Chap.3 Knowledge Representation Issues Chap.4 Inference in First Order Logic

Artificial Intelligence
Chap.3 Knowledge Representation
Issues
Prepared by:
Prof. Khushali B Kathiriya

Outline for 8th semester
 Representations And Mappings
 Approaches To Knowledge Representation
Prepared by: Prof. Khushali B Kathiriya
2

Chap.3 Logical Agents & First
Order Logic
Prepared by:

Outline for 6th semester
 Logical Agents:
 Knowledge–based agents
 The Wumpus world
 Logic
 Propositional logic
 Propositional theorem proving
 Effective propositional model checking
 Agents based on propositional logic
 First Order Logic:
 Representation Revisited
 Syntax and Semantics of First Order logic
 Using First Order logic
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What is Knowledge?
 Knowledge is a general term. Knowledge is a progression that starts with
data which is of limited utility.
 By organizing or analyzing the data, we understand what the data means
and this becomes information.
 The interpretation or evaluation of information yield knowledge.
 An understanding of the principles embodied within the knowledge is
wisdom.
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What is Knowledge representation ?
 Humans are best at understanding, reasoning, and interpreting knowledge.
Human knows things, which is knowledge and as per their knowledge they
perform various actions in the real world. But how machines do all these
things comes under knowledge representation and reasoning.
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What is Knowledge representation ?
(Cont.)
 Hence we can describe Knowledge representation as following:
1. Knowledge representation and reasoning (KR, KRR) is the part of Artificial
intelligence which concerned with AI agents thinking and how thinking
contributes to intelligent behavior of agents.
2. It is responsible for representing information about the real world so that a
computer can understand and can utilize this knowledge to solve the complex
real world problems such as diagnosis a medical condition or communicating
with humans in natural language.
3. It is also a way which describes how we can represent knowledge in artificial
intelligence. Knowledge representation is not just storing data into some
database, but it also enables an intelligent machine to learn from that
knowledge and experiences so that it can behave intelligently like a human.
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What to represent?
 Object: All the facts about objects in our world domain. E.g., Guitars
contains strings, trumpets are brass instruments.
 Events: Events are the actions which occur in our world.
 Performance: It describe behavior which involves knowledge about how to
do things.
 Meta-knowledge: It is knowledge about what we know.
 Facts: Facts are the truths about the real world and what we represent.
 Knowledge-Base: The central component of the knowledge-based agents
is the knowledge base. It is represented as KB. The Knowledgebase is a
group of the Sentences (Here, sentences are used as a technical term and
not identical with the English language).
 Knowledge: Knowledge is awareness or familiarity gained by experiences
of facts, data, and situations.
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Framework of Knowledge Representation
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Knowledge-Based Agent in AI
Prepared by:

Knowledge-Based Agent in AI
 An intelligent agent needs knowledge about the real world for
taking decisions and reasoning to act efficiently.
 Knowledge-based agents are those agents who have the
capability of maintaining an internal state of knowledge, reason
over that knowledge, update their knowledge after observations
and take actions. These agents can represent the world with some
formal representation and act intelligently.
 Knowledge-based agents are composed of two main parts:
1. Knowledge-base and
2. Inference system.
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The architecture of knowledge-based
agent
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The architecture of knowledge-based
agent (Cont.)
 The diagram is representing a generalized architecture for a knowledge-
based agent. The knowledge-based agent (KBA) take input from the
environment by perceiving the environment. The input is taken by the
inference engine of the agent and which also communicate with KB to
decide as per the knowledge store in KB. The learning element of KBA
regularly updates the KB by learning new knowledge.
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Why use knowledge base?
 Knowledge base: Knowledge-base is a central component of a
knowledge-based agent, it is also known as KB. It is a collection of
sentences (here 'sentence' is a technical term and it is not identical to
sentence in English). These sentences are expressed in a language which is
called a knowledge representation language. The Knowledge-base of KBA
stores fact about the world.
 Knowledge-base is required for updating knowledge for an agent to learn
with experiences and take action as per the knowledge.
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Inference System
 Inference means deriving new sentences from old. Inference system
allows us to add a new sentence to the knowledge base. A
sentence is a proposition about the world. Inference system applies
logical rules to the KB to deduce new information.
 Inference system generates new facts so that an agent can update
the KB. An inference system works mainly in two rules which are
given as:
1. Forward chaining
2. Backward chaining
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Techniques of Knowledge Representation
Prepared by:

Techniques/ Approaches of Knowledge
Representation
 Knowledge can be represented using the following
approaches/techniques:
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Logical Representation
Prepared by:

1. Logical Representation
 Logical representation is a language with some concrete rules which
deals with propositions and has no ambiguity in representation. Logical
representation means drawing a conclusion based on various
conditions.
 This representation lays down some important communication rules. It
consists of precisely defined syntax and semantics which supports the
sound inference. Each sentence can be translated into logics using
syntax and semantics.
 Logical representation can be categorized into:
1. Propositional Logic
2. First Order Predicate Logic
3. Higher order Predicate Logic
4. Fuzzy Logic
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1. Logical Representation (Cont.)
1. Propositional Logics:
 All propositions either true/false (1/0).
 We can not identify relation between 2 sentences.
 For example…..
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Sentences Truth value Proposition value
Sky is blue True True
Roses are red True True
2+2=5 False True

2. First Order Predicated Logic:
 These are much more expressive and make use of variables, constants,
predicates, functions and quantifiers along with the connective explained
already in previous section.
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Advantages of logical representation Disadvantages of logical representation
Logical representation enables us to
do logical reasoning.
Logical representations have some
restrictions and are challenging to work
with.
Logical representation is the basis for
the programming languages.
Logical representation technique may
not be very natural, and inference may
not be so efficient.
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Semantic net representation
Prepared by:

