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Artificial IntelligenceCourse Code: ECE434 UNIT -IV

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SyllabusUNIT

1. Introduction to Artificial Intelligence

2. Problem Solving : state-space Search And Control Strategies

3. Problem Reduction And Game Playing

4. Logic Concept And Logic Programming

5. Prolog Programming Language

MID TERM

1. Knowledge Representations

2. Expert Systems And Applications

3. Uncertainty Measure: Probability Theory And Fuzzy Logic

4. Machine Learning, Ann And Evolutionary Computation

5. Introduction To Intelligent Agents And Natural Language

Processing

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References

Text

1. Artificial Intelligence Saroj Kaushik 1st Edition

Cengage Learning 2011

Reference Books

2. Introduction to Artificial Intelligence and ExpertSystems Dan. W. Patterson 1st Edition 1990 PHI(Pretice Hall India).

3. Artificial Intelligence- A Modern Approach

Stuart Russel Peter Norvig 3rd Edition Pearson, 2009 .

4. Artificial Intelligence Elaine Rich Kevin Knight3rd Edition 2008 Tata McGraw Hill, India

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Topics to be covered in Unit IV

Logic & its Concepts

Meaning of Logic & its need

What is Symbolic Logic & its Types

Propositional Logic and Predicate Logic

Various methods for proving the validity of the formulae: Natural Deduction System

Axiomatic System

Semantic Tableau Method

Resolution Refutation Method

Predicate Logic

Logic Programming

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INTRODUCTION TO LOGIC CONCEPTS &LOGIC PROGRAMMING

Logic was to be a branch of philosophy;however formal logic has been studied in the

context of foundation of mathematics where it

is referred to as symbolic logic. Logic is concerned with the principles of

drawing valid inferences from a given set of

true statements. Hence, it is referred to as symbolic logic.

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LOGICLogic is concerned with the truth of statements about the world.

Generally each statement is either TRUE or FALSE. Logic includes : Syntax ,

Semantics and Inference Procedure.

Syntax :

Specifies the symbols in the language about how they can be combined toform sentences. The facts about the world are represented as sentences in

logic.

Semantic :

Specifies how to assign a truth value to a sentence based on its meaning in

the world. It Specifies what facts a sentence refers to. A fact is a claim about

the world, and it may be TRUE or FALSE.

Inference Procedure :

Specifies methods for computing new sentences from an existing sentences.

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Branches of Symbolic logic

It is divided into 2 branches: Propositional logic

Predicate logic

A Proposition refers to a declarative statementthat is either true or false (but not both) in agiven context.

One can infer a new proposition from a givenset of propositions in the same context usinglogic.

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Example of Propositional Logic

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Propositional Calculus

Propositional Calculus (PC) refers to a languageof propositions in which a set of rules are used

to combine simple propositions to form

compound propositions with the help ofcertain logical operators.

These logical operators are also called as

connectives; These operators are: not(~), or (v), and (^),

implies ( ), and equivalence ( ).

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Well-formed formula (wff)

A wff is defined is defined as a symbol or a stringof symbols generated by the formal grammar of aformal language of a formal language.

Properties of wff:

The smallest unit (or an atom) is considered to beal wff.

If is a wff , then ~ is also a wff. If and are wff, then (^ ), ( ), ( ) and

( ) are also wff.

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Truth Table to prove PC

In Propositional Calculus (PC) is used to provide

operational definitions of important logical

operators.

The logical constants in PC are true or False

and these are represented as T or F.

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Truth Table for PC

A B ~A A^B AB AB AB

T T F T T T T

T F F F T F F

F T T F T T F

F F T F F T T

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Do it yourself

Compute the truth value of using truth table

approach.

: (A V B) ^ (~B A)

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Equivalence Laws

NAME OF THE RELATION EQUIVALENCE RELATIONS

Commutative Law A v B B v A

A ^ B B ^ A

Associative Law A v (B v C) (A v B) v C

A ^ (B ^ C) (A ^ B) ^ C

Double Negation ~ (~A) A

Distributive Laws A v (B ^ C) (A v B) ^ (A v C)

A ^ (B v C) (A ^ B) v (A ^ C)

De Morgans Laws ~(A v B) ~A ^ ~B

~(A ^ B) ~A v ~B

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Equivalence Laws

NAME OF THE RELATION EQUIVALENCE RELATIONS

Absorption Laws A v (A ^ B) A

A ^ (A v B) A

A v (~A ^ B) A v BA ^ (~A v B) A ^ B

Idempotence A v A A

A ^ A A

Excluded Middle Law A v ~A T (True)

Contradiction Law A ^ ~ A F (False)

Commonly Used Equivalence Relations A B ~A v B

A B (A B) ^ (B A)

(A ^ B) v (~A ^ ~B)

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Propositional Logic

Propositional logic deals with the validity,

satisfiability (also called consistency) and

unsatisfiability (inconsistency) of a formula andthe derivation of a new formula using

equivalence laws.

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Propositional Logic

The validity, satisfiability, and unsatisfiability ofa formula may be determined on the basis of

the following conditions:

A formula is said to be valid if and only if it isa tautology.

A formula is said to be satisfiable if there

exists at least one interpretation for which istrue.

A formula is said to be unsatisfiable if the

value of is false under all interpretations.

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Example

Show that the following is a valid argument:

If it is humid then it willrain and since it is humidtoday it will rain

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Solution to the Example

Let us symbolize each part of the English

sentence by propositional atoms as follows:

A: It is humid

B: It will rain

Formula can be represented as:

: *(A B) ^ A+ B

Now Solve it to calculate the value of this formula

using Truth Table method

Ad t & Di d t f thi

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Advantages & Disadvantage of thismethod for evaluating any formula

Advantage:

The Truth Table approach is simple and straightforward method by

providing truth values in a given situation.

It is an easy method for evaluating consistency, inconsistency, or validity of

a formula.

Disadvantage:

As no. of values grows, the size of truth table grows exponentially.

For example: in order to validate : (A ^ B ^ C ^ D) (B v E), we require 32

rows and compute the value of for all the 32 interpretations.

Hence, use of truth table approach proves to be wastage of time for various

cases. Therefore, we require some other methods which can help in proving

the validity of the formula directly.

V i th d f f &

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Various methods for proofs &Deduction

Some methods apart from the Truth Table

method that are concerned with proofs and

deductions are as follows:

Natural Deduction System

Axiomatic System

Semantic Tableau Method Resolution Refutation Method

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Natural Deduction system

Natural deduction system (NDS) is called so

because of the fact that it mimics the pattern

of natural reasoning.

NDS is based on a set of deductive inference

rules.

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NDS Rules Table

RULE NAME SYMBOL RULE DESCRIPTION

Introducing ^ (I: ^) If A1, ..An then

A1^.^An

IfA1, A2,An is true, then

their conjunction is also true.

Eliminating ^ (E:^) IfA1^ A2 ^^An

then Ai(1

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NDS Rules Table

RULE NAME SYMBOL RULE DESCRIPTION

Introducing (I: )If from 1,. n infer is proved then

1,n is proved

If given that 1, n are true andfrom these we deduce then 1^

2^n is also true

Eliminating (E:) If A1A, A1 , then A If A1A is true and A1 is also true,then A is also true. This is called as

Modus Ponen (MP) rule

Introducing (I: ) If A1A2, