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First-Order Logic
ECE457 Applied Artificial Intelligence
Fall 2007
Lecture #7
Outline


First-order logic (FOL)
Sentences


Russell & Norvig, chapter 8
Inference

Russell & Norvig, chapter 9
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 2
First-Order Logic

Propositional logic is limited

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Cannot represent information concisely
We want something more expressive
First-order logic

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Allows the representation of objects,
functions on objects and relations between
objects
Allows us to represent almost any English
sentence
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 3
New Symbols in FOL

Constant symbol
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Predicate symbol
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A relation between two objects that can be true or
false
Brother(Bob, James), OnHead(Hat, Bob)
Function symbol

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
An individual object in the world
Bob, James, Hat
Special type of relation that maps one object to
another
Head(Bob)
All symbols begin with uppercase letters
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 4
New Variables in FOL


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Begin with lowercase letters
Stand-in for any symbol
Brother(x,y)

x is the brother of y
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 5
New Quantifiers in FOL

Universal quantifier 

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x means “For all x…”
Always true
Usually used with 
x means “Not all x…”
Existential quantifier 




x means “There exists an x…”
True for at least one interpretation
Usually used with 
x means “There exists no x…”
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 6
Properties of New Quantifiers

Nesting
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x y P(x,y) same as y x P(x,y)
x y P(x,y) same as y x P(x,y)
x y P(x,y) not same as y x P(x,y)
Duality

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x P(x) same as x P(x)
x P(x) same as x P(x)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 7
Sentences in FOL

Term
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Atom (atomic sentence)
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Constant symbol, function symbol, or
variable
Predicate symbol with value true or false
Represents a relation between terms
Sentence (complex sentence)

Atom(s) joined together using logical
connectives and/or quantifiers
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 8
Example Sentences

Purple mushrooms are poisonous

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Some purple mushrooms are poisonous
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x Purple(x)  Mushroom(x)  Poisonous(x) (bad!)
x Purple(x)  Mushroom(x)  Poisonous(x)
No purple mushrooms are poisonous
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Purple(x)  Mushroom(x)  Poisonous(x)
x Purple(x)  Mushroom(x)  Poisonous(x)
x Purple(x)  Mushroom(x)  Poisonous(x)
There are exactly two purple mushrooms

x y z Purple(x)  Mushroom(x)  Purple(y) 
Mushroom(y)  (x=y)  Purple(z)  Mushroom(z)
 (x=z)  (y=z)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 9
Implication Truth Table

α
T
T
F
F
β αβ
T
T
F
F
T
T
F
T

If α is true, then β must be true
for the implication to be true
If α is false, then we have no
information about β
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β might be true or false without
affecting the implication
β is not necessarily true
If implication is true and α is
true, then β must be true
(Modus Ponens)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 10
Universal Implication
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x={
,
, }
All purple mushrooms are poisonous
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Universal: all three implications are true
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x Purple(x)  Mushroom(x)  Poisonous(x)
is purple and a mushroom  is poisonous
is purple and a mushroom  is poisonous
is purple and a mushroom  is poisonous
But the first one is the only one where the
premise is true, therefore the only one where
we know the consequent is true (Modus
Ponens)

Otherwise, we don’t know whether x is poisonous
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 11
Existential Implication
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x={
,
, }
Some purple mushrooms are poisonous
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Existential: at least one implication is true
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x Purple(x)  Mushroom(x)  Poisonous(x)
is purple and a mushroom  is poisonous
is purple and a mushroom  is poisonous
is purple and a mushroom  is poisonous
But all those where the premise is false are
true

The existential implication doesn’t give us any
information
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 12
Existential Conjunction
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x={
,
, }
Some purple mushrooms are poisonous
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Existential: at least one statement is true
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x Purple(x)  Mushroom(x)  Poisonous(x)
is purple and a mushroom and is poisonous
is purple and a mushroom and is poisonous
is purple and a mushroom and is poisonous
The first statement is true, the other two are
false
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The existential quantifier is satisfied
The situation is consistent with our original meaning
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 13
Universal Conjunction
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x={
,
, }
All purple mushrooms are poisonous
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Universal: all three statements are true
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x Purple(x)  Mushroom(x)  Poisonous(x)
is purple and a mushroom and is poisonous
is purple and a mushroom and is poisonous
is purple and a mushroom and is poisonous
“Everything is a poisonous purple mushroom”