2. Semantic net representation
 Semantic networks are alternative of predicate logic for knowledge
representation. In Semantic networks, we can represent our knowledge in
the form of graphical networks. This network consists of nodes representing
objects and arcs which describe the relationship between those objects.
Semantic networks can categorize the object in different forms and can
also link those objects. Semantic networks are easy to understand and can
be easily extended.
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2. Semantic net representation (Cont.)
 Following are some statements which we need to represent in the form of
nodes and arcs.
1. Jerry is a cat.
2. Jerry is a mammal
3. Jerry is owned by Priya.
4. Jerry is brown colored.
5. All Mammals are animal.
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 Drawbacks in Semantic representation:
1. Semantic networks take more computational time at runtime as we need to
traverse the complete network tree to answer some questions. It might be
possible in the worst case scenario that after traversing the entire tree, we
find that the solution does not exist in this network.
2. Semantic networks try to model human-like memory (Which has 1015
neurons and links) to store the information, but in practice, it is not possible
to build such a vast semantic network.
3. These types of representations are inadequate as they do not have any
equivalent quantifier, e.g., for all, for some, none, etc.
4. Semantic networks do not have any standard definition for the link names.
5. These networks are not intelligent and depend on the creator of the system.
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 Advantages of Semantic network:
1. Semantic networks are a natural representation of knowledge.
2. Semantic networks convey meaning in a transparent manner.
3. These networks are simple and easily understandable.
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 Represent following sentences using semantic networks.
 Isa(person, mammal)
 Instance(Mike-Hall, person)
 Team(Mike-Hall, Cardiff)
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Frame Representation
Prepared by:

3. Frame Representation
 This concept was introduced by Marvin Minsky in 1975. they are mostly used
when the task becomes quite complex and needs more structured
representation.
 More structured the system becomes more would be the requirement of
using frames which would prove beneficial. Generally frames are record
like structures that consists of a collection of slots or attributes and the
corresponding slot values.
 Slots can be of any size and type. The slots have names and values called
as facts. Facets can have names or numbers too. A simple frame is shown
in fig for person ram.
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3. Frame Representation (Cont.)
33
Ram Brothers Laxman Cat
Bharat
Grey
Color

Sr. No. Slot Value
1 Ram -
2 Profession Professor
3 Age 50
4 Wife Sita
5 Children Luv Kush
6 Address 4C gb Road
7 City Banaras
8 State UP
9 Zip 400615
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 Advantages of frame representation:
1. The frame knowledge representation makes the programming easier by
grouping the related data.
2. The frame representation is comparably flexible and used by many
applications in AI.
3. It is very easy to add slots for new attribute and relations.
4. It is easy to include default data and to search for missing values.
5. Frame representation is easy to understand and visualize.
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 Disadvantages of frame representation:
1. In frame system inference mechanism is not be easily processed.
2. Inference mechanism cannot be smoothly proceeded by frame
representation.
3. Frame representation has a much generalized approach.
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4. Production Rules
 Production rules system consist of (condition, action) pairs which
mean, "If condition then action". It has mainly three parts:
 The set of production rules
 Working Memory
 The recognize-act-cycle
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4. Production Rules (Cont.)
 Example:
 IF (at bus stop AND bus arrives) THEN action (get into the bus)
 IF (on the bus AND paid AND empty seat) THEN action (sit down).
 IF (on bus AND unpaid) THEN action (pay charges).
 IF (bus arrives at destination) THEN action (get down from the bus).
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4. Production Rules (Cont.)
 Advantages of Production rule:
1. The production rules are expressed in natural language.
2. The production rules are highly modular, so we can easily remove, add
or modify an individual rule.
 Disadvantages of Production rule:
1. Production rule system does not exhibit any learning capabilities, as it
does not store the result of the problem for the future uses.
2. During the execution of the program, many rules may be active hence
rule-based production systems are inefficient.
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Issues in Knowledge Representation
Prepared by:

 Important Attributes:
 There can be attributes that occur in many different types of problem with
different names.
 For example, instance and isa and each is important because each supports
property inheritance.
 Relationships:
 The relationships, such as, inverses, existence; among various attributes of an
object need to be represented without any ambiguity.
 For example, band(John, NewYork City)
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(Cont.)
 Granularity:
 This represents at what level should the knowledge be represented and what are
primitives. Choosing the granularity of representation primitives are fundamental
concepts such as holding, seeing, playing and as English is a very rich language
with over half a million words to choose as our primitives in a series of situations.
 For example, if Tom feeds a dog then it could become: feeds(tom,dog)
 If tom gives the dog a bone like: gives(tom,dog,bone) are these the same?
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Propositional Logic
Prepared by:

Propositional Logic
 Propositional logic (PL) is the simplest form of logic where all the
statements are made by propositions. A proposition is a declarative
statement which is either true or false. It is a technique of
knowledge representation in logical and mathematical form.
 Example:
 It is Sunday. T
 The Sun rises from West . F
 3+3= 7. F
 Some students are intelligent. T/F both
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PL
Semantic
Syntax

Propositional Logic (Cont.)
 Following are some 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.
 Propositional logic consists of an object, relations or function,
and logical connectives.
 These connectives are also called logical operators.
45
PL
Complex
Atomic

Syntax in Propositional Logic (Cont.)
 Rules for conjunction: NEGATIVE operator
 A sentence such as ¬ P is called negation of P. A literal can be
either Positive literal or negative literal.
 Example,
P= Today is Sunday.
 1 is represent as a true
 0 is represent as a false
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P ¬ P
1 0
0 1

 Rules for conjunction: AND operator
 A sentence which has ∧ connective such as, P ∧ Q is called a
conjunction.
 Example: Rohan is intelligent and hardworking.
 It can be written as,
P= Rohan is intelligent.
Q= Rohan is hardworking.
 So we can write it as P ∧ Q.
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P Q P ^ Q
1 1 1
1 0 0
0 1 0
0 0 0

