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```					    4.0 Knowledge Representation
Outline
4.1 Introduction
4.2 Knowledge Representation schemes.
4.3 Propositional Calculus (PC)
4.3.1 PC Symbols and Sentences
4.3.2 Syntax of the propositional calculus
4.3.3 Semantics of PC
4.3.3.1 logic Laws and inference rules
4.3.3.2 Classification of Compound Propositions
4.3.4 Exercises
4.4 Predicate Calculus
4.4.1          Syntax of predicate calculus
4.4.2          Some predicate calculus equivalences
4.4.3          Inference rules
4.5 Semantic Networks
4.5.1          Representation of Some Relations
4.5.2          Inheritance in Semantic Networks
4.6 Conceptual Graphs
4.6.1          Types and individuals
1     4.7 Frame Structure
4.7.1          Examples
4.1 Introduction
AI System

Knowledge Base (KB)               Inference Mechanism

Facts about objects:   Rules,Procedures,   Set of procedures to
properties and             Theories        examine the KB in an
relations                                  orderly manner to
solve problems, or
make decisions within
the KB domain

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4.2 Knowledge Representation schemes

Logic            Network                 Structural
Representation    Representation           Representation

Propositional    Predicate   Semantic     Conceptual          Frame
Calculus       Calculus    Networks      Graphs           Structures

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4.3 Propositional Calculus
•   Proposition:
is a statement about the world that is either true or false.
It is a declarative sentence.
Examples : P= “It is hot”& Q =“The earth orbits the sun” are
propositions. R=“can you play tennis” is not a proposition.
•   Premise:
is a proposition which is assigned a logical value( true or false).
Example : P is true       Interpretation of P = assertion of its truth value

•   Rules are used to draw new propositions from              known premises
•   Example Q=“We are in summer”, P=“It is hot”
•   Rule: QèP
conclusion or consequent
Premise or antecedent
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4.3.1 Propositional Calculus Symbols and
Sentences
4.3.1.1 Symbols:         P Q P^Q PvQ PàQ P↔Q
•   Propositional
symbols:
P,Q,R,S,…..          T   T    T        T        T          T
•   Truth symbols:
true, false
•   Connective           T   F    F        T        F          F
symbols:
Not(┐,!,~), AND(^)
OR(v), Imply(è)       F   T    F        T        T          F
Equivalence
(↔)
The shown truth table
gives their meaning
F   F    F        F        T          T

A sentence in PC can be a symbol or a collection of symbols
connected legally by the connective symbols (^,v,!,à, ↔). Symbols ( ) and
]
[ 5 are used to group symbols into subexepressions.
4.3.1.2 Examples for using connectives
•  1-Although Ali and Samy are not young, Ali has a better chance of
winning the next tennis match, despite Samy’s considerable
experience.
• The above is a complex sentence that can be broken into the
following propositions:
• S =“Ali is young”
• G =“Samy is young”
• B =“Ali has a better chance of winning the next tennis match”
• E =“Samy has a considerable experience in tennis”
• Formalization is done using connectives as follows:
• (~S) ^(~G) ^B^E
• 2- If Ali is not in a hospital then Ali is well
C =“Ali is in a hospital”
W =“Ali is well”
Formalization:
(~C)èW
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4.3.2 Syntax of the propositional calculus
•   The syntax provides us with a set of rules for generating complex
formulae as follows:
•   1- An identifier is a proposition (e.g. A,B, p, q).
•   2-If p and q are propositions then so are ~P, (P), p^q, pvq, pèq,
póq.

•   Example: if A, B and C are propositions then
•   Aè C is a grammatically correct formula and, therefore
•   AèC v B is also correct.