This is false, and not the original meaning of our
sentence
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 14
Summary
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Universal quantifier 
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Usually used with  to create universal
rules
Using it with  makes blanket statements
about all objects in the world
Existential quantifier 
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Usually used with  to list properties of an
object
Using it with  creates sentences that do
not say much
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 15
Example Sentences

You can fool some people all the time
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Everyone who loves all animals is loved
by someone
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x y Person(x)  Time(y) 
CanBeFooled(x,y)
x y Animal(y)  Love(x,y)  z Love(z,x)
No living man can kill the Witch-King

x Living(x)  Man(x) 
CanKill(x, Witch-King)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 16
Wumpus World in FOL
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Predicate to represent adjacent rooms
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x,y,a,b Adjacent([x,y],[a,b]) 
[a,b]  {[x+1,y], [x-1,y], [x,y+1], [x,y-1]}
Sentences to represent the relationship
between breezy rooms and pits
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r Breeze(r)  s Adjacent(r,s)  Pit(s)
s Pit(s)  (r Adjacent(r,s)  Breeze(r))
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 17
Inference in FOL


We know how to do inference in
propositional logic
Can we reduce FOL to propositional
logic and infer from there?
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Propositionalization
Eliminate universal quantifiers
Eliminate existential quantifiers
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 18
Universal Instantiation
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A universal variable x in a true sentence P(x)
can be substituted by any (and all) ground
terms g without variables in the domain of x,
and P(g) will be true
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SUBST({x/g},P(x)) = P(g)
Example
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x  {Bob, James, Tom}
x Strong(x)  Fast(x)  Athlete(x)
Strong(Bob)  Fast(Bob)  Athlete(Bob)
Strong(James)  Fast(James)  Athlete(James)
Strong(Tom)  Fast(Tom)  Athlete(Tom)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 19
Existential Instantiation

Any existential variable x in a true
sentence P(x) can be replaced by a
constant symbol not already in the KB

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Skolem constant
Example
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x Purple(x)  Mushroom(x) Poisonous(x)
Purple(C)  Mushroom(C)  Poisonous(C)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 20
Inference Example
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We know the following
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Translate into FOL KB
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All of Bob’s dogs are brown. Fido is a dog. Bob owns Fido.
Can we infer that Fido is brown?
x Dog(x)  Own(Bob,x)  Brown(x)
Dog(Fido)
Own(Bob,Fido)
Apply inference rules
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Universal Instantiation with {x/Fido}:
Dog(Fido)  Own(Bob,Fido)  Brown(Fido)
And-Introduction: Dog(Fido)  Own(Bob,Fido)
Modus Ponens: Brown(Fido)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 21
Problem with Propositionalization
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The set of ground substitution can be infinite
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Algorithms can prove sentence is entailed
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Function symbols can be nested infinitely
Example: Father(x) = Father(Bob),
Father(Father(Bob)), Father(Father(Father(Bob))), …
Search every depth successively until sentence is
found
Algorithms cannot prove sentence is not
entailed
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Will search deeper and deeper, forever
Entailment in FOL is semidecidable
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 22
Problem with Propositionalization
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Modify the previous example
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x Dog(x)  Own(Bob,x)  Brown(x)
Dog(Fido), Dog(Barky), Dog(Dogbert), Dog(Princess),
Dog(Lassie), Dog(K-9)
Own(Bob,Fido)
Inference with Propositionalization
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Universal Instantiation:
Dog(Fido)  Own(Bob,Fido)  Brown(Fido)
Dog(Barky)  Own(Bob,Barky)  Brown(Barky)
Dog(Dogbert)  Own(Bob,Dogbert)  Brown(Dogbert)
Dog(Princess)  Own(Bob,Princess)  Brown(Princess)
Dog(Lassie)  Own(Bob,Lassie)  Brown(Lassie)
Dog(K-9)  Own(Bob,K-9)  Brown(K-9)
And-Introduction: Dog(Fido)  Own(Bob,Fido)
Modus Ponens: Brown(Fido)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 23
Problem with Propositionalization
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Brown(Fido) is obvious to us
Universal Instantiation generates a lot
of useless substitutions which are
obviously unnecessary (to us)
Recall Modus Ponens:

