Knowledge Representation techniques in Artificial Intelligence

ARTIFICIAL
INTELLIGENCE
(Unit III)
VI Sem B.E. Computer Sc. &
Engg
Department of Computer Science & Engineering
Priyadarshini Bhagwati College of Engineering,
Nagpur
(Even 2021)
Kapil N Hande
Head of the Department

Contents
Knowledge representation: Knowledge
representation Issues: First order logic,
Predicate Logic,
Structured Knowledge Representation:
Backward Chaining , Backward Chaining
Backward Chaining , Backward Chaining
Resolution
Semantic Nets
Frames, and Scripts
Ontology
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Representations and Mappings
In order to solve complex problems in AI, we
need both large amounts of knowledge and
some mechanism for manipulating that
knowledge.
Representing the knowledge is called as facts
Representing the knowledge is called as facts
We are dealing with two different kinds of
entities
Facts: truths in some relevant world. These are
the things we want to represent
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Representation of facts in some chosen
formalism. Things we will actually be able to
manipulate.
These entities are structured at two levels:
The knowledge level, at which facts are
The knowledge level, at which facts are
described.
The symbol level, at which representation of
objects are defined in terms of symbols that can
be manipulated by programs.
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Figure: Mapping between Facts and Representation

One representation of the facts is so common
that is natural language(English) sentences.
English representation of the facts in order to
facilitate getting information in and out of the
system.
Lets look at the simple example using
Lets look at the simple example using
mathematical logic as representational
formalism.
Consider the sentence
Spot is a dog
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This fact can be represented in logic as
dog(Spot)
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Available mapping functions are many to many
relations.
For example two sentences “All dogs have
tails” and “Every dog has a tail” could both
tails” and “Every dog has a tail” could both
represent same fact namely that every dog has
atleast one tail.
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Knowledge Representation techniques in Artificial Intelligence

If no good mapping can be defined for the
problem, then no matter how good the
program to solve the problem is, it will not be
able to produce answers that corresponds to
able to produce answers that corresponds to
real answers of the problem.
We need to find the concrete implementation
of the abstract concepts.
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Approaches to Knowledge Representation
A good knowledge representation system must possess the
following properties.
1. Representational Accuracy:
KR system should have the ability to represent all kind of
required knowledge.
2. Inferential Adequacy:
KR system should have ability to manipulate the
representational structures to produce new knowledge
representational structures to produce new knowledge
corresponding to existing structure.
3. Inferential Efficiency:
The ability to direct the inferential knowledge mechanism
into the most productive directions by storing appropriate
guides.
4. Acquisitional efficiency- The ability to acquire the new
knowledge easily using automatic methods.
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Simple Relational Knowledge
It is the simplest way of storing facts which uses
the relational method, and each fact about a
set of the object is set out systematically in
columns.
This approach of knowledge representation is
This approach of knowledge representation is
famous in database systems where the
relationship between different entities is
represented.
This approach has little opportunity for
inference.
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Inheritable Knowledge
Structure must have inference mechanism
One of the most useful form of inference is
property inference, in which elements of
specific classes inherit attributes and values
from more general classes in which they are
from more general classes in which they are
included.
Objects must be organized into classes, and
classes must be arranged in generalization
hierarchy
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Following figure shows some additional
baseball knowledge inserted into a structure.
Lines represent attributes
Boxed nodes represent objects and values of
attributes of object.
attributes of object.
These values can also be viewed as objects with
values and attributes and so on.
The arrows on the line points from an object to
its value
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The structure shown in the figure is a slot-and-
filler structure, also called as semantic network
or collection of frames.
Each individual frame represent the collection
Each individual frame represent the collection
of attributes and values associated with a
particular node.
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A semantic network is a graphic notation for
representing knowledge in patterns of interconnected
nodes. Semantic networks became popular
in artificial intelligence and natural language
processing only because it represents knowledge or
supports reasoning.
Frames are an artificial intelligence data structure
Frames are an artificial intelligence data structure
used to divide knowledge into substructures by
representing stereotyped situations. ... Frames are
the primary data structure used in artificial
intelligence frame language; they are stored as
ontologies of sets.
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The attribute isa, is being used to show class
inclusion and attribute instance is being used to
show class membership.
These two specific attributes provides basis for
These two specific attributes provides basis for
property inheritance as an inference technique.
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Inferential Knowledge
Inferential knowledge approach represents
knowledge in the form of formal logics.
This approach can be used to derive more facts.
It guaranteed correctness.
Sometimes all the power of traditional logic is
needed to describe the inferences that are
Sometimes all the power of traditional logic is
needed to describe the inferences that are
needed.
Following figure shows examples of the use of
first order predicate logic to represent
additional knowledge about baseball
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We need a inference procedure that can exploit
this knowledge.
The required inference procedure is the one that
implements the standard rules of inference.
implements the standard rules of inference.
One of the most commonly use procedure is the
resolution, which exploits the proof by
contradiction.
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Procedural Knowledge
Operational or procedural knowledge describes
what to do when.
Procedural knowledge can be represented in
the programs by coding.
the programs by coding.
Procedural knowledge can be represented by
production rules.
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Procedural Knowledge
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Issues in
Knowledge
Knowledge
Representation

