Predicate Logic in Artificial Intelligence
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ARTIFICIAL INTELLIGENCE
S.MANIMOZHI
ASSISTANT PROFESSOR,
DEPARTMENT OF CA
BON SECOURS COLLEGE FOR WOMEN,
THANJAVUR
Ch
apter
5: Using Predicate
Logic
•
Representing simple facts in
Logic
•
Representing Instance and Isa
relationships
Slide
1
Using Predicate
Logic
L
o
g
i
c
Logic
is
concerned with the truth of statements
about
the world.
Generally each statement
is
either
TRUE
or
FALSE
.
Logic includes :
Syntax , Semantics and Inference
Procedure
.
S
y
n
t
a
x
:
S
p
e
c
i
f
i
e
s
t
h
e
s
y
m
b
o
l
s
i
n
t
h
e
l
a
n
g
u
a
g
e
a
b
o
u
t
h
o
w
t
h
e
y
c
a
n
b
e
c
o
m
b
i
n
e
d
t
o
f
o
r
m
s
e
n
t
e
n
c
e
s
.
T
h
e
f
a
c
t
s
a
b
o
u
t
t
h
e
w
o
r
l
d
a
r
e
r
e
p
r
e
s
e
n
t
e
d
a
s
s
e
n
t
e
n
c
e
s
i
n
l
o
g
i
c
.
Slide
2
Using Predicate
Logic
◊
S
e
m
a
n
t
i
c
:
Specifies how
to
assign a truth value
to
a sentence based on
its
meaning
in the world.
It
Specifies what facts a sentence refers to.
A fact
is
a claim about the world, and
it
may be
TRUE
or
FALSE
.
◊
I
n
f
e
r
e
n
c
e
P
r
o
c
e
d
u
r
e
:
Specifies
methods
for
computing
new
sentences
from
the
existing
sentences
.
Slide
2
Representing Simple Facts in
Logic
Using propositional
logic
–Rea
l-
w
o
rld
facts
are repre
s
e
n
ted
as
logic
a
l
pro
p
o
s
it
i
o
n
s
wr
itten
as well-formed formulas
(wff’s)
–Example
1:
Representing Simple Facts in
Logic
Example
2:
Slide
6
=> Since the assertions are separate,
it
is not possible to dra
w any conclusion about similarities between Socrates and
Pl
ato.
Representing Simple Facts in
Logic
It would be much better to represent these facts
as
•
This
fails
to
capture the relationship between any individual being a man
and
that
individual being a
mortal.
•
Therefore
it is necessary to move to first
order predicate
logic
as a way of representing
knowledge because
it
permits representation o f things that cannot reasonably be
represented in
prepositional logi c.
•
In
predicate logic, real world facts are represented as
statements
written as
wff’s
.
Slide
5
Representing Simple Facts in
Logic
Slide
8
Q
Propositional logic
vs.
predicate
logic
–Using propositional
logic
•
Theorem proving is
decidable
•
Cannot represent objects and
quantification
–Using predicate
logic
•
Can represent objects and
quantification
•
Theorem proving is
semi-decidable
Representing Simple Facts in
Logic
Slide
9
Consider the following set of
sentences.
1.
Marcus was a
man.
2.
Marcus was a
Pompeian.
3.
All Pompeians were
Romans.
4.
Caesar was a
ruler.
5.
All Romans were either loyal to Caesar or hated
him.
6.
Every one is loyal to
someone.
7.
Pe
o
ple
o
n
ly try
t
o
a
s
s
a
s
s
i
n
a
te
rulers
they
are not loy
a
l
to.
8.
Marcus
tried to
assassinate
Caesar
Representing Simple Facts in
Logic
Slide
10
1.
Mar
c
us
w
a
s
a ma
n
.
m
a
n(M
a
r cus)
2.
Mar
c
us
w
a
s
a Po
m
p
e
ian.
Po mpeian(Marcus)
3.
All Pompeians were
Romans. x:
Pompeian(x)
Roman(x)
4.
C
a
e
s
ar
w
a
s
a ruler.
ruler(Cae sar)
Representing Simple Facts in
Logic
Slide
11
5.
All Romans were either loyal to Caesar or hated
him.
In English the word
‘or’
means the logical inclusive-or and
sometimes means the logical
exclusive-or(XOR)
hate(x,
Caesar)
inclusive-or
x:
Roman(x)
loyalto(x,
Caesar)
exclusive-or
(XOR)
x:
Roman(x)
(loyalto(x,
Caesar)
hate(x,
Caesar))
(
loyalto(x,
Caesar)
hate(x,
Caesar))
Representing Simple Facts in
Logic
Slide
12
6.
Every one 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.
Mar
c
us
tr
i
ed
t
o
a
s
s
a
s
s
ina
t
e
C
a
e
s
ar.
trya
s
s
a
s
s
ina
t
e(
Marcus,
Caesar)
Representing Simple Facts in
Logic
Slide
13
To answer the
question
Was Marcus loyal to
Caesar?
To produce a formal proof , reasoning backward from
th
e desired
goal
loyalto(Marcus,
Caesar)
To prove the goal, rules of inference
are to
be used
to
trans
form into another goal that
in
turn be transformed and s
o on, until there are no insatisfied goals
remaining.
Representing Simple Facts in
Logic
Slide
14
•
This attempt fails , since there is no way to satisfy the goal perso
n(Marcus)with the statements
available.
•
The problem is although its known that Marcus was a man there
is no way to conclude
it.
•
Therefore another representation is added
namely
Representing Simple Facts in
Logic
9.All men are people
x:m
a
n(x)
pers
o
n(x)
This satisfies the last goal and produce a proof that Marcus w as not loyal to
ceasar.
Three
important
issues
to
be addressed
in
this process of converting
English
sentences
to logical
statements
and
the
n using these statements
to
deduce new
ones.
•
Many English sentences are
ambiguous.
Choosing the
corre ct interpretation
may be
difficult.
•
There is often a choice of how
to
represent knowledge
•
Obvious information may be necessary
for
reasoning
Slide
13
Representing Instance & Isa
Relationships
•
Attributes “ IsA ”
and
“ Instance ” support property inheritance and play important
role
in
knowledge representation.
•
The
ways
these two attributes "instance" and "isa", are logically expresse d are
shown
in
the example below
:
Example : A simple sentence like "Joe
is
a
musician"
H
e
r
e
"
i
s
a
"
(
c
a
l
l
e
d
I
s
A
)
i
s
a
w
a
y
o
f
e
x
p
r
e
s
s
i
n
g
w
h
a
t
l
o
g
i
c
a
l
l
y
i
s
c
a
l
l
e
d
a
c
l
a
s
s
-
i
n
s
t
a
n
c
e
r
e
l
a
t
i
o
n
s
h
i
p
b
e
t
w
e
e
n
t
h
e
s
u
b
j
e
c
t
s
r
e
p
r
e
s
e
n
t
e
d
b
y
t
h
e
t
e
r
m
s
"
J
o
e
"
a
n
d
"
m
u
s
i
c
i
a
n
"
.
◊ "Joe"
is
an
instance
of the class of things
called
"musician". " Joe" plays the
role of
instance,
"musician" plays the role of class
in
that
sentence.
◊ Note : In such a sentence, while for a human there
is
no con fusion, but for
computers
each
relationship have
to
be defined explicitly.
Representing Instance & Isa
Relationships
Slide
17
Representing
Instance
& Isa
Relationships
Slide
18
•
The
first part
of the figure contains the representations,
in whic
h the
class membership
is
represented with unary predicates, each
corresponding
to
a class. Asserting that p(x)
is
true
is
equi valent
to
asserting that X
is
an instance of p.
•
The second part uses the instance
predicate
explicitly. It
is
a b inary
one, whose
first
argument
is
an object and whose second argument
is
a class
to
which the object belongs.
The
implicatio
n rule
in
statement 3 states that object
is
an instance of the sub class
pompeian then
it is
an instance of the superclass
Roman.
Representing
Instance
& Isa
Relationships
Slide
19
•
The
third part contains representations that use both the instan ce
and
isa
predicates
explicitly.
Use of
isa simplifies
the
repres
entation
of sentence 3 but requires one additional
axiom.It
desc
ribes how an
instance relation and an isa relation combined
to
derive a new
instance
relation.
In the world of artificial intelligence, predicate logic plays a crucial role in representing simple facts. It involves syntax, semantics, and inference procedures to determine the truth value of statements. Real-world facts are represented using propositions in logic, allowing for structured knowledge representation. Utilizing first-order predicate logic enhances the ability to capture relationships between entities, moving beyond the limitations of propositional logic.
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Presentation Transcript
ARTIFICIAL INTELLIGENCE S.MANIMOZHI ASSISTANT PROFESSOR, DEPARTMENT OF CA BON SECOURS COLLEGE FOR WOMEN, THANJAVUR
Chapter 5: Using Predicate Logic Representing simple facts in Logic Representing Instance and Isa relationships Slide 1
Using Predicate Logic Logic Logic is concerned with the truth of statements about the world. Generally each statement is either TRUE or FALSE. Logic includes :Syntax , Semantics and Inference Procedure. Syntax : Specifies the symbols in the language about how they can be co mbined to form sentences. The facts about the world are repre sented as sentences in logic. Slide 2
Using Predicate Logic Semantic : Specifies how to assign a truth value to a sentence based on its meaning in the world. It Specifies what facts a sentence refers to. A fact is a claim about the world, and it may be TRUE or FALSE. Inference Procedure : Specifies methods for computing new sentences from the existing sentences Slide 2 .
Representing Simple Facts in Logic Using propositional logic Real-world facts are represented as logical propositions wr itten as well-formed formulas (wff s) Example 1: It is raining RAINING It is sunny SUNNY It is Windy WINDY RAINING SUNNY If it is raining, then it is notsunny
Representing Simple Facts in Logic Example 2: Socrates is a man SOCRATESMAN Plato is a man PLATOMAN All men are mortal MORTALMAN => Since the assertions are separate, it is not possible to dra w any conclusion about similarities between Socrates and Pl ato. Slide 6
Representing Simple Facts in Logic It would be much better to represent these facts as Socrates is a man man(socrates) Plato is a man man(plato) All men are mortal mortalman This fails to capture the relationship between any individual being individual being a mortal. Therefore it is necessary to move to first order predicate logic as a way of representing knowledge because it permits representation o represented in prepositional logi c. In predicate logic, real world facts are represented as statements written as wff s a man and that f things that cannot reasonably be . Slide 5
Representing Simple Facts in Logic Q Propositional logic vs. predicate logic Using propositional logic Theorem proving is decidable Cannot represent objects and quantification Using predicate logic Can represent objects and quantification Theorem proving is semi-decidable Slide8
Representing Simple Facts in Logic Consider the following set of sentences. 1. Marcus was a man. 2. Marcus was a Pompeian. 3. All Pompeians were Romans. 4. Caesar was a ruler. 5. All Romans were either loyal to Caesar or hated him. 6. Every one is loyal to someone. 7. People only try to assassinate rulers they are not loyal to. 8. Marcus tried to assassinate Caesar Slide9
Representing Simple Facts in Logic 1. Marcus was a man. man(Mar cus) 2. Marcus was a Pompeian. Po mpeian(Marcus) 3.All Pompeians were Romans. x: Pompeian(x) Roman(x) 4. Caesar was a ruler. ruler(Cae sar) Slide10
Representing Simple Facts in Logic 5. All Romans were either loyal to Caesar or hated him. In English the word or means the logical inclusive-or and sometimes means the logical exclusive-or(XOR) inclusive-or x: Roman(x) loyalto(x, Caesar) hate(x, Caesar) exclusive-or (XOR) x: Roman(x) ( loyalto(x, Caesar) hate(x, Caesar)) (loyalto(x, Caesar) hate(x, Caesar)) Slide11
Representing Simple Facts in Logic 6.Every one 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. Marcus tried to assassinate Caesar. tryassassinate( Marcus, Caesar) Slide12
Representing Simple Facts in Logic To answer the question Was Marcus loyal to Caesar? To produce a formal proof , reasoning backward from th e desired goal loyalto(Marcus, Caesar) To prove the goal, rules of inference are to be used to trans form into another goal that in turn be transformed and s o on, until there are no insatisfied goals remaining. Slide13
Representing Simple Facts in Logic This attempt fails , since there is no way to satisfy the goal perso n(Marcus)with the statements available. The problem is although its known that Marcus was a man there is no way to conclude it. Therefore another representation is added namely Slide14
Representing Simple Facts in Logic 9.All men are people x:man(x) person(x) This satisfies the last goal and produce a proof that Marcus w as not loyal to ceasar. Three important issues to be addressed in this process of converting English sentences to logical statements and the n using these statements to deduce new ones. Many English sentences are ambiguous. Choosing the corre ct interpretation may be difficult. There is often a choice of how to represent knowledge Obvious information may be necessary for reasoning Slide 13
Representing Instance & IsaRelationships Attributes IsA and Instance support property inheritance and play important role in knowledge representation. The ways these two attributes "instance" and "isa", are logically expresse d are shown in the example below : Example : A simple sentence like "Joe is a musician" Here "is a" (called IsA) is a way of expressing what logically is called a class-instance relationship between the su "Joe" and "musician". "Joe" is an instance of the class of things called "musician". " Joe" plays the role of instance, "musician" plays the role of class in that sentence. Note : In such a sentence, while for a human there is no con fusion, but for computers each relationship have to be defined explicitly. bjects represented by the terms Slide 14
Representing Instance & IsaRelationships The first part of the figure contains the representations, in which the class membership is represented with unary predicates, corresponding to a class. Asserting that p(x) is true is equi valent to asserting that X is an instance of p. The second part uses the instance predicate explicitly. It is a b inary one, whose first argument is an object and whose second argument is a class to which the object belongs. The implicatio statement 3 states that object is an instance of the sub pompeian then it is an instance of the superclass Roman. each n rule in class Slide18
Representing Instance & IsaRelationships The third part contains representations that use both the instan ce and isa predicates explicitly. Use of isa simplifies the repres entation of sentence 3 but requires one additional axiom.It desc ribes how an instance relation and an isa relation combined to instance relation. derive a new Slide19
