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Inference in First-Order LogicChapter 9
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Some Basic Definitions
A free variable is a variable thatisn't bound by a quantifier.– y Likes(x,y)
x is free, y is bound
A sentence is a well-formed formula whereall variables are quantified.
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Rules of Inference
x P(x)__________ P(c) if cU
Universal Universal instantiationinstantiation
P(c) for an arbitrary cP(c) for an arbitrary cUU______________________________________
x P(x)x P(x)
Universal Universal generalizationgeneralization
x P(x)x P(x)____________________________________________
P(c) for some element cP(c) for some element cUU
Existential Existential instantiationinstantiation
P(c) for some element cP(c) for some element cUU________________________________________
x P(x) x P(x)
Existential Existential generalizationgeneralization
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An example of U.G.
1. x(Ontable(x, T1) Upturned(x, T2))
2. xy(Upturned(x, y) Empty(x, y))
T1 T2
b1 b2 b3b1 b2 b3
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An example of U.G.
3. Ontable(b2, T1) Assumption4. Ontable(b2, T1) Upturned(b2, T2) - , 15. Upturned (b2, T2) - , 3, 46. Upturned(b2, T2) Empty(b2, T2) - , 2 7. Empty(b2, T2) - , 5, 68. Ontable(b2, T1) Empty(b2, T2) + , 7 9. x(Ontable(x, T1) Empty(x, T2)) + , 8
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An example of E.I.
1. x(Bottle(x,T1) Upturned(x, T2))
2. xy(Upturned(x, y) Empty(x, y))
3. x(Full(x, T1) & Empty(x, T2) Wet(Floor))
4. x (Bottle(x, T1) & Full(x, T1))
T1 T2
b1 b2 b3b1 b2 b3
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An example of E.I.
5. Bottle(b1, T1) & Full(b1, T1)) - EI assumption
6. Bottle(b1, T1) - & , 5
7. Full(b1, T1) - & , 5
8. Upturned (b1, T2) - , 1 , - , 6
9. Empty(b1, T2) - , 2 , - , 7
10. Full(b1, T1) & Empty (b1, T2) + & , 7, 9
11. Wet(Floor) - , 3, - , 10 Wet(Floor) - EI conclusion
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1. x(P(x) Q) Assumption2. xP(x)
Assumption3. P(x) - Ass.4. P(x) Q + , 1 5. Q - 3, 46. Q - Conc. 7. xP(x) Q + 2, 68. (x(P(x) Q)) (xP(x) Q) + 1, 7
Prove (x(P(x) Q)) (xP(x) Q)
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1. (xP(x) Q) Assumption2. P(x)
Assumption3. xP(x) + 4. Q - 1, 35. P(x) Q - 2, 4 6. x(P(x) Q) + 57. (xP(x) Q) (x(P(x) Q)) + 1, 6
Prove (xP(x) Q) (x(P(x) Q))
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Unification
Can we find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y)
= {x/John,y/John} is such a substitution
Unify( ,) = if = p q Knows(John,x) Knows(John,Jane) Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y))Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Unification
Can we find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y)
= {x/John,y/John} is such a substitution
Unify( ,) = if = p q Knows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y))Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Unification
Can we find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y)
= {x/John,y/John} is such a substitution
Unify( ,) = if = p q Knows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ) {x/OJ,y/John}Knows(John,x) Knows(y,Mother(y))Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Unification
Can we find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y)
= {x/John,y/John} is such a substitution
Unify( ,) = if = p q Knows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ) {x/OJ,y/John}Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}Knows(John,x) Knows(x,OJ)
Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Unification
Can we find a substitution such that King(x) and Greedy(x) match King(John) and Greedy(y)
= {x/John,y/John} is such a substitution
Unify( ,) = if = p q Knows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ) {x/OJ,y/John}Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}Knows(John,x) Knows(x,OJ) {fail}
Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)
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Unification
To unify Knows(John,x) and Knows(y,z),θ = {y/John, x/z } or θ = {y/John, x/John, z/John}
The first unifier is more general than the second. There is a single most general unifier (MGU) that is
unique up to renaming of variables.MGU = { y/John, x/z }
–
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The unification algorithm
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The unification algorithm
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Example Knowledge Base
Mark was a man. Mark was a Pompeian. All Pompeians were Romans. Caesar was a ruler. All Romans were either loyal to Caesar or hated him. Everyone is loyal to someone. People only try to assassinate rulers they are not loyal to. Mark tried to assassinate Caesar.