•   Terminology
•   Formula that contains only symbols of the agreed alphabet and
obey grammatical rules given in the definition are called well-formed
formulae (wffs).
•   Note that when mixing connectives, the rules of precedence should
be taken into consideration.
•   Precedence from higher to lower :(),~,^,v, è ,ó
•   Hence A^BvC means (A^B)vC not A^(BvC).
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4.3.3 Semantics of PC
• Semantics deal with the meaning of compound
propositions.
• Meaning of a propositional sentence is
The determination of its truth value
≡ Mapping it into T or F ≡Interpretation of it
• This is done by using truth tables or by applying the
laws and rules of propositional logic listed in the
following section.
4.3.3.1 Logic laws and inference rules
Cumulative laws: (P^Q) ≡(Q^P) and (PvQ) ≡(QvP)
Associative Laws:      ( (PvQ)vR) ≡(Pv(QvR)) and
( (P^Q)^R) ≡(P^(Q^R))
Distributive laws:     Pv(Q^R) ≡(PvQ)^(PvR) and
P^(QvR) ≡(PvQ)v(P^R)
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~(~P) ≡ P
(PvQ) ≡(~PàQ) and
(~PvQ) ≡ PàQ
Contrapositive law: (PàQ)
≡(~Qà~P)
de Morgan’s law:
~(PvQ) ≡~P^~Q and
~(P^Q) ≡~Pv~Q

An example of an inference rule:
Modus Ponens rule:
if P is true and
PàQ                                 Prime              Premises Conclusion
propositions
Then Q is true
The validity (correctness or        p         q pèq         p   q
soundness) of this rule can be
proven using truth tables as     T    T         T        T   T
shown.                           T    F         F        T   F
F    T         T        F   T
9                                   F    F         T        F   F
List of inference rules: each introduces or eliminates a connective
Note that : _I means introduction of a connective.
_E means Elimination of a connective.
• Conjunction: ^_I                   • Modus ponens: è_E
p                                    Pèq
q                                    p
p^q                                  q
• Simplification: ^_E                • Modus tollens: è_E
p^q                                  Pèq
p                                    ~q
p                                    • Double negation: ~_E
pvq                                  ~~p
• Disjunctive syllogism: v_E         p
pvq                                  • Transivity of equivalence: ó_E
~p                                   p óq            p óq
q                                    pèq             qèp
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4.3.3.2 Classification of Compound Propositions
•    Compound propositions can be classified based on the
truth value each proposition can take under all possible
interpretations of its constituent prime proposition as
follows:

•    Tautology: is true under all possible assignments of
truth values to its prime propositions èalways true
–   Example p v ~p is a tautology.

–   Example: p ^ ~p is a contradiction.

•    Contingent: sometimes true and sometimes false.
–   Example: pè~p

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4.3.4 Exercises
1- Build the truth table for the following expression and
deduce an expression equivalent to it:
(PèQ)^(QèP) ≡ P….Q

P   Q   PàQ     QàP    (PèQ)^(QèP)

2-Determine the type of the following expressions:
a- p v qóp ^ q            b- pèqóqèp
c-~p v ~qó ~(p^q)         d-~p ^ ~q v p ^ q
3-Demonestrate the validity of modes tollens’ rule:
If pèq and ~q then ~p
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4.4 Predicate Calculus
•   In Propositional Calculus, each symbol denotes a proposition. There
is no way to access the components of an individual assertion.
Predicate calculus provides this ability.
•   For Example in Propositional Calculus the symbol P may represent
the proposition:
P=“Ali is the father of Salem”
•   In Predicate calculus, the same proposition can be represented
as
father (ali , salem).
By dividing the proposition into its internal components, predicate
calculus allows inference of another information such as :
Ali is a father,
Salem is the son of Ali,
And many other conclusions can be derived if combined with other
predicates.
• Predicate calculus also allows expressions to
contain variables.
Ex: "X
13           alive(X)àdrinks(X, water).
4.4.1 Syntax of predicate calculus
• Constatnsà specific names of objects, properties, or
relations.
Must begin with small letters:
Ex: ali, book, red, father, above.
• Variables à unknown or general classes of objects.
Represented by symbols beginning with capital letters.
X, Y, Z, Man…
• Functions:à maps its arguments from the domain into
a unique object in the range (co-domain):
Examples:                                       1-arity function
fatherof(ali): maps ali to his father.
plus(3,5): maps 3,5à8                  2-arity function
price (banans): maps banans to price value.