Issues in Knowledge Representation
1. Important Attributes:
There are two general attributes that are very
general significance: isa and instance.
They support property inheritance
They support property inheritance
They represent class membership and class
inclusion
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2. Relationships among Attributes
The attributes that we used to describe objects are
themselves entities that we represent.
Properties of the attributes are:
Inverses
Inverses
Existence in an isa hierarchy
Techniques for reasoning about values
Single-valued attributes
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Inverses
As soon as we describe the entities in the world
and its relationship as attributes, we look for
binary relationship with others.
Attributes are those relationships
In semantic nets it was shown as directed arrow
originating at the object that was being
originating at the object that was being
described and terminating at the object
representing the value of specified attribute.
There are two ways to show the other way of
relationships
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Inverses
The first is to represent both relationships in a
single representation that ignores focus. For ex:
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The second approach is to use attributes that
focus on single entity but to use them in pairs, one
the inverse of the other.

An isa hierarchy of attributes
Just as we have classes of objects and specialized subsets of
those classes, there are attributes and specializations of the
attributes.
For example attribute height is specialization of the more
general attribute physical-size, which in turn specialization of
general attribute physical-size, which in turn specialization of
physical-attribute.
These generalization-specialization relationships are
important as they support inheritance
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Sometimes the values of the attributes are
specified explicitly when a knowledge base is
created.
But the reasoning system must reason about
But the reasoning system must reason about
values it has not been given explicitly.
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Single-valued attributes
A specific but very useful kind of attribute is one
that is guaranteed to take unique value.
If there is already the value present for one of the
attributes and a different value is asserted, then
one of the two things has happened.
Either a change has occurred in the world or there
is an contradiction in the knowledge base that
needs to be resolved.
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3. Choosing the Granularity of the
Representation
“At what level of detail should the world be
represented?”
“What should be our primitives?”
In computer science, granularity refers to a
ratio of computation to communication – and
ratio of computation to communication – and
also, in the classical sense, to the breaking
down of larger holistic tasks into smaller, more
finely delegated tasks.
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Choosing the Granularity of the Representation
Primitive is the simplest type of programming
language item. It may also refer to the smallest
processing unit accessible by a programmer.
In computer programming, a primitive
In computer programming, a primitive
(pronounced PRIH-muh-teev ) is a basic interface
or segment of code that can be used to build more
sophisticated program elements or interfaces.
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Lets take the following example.
Suppose we are interested in following fact
John spotted Sue
We could represent this as
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So here we can easily answer the question
Who spotted Sue?
But suppose we wanted to know
Did John see Sue?
The answer is “Yes” but given only one fact we cant give the
answer.