 Rules for Disjunction: OR operator
 A sentence which has ∨ connective, such as P ∨ Q. is called
disjunction, where P and Q are the propositions.
 Example: "Ritika is a doctor or Engineer",
 It can be written as,
P= Ritika is Doctor.
Q= Ritika is Engineer,
 So we can write it as P ∨ Q.
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P Q P v Q
1 1 1
1 0 1
0 1 1
0 0 0

 Rules for conjunction: CONDITIONAL
 A sentence such as P → Q, is called an implication. Implications
are also known as if-then rules.
 Example: If it is raining, then the street is wet.
Let P= It is raining,
Q= Street is wet,
 so it is represented as P → Q
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P Q P → Q
1 1 1
1 0 0
0 1 1
0 0 1

 Rules for conjunction: BICONDITIONAL
 A sentence such as P⇔ Q is a BiConditional sentence,
 Example: If I am breathing, then I am alive
P= I am breathing
Q= I am alive
 It can be represented as P ⇔ Q.
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P Q P ⇔ Q
1 1 1
1 0 0
0 1 0
0 0 1

Summarized table Propositional Logic
Connectives (Cont.)
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Example of Propositional Logic
Prepared by:

Properties of Operators
 Commutatively
 P∧ Q= Q ∧ P, or
 P ∨ Q = Q ∨ P.
 Associativity
 (P ∧ Q) ∧ R= P ∧ (Q ∧ R),
 (P ∨ Q) ∨ R= P ∨ (Q ∨ R)
 Identity element
 P ∧ True = P,
 P ∨ True= True.
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 Distributive
 P∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R).
 P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).
 DE Morgan's Law
 ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
 ¬ (P ∨ Q) = (¬ P) ∧ (¬Q).
 Double-negation elimination
 ¬ (¬P) = P.

What is Propositional Logic?
 A ^ B and B ^ A should have same meaning but in natural language words
and sentences may have different meaning
 Example,
1. Radha started feeling feverish and Radha went to the doctor.
2. Radha went to doctor and Radha stared feeling feverish.
 Here, sentence 1 and sentence 2 have different meaning
 In AI propositional logic is a relation between the truth value of one
statement to that of the truth table of other statement.
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Example of Propositional Logic
1. ¬(P ^ Q) , P → ¬Q
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P Q ¬Q (P ^ Q) ¬(P ^ Q) P → ¬Q
1 1 0 1 0 0
1 0 1 0 1 1
0 1 0 0 1 1
0 0 1 0 1 1
P Q IF P → Q
1 1 1
1 0 0
0 1 1
0 0 1
P Q IF P ^ Q
1 1 1
1 0 0
0 1 0
0 0 0
P Q IF P v Q
1 1 1
1 0 1
0 1 1
0 0 0

Example of Propositional Logic (Cont.)
2. ¬P v ¬Q v R , Q v R, P→R
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P Q R ¬P ¬Q ¬P v ¬Q v R (Q v R) P → R
0 0 0 1 1 1 0 1
0 0 1 1 1 1 1 1
0 1 0 1 0 1 1 1
0 1 1 1 0 1 1 1
1 0 0 0 1 1 0 0
1 0 1 0 1 1 1 1
1 1 0 0 0 0 1 0
1 1 1 0 0 1 1 1

3. (P v Q) v ~(P v (Q ^ R))=1
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P Q R P v Q Q ^ R P v (Q ^ R) ~(P v (Q ^ R)) (P v Q) v ~(P v (Q ^ R))
0 0 0 0 0 0 1 1
0 0 1 0 0 0 1 1
0 1 0 1 0 0 1 1
0 1 1 1 1 1 0 1
1 0 0 1 0 1 0 1
1 0 1 1 0 1 0 1
1 1 0 1 0 1 0 1
1 1 1 1 1 1 0 1

4. ~[{~P v ~(Q ^ R)} v (P ^ R)]
59
I1
I2
I3
P Q R ~ P I1 ~ I1 I2 I3 I2 v I3 ~ (I2 v I3)
0 0 0 1 0 1 1 0 1 0
0 0 1 1 0 1 1 0 1 0
0 1 0 1 0 1 1 1 1 0
0 1 1 1 1 0 1 0 1 0
1 0 0 0 0 1 1 0 1 0
1 0 1 0 0 1 1 1 1 0
1 1 0 0 0 1 1 0 1 0
1 1 1 0 1 0 0 1 1 0

5. (P→(Q→R)) → ((P→Q) → (P→R))
60
P Q R Q→R I1 I2 I3 I4 I1→I4
0 0 0 1 1 1 1 1 1
0 0 1 1 1 1 1 1 1
0 1 0 0 1 1 1 1 1
0 1 1 1 1 1 1 1 1
1 0 0 1 1 0 0 1 1
1 0 1 1 1 0 1 1 1
1 1 0 0 0 1 0 0 1
1 1 1 1 1 1 1 1 1
I1 I2 I3
I4
P Q IF P → Q
1 1 1
1 0 0
0 1 1
0 0 1

 Try out your self:
1. (P ⇔ (Q→ R)) ⇔ ((P ⇔ Q)→ R)
2. ((P ⇔ Q) ^ (~ Q → R)) ⇔ (~ (P ⇔ R) → Q)
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Limitation of Propositional Logic
 We cannot represent relations like ALL, some, or none with
propositional logic.
 Example:
 All the girls are cute.
 Some apples are sweet.
 Few students are intelligent.
 Propositional logic has limited expressive power.
 In propositional logic, we cannot describe statements in terms of
their properties or logical relationships.
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Inference Rules
Prepared by:

Inference Rules
 Inference:
 In artificial intelligence, we need intelligent computers which can
create new logic from old logic or by evidence, so generating the
conclusions from evidence and facts is termed as Inference.
 Inference Rule:
 Inference rules are the templates for generating valid arguments.
Inference rules are applied to derive proofs in artificial intelligence, and
the proof is a sequence of the conclusion that leads to the desired
goal.
 In inference rules, the implication among all the connectives plays an
important role.
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Inference Rules (Cont.)
1. Modus Ponens
2. Modus Tollens
3. Hypothetical Syllogism
4. Disjunctive Syllogism
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1. Modus Ponens
 The Modus Ponens rule is one of the most important rules of inference,
and it states that if P and P → Q is true, then we can infer that Q will be
true. It can be represented as:
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1. Modus Ponens
 Example:
 Statement-1: "If I am sleepy then I go to bed" ==> P → Q
 Statement-2: "I am sleepy" ==> P
 Conclusion: "I go to bed." ==> Q
 Hence, we can say that, if P→ Q is true and P is true then Q will be true.
67

2. Modus Tollens
 The Modus Tollens rule state that if P→ Q is true and ¬ Q is true, then
¬ P will also true. It can be represented as:
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2. Modus Tollens
 Statement-1: "If I am sleepy then I go to bed" ==> P→ Q
 Statement-2: "I do not go to the bed."==> ~Q
 Statement-3: Which infers that "I am not sleepy" => ~P
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 The Hypothetical Syllogism rule state that if P→R is true whenever P→Q
is true, and Q→R is true. It can be represented as the following
notation:
 Notation is : P → Q, Q → R
P → R
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 Statement-1: If you have my password, then you can log on to
my face book.
 Statement-2: If you can log onto my face book account then you
can delete my face book account.
 Statement-3: If you have my password then you can delete my
face book account.
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 The Disjunctive syllogism rule state that if P∨Q is true, and ¬P is true,
then Q will be true. It can be represented as:
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 Statement-1: Today is Sunday or Monday. ==>P∨Q
 Statement-2: Today is not Sunday. ==> ¬P
 Conclusion: Today is Monday. ==> Q
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 In short inference rule says that new sentence can be create by logically
following the set of sentences of knowledge base.
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Sr. No. Inference Rules Premises (KB) Conclusion
1 Modus Ponens X, X → Y Y
2 Substitution X → Y & Y → Z X = Y
3 Chain Rule X → Y, Y → Z X → Z
4 AND introduction X, Y X ^ Y
5 Transposition X →Y ~X → ~Y

Horn Clause
Prepared by:

Horn Clause
 A Horn clause is a clause (a disjunction of literals) with at most one positive
literal.
~A1 V ~A2 V ~A3 V . . . . . V ~An V B
 Lets take one example,
 (A1 ^ A2 ^ A3 ^ . . . . . ^ An) → B
 Apply DE Morgan's Law on given equation,
 ~(A1 ^ A2 ^ A3 ^ . . . . . ^ An) V B
 Apply Distributive Law,
 ~A1 V ~A2 V ~A3 V . . . . . V ~An V B
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Horn clause

Properties of Operators
 Commutatively
 P∧ Q= Q ∧ P, or
 P ∨ Q = Q ∨ P.
 Associativity
 (P ∧ Q) ∧ R= P ∧ (Q ∧ R),
 (P ∨ Q) ∨ R= P ∨ (Q ∨ R)
 Identity element
 P ∧ True = P,
 P ∨ True= True.
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 Distributive
 P∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R).
 P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).
 DE Morgan's Law
 ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
 ¬ (P ∨ Q) = (¬ P) ∧ (¬Q).
 Double-negation elimination
 ¬ (¬P) = P.

Horn Clause (Cont.)
1. (A ^ B ) → C
= (A ^ B) → C
Apply DE Morgan's Law
= ~ (A ^ B) V C
Apply Distributive Law
= ~A V ~B V C
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Horn clause

Horn Clause (Cont.)
2. (A V B) → C
= (A V B ) → C
= ~(A V B ) V C
= (~A ^ ~B) V C
= (~A V C) ^ (~A V ~B)
= (~A V C) , (~B V C)
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Horn Clause (Cont.)
3. ( A ^ ~B ) → C
= ( A ^ ~B ) → C
= ~(A ^ ~B) V C
= (~A V B) V C
= ~A V B V C
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Not Possible

Wampus World
Prepared by:

Wampus World
 The Wumpus world is a cave which has 4/4 rooms connected with
passageways. So there are total 16 rooms which are connected with each
other. We have a knowledge-based agent who will go forward in this
world. The cave has a room with a beast which is called Wumpus, who eats
anyone who enters the room. The Wumpus can be shot by the agent, but
the agent has a single arrow.
 In the Wumpus world, there are some Pits rooms which are bottomless, and
if agent falls in Pits, then he will be stuck there forever. The exciting thing
with this cave is that in one room there is a possibility of finding a heap of
gold. So the agent goal is to find the gold and climb out the cave without
fallen into Pits or eaten by Wumpus. The agent will get a reward if he comes
out with gold, and he will get a penalty if eaten by Wumpus or falls in the
pit.
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Wampus World (Cont.)
 Following is a sample diagram for representing the Wumpus world. It is
showing some rooms with Pits, one room with Wumpus and one agent at
(1,1) square location of the world.
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 There are also some components which can help the agent to navigate
the cave. These components are given as follows:
1. The rooms adjacent to the Wumpus room are smelly, so that it would have some
stench.
2. The room adjacent to PITs has a breeze, so if the agent reaches near to PIT, then
he will perceive the breeze.
3. There will be glitter in the room if and only if the room has gold.
4. The Wumpus can be killed by the agent if the agent is facing to it, and Wumpus
will emit a horrible scream which can be heard anywhere in the cave.
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 PEAS description of Wumpus world
 To explain the Wumpus world we have given PEAS description as
below:
 Performance measure
 Environment
 Actuators
 Sensors
 The Wumpus world Properties
 Exploring the Wumpus world
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 Performance measure:
 +1000 reward points if the agent comes out of the cave with the gold.
 -1000 points penalty for being eaten by the Wumpus or falling into the
pit.
 -1 for each action, and -10 for using an arrow.
 The game ends if either agent dies or came out of the cave.
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 Environment:
 A 4*4 grid of rooms.
 The agent initially in room square [1, 1], facing toward the right.
 Location of Wumpus and gold are chosen randomly except the first
square [1,1].
 Each square of the cave can be a pit with probability 0.2 except the
first square.
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 Actuators:
 Left turn,
 Right turn
 Move forward
 Grab
 Release
 Shoot
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 Sensors:
 The agent will perceive the stench if he is in the room adjacent to the
Wumpus. (Not diagonally).
 The agent will perceive breeze if he is in the room directly adjacent to
the Pit.
 The agent will perceive the glitter in the room where the gold is present.
 The agent will perceive the bump if he walks into a wall.
 When the Wumpus is shot, it emits a horrible scream which can be
perceived anywhere in the cave.
 These percepts can be represented as five element list, in which we will
have different indicators for each sensor.
 Example if agent perceives stench, breeze, but no glitter, no bump, and
no scream then it can be represented as:
[Stench, Breeze, None, None, None].
89

 The Wumpus world Properties:
 Partially observable: The Wumpus world is partially observable because
the agent can only perceive the close environment such as an
adjacent room.
 Deterministic: It is deterministic, as the result and outcome of the world
are already known.
 Sequential: The order is important, so it is sequential.
 Static: It is static as Wumpus and Pits are not moving.
 Discrete: The environment is discrete.
 One agent: The environment is a single agent as we have one agent
only and Wumpus is not considered as an agent.
90

 Exploring the Wumpus world:
 Agent's First step:
 Initially, the agent is in the first room or on
the square [1,1], and we already know
that this room is safe for the agent, so to
represent on the below diagram (a) that
room is safe we will add symbol OK.
Symbol A is used to represent agent,
symbol B for the breeze, G for Glitter or
gold, V for the visited room, P for pits, W
for Wumpus.
 At Room [1,1] agent does not feel any
breeze or any Stench which means the
adjacent squares are also OK.
91

92

 Agent's second Step:
 Now agent needs to move forward, so it will either move to [1, 2], or
[2,1]. Let's suppose agent moves to the room [2, 1], at this room agent
perceives some breeze which means Pit is around this room. The pit can
be in [3, 1], or [2,2], so we will add symbol P? to say that, is this Pit room?
 Now agent will stop and think and will not make any harmful move. The
agent will go back to the [1, 1] room. The room [1,1], and [2,1] are
visited by the agent, so we will use symbol V to represent the visited
squares.
93

 Agent's third step:
 At the third step, now agent will move to the room [1,2] which is OK. In
the room [1,2] agent perceives a stench which means there must be a
Wumpus nearby. But Wumpus cannot be in the room [1,1] as by rules of
the game, and also not in [2,2] (Agent had not detected any stench
when he was at [2,1]). Therefore agent infers that Wumpus is in the room
[1,3], and in current state, there is no breeze which means in [2,2] there
is no Pit and no Wumpus. So it is safe, and we will mark it OK, and the
agent moves further in [2,2].
94

95

 Agent's fourth step:
 At room [2,2], here no stench and no breezes present so let's suppose
agent decides to move to [2,3]. At room [2,3] agent perceives glitter, so
it should grab the gold and climb out of the cave.
96

Chap.3 First Order Predicate Logic
Prepared by:

First Order Predicate Logic
 In the topic of Propositional logic, we have seen that how to represent
statements using propositional logic. But unfortunately, in propositional
logic, we can only represent the facts, which are either true or false.
 PL is not sufficient to represent the complex sentences or natural language
statements. The propositional logic has very limited expressive power.
Consider the following sentence, which we cannot represent using PL logic.
 "Some humans are intelligent“.
 "Sachin likes cricket."
 To represent the above statements, PL logic is not sufficient, so we required
some more powerful logic, such as first-order logic (FOL).
98

First Order Predicate Logic (Cont.)
 First-order logic 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 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.
99

First Order Predicate Logic (Cont.)
 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:
1. Constant term: A, B, people, numbers, colors, wars, theories, squares, pits,
Wumpus, etc.
2. Variable term : It can be unary relation such as: red, round, is adjacent, or
n-any relation such as: the sister of, brother of, has color, comes between
3. Function: Father of, best friend, third inning of, end of, etc.
 As a natural language, first-order logic also has two main parts:
1. Syntax
2. Semantics
100

Atomic Sentences
 Atomic sentences are the most basic sentences of first-order logic. These
sentences are formed from a predicate symbol followed by a parenthesis
with a sequence of terms.
 We can represent atomic sentences as Predicate (term1, term2, temp3,
......, term n).
 Example:
 Ravi and Ajay are brothers: => Brothers(Ravi, Ajay).
 Chinky is a cat: => cat (Chinky).
101

Complex Sentences
 Complex sentences are made by combining atomic sentences using
connectives.
 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.
 Example:
 "x is an integer."
102

Quantifiers in First-Order Logic
 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:
1. Universal Quantifier, (for all, everyone, everything)
2. Existential quantifier, (for some, at least one).
103

Universal Quantifier
 Universal quantifier is a symbol of logical representation, which specifies
that the statement within its range is true for everything or every instance of
a particular thing.
 The Universal quantifier is represented by a symbol ∀, which resembles an
inverted A.
 If x is a variable, then ∀x is read as:
• For all x
• For each x
• For every x.
104

Universal Quantifier (Cont.)
105

Existential Quantifier
 Existential quantifiers are the type of quantifiers, which express that the
statement within its scope is true for at least one instance of something.
 It is denoted by the logical operator ∃, which resembles as inverted E.
When it is used with a predicate variable then it is called as an existential
quantifier.
 If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will be
read as:
• There exists a 'x.'
• For some 'x.'
• For at least one 'x.'
106