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• Term: is a constant, variable, or function:
Examples: cat,        X, times(2,3),
• Predicate: names a property of an object or a
relationship between zero or more objects in the world:
Examples: tall, likes, equal, on, near , part_of,….etc.
• A predicate expression consists of a predicate symbol
followed by n terms equal to its arity:
predicate_symbol( term 1, term 2,… term n)
Examples:
likes (ali, football)
friends (ali, father_of(ibrahim))
likes (X,Y)
• Atomic sentence: includes truth symbols or predicate
expression:
true , false
friends (ali, khaled)
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• Sentences:
§ Every atomic sentence is a sentence.
§ A legal combination of sentences using logical
operators ( connectives of propositional calculus).
• Quantifiers:
– constrains the meaning of a sentence containing a
variableà to make it a proposition (can be T or F).
– If X is a variable and S is a sentence then
• "X S is a sentence, and \$X S is a sentence.
– The universal quantifier " , for all: the sentence is true
for all values of its variable.
– The existential quantifier \$, there exist: the sentence
is true for at least one value of its variable.
– For complex expressions “()” are used to indicate the
scope of quantification.
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– Free variables are not allowed in quantified
expressions.
• Examples:
– Ali is a teacher and a good man
• goodman (ali) ^ isa (ali, teacher)
– Nobody likes taxes:
• Ø \$X likes (X , taxes)
– Every dog is an animal:
• "X ( dog (X) è animal (X))
– Every student has a pen:
• "X (\$Y(student(X) è (pen (Y) ^ owns (X,Y))))

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More examples
• Everybody is an adult or a child
• "X ( adult(X) v child (X))
• Everybody is an adult or there are children
• ("X adult(X)) v (\$X child (X))
• Everybody is tall but there are children
• ("X tall(X)) ^ (\$X child (X))
• Although all adults are tall, they are not clever.
• All first-year students are clever.
• "X ( first_year(X)^student (X)èclever(X) )
• No one can be clever without being hardworking
• ~\$X (clever (X) ) ^ ~hardworking (X))
• Not being lazy is equivalent to being hardworking.
• "X (~lazy(X)óhardworking (X))
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Examples for expressing predicate formulae in English
• "X (child(X)^ clever(X)è\$Y loves(Y,X) ):
– Clever children are loved.
• "X (child(X)^ clever(X)è\$Y (adult (Y) ^loves(Y,X))):
– Clever children are loved by adults.
• "X (clever(X)è\$Y cleverer(X,Y) ):
– One can not be clever without being cleverer than someone.
• Tall(samir) ^ ~\$Y loves(Y,samir) ^ "X (respects( X,
samir)) :
– Samir is tall and nobody loves him although everybody
respects him.
• ~ \$X ( person (X) ^ (X/=samir) ^ speaks_slowly(X):
– Nobody, except Samir, speaks slowly.
•

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Important notes
•        Scope of the quantifier
– the variable X may refer to different objects in the same argument as
follows:
– "X distant(X)^ planet (X)è\$X moon(X)

XàOne object         Xàanother object
•        Bound and free variables:
• "X .p(X,Y)

bound variable    free variable

• Types of values
• When a variable is quantified, we should associate it with
a type or domain as follows:
• "Y:person. mortal (Y)
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Semantics of
4.4.2 of universal quantification predicate logic
Interpretation
•        Assuming we deal with finite domain of individuals
• "X.p(X)ó p(x1)^p(x2)^….^p(Xn)
•    Example: assume the human society H consist of the individuals samir, samy, ali, salma,
amira, and let the following propositions hold true :
• male(samir)         greedy (samir)        kind(ali)
• male (samy)         greedy (samy)         kind (amira)
• male (ali)
Let these predicate be false for individuals not mentioned above.
Is the following formula (Z) true in the given context?
"X:H.male(X)è greedy(X) v kind (X)
To validate this argument we build a truth table as follows:
(1)     (2)        (3)       (4)          (5)         (6)
X       male (X)   greedy      Kind (X)   (3) v (4)   (2)è(5)
(X)
samir   T          T         F            T           T
samy    T          T         F            T           T
ali     T          F         T            T           T
Z is T
salma   F          F         F            F           T
amira   F          F         T            T           T

