An Attempt to Prove
not loyalto(Marcus,Caesar)
Chapter 5
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Representing Instance and Isa
Relationships
1. man(Marcus)
2. Pompiean(Marcus)
3. ν x: Pompiean(x) -> Roman(x)
4. ruler(Caesar )
5. ν x: Roman(x) -> loyalto(x,Caesar) ν hate(x,Caesar)
1. instance(Marcus,man)
2. instance(Marcus, Pompiean)
3. ν x: instance(x, Pompiean)->instance(x,Roman)
4. instance(Caesar,ruler)
5. ν x: instance(x, Roman)->loyalto(x,Caesar) ν hate(x,Caesar)
1. instance(Marcus,man)
2. instance(Marcus, Pompiean)
3. isa(Pompiean,Roman)
4. instance(Caesar,ruler)
5. ν x: instance(x, Roman)->loyalto(x,Caesar) hate(x,Caesar)
6.
ν
x: ν y: ν z: instance(x,y)Λ isa(y,z)-> instance(x,z)
Three Ways of Representing Class
Membership
Chapter 5
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Conversion to Clause Form
ν x: [R(x)Λ k(x,M)] ->
[h(x,C) ν ( ν y: ∋ z: h(y,z) -> t(x,y))]
1.Eliminate -> using the fact a->b is eq. to not a ν b
ν x: not [R(x) Λ k(x,M)]
ν [h(x,C) ν ( ν y:not(∋ z: h(y,z))ν t(x,y))]
2.Reduce the scope of each not to a single
term using not (not p) = p, deMorgan's law, and the
standard correspondence between quantifiers
[not ν x:P(x)= ∋ x: not P(x) and
not ∋ x:P(x)= ν x:-P(x)]
ν x: [not R(x) ν not k(x,M)]
ν [h(x,C) ν ( ν y:ν z: not h(y,z)ν t(x,y))]
3.Standardize variables
For example
ν x: P(x) ν ν x:Q(x)
would be converted to
νx: P(x) ν ν y:Q(y)
Chapter 5
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4.Move all quantifiers to the left of the formula
ν x:ν y: ν z: [ not R(x) ν not k(x,M)]
ν [h(x,C) ν (not h(y,z)ν t(x,y))] (prenex normal
form)
5.Eliminate existential quantifier
∋ y: President(y) can be transformed into the formula
President(S1)
where S1 is a function with no argument that somehow
produces a value that satisfies President
ν x: ∋ y: father-of(y,x) can be transformed into
ν x: father-of(S2(x),x))
6.Drop the prefix
not R(x) ν not k(x,M) ν h(x,C) ν not h(y,z)ν t(x,y)
7.Convert the matrix into a conjunction of disjuncts
You can use the distribution law in this step but
we don have in this example
8.Create a separate clause for each conjunct
We have only one clause
Chapter 5
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9. Standardize apart the variables in the set of clauses
generated in step 8
Chapter 5
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Resolution in Propositional Logic
Given Axioms
Converted to Clause Form
P
(P Λ Q) -> R
(S ν T) -> Q
P
not P ν not Q ν R
not S ν Q
not T ν
Q
T
T
A Few Facts in Propositional Logic
-Pν-Qν R
-R
-P ν -Q
-T
Q
P
-Q
-T
T
Resolution in Propositional Logic
Chapter 5
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Resolution in Predicate Logic
Axioms in clause form:
1.man(Marcus)
2.Pompiean(Marcus)
3.- Pompiean(x1) ν Roman(x1)
4.ruler(Caesar )
5.- Roman(x2) ν loyalto(x2,Caesar) ν hate(x2,Caesar)
6. loyal(x3,f(x3))
7.- man(x4) ν - ruler(y1) ν - tryassassinate(x4,y1) ν
loyalto(x4,y1)
8.tryassassinate(Marcus,Caesar)
Prove: hate(Marcus,Caesar)
- hate(Marcus,Caesar)
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- Roman(Marcus) ν loyalto(Marcus,Caesar)
Pompiean(Marcus) ν loyalto(Marcus,Caesar)
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2
loyalto(Marcus,Caesar)
1 - man(Marcus) ν- ruler(Caesar)νtryassassi..(Marcus,Caesar)
- ruler(Caesar) ν - tryassassinate(Marcus,Caesar)
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- tryassassinate(Marcus,Caesar) 8
A Resolution Proof
Chapter 5
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Prove: loyalto(Marcus,Caesar)
- loyalto(Marcus,Caesar)
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- Roman(Marcus) ν hate(Marcus,Caesar)
3
- Pompiean(Marcus) ν hate(Marcus,Caesar)
10
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hate(Marcus,Caesar)
persecute(Caesar,Marcus)
hate(Marcus,Caesar)
An Unsuccessful Attempt at
Resolution
Chapter 5
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2
Given
1.
2.
3.
4.
1
father (x,y) ν - women(x)
mother(x,y) ν women(x)
mother(Chris, Mary)
father(Chris, Bill)
2
- father(x,y) ν - mother(x,y)
3
- father(Chris,Mary)
The need to Standardize Variables
1
2
- father(a,y) ν - mother(a,b)
- father(Chris,y)
Chapter 5
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4
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Prove:∋x: hate(Marcus,x)Λ ruler(x)
(negate):- ∋ x: hate(Marcus,x) Λ ruler(x)
(clausify): - hate(Marcus,x) ν - ruler(x)
- hate(Marcus,x) ν - ruler(x)
hate(Marcus, Paulus)
- ruler(x)
(a)
- hate(Marcus,x) ν - ruler(x)
hate(Marcus, Julian)
- ruler(Julian)
(b)
- hate(Marcus,x) ν - ruler(x)
hate(Marcus, Caesar)
- ruler(Caesar)
ruler(Caesar)
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(c)
Trying Several Substitution
Chapter 5
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- ∋ t: died(Marcus,t) = - died(Marcus,t)
- Pompeian(x1) ν died(x1,79)
- died(Marcus,t)
79/t,Marcus/x1
- Pompeian(Marcus)
Pompeian(Marcus)
(a)
- Pompeian(x1) νdied(x1,79) - died(Marcus,t)νdied(Marcus,t)
79/t,Marcus/x1
- Pompeian(Marcus) ν died(Marcus,t)
Pompeian(Marcus)
died(Marcus,79)
Answer Extraction Using Resolution
Chapter 5
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