We could add other facts:
spotted(x,y)

saw (x,y)
We need to represent the fact that spotting is
special type of seeing explicitly in the
representation of the fact. We might write it
representation of the fact. We might write it
like:
saw(agent(John),
object(Sue),
timespan(briefly)
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Here we have used more primitive concepts as
seeing and time span.
The advantages of using small set of primitives is
that rules that are used to define inferences from
that knowledge need to be written in terms of
inferences.
Simple high level facts may require a lot of storage
when broken down into primitives.
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A second but related problem is that if knowledge
is initially presented to the system in a relatively
high level form, then substantial work must be
done to reduce the knowledge in primitive form.
done to reduce the knowledge in primitive form.
A third problem with the use of low-level
primitives is that in many domains it is not all clear
what the primitives should be.
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There may not be enough information present
in each use of high level constructs to be enable
them to be converted into there primitive
components.
The classical example of this is provided by
The classical example of this is provided by
Kinship terminology.
Consider one obvious set of primitives:
mother, father, son, daughter, brother and
sister.
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But now we are told that Mary is Sue’s cousin
An attempt to describe the cousin relationship
in terms of the primitives could produce any of
the following interpretations:
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The way to solve this problem is to change our
primitives.
We could use the set:
parent, child, sibling, male and female
Then the fact that Mary is Sue’s cousin can be
represented as
represented as
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4. Representing the Sets of Objects
It is important to represent the sets of objects
for several reasons.
There are several properties that are true of
sets that are not true of individual members of
set.
set.
As an example consider the assertions being
made in the two sentences, “There are more
sheep than people in Australia” and “English
speakers can be found all over the World”
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Representing the Sets of Objects
We can attach assertions here to people,
sheep's and English speakers.
If a property is true of all elements of a set,
then it is more efficient to associate it with set
rather than explicitly with every element of the
rather than explicitly with every element of the
set.
Example is Slot-and-filler structure where we
have used nodes to represent the sets and
inheritance to propagate set level assertions
down to individuals.
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There are three ways in which sets may be
represented
The simplest is by name
For example node named baseball player in
semantic net
semantic net
There are two ways to state the definition of set
and its element.
The first is to list the members. Such a
specification is called extensional definition
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The second is to provide the rule that, when a
particular object is evaluated, returns true or false
depending on whether the object is in set or not.
Such a rule is called intentional definition.
Such a rule is called intentional definition.
For example, an extensional description of the set
of our Sun’s planet on which people live is {Earth}
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An intentional description is
One way in which extensional and intentional
representation differs is that they do not
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representation differs is that they do not
necessarily correspond one-to-one with each
other

5. Finding the Right Structure as needed
Locating the appropriate knowledge structures
that have been stored in memory
For example suppose we have a script
(description of class of events in terms of
contexts, participants and sub events) that
contexts, participants and sub events) that
describes the typical sequence of events in a
restaurant.
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Finding the Right Structure as needed
The answer to the question Did John ate last
night? Is “yes”
In order to be able to reason about many things
a system must have many scripts about
a system must have many scripts about
everything.
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Finding the Right Structure as needed
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Selecting an Initial structure
Selecting candidate knowledge structures to
match a particular problem solving situation is
hard problem.
There are three approaches to do it
Index the structures directly by the significant
Index the structures directly by the significant
English words that can be used to describe
them.
Let each verb have associated with it structure
that describes its meaning
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This approach may not be adequate as many words
may have several distinct meaning.
For ex the word “fly” has different meaning in each
of the following sentences
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Consider each major concept as a pointer to all
structures in which it might be involved.
For ex the concept Steak might point to two
scripts, one for restaurant and one for super
market.
market.
The concept Bill may point to restaurant and a
shopping script.
Take the intersection of those sets to get the
structures, preferably precisely one, that
involves all the content words
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Locate one major clue in the problem
description and use it to select an initial
structure.
As other clues appear, use them to refine the
As other clues appear, use them to refine the
initial selection or to make the completely new
one if necessary.
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Revising the choice when necessary
The process of instantiating a structure in
particular situation often does not proceed
smoothly.
There are variety of things that can be done
Select the fragments of the current structure
Select the fragments of the current structure
that do correspond to the situation and match
them against candidate alternatives.
Make an excuse for current structures failure
and continue to use it.
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Revising the choice when necessary
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Predicate
Logic /
First Order
First Order
Logic (FOL)