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From the same table we can deduce that "X:H. greedy(X) v kind (X) is false.
Semantics of predicate logic
Interpretation of Existential quantification
• Assuming we deal with finite domain of individuals
• \$X.p(X)ó p(x1) v p(x2) v….v p(Xn)
•        Example: assume the human society H mensioned in the previous example
and consider the truth value of the formula Z

• \$ X:H.male(X)^(~ greedy(X) è kind (X))
To validate this argument we build a truth table as
follows:
(1)      (2)        (3)      (4)          (5)    (6)       (7)
X        male (X)   greedy     Kind (X)   ~(3)   (5)è(4)   (2) ^(6)
(X)
samir    T          T        F            F      T         T
samy     T          T        F            F      T         T          Z is T
ali      T          F        T            T      T         T
salma    F          F        F            T      F         F
amira    F          F        T            T      T         F

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4.4.2 Some predicate calculus equivalences
Assume p and q are predicates and X,Y are variables:
Ø   Ø \$X p(X) ó "X Ø p(X)
Ø   Ø "X p(X) ó \$X Ø p(X)
Ø   \$X p(X)      ó \$Y p(Y)
Ø   "X p(X)      ó "Y p(Y)
Ø   "X ( p(X) ^ q(X) ) ó "X p(X) ^ "Y q(Y)
Ø   \$X ( p(X) v q(X) ) ó \$X p(X) v \$Y q(Y)

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4.4.3 Inference rules
Predicate calculus
equivalence:
•    Modus Ponens:
Given :PèQ and P are true                            Given:
Conclusion: Q is true                                "X p(X) èq(X) and
Example:                                             p(a)
PèQ: “If I am in Jeddah then I’m in Saudi Arabia”
P=“I am in Jeddah “                                  Conclusion: q(a)
Q=“I’m in Saudi Arabia” added to PèQ and P           Using substitution
•    Modus Tolens:                                   X=a
Given :PèQ and Q is false                           Predicate calculus
Conclusion: P is false                              equivalence:
Example:
Given:
PèQ: “If I am in Jeddah then I’m in Saudi Arabia”
Ø Q=“I’m not in Saudi Arabia”                       "X p(X) èq(X) and
Ø P=“I am not in Jeddah “ added to PèQ and Ø Q      Ø q(a)
Conclusion: Ø p(a)
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Using substitution X=a
4.5 Semantic Networks
• Network representation provides a means of structuring
knowledge.
• Semantic networks are :
Directed graphs with labeled nodes and arcs ( arrows, or
• Nodes corresponds to objects (physical as book or
nonphysical such as concepts, events, or actions)
• Links corresponds to relationships between objects.
Examples: IS_A, MEMBER_OF, SUBSET_OF, AKO (a
kind of), HAS_PARTS, INSTANCE_OF, SHAPED_LIKE,
OWNS

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4.5. 1 Representation of some
Relations
• In our course we will limit the use of these
• IS_A : to represent inheritance :
subclass to superclass relationship:
Example: a lion is an animal
lion       IS_A       animal

• Instance_of: To represent object to its class
relationship:
Example :Ali is a student
Instance_of
ali                   student
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• HAS_PARTS: to represent composition or
aggregation
• Example: A computer is an electronic
device consisting of CPU unit, screen,
keyboard          Electronic Device

IS_A
HAS_PARTS               HAS_PARTS
screen               computer                keyboard