Existential Quantifier (Cont.)
107

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.
108

Example of FOL
Prepared by:

Example of FOL
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).
110

Example of FOL
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).
111

Example of FOL (Cont.)
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).
112

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 representation for this:
 ¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)].
113

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 representation for this:
 ∃(x) [ student(x) → failed (x, Mathematics) ∧∀ (y) [¬(x==y) ∧ student(y)
→ ¬failed (x, Mathematics)].
114

Representation Simple Facts in logic
Prepared by:

1. All students are smart.
 ∀x (Student(x) ⇒ Smart(x))
2. There exists a student.
 ∃x Student(x)
3. There exists a smart student.
 ∃x (Student(x) ∧ Smart(x))
117

(Cont.)
4. Every student loves some student.
 ∀x (Student(x) ⇒ ∃y (Student(y) ∧ Loves(x,y)))
5. Every student loves some other student.
 ∀x (Student(x) ⇒ ∃y (Student(y) ∧¬(x=y) ∧ Loves(x,y)))
6. There is a student who is loved by every other student.
 ∃x (Student(x) ∧∀y (Student(y) ∧¬(x=y) ⇒Loves(y,x)))
118

(Cont.)
1. Bill is a student.
 Student(Bill)
2. Bill takes either Analysis or Geometry (but not both).
 Takes(Bill,Analysis) ⇔ ¬Takes(Bill,Geometry)
3. Bill takes Analysis or Geometry (or both).
 Takes(Bill,Analysis) ∨ Takes(Bill,Geometry)
4. Bill takes Analysis and Geometry.
 Takes(Bill,Analysis) ∧ Takes(Bill,Geometry)
5. Bill does not take Analysis.
 ¬Takes(Bill,Analysis)
119

(Cont.)
6. No student loves Bill.
 ¬∃x (Student(x) ∧ Loves(x,Bill)
7. Bill has at least one sister.
 ∃x SisterOf(x,Bill)
8. Bill has no sister.
 ¬∃x SisterOf(x,Bill)
9. Bill has at most one sister.
 ∀x,y (SisterOf(x,Bill) ∧ SisterOf(y,Bill) ⇒x=y))
10.Bill has exactly one sister.
 ∃x (SisterOf(x,Bill) ∧∀y (SisterOf(y,Bill) ⇒x=y))
11.Bill has at least two sisters.
 ∃x,y (SisterOf(x,Bill) ∧ SisterOf(y,Bill) ∧¬(x=y))
120

(Cont.)
1. Anyone whom Mary loves is a football star.
2. Any student who does not pass does not play.
3. John is a student.
4. Any student who does not study does not pass.
5. Anyone who does not play is not a football star.
6. If John does not study, then Mary does not love John.
121

(Cont.)
1. Anyone whom Mary loves is a football star.
 ∀ x (LOVES(Mary,x) → STAR(x))
2. Any student who does not pass does not play.
 ∀ x (STUDENT(x) ∧ ¬ PASS(x) → ¬ PLAY(x))
3. John is a student.
 STUDENT(John)
4. Any student who does not study does not pass.
 ∀ x (STUDENT(x) ∧ ¬ STUDY(x) → ¬ PASS(x))
5. Anyone who does not play is not a football star.
 ∀ x (¬ PLAY(x) → ¬ STAR(x))
6. If John does not study, then Mary does not love John.
 ¬ STUDY(John) → ¬ LOVES(Mary,John)
122

(Cont.)
1. Every child loves Santa.
2. Everyone who loves Santa loves any reindeer.
3. Rudolph is a reindeer, and Rudolph has a red nose.
4. Anything which has a red nose is weird or is a clown.
5. No reindeer is a clown.
6. Scrooge does not love anything which is weird.
7. Scrooge is not a child.
123

(Cont.)
1. Every child loves Santa.
 ∀ x (CHILD(x) → LOVES(x, Santa))
2. Everyone who loves Santa loves any reindeer.
 ∀ x (LOVES(x, Santa) → ∀ y (REINDEER(y) → LOVES(x, y)))
3. Rudolph is a reindeer, and Rudolph has a red nose.
 REINDEER(Rudolph) ∧ REDNOSE(Rudolph)
4. Anything which has a red nose is weird or is a clown.
 ∀ x (REDNOSE(x) → WEIRD(x) ∨ CLOWN(x))
124

(Cont.)
5. No reindeer is a clown.
 ¬ ∃ x (REINDEER(x) ∧ CLOWN(x))
6. Scrooge does not love anything which is weird.
 ∀ x (WEIRD(x) → ¬ LOVES(Scrooge, x))
7. Scrooge is not a child.
 ¬ CHILD(Scrooge)
125

(Cont.)
1. Anyone who buys carrots by the bushel owns either a rabbit or a grocery
store.
2. Every dog chases some rabbit.
3. Mary buys carrots by the bushel.
4. Anyone who owns a rabbit hates anything that chases any rabbit.
5. John owns a dog.
6. Someone who hates something owned by another person will not date
that person.
7. If Mary does not own a grocery store, she will not date John.
126

(Cont.)
1. Anyone who buys carrots by the bushel owns either a rabbit or a grocery
store.
 ∀ x (BUY(x) → ∃ y (OWNS(x,y) ∧ (RABBIT(y) ∨ GROCERY(y))))
2. Every dog chases some rabbit.
 ∀ x (DOG(x) → ∃ y (RABBIT(y) ∧ CHASE(x,y)))
3. Mary buys carrots by the bushel.
 BUY(Mary)
4. Anyone who owns a rabbit hates anything that chases any rabbit.
 ∀ x ∀ y (OWNS(x,y) ∧ RABBIT(y) → ∀ z ∀ w (RABBIT(w) ∧ CHASE(z,w) → HATES(x,z)))
127