(α  β), α
β
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 24
Generalized Modus Ponens
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Combine And-Introduction, Universal
Elimination, and Modus Ponens
p1’, p2’,…, pn’, (p1  p2  …  pn  q)
SUBST(, q)
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p1 = Dog(x)
p2 = Own(Bob,x) q = Brown(x)
p1’ = Dog(Fido) p2’ = Own(Bob,Fido)
 = {x/Fido}
SUBST(, q) = Brown(Fido)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 25
Inference Example
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FOL KB
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Inference with Generalized Modus Ponens
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x Dog(x)  Own(Bob,x)  Brown(x)
Dog(Fido), Dog(Barky), Dog(Dogbert),
Dog(Princess), Dog(Lassie), Dog(K-9)
Own(Bob,Fido)
Dog(Fido), Own(Bob,Fido),
Dog(x)  Own(Bob,x)  Brown(x)
 = {x/Fido}
Brown(Fido)
Requires some form of pattern matching to
discover 
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 26
Unification
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Unification algorithm takes two
sentences p and q
Returns the substitutions to make both
sentences match, or failure if none is
possible
UNIFY(p,q) = , where  is the list of
substitutions in p and q
Find the matching that places the least
restrictions on the variables

Most general unifier (MGU)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 27
Unification
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Given sentences p and q,  is empty
Scan p and q left to right
If they disagree on terms r and s
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If r is a variable
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If s is a complex term and r occurs in s
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Else
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Add r/s to 
Else if s is a variable (but r is a complex term)
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If s occurs in r
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Else
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Fail
Fail
Add s/r to 
Else (both r and s are complex term)

Apply unification to r and s
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 28
Unification Rules
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Function symbols and predicate symbols unify
with function symbols and predicate symbols
that have identical names and number of
arguments
Constant symbols can only unify with identical
constant symbols
Variables can unify with other variables,
constant symbols, or function symbols
Variables cannot unify with a term in which
that variable occurs.


x cannot unify with P(x), as that leads to
P(P(P(P(…P(x)…))))
That is an occur check error
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 29
Unification Examples

p = Parents(x, Father(x), Mother(Bob))
q = Parents(Bob, Father(Bob), y)

Consider first term of each sentence
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Disagreement
r and s are predicate symbols with identical names
and number of arguments, so unification is
possible
Unify arguments
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r = Parents(x, Father(x), Mother(Bob))
s = Parents(Bob, Father(Bob), y)
r = x, s = Bob, x/Bob
r = Father(Bob), s = Father(Bob), no substitution
r = Mother(Bob), s = y, y/Mother(Bob)
 = {x/Bob, y/Mother(Bob)}
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 30
Unification Examples

p = Parents(x, Father(x), Mother(Bob))
q = Parents(Bob, Father(y), z)

Unify arguments
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

r = x, s = Bob, x/Bob
r = Father(Bob), s = Father(y), y/Bob
r = Mother(Bob), s = z, z/Mother(Bob)
 = {x/Bob, y/Bob, z/Mother(Bob)}
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 31
Unification Examples

p = Parents(x, Father(x), Mother(Jane))
q = Parents(Bob, Father(y), Mother(y))

Unify arguments
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

r = x, s = Bob, x/Bob
r = Father(Bob), s = Father(y), y/Bob
r = Mother(Jane), s = Mother(Bob), Fail
Fail
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 32
Unification Exercises

p = Parents(x, Father(x), Mother(x))
q = Parents(Bob, SpermDonor(y), Mother(Bob))


p = Parents(x, Father(x), Mother(x))
q = Parents(Anakin, Mother(Anakin))

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Fail
p = Parents(Bob, x, Mother(y))
q = Parents(Bob, Father(y), Mother(Bob))


Fail
{x/Father(Bob), y/Bob}
p = Parents(Bob, x, Mother(y))
q = Parents(Bob, Father(x), Mother(Bob))

Fail
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 33
Forward / Backward Chaining

Unification algorithm finds  that unifies
(p1’, …, pn’) and (p1  …  pn)

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Recall from Propositional Logic



Makes Generalized Modus Ponens possible
Inference using Modus Ponens
Either forward chaining or backward chaining
We have Generalized Modus Ponens for
FOL


Inference is possible!
Forward chaining or backward chaining
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 34
Forward / Backward Chaining
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Forward chaining
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Data-driven reasoning
Start with known symbols
Infer new symbols and add to KB
Use new symbols to infer more new symbols
Repeat until query proven or no new symbols can be inferred
Backward chaining





Goal-driven reasoning
Start with query, try to infer it
If there are unknown symbols in the premise of the query,
infer them first
If there are unknown symbols in the premise of these
symbols, infer those first
Repeat until query proven or its premise cannot be inferred
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 35
Resolution

Recall from Propositional Logic



(αβ), (¬βγ)
(α  γ)
Resolution rule is both sound and complete
Proof by contradiction




Convert KB to conjunctive normal form (CNF)
Add negation of query
Then perform inference until we reach a
contradiction resulting from the negated query
Reaching contradiction means query is true
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 36
Resolution Steps
1.
2.
3.
4.
5.
Convert problem into FOL KB
Convert FOL statements in KB to CNF
Add negation of query (in CNF) in KB
Use FOL resolution rule to infer new
clauses from KB
Produce a contradiction that proves our
query
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 37
Conversion to CNF

Three new steps (4, 5, 6) & rules in step 3
1.
2.
3.
4.
5.
6.
7.
Eliminate biconditionals: (αβ)  ((αβ)(βα))
Eliminate implications: (α  β)  (¬α  β)
Move/Eliminate negations: ¬(¬α)  α,
¬(α  β)  (¬α  ¬β), ¬(α  β)  (¬α  ¬β),
¬x P(x)  x ¬P(x), ¬x P(x)  x ¬P(x)
Standardize variables:
x P(x)  x Q(x)  x P(x)  y Q(y)
Skolemize: x P(x)  P(C)
Drop universal quantifiers: x P(x)  P(x)
Distribute  over : (α  (βγ))  ((αβ)  (αγ))
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 38
Conversion to CNF







Everyone who loves all animals is loved by
someone
x [y Animal(y)  Love(x,y)]  y Love(y,x)
x ¬[y Animal(y)  Love(x,y)]  y Love(y,x)
x y ¬[Animal(y)  Love(x,y)]  y Love(y,x)
x y ¬Animal(y)  ¬Love(x,y)  y Love(y,x)
x y ¬Animal(y)  ¬Love(x,y)  z Love(z,x)
x ¬Animal(C1)  ¬Love(x,C1)  Love(C2,x)
¬Animal(C1)  ¬Love(x,C1)  Love(C2,x)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 39
Conversion to CNF









There is a book which everyone buys only if there is a class
that requires it
x Book(x)  [y Buy(y,x)]  [y Class(y)  Requires(y,x)]
x Book(x)  {[y Buy(y,x)]  [y Class(y)  Requires(y,x)]}
 {[y Class(y)  Requires(y,x)]  [y Buy(y,x)]}
x Book(x)  {¬[y Buy(y,x)]  [y Class(y)  Requires(y,x)]}
 {¬[y Class(y)  Requires(y,x)]  [y Buy(y,x)]}
x Book(x)  {[y ¬Buy(y,x)]  [y Class(y)  Requires(y,x)]}
 {[y ¬Class(y)  ¬Requires(y,x)]  [y Buy(y,x)]}
x Book(x)  {y ¬Buy(y,x)  [z Class(z)  Requires(z,x)]} 
{[z ¬Class(z)  ¬Requires(z,x)]  y Buy(y,x)}
Book(C1)  {¬Buy(C2,C1)  [Class(C3)  Requires(C3,C1)]} 
{[z ¬Class(z)  ¬Requires(z,C1)]  y Buy(y,C1)}
Book(C1)  {¬Buy(C2,C1)  [Class(C3)  Requires(C3,C1)]} 
{¬Class(z)  ¬Requires(z,C1)  Buy(y,C1)}
Book(C1)  [¬Buy(C2,C1)  Class(C3)]  [¬Buy(C2,C1) 
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 40
Requires(C
,C
)]

[¬Class(z)

¬Requires(z,C
)

Buy(y,C
3 1
1
1)]
FOL Resolution

(αβ’), (¬βγ)
SUBST(, αγ)


 = UNIFY(β’, ¬β)
Example




Animal(Dog(x))  Love(Bob,x)
¬Love(y,z)  ¬Hunt(y,z)
 = {y/Bob, z/x}
Animal(Dog(x))  ¬Hunt(Bob,x)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 41
Example of Resolution

