Knowledge Representation Using Predicate Logic Representing Simple - - PDF document

Chapter 5 1

Knowledge Representation Using Predicate Logic

• Representing Simple Facts in Logic
• Representing Instance and Isa

Relationships

• Computable Functions and

Predicates

• Resolution
• Natural Deduction

Chapter 5 2

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 3

• 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 4

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 5

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 6

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 7

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 8

• 9. Standardize apart the variables in the set of clauses

generated in step 8

Chapter 5 9

Resolution in Propositional Logic

Given Axioms Converted to Clause Form P P (P Λ Q) -> R not P ν not Q ν R (S ν T) -> Q not S ν Q not T ν Q T T

A Few Facts in Propositional Logic

• P ν - Q ν R
• R
• P ν - Q

P

• T Q
• Q
• T

T

Resolution in Propositional Logic

Chapter 5 10

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)

5 3

• Roman(Marcus) ν loyalto(Marcus,Caesar)

Pompiean(Marcus) ν loyalto(Marcus,Caesar) 2 7 loyalto(Marcus,Caesar) 1 - man(Marcus) ν- ruler(Caesar)νtryassassi..(Marcus,Caesar)

• ruler(Caesar) ν - tryassassinate(Marcus,Caesar)

4

• tryassassinate(Marcus,Caesar) 8

A Resolution Proof

Chapter 5 11

Prove: loyalto(Marcus,Caesar)

• loyalto(Marcus,Caesar)

5 3

• Roman(Marcus) ν hate(Marcus,Caesar)
• Pompiean(Marcus) ν hate(Marcus,Caesar)

2 10 hate(Marcus,Caesar) 9 persecute(Caesar,Marcus) hate(Marcus,Caesar)

An Unsuccessful Attempt at Resolution

Chapter 5 12

Given 1. father (x,y) ν - women(x) 2. mother(x,y) ν women(x) 3. mother(Chris, Mary) 4. father(Chris, Bill) 1 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)

3

• father(Chris,y)

4

Chapter 5 13

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)

• (c)

Trying Several Substitution

Chapter 5 14

• ∋ 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