(Cont.)
5. John owns a dog.
 ∃ x (DOG(x) ∧ OWNS(John,x))
6. Someone who hates something owned by another person will not date
that person.
 ∀ x ∀ y ∀ z (OWNS(y,z) ∧ HATES(x,z) → ¬ DATE(x,y))
7. (Conclusion) If Mary does not own a grocery store, she will not date John.
 (( ¬ ∃ x (GROCERY(x) ∧ OWN(Mary,x))) → ¬ DATE(Mary,John))
128

Chap.4 Inference in First Order
Logic
Prepared by:

Comparison between Propositional
Logic and First Order Logic
Prepared by:

Logic and First Order Logic (Cont.)
Propositional Logic (PL) First Order Logic (FOL)
PL can not represent small worlds like
vacuum cleaner world.
FOL can very well represent small
world’s problems.
PL is a weak knowledge
representation language.
FOL is a strong way of representing
language.
PL uses propositions in which the
complete sentence is denoted by a
symbol.
FOL uses predicated which involve
constants, variables, functions,
relations.
PL can not directly represent
properties of individual entities or
relation between entities.
i.e., Hiral is short.
FOL can directly represent properties
of individual entities or relation
between entities.
i.e., Hiral is short.
Ans is: short (Hiral)
131

Logic and First Order Logic
Propositional Logic (PL) First Order Logic (FOL)
PL can not express specialization,
generalization, or patterns, etc.
i.e., all rectangle have 4 sides
FOL can express specialization,
generalization, or patterns, etc.
i.e., all rectangle have 4 sides
Ans. is: no of size(rectangle,4)
Foundation level language Higher level language
PL is not sufficiently expressive to
represent complex statements.
FOL is represent complex statements.
In PL meaning of the facts is context-
independent unlike natural language.
In FOL meaning of the sentences is
context dependent like natural
language.
Declarative in nature Derivative in nature
132

Inference in First Order Logic
Prepared by:

Inference in First Order Logic
 The inference engine is the component of the intelligent system in artificial
intelligence, which applies logical rules to the knowledge base to infer new
information from known facts. The first inference engine was part of the
expert system. Inference engine commonly proceeds in two modes, which
are:
1. Forward chaining
2. Backward chaining
134

Forward Chaining/ Resolution
Prepared by:

1. Forward Chaining
 Forward chaining is also known as a forward deduction or forward
reasoning method when using an inference engine. Forward chaining is a
form of reasoning which start with atomic sentences in the knowledge base
and applies inference rules (Modus Ponens) in the forward direction to
extract more data until a goal is reached.
 The Forward-chaining algorithm starts from known facts, triggers all rules
whose premises are satisfied, and add their conclusion to the known facts.
This process repeats until the problem is solved.
 Forward chaining is called as a data driven inference technique.
136

1. Forward Chaining (Cont.)
 Properties of Forward-Chaining:
 It is a down-up approach, as it moves from bottom to top.
 It is a process of making a conclusion based on known facts or data, by
starting from the initial state and reaches the goal state.
 Forward-chaining approach is also called as data-driven as we reach
to the goal using available data.
 Forward -chaining approach is commonly used in the expert system,
such as CLIPS, business, and production rule systems.
137

1. Forward Chaining / Resolution
(Cont.)
 Example:
 If it is raining then, we will take umbrella.
Data: It is raining
Decision: we will take umbrella
 That’s means we already known that it’s raining that’s why it decided to
take umbrella.
138
Data
Decision

(Cont.)
 Facts:
1. It is a crime for an American to sell weapons to enemy of America.
2. Country Nono is an enemy of America.
3. Nono has a some missiles.
4. All of the missiles were sold to Nono by colonel west.
5. Missiles are weapons.
6. Colonel west is American.
 We have prove that west is a criminal.
139

(Cont.)
 Facts conversion into FOL:
1. American (X) ∧ weapon(Y) ∧ sells (X,Y,Z) ∧ enemy(Z, America) →
Criminal(X)
2. Enemy (Nono, America)
3. Owns ( Nono, X), Missile (X)
4. Missile (X) ^ owns (Nono, X)→ Sell (West, X, Nono)
5. Missile (X) → Weapon (x)
6. American (West)
140

(Cont.)
 Forward chaining proof:
 STEP: 1
141
American (West) Missiles (X) Owns (Nono, X)
Enemy (Nono,
America)

(Cont.)
 STEP: 2 apply rule number 4 and 5
142

(Cont.)
143
Enemy (Nono,
America)
Weapon (X) Sell (West, X, Nono)
5
4

(Cont.)
 STEP: 3 apply rule number 1
Criminal(X)
144

(Cont.)
145
Enemy (Nono,
America)
Weapon (X) Sell (West, X, Nono)
5
4
Criminal (West)
1

Backward Chaining/ Resolution
Prepared by:

2. Backward Chaining/ Resolution
 Backward-chaining is also known as a backward deduction or backward
reasoning method when using an inference engine. A backward chaining
algorithm is a form of reasoning, which starts with the goal and works
backward, chaining through rules to find known facts that support the goal.
147
Decision
Data

(Cont.)
 Properties of backward chaining:
 It is known as a top-down approach.
 Backward-chaining is based on modus ponens inference rule.
 In backward chaining, the goal is broken into sub-goal or sub-goals to
prove the facts true.
 It is called a goal-driven approach, as a list of goals decides which rules
are selected and used.
 Backward -chaining algorithm is used in game theory, automated
theorem proving tools, inference engines, proof assistants, and various
AI applications.
 The backward-chaining method mostly used a depth-first
search strategy for proof.
148

(Cont.)
 Facts:
 We have know that west is a criminal.
149

(Cont.)
 Facts conversion into FOL:
Criminal(X)
6. American (West)
150

(Cont.)
 Backward chaining proof:
 STEP: 1 Take conclusion (leaf node)
151
Criminal (West)

(Cont.)
 STEP: 2 Apply 1st rule
152
Criminal (West)
American (West) Missiles (X) Sell (West, X, Z)
Enemy (Nono,
America)
1

