Knowledge Representation Using Predicate Logic Representing Simple - PDF document

Knowledge Representation Using Predicate Logic • Representing Simple Facts in Logic • Representing Instance and Isa Relationships • Computable Functions and Predicates • Resolution • Natural Deduction Chapter 5 1

Representing Simple Facts in Logic 1. Marcus was a man man(Marcus) 2. Marcus was a Pompiean Pompiean(Marcus) 3. All Pmpien are Romans ν x: 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 2

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 3

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 4

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 5

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 ν x: P(x) ν ν y:Q(y) Chapter 5 6

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 7

9. Standardize apart the variables in the set of clauses generated in step 8 Chapter 5 8

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 9

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 - Roman(Marcus) ν loyalto(Marcus,Caesar) 3 Pompiean(Marcus) ν loyalto(Marcus,Caesar) 2 7 loyalto(Marcus,Caesar) 1 - man(Marcus) ν - ruler(Caesar) ν tryassassi..(Marcus,Caesar) - ruler(Caesar) ν - tryassassinate(Marcus,Caesar) 4 - tryassassinate(Marcus,Caesar) 8 A Resolution Proof Chapter 5 10

Prove: loyalto(Marcus,Caesar) - loyalto(Marcus,Caesar) 5 - Roman(Marcus) ν hate(Marcus,Caesar) 3 - 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 11

Given father (x,y) ν - women(x) 1. mother(x,y) ν women(x) 2. 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 12

∋ x: hate(Marcus,x) Λ ruler(x) Prove: (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 13

- ∋ 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) ν died(x1,79) - died(Marcus,t) ν died(Marcus,t) 79/t,Marcus/x1 - Pompeian(Marcus) ν died(Marcus,t) Pompeian(Marcus) died(Marcus,79) Answer Extraction Using Resolution Chapter 5 14