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Example Knowledge Base
(1) Mark was a man.man(Mark) Was Mark loyal to Caesar?
(2) Mark was a Pompeian.Pompeian(Mark)
(3) All Pompeians were Romans.x (Pompeian(x) Roman(x))
(4) Caesar was a ruler.ruler(Caesar)
(5) All Romans were either loyal to Caesar or hated him. x (Roman(x) loyalto(x, Caesar) hate(x, Caesar))
(6) Everyone is loyal to someone. x y loyalto(x, y)
(7) People only try to assassinate rulers they are not loyal to. x y (person(x) ruler(y) tryassassinate(x, y) loyalto(x, y))
(8) Mark tried to assassinate Caesar.tryassassinate(Mark, Caesar)
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Reasoning Backward
(1) man(Mark)(2) Pompeian(Mark)(3) x (Pompeian(x) Roman(x))(4) ruler(Caesar)(5) x (Roman(x) loyalto(x, Caesar) hate(x, Caesar))(6) x y loyalto(x, y)(7) x y (person(x) ruler(y) tryassassinate(x, y) loyalto(x, y))(8) tryassassinate(Mark, Caesar)
loyalto(Mark, Caesar)
(x/Mark) (y/Caesar)
person(Mark) ruler(Caesar) tryassassinate(Mark, Caesar)
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Reasoning Backward
(1) man(Mark)(2) Pompeian(Mark)(3) x (Pompeian(x) Roman(x))(4) ruler(Caesar)(5) x (Roman(x) loyalto(x, Caesar) hate(x, Caesar))(6) x y loyalto(x, y)(7) x y (person(x) ruler(y) tryassassinate(x, y) loyalto(x, y))(8) tryassassinate(Mark, Caesar)(9) x (man(x) person(x))
loyalto(Mark, Caesar)
(x/Mark) (y/Caesar)
person(Mark) ruler(Caesar) tryassassinate(Mark, Caesar)
(x/Mark)
Man(Mark)
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Functions and Predicates
(1) Mark was a man.man(Mark)
(2) Mark was a Pompeian.Pompeian(Mark)
(3) Mark was born in 40 A.D.born(Mark, 40)
(4) All men are mortal.x (man(x) mortal(x))
(5) A volcano erupted in 79 A.D. erupted(volcano, 79)
(6) AllPompeians died in 79 A.D. x (Pompeian(x) died(x,79))
(7) No mortal lives longer than 150 years. x t1 t2 (mortal(x) born(x, t1) gt(t2-t1, 150) dead(x, t2))
(8) It is now 2010.now = 2010
Is Mark alive?
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Functions and Predicates
(1) man(Mark)(2) Pompeian(Mark)(3) born(Mark, 40)(4) x (man(x) mortal(x))(5) erupted(volcano, 79) (6) x (Pompeian(x) died(x,79)) (7) x t1 t2 (mortal(x) born(x, t1) gt(t2-t1, 150) dead(x, t2))(8) now = 2010
alive(Mark, now)
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Functions and Predicates - Filling in the Blanks
(1) man(Mark)(2) Pompeian(Mark)(3) born(Mark, 40)(4) x (man(x) mortal(x))(5) erupted(volcano, 79) (6) x (Pompeian(x) died(x,79)) (7) x t1 t2 (mortal(x) born(x, t1) gt(t2-t1, 150) dead(x, t2))(8) now = 2010(9a) x t (dead(x,t) alive(x, t))(9b) x t (alive(x, t) dead(x, t))(10) x t1 t2 (died(x, t1) gt(t2, t1) dead(x, t2))
alive(Mark, now)
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Showing that Mark is Not Alive
alive(Mark, now) (9a) (x/Mark) (t/now)
dead(Mark, now)
(10)(x/Mark)(t2/now) (7) (Mark/x)(now/t2) died(Mark, t1) gt(now, t1) mortal(Mark) born(Mark, t1) gt(now-t1,150)
(6)(x/Mark) (t1/79) (4) (x/Mark) (3) (t1/40)
Pompeian(Mark) gt(now, 79) man(Mark) born(Mark, 40) gt(now-40,150) (2) subst (1) subst
T gt(2010, 79) T T gt(2010-40,150) eval eval
T T
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A Harder One
Given:x [Roman(x) know(x, Mark)]
[hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
Roman(Isaac)hate(Isaac, Caesar)hate(Paulus, Mark)thinkcrazy(Isaac, Paulus)
Prove:know(Isaac, Mark)
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Clause Form Simplifies the Process
Standard:x [Roman(x) know(x, Mark)]
[hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
Clausal:Roman(x) know(x, Mark) hate(x, Caesar) hate(y, z) thinkcrazy(x, z)
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Conversion to Clause Form - Step 1
1. Eliminate , using the fact that a b is equivalent to a b
x [Roman(x) know(x, Mark)] [hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
x [Roman(x) know(x, Mark)] [hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
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Conversion to Clause Form - Step 2
2. Reduce the scope of each to a single term, using:(p) = p•deMorgan’s lawsx P(x) x P(x)x P(x) x P(x)
x [Roman(x) know(x, Mark)] [hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
x [ Roman(x) know(x, Mark)] [hate(x, Caesar) (y z hate(y, z) thinkcrazy(x, y))]
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Conversion to Clause Form - Step 3