(Cont.)
 STEP: 3 Apply rules
153

(Cont.)
154
Criminal (West)
American (West) Weapon (X) Sell (West, X, Z)
Enemy (Nono,
America)
1
Missiles (X) Owns (Nono, X)
Missiles (X)

Unification Algorithm
Prepared by:

Unification Algorithm
 Algorithm Unify(L1,L2):
1. If L1 and L2 is a variable or constant then,
a. If L1 and L2 are identical return NIL,
b. Else if L1 is a variable, then if L1 occurs in L2 then return fail, else return { (L1/L2) }
c. Else if L2 is a variable, then if L2 occurs in L1 then return fail, else return { (L2/L1) }
d. Else return fail
2. If the initial predicate symbols in L1 and L2 are identical, then return Fail
3. If L1 and L2 have different number of arguments, then fails
a. i.e., P(L1,L2) and P(L1,L2,L3)
4. Set SUBST to nil.
5. Loop (apply for all variable/constant)
6. Return SUBST.
156

Unification Example
 Unification is all about making the expressions look identical. So, for the
given expressions to make them look identical we need to do substitution.
 (x, y)=(2,3)
 So we can say like,
 x=2, {2 for X}
 y=3 {3 for y}
 P(x, F(y)) = P (a, F(g(z))
 In our case, P(x, F(y)) and P (a, F(g(z))
 x=a (a for x)
 y=g(z) (g(z) for y)
 [a/x , g(z)/y]
157

Unification Example(Cont.)
 Food (Peanuts) and Food (x)
 So here we can say like this,
 Peanuts for X
 {X / Peanuts}
 Like (Ravi, P) and like (Ravi, Apple)
 So here we can say like this,
 Apple for P
 {P / Apple)
158

Resolution
Prepared by:

Resolution
 Resolution is used, if there are various statements are given, and we need
to prove a conclusion of those statements. Unification is a key concept in
proofs by resolutions. Resolution is a single inference rule which can
efficiently operate on the conjunctive normal form or clausal form.
160

Conversion from FOL Clausal Normal
Form (CNF)
 Steps for Resolution :
1. Conversion of facts into FOL.
2. Convert FOL to CNF
A. Elimination of implication
 Eliminate all ”→” sign
 Replace P → Q with ~P V Q
B. Distribute negations
 Replace ~~P with P
 Replace ~(P V Q) with ~P ^ ~Q (pVQ) → R
161

Form (CNF) (Cont.)
2. Convert FOL to CNF. (Cont.)
C. Eliminate existential quantifiers by replacing with skolem constants or
skolem function
 ∀X ∃Y ((P1(X,Y) v (P2(X,Y))) ≡ ∀X ((P1(X,F(Y)) v (P2(X,F(Y)))
D. Rename variables/ use standard variable to avoid duplicate
quantifiers.
E. Drop all universal quantifiers.
 (P1(X,F(Y)) v (P2(X,F(Y))
162

Form (CNF) (Cont.)
3. Negate the statement which needs to prove (proof by
contradiction)
 In this statement, we will apply negation to the conclusion statements,
which will be written as
 ¬ likes(John, Peanuts)
4. Draw Resolution graph
 Now in this step, we will solve the problem by resolution tree using
substitution.
163

Form (CNF) (Cont.)
 Facts:
 Prove: We have know that west is a criminal.
164

Form (CNF) (Cont.)
 Step 1: Convert English to FOL
Criminal(X)
6. American (West)
165

Form (CNF) (Cont.)
 Step 2: Remove → sign from the FOL
1. American (X) ∧ weapon(Y) ∧ sells (X,Y,Z) ∧ enemy(Z, America) → Criminal(X)
~(American (X) ∧ weapon(Y) ∧ sells (X,Y,Z) ∧ enemy(Z, America)) V Criminal(X)
Apply ~ sign in whole sentence ,
 ~American (X) V ~weapon(Y) V ~sells (X,Y,Z) V ~enemy(Z, America) V Criminal(X)
Owns ( Nono, X)
Missile (X)
166

Form (CNF) (Cont.)
 Step 2: Remove ‘→’ sign from the FOL
4. Missile (X) ^ owns (Nono, X) → Sell (West, X, Nono)
~(Missile (X) ^ owns (Nono, X)) V Sell (West, X, Nono)
Apply ~ sign in whole sentence ,
~Missile (X) V ~owns (Nono, X) V Sell (West, X, Nono)
~Missile (X) V Weapon (x)
6. American (West)
7. Criminal (West)
167

Example for Conversion from FOL
Clausal Normal Form (CNF)
Prepared by:

Clausal Normal Form (CNF)
 Step 1: Convert English to FOL
169

Clausal Normal Form (CNF) (Cont.)
 Step 2: Eliminate all implication (→) and rewrite
170

 Step 3: Move negation (¬)inwards and rewrite
171

 Step 4: Rename variables or standardize variables
172

 Step 5: Eliminate existential instantiation quantifier by elimination.
 In this step, we will eliminate existential quantifier ∃, and this process is
known as Skolemization. But in this example problem since there is no
existential quantifier so all the statements will remain same in this step.
173

 Step 6: Drop Universal quantifiers
174

175
¬ likes(John, Peanuts) ¬ food(x) V likes(John, x)
¬ food(Peanuts)
{Peanuts/x}
¬ eats(y, z) V killed(y) V food(z)
{Peanuts/z}
¬ eats(y, z) V killed(y) eats (Anil, Peanuts)
{Anil/y}
killed(Anil) ¬ alive(k) V ¬ killed(k)
¬ alive(Anil)
{Anil/k}
alive(Anil)
{ }
Step
7:
Draw
Resolution
graph

Chap.3 Knowledge Representation Issues Chap.4 Inference in First Order Logic

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Chap.3 Knowledge Representation Issues Chap.4 Inference in First Order Logic