Introduction
Representing simple facts in Logic
Representing Instance and Isa
relationships
relationships
Computable functions and predicates
Resolution
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Introduction
Logic, as per the definition of the Oxford
dictionary, is the reasoning conducted or
assessed according to strict principles and
validity“
Logic can be defined as the proof or validation
behind any reason provided
behind any reason provided
It was important to include logic in Artificial
Intelligence because we want
our agent (system) to think and act humanly,
and for doing so, it should be capable of taking
any decision based on the current situation.
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First Order Logic (FOL)
First-order logic(FOL) is another way of knowledge
representation in artificial intelligence. It is an
extension to propositional logic.
FOL is sufficiently expressive to represent the
natural language statements in a concise way.
First-order logic is also known as Predicate logic or
First-order logic is also known as Predicate logic or
First-order predicate logic. First-order logic is a
powerful language that develops information
about the objects in a more easy way and can also
express the relationship between those objects.
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First Order Logic (FOL)
First-order logic (like natural language) does not only
assume that the world contains facts like propositional
logic but also assumes the following things in the world:
Objects: A, B, people, numbers, colors, wars, theories,
squares, pits, wumpus, ......
Relations: It can be unary relation such as: red, round, is
adjacent, or n-any relation such as: the sister of, brother of,
adjacent, or n-any relation such as: the sister of, brother of,
has color, comes between
Function: Father of, best friend, third inning of, end of, ......
As a natural language, first-order logic also has two main
parts:
Syntax
Semantics
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Syntax of First-Order logic
The syntax of FOL determines which collection
of symbols is a logical expression in first-order
logic. The basic syntactic elements of first-order
logic are symbols. We write statements in
logic are symbols. We write statements in
short-hand notation in FOL.
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Basic Elements of First-order logic
Constant 1, 2, A, John, Mumbai, cat,....
Variables x, y, z, a, b,....
Predicates Brother, Father, ,....
Function sqrt, LeftLegOf, ....
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Function sqrt, LeftLegOf, ....
Connectives ∧
∧
∧
∧, ∨
∨
∨
∨, ¬, ⇒
⇒
⇒
⇒, ⇔
⇔
⇔
⇔
Equality ==
Quantifier ∀
∀
∀
∀, ∃
∃
∃
∃

Predicate Logic
A predicate is an expression of one or more
variables determined on some specific domain.
A predicate with variables can be made a
proposition by either authorizing a value to the
variable or by quantifying the variable.
variable or by quantifying the variable.
The variable of predicates is quantified by
quantifiers. There are two types of quantifier in
predicate logic - Existential Quantifier and
Universal Quantifier.
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Predicate Logic Symbols
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Propositional Logic
Propositional logic is a simple form of logic
which is also known as Boolean logic.
A proposition has TRUTH values (0 and 1) which
means it can have one of the two values i.e.
True or False.
True or False.
Propositional logic is used in artificial
intelligence for planning, problem-solving,
intelligent control and most importantly for
decision-making.
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Propositional Logic
A proposition is basically a declarative sentence
that has a truth value.
The purpose of using propositional logic is to
analyze a statement, individually or
analyze a statement, individually or
compositely.
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Propositional Logic
For example :
The following statements :
If x is real, then x2 0
What is your name?
(a+b)2 = 100
(a+b)2 = 100
This statement is false.
This statement is true.
Are not propositions because they do not have a
truth value. They are ambiguous.
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Propositional Logic
But the following statements :
(a+b)2 = a2 + 2ab + b2
If x is real, then x2 = 0
The sun rises in the east.
The sun rises in the west.
Are all propositions because they have a specific
truth value, true or false.
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Basic facts about Propositional Logic
Propositional logic is also called Boolean logic as it works
on 0 and 1.
In propositional logic, we use symbolic variables to
represent the logic, and we can use any symbol for a
representing a proposition, such A, B, C, P, Q, R, etc.
Propositions can be either true or false, but it cannot be
both.
both.
Propositional logic consists of an object, relations or
function, and logical connectives.
These connectives are also called logical operators.
The propositions and connectives are the basic elements
of the propositional logic.
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Basic facts about Propositional Logic
Connectives can be said as a logical operator which
connects two sentences.
A proposition formula which is always true is
called tautology, and it is also called a valid sentence.
A proposition formula which is always false is
called Contradiction.
called Contradiction.
Statements which are questions, commands, or
opinions are not propositions such as Where is
Rohini, How are you, What is your name, are not
propositions.
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Differences
Sr No Propositional Logic Predicate Logic
1
Propositional logic is the logic that deals with
a collection of declarative statements which
have a truth value, true or false.
Predicate logic is an expression consisting
of variables with a specified domain. It
consists of objects, relations and functions
between the objects.
2
It is the basic and most widely used logic.
Also known as Boolean logic.
It is an extension of propositional logic
covering predicates and quantification.
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3
A proposition has a specific truth value,
either true or false.
A predicate’s truth value depends on the
variables’ value.
4
Scope analysis is not done in propositional
logic.
Predicate logic helps analyze the scope of
the subject over the predicate. There are
three quantifiers : Universal Quantifier (∀)
depicts for all, Existential Quantifier (∃)
depicting there exists some and
Uniqueness Quantifier (∃!) depicting
exactly one.

Differences
Sr No Propositional Logic Predicate Logic
5
Propositions are combined with Logical
Operators or Logical Connectives like
Negation(¬), Disjunction(∧),
Conjunction(∨), Exclusive OR(⊕),
Implication(⇒), Bi-Conditional or
⇔
Predicate Logic adds by
introducing quantifiers to the
existing proposition.
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∧
∨ ⊕
Implication(⇒), Bi-Conditional or
Double Implication(⇔).
6 It is a more generalized representation.
It is a more specialized
representation.
7 It cannot deal with sets of entities.
It can deal with set of entities
with the help of quantifiers.

Representing Simple Facts in Logic
First-order logic statements can be divided into two
parts:
Subject: Subject is the main part of the statement.
Predicate: A predicate can be defined as a relation,
which binds two atoms together in a statement.
Consider the statement: x is an integer., it
Consider the statement: x is an integer., it
consists of two parts, the first part x is the
subject of the statement and second part is an
integer, is known as a predicate.
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A quantifier is a language element which generates quantification, and
quantification specifies the quantity of specimen in the universe of discourse.
These are the symbols that permit to determine or identify the range and scope of
the variable in the logical expression. There are two types of quantifier:
Universal Quantifier, (for all, everyone, everything)
Existential quantifier, (for some, at least one).

Properties of Quantifiers:
In universal quantifier, ∀
∀
∀
∀x∀
∀
∀
∀y is similar to ∀
∀
∀
∀y∀
∀
∀
∀x.
In Existential quantifier, ∃
∃
∃
∃x∃
∃
∃
∃y is similar to ∃
∃
∃
∃y∃
∃
∃
∃x.
∃
∃
∃
∃ ∀
∀
∀
∀ ∀
∀
∀
∀ ∃
∃
∃
∃
∃
∃
∃
∃ ∃
∃
∃
∃ ∃
∃
∃
∃ ∃
∃
∃
∃
∃
∃
∃
∃x∀
∀
∀
∀y is not similar to ∀
∀
∀
∀y∃
∃
∃
∃x.
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Some Examples of FOL using quantifier
1. All birds fly.
In this question the predicate is fly(bird).
And since there are all birds who fly so it will
be represented as follows.
∀
∀
∀
∀x bird(x) →ﬂy(x).
2. Every man respects his parent.
∀
∀
∀
∀
2. Every man respects his parent.
In this question, the predicate is respect(x,
y), where x=man, and y= parent.
Since there is every man so will use ∀
∀
∀
∀, and it
will be represented as follows:
∀
∀
∀
∀x man(x) → respects (x, parent).
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3. Some boys play cricket.
In this question, the predicate is play(x, y),
where x= boys, and y= game. Since there are
some boys so we will use ∃
∃
∃
∃, and it will be
represented as:
∃
∃
∃
∃x boys(x) → play(x, cricket).
∃
∃
∃
∃
∃
∃
∃
∃x boys(x) → play(x, cricket).
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4. Not all students like both Mathematics and
Science.
In this question, the predicate is like(x, y),
where x= student, and y= subject.
Since there are not all students, so we will
use ∀
∀
∀
∀ with negation, so following is the
use ∀
∀
∀
∀ with negation, so following is the
representation for this:
¬∀
∀
∀
∀ (x) [ student(x) → like(x, Mathemacs) ∧
∧
∧
∧ like(x,
Science)].
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5. Only one student failed in Mathematics.
In this question, the predicate is failed(x, y),
where x= student, and y= subject.
Since there is only one student who failed in
Mathematics, so we will use following
∃
∃
∃
∃
∃
∃
∃
∃(x) [ student(x) → failed (x,
Mathematics) ∧∀
∧∀
∧∀
∧∀ (y) [¬(x==y) ∧
∧
∧
∧ student(y) →
¬failed (x, Mathematics)].
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We can represent real world facts as logical
propositions written as well formed formulas
(wff) in propositional logic.
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Suppose we want to represent the fact
Socrates is a Man
We could write
SOCRATESMAN
But if we also want to represent
But if we also want to represent
Plato is a Man
We would have to write something such as
PLATOMAN
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It would be much better to represent these facts
as:
MAN(SOCRATES)
MAN(PLATO)
Since now the structure of representation reflects
Since now the structure of representation reflects
the structure of the knowledge itself.
Suppose we want to represent the fact
All men are mortal.
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We could represent this as
MORTALMAN
But that fails to capture the relationship
between any individual being a man and that
individual being a mortal.
individual being a mortal.
In predicate logic we can represent real world
facts as statements written as wff’s
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Lets now see the use of predicate logic as a way
of representing knowledge by looking at
specific example.
Consider the following set of sentences:
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These sentences can be represented as wff in
Predicate logic as follows:
1. Marcus was a man
man(Marcus)
It fails to capture the notion of a past tense.
2. Marcus was a Pompeian
Pompeian(Marcus)
3. All Pompeian's were Romans
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4. Caesar was a ruler
ruler(Caesar)
5. All Romans were either loyal to Caesar or hated
him
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6. Everyone is loyal to someone
7. People only try to assassinate rulers they are not
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7. People only try to assassinate rulers they are not
loyal to

7. People only try to assassinate rulers they are
not loyal to
8. Marcus try to assassinate Caesar.
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8. Marcus try to assassinate Caesar.

Now suppose we want to use these statements
to answer the question.
“Was Marcus loyal to Caesar?”
Using statement 7 and 8 we should be able to
prove that Marcus was not loyal to Caesar.
prove that Marcus was not loyal to Caesar.
Now let us try to produce a formal proof
reasoning backward from the desired goal.
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We need to add another representation of the
fact in our system.
9. All men are people.
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Representing instance and isa relationships
Class membership and Class inclusion
Class membership is represented with unary
predicates.
Asserting that P(x) is true is equivalent to asserting
Asserting that P(x) is true is equivalent to asserting
that x is an instance or element of P.
Predicate instance is binary one whose first
argument is an object and whose second argument
is a class to which object belongs.
K N Hande 99

Computable
Functions and
Functions and
Predicates

Computable Functions and Predicates
All the simple facts were represented as a
combination of individual predicates.
This is fine if the no of facts is not very large
But suppose we want to express simple facts
such as below
such as below
K N Hande 102

We cant write here representation of each of the
fact individually.
So here we need computable predicates.
We will simply invoke a procedure to find out the
We will simply invoke a procedure to find out the
value of the predicate, that will evaluate it and
return either true or false.
We need computable functions as well as
computable predicates.
K N Hande 103

Computable functions are the formalized analogue of
the intuitive notion of algorithms, in the sense that a
function is computable if there exists an algorithm that
can do the job of the function, i.e. given an input of
the function domain it can return the corresponding
output.
Ex: If we want to evaluate the truth value of
Ex: If we want to evaluate the truth value of
gt(2+3,1)
Here we need computable function for + first,
and then we can find the gt as computable
predicate
K N Hande 104

Lets take an example to see how these
functions and predicates are useful.
Consider the following set of facts.
1. Marcus was a man
man(Marcus)
man(Marcus)
Pompeian(Marcus)
3. Marcus was born in 40 A.D.
born(Marcus,40)
K N Hande 105

4. All men are mortal
K N Hande 106
6. No mortal lives longer than 150 years

7. It is now 1991
now = 1991
Now suppose we want to answer the question,
“Is Marcus Alive?”
To answer this question we need some additional
To answer this question we need some additional
knowledge.
8. Alive means not dead
K N Hande 107

This is an example of the fact that rarely do two
expressions have truly identical meanings in all
circumstances.
9. If someone dies, then he is dead at all later
times.
times.
K N Hande 108

Now lets attempt to answer the question,
“Is Marcus alive?” by proving
K N Hande 110
Two such proofs are shown below.
The term nil at the end of each proof indicates that
the list of conditions remaining to be proved is
empty and so the proof has succeeded.

Resolution
Resolution operates on the statements that have
been converted to a very convenient standard
form.
Resolution produces proofs by refutation.
Refutation is simply disproving an opposing
argument
It is the action of proving a statement or theory to
be wrong or false.
K N Hande 114

Resolution
Resolution attempts to show that negation of
the statement produces the contradiction with
the known statements.
That is it is unstaisfiable.
That is it is unstaisfiable.
K N Hande 115

Conversion to Clause Form
Suppose we know that all Romans who know
Marcus either hate Caesar or thinks that
anyone who hates anyone is crazy.
We could wff as
K N Hande 116
To use this formula in a proof requires complex matching
process.

The formula would be easier to work with if
It were flattered, that is there was less
embedding of components
The quantifiers were separated from the rest
of the formula so that they did not need to be
of the formula so that they did not need to be
considered.
Conjunctive Normal Form has both of these
properties.
K N Hande 117

Conjunctive normal form (CNF) is an approach to
Boolean logic that expresses formulas as conjunctions
of clauses with an AND or OR. Each clause connected
by a conjunction, or AND, must be either a literal or
contain a disjunction, or OR operator. CNF is useful for
automated theorem proving.
automated theorem proving.
Remember that we also called “or” “disjunction” and
“and” “conjunction”.
A clause that contains only ∨
∨
∨
∨ is called a disjunctive
clause and only ∧
∧
∧
∧ is called a conjunctive clause
K N Hande 118

• If we put a bunch of disjunctive clauses
together with ∧, it is called conjunctive normal
form.
– For example:
– (p∨
∨
∨
∨r)∧
∧
∧
∧(¬q∨
∨
∨
∨¬r)∧
∧
∧
∧q is in conjunctive normal form.
– (p∨
∨
∨
∨r)∧
∧
∧
∧(¬q∨
∨
∨
∨¬r)∧
∧
∧
∧q is in conjunctive normal form.
K N Hande 119

Algorithm: Convert To Clause Form
K N Hande 120

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K N Hande 125

Knowledge Representation techniques in Artificial Intelligence

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