Consider the story


Anyone passing their ECE457 exam and
winning the lottery is happy. Anyone who
studies or is lucky can pass all their exams.
Bob did not study but is lucky. Anyone
who’s lucky can win the lottery.
Is Bob happy?
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 42
Example of Resolution

Convert to FOL





Anyone passing their ECE457 exam and winning
the lottery is happy.
x Pass(x, ECE457)  Win(x,Lottery)  Happy(x)
Anyone who studies or is lucky can pass all their
exams.
x y Study(x)  Lucky(x)  Pass(x,y)
Bob did not study but is lucky.
¬Study(Bob)  Lucky(Bob)
Anyone who’s lucky can win the lottery.
x Lucky(x)  Win(x,Lottery)
Is Bob happy?
Happy(Bob)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 43
Example of Resolution

Convert to CNF


x Pass(x, ECE457)  Win(x,Lottery)  Happy(x)
x ¬(Pass(x, ECE457)  Win(x,Lottery)) Happy(x)
x ¬Pass(x, ECE457)  ¬Win(x,Lottery)  Happy(x)
¬Pass(x, ECE457)  ¬Win(x,Lottery)  Happy(x)
x y Study(x)  Lucky(x)  Pass(x,y)
x y ¬(Study(x)  Lucky(x))  Pass(x,y)
x y (¬Study(x)  ¬Lucky(x))  Pass(x,y)
(¬Study(x)  ¬Lucky(x))  Pass(x,y)
(¬Study(x)  Pass(x,y))  (¬Lucky(x)  Pass(x,y))
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 44
Example of Resolution

Convert to CNF



¬Study(Bob)  Lucky(Bob)
x Lucky(x)  Win(x,Lottery)
x ¬Lucky(x)  Win(x,Lottery)
¬Lucky(x)  Win(x,Lottery)
Negation of query for resolution
¬Happy(Bob)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 45
Example of Resolution

Content of KB







¬Pass(x1, ECE457)  ¬Win(x1,Lottery) 
Happy(x1)
¬Study(x2)  Pass(x2,y1)
¬Lucky(x3)  Pass(x3,y2)
¬Study(Bob)
Lucky(Bob)
¬Lucky(x4)  Win(x4,Lottery)
¬Happy(Bob)
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 46
Example of Resolution


(¬Pass(x1, ECE457)  ¬Win(x1,Lottery)  Happy(x1))
 (¬Study(x2)  Pass(x2,y1))  (¬Lucky(x3) 
Pass(x3,y2))  ¬Study(Bob)  Lucky(Bob) 
(¬Lucky(x4)  Win(x4,Lottery))  ¬Happy(Bob)
(¬Pass(x1, ECE457)  ¬Win(x1,Lottery)  Happy(x1))
 (¬Study(x2)  Pass(x2,y1))  (¬Lucky(x3) 
Pass(x3,y2))  ¬Study(Bob)  Lucky(Bob) 
(¬Lucky(x4)  Win(x4,Lottery))  ¬Happy(Bob)


{x1/Bob}
(¬Pass(Bob, ECE457)  ¬Win(Bob,Lottery)) 
(¬Study(x2)  Pass(x2,y1))  (¬Lucky(x3) 
Pass(x3,y2))  ¬Study(Bob)  Lucky(Bob) 
(¬Lucky(x4)  Win(x4,Lottery))  ¬Happy(Bob)

{x4/Bob}
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 47
Example of Resolution

(¬Pass(Bob, ECE457)  ¬Lucky(Bob)) 
(¬Study(x2)  Pass(x2,y1))  (¬Lucky(x3) 
Pass(x3,y2))  ¬Study(Bob)  Lucky(Bob) 
¬Happy(Bob)


¬Pass(Bob, ECE457)  (¬Study(x2) 
Pass(x2,y1))  (¬Lucky(x3)  Pass(x3,y2)) 
¬Study(Bob)  Lucky(Bob)  ¬Happy(Bob)


No substitution needed
{x3/Bob, y2/ECE457}
¬Pass(Bob, ECE457)  (¬Study(x2) 
Pass(x2,y1))  ¬Lucky(Bob)  ¬Study(Bob) 
Lucky(Bob)  ¬Happy(Bob)

No substitution needed
ECE457 Applied Artificial Intelligence
R. Khoury (2007)
Page 48