HAS_PARTS

CPU UNIT

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• HAS_ATTRIBUTE: to link an object to a
property.
Example : A person named Ahmad has a red
car       person

instance_of
owns

has_color
semantic networks                 red

can represent binary
relations only.
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4.5.2 Inheritance in Semantic
Networks
Inheritance is represented with the IS_A And
information to be inferred.
Example:
Has_parts              drinks
hair                     mammal              milk

IS_A
From the shown net,
using inheritance, we    rodent
can infer that a
mouse has hair and           IS_A
drinks milk.
mouse
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4.6 Conceptual Graphs
• It is a Network scheme in which both
concepts and conceptual relations are
represented by nodes in a directed graph.
The links are not labelled. They indicate
the direction of the relation only.
• Concepts are represented with labelled
rectangles.
concepts
• Relations are represented by labelled
ellipsis.
pen    color red
• Example: the pen is red
30                            2-ary conceptual relation
• A concept may represent concrete or
abstract objects.
• Concrete concepts are like cat, telephone ,
or anything that we can imagine its shape.
• Abstract concepts include concepts such
as love, beauty….that do not correspond
to images in our minds.
• A conceptual relation has 1 to n- ary.
flies is a 1-ary relation   bird      flies

parents is a 3-ary relation             father
child     parents
31                                   mother
4.6.1 Types and individuals
• A concept symbol may be individual (as instance
objects) or generic) as classes).
• Individual concepts have a type field followed by a
referent field.
• Example: the concept Person: ali
has type ‘person’ and referent ‘ali’
• Each conceptual graph represents a single proposition.
The shown graph represents the proposition “ali gave
Person: ali     agent             object
give

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4.7 Frame Structure
• Extends network by allowing each node to be a
complex data structure.
object or event is stored together in memory as a
unit.
• Each frame consists of named slots with
attached values.
• The slot values may be
–   Numeric data.
–   Symbolic data
–   Pointers to other frames
–   Procedures for performing a particular task.

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•    Frames are general record-like structures, consisting of a collection
of slots.
•    The slots may be of any size and type
•    Slots also may have subfields called facets.
•    Facets may also have names and any number of values.
•    A general frame structure is as follows:
•    (<frame name>
(<slot 1> (<facet 1> <value 1> ……<value k1>)
(<facet 2> <value 1> ……<value k2>)
--
)
(<slot 2> (<facet 1> <value 1> ……<value km>)
--
--
)

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4.7.1 Examples
• To define a frame which describes the shown
screen: Frame name
) screen
(object 1 (shape (value pyramid))
(color (value yellow))
slots          (supported_by (value object2)))
(object 2 (shape (value cube))
(color (value red))
(supported_by (value object3)))
(object 3 (shape (value cube))
(color (value blue))
slots
facets    (supported_by (value nil)))).

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• Another example: to describe a type
(Cressida ( ako ( value car))
( color ( value white, blue, black))
Pointer to    ( speed (value 220 km/hr))
another frame
to allow      ( model ( value 4-doors)).
inheritance
(subclass-
superclass
relationship)

( Mycar ( instance_of (value Cressida ))
Pointer to
( color ( value white))).
another frame
to allow object
-class
relationship
36
•
Problem Set 2
Construct truth tables for the compound proposition:
– p v (q ^ r) è (p ^q) v r
•    Test Validity of the following rule
– (p v q) ^ ~pèq
•    Express the following in predicate logic
–   Although the tree is tall, it has no leaves.
–   If x is the father of y or mother of y then x is a parent of y
–   If y and x have same parent then they are sibling
–   If x and y are parents of z then they are married.
–   Every Muslim does prayer.
–   Nobody likes taxes
–   All students like AI
•    Evaluate the truth of the following, assuming the data is given in
section 4.2
– \$ X:H.male(X)^(~ greedy(X) v ~kind (X))
– (\$ X:H.~male(X) ^ kind (X))è("X:H.male(X)è~kind (X))

37

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