3. Standardize variables so that each quantifier binds a unique variable.
x P(x) x Q(x)
x P(x) y Q(y)
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Conversion to Clause Form - Step 4
4. Move all quantifiers to the left without changing their relative order.
x [ Roman(x) know(x, Mark)] [hate(x, Caesar) (y z hate(y, z) thinkcrazy(x, y))]
x y z [ Roman(x) know(x, Mark)] [hate(x, Caesar) ( hate(y, z) thinkcrazy(x, y))]
At this point, we have prenex normal form.
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Conversion to Clause Form - Step 5
5. Eliminate existential quantifiers through the use of Skolem functions and constants.
x Roman(x) Roman(S1)
x z father-of(x, z) x father-of(x, S2(x))
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Conversion to Clause Form - Step 6
6. Drop the prefix since all remaining quantifiers are universal.
x y z [ Roman(x) know(x, Mark)] [hate(x, Caesar) ( hate(y, z) thinkcrazy(x, y))]
[ Roman(x) know(x, Mark)] [hate(x, Caesar) ( hate(y, z) thinkcrazy(x, y))]
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Conversion to Clause Form - Step 7
7. Convert the matrix into a conjunction of disjuncts by using:•Associative properties of and .
[ Roman(x) know(x, Mark)] [hate(x, Caesar) ( hate(y, z) thinkcrazy(x, y))]
Roman(x) know(x, Mark) hate(x, Caesar) ( hate(y, z) thinkcrazy(x, y))
•Distribution of over .
(P(x) Q(x)) T(x)
(P(x) T(x)) (Q(x)) T(x))
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Conversion to Clause Form - Step 8
8. Create a separate clause for each conjunct.
(P(x) T(x)) (Q(x) T(x))
(P(x) T(x)) (Q(x) T(x))
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Conversion to Clause Form - Step 9
9. Standardize apart the variables.
(P(x) T(x)) (Q(x) T(x))
(P(x) T(x)) (Q(y) T(y))
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Resolution in Predicate Logic
p’ v q ~p v r (q v r)
p(x) v q(x) ~p(a) v r(b) (q(x) v r(b)) [x/a] => q(a) v r(b)
All variables assumed universally quantified.
where p' = p
where = [x/a]
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Importance of Standardizing Variables Apart
Consider:1. father(x, y) woman(x)
{father(x, y) woman(x)}2. mother(x, y) woman(x)
{mother(x, y) woman(x)}3. mother(Chris, Mary)4. father(Chris, Bill)
1 2
* father(x, y) mother(x, y) 3
x/Chris, y/Mary
father(Chris, Mary)
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Back to Mark
(1) man(Mark)(2) Pompeian(Mark)(3) Pompeian(x1) Roman(x1)(4) ruler(Caesar)(5) Roman(x2) loyalto(x2, Caesar) hate(x2, Caesar)(6) loyalto(x3, S1(x3))(7) man(x4) ruler(y1) tryassassinate(x4, y1) loyalto(x4, y1)(8) tryassassinate(Mark, Caesar)
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Proving Mark Not Loyal to Caesar
loyalto(M, C) man(x4) ruler(y1) tryassassinate(x4, y1) loyalto(x4, y1)
(x4/M)(y1/C)
man(M) ruler(C) tryassassinate(M,C) man(M)
ruler(C) tryassassinate(M,C) ruler(C)
tryassassinate(M,C) tryassassinate(M,C)
nil
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Does Isaac Know Mark?
Given:1. x [Roman(x) know(x, Mark)]
[hate(x, Caesar) (y (z hate(y, z)) thinkcrazy(x, y))]
Roman(x) know(x, Mark) hate(x, Caesar) hate(y, z) thinkcrazy(x, y)
2. Roman(Isaac)3. hate(Isaac, Caesar)4. hate(Paulus, Mark)5. thinkcrazy(Isaac, Paulus)
Prove:know(Isaac, Mark) * know(Isaac, Mark)
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Proving Issac Doesn’t know Mark
know(Isaac, Mark) 1
(x/I)
Roman(I) hate(I, C) hate(y, z) thinkcrazy(I, y) 2: Roman(I)
hate(I, C) hate(y, z) thinkcrazy(I, y) 3: hate(I, C)
hate(y, z) thinkcrazy(I, y) 4: hate(P, M)
(y/P)(z/M)
thinkcrazy(I, P) 5: thinkcrazy(I, P)
nil
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Question Answering
When did Mark die?
died(Mark, ??)
Mark must have died sometime, so we could try to prove:
x died(Mark, x) by adding to the KB:
x died(Mark, x). Converting to clause form:
x died(Mark, x)
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When Did Mark Die?
died(Mark, x) Pompeian(x1) died(x1,79)
(x1/Mark)(x/79)
Pompeian(Mark) Pompeian(Mark)
nil
But how do we retrieve the answer 79?
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When Did Mark Die?
died(Mark, x) died(Mark, x) Pompeian(x1) died(x1,79)
(x1/Mark)(x/79)
Pompeian(Mark) died(Mark, 79) Pompeian(Mark)
died(Mark, 79)
The underlined literal will get carried along and collect all the substitutions. We stop when we generate a clause that contains only it.
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Resolution Strategies
Unit Preference:– prefer to do resolutions where 1 sentence
is a single literal, a unit clause
– goal is to produce the empty clause, focus searchby producing resolvents that are shorter
complete but too slow for medium sized problems
Unit Resolution:– requires at least 1 to be a unit clause complete for Horn Clauses
1. p q KB2. ~p r KB3. ~q r KB4. ~r S5. ~p 2, 4
6. ~q 3, 47. q 1, 5
8. p 1, 6
9. r 3, 7
10. nil 6, 7
11. r 2, 8
12. nil 5, 8
Unit Resolution Example (1)
1. p q
2. ~p q
3. p ~q
4. ~p ~q
Cannot perform a single unit resolution since all clauses are
of size 2!
Unit Resolution Example(2)
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Resolution Strategies (Cont.)
Set-of-Support (SoS):– identify some subset of sentences, called SoS
– P and Q can be resolved if one is from SoS
– resolvent is added to the SoS
– common approach: query is the initial SoS, resolvents are added assumes KB is satisfiable
complete if KB-SoS is satisfiable
1. p q KB2. ~p r KB3. ~q r KB4. ~r S5. ~p 2, 4 add to S6. ~q 3, 4 add to S7. q 1, 5 add to S8. p 1, 6 add to S9. r 3, 710. nil 6, 711. r 2, 812. nil 5, 8
Set of Support resolution example
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Resolution Strategies (Cont.)
Input Resolution:– P and Q can be resolved if at least one is from
the set of original clauses, i.e., KB and query complete for Horn Clauses
Linear Resolution:– a slight generalization of input resolution– P and Q can be resolved if:
– P is from the set of the original clauses, or – P is an ancestor of Q, in the proof tree
complete
Ex: 1. p q KB
2. ~p q KB
3. p ~q KB
4. ~p ~q KBCannot performinput resolution!
Input Resolution
p q ~p q p ~q ~p ~q
q
p
~q q
nil
Linear Resolution
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Natural Deduction vs. Resolution
• CNF is simpler to work with• But, we loose readability and some useful control
information
x(judge(x) & ~crooked(x) educated(x))
~judge(x) v crooked(x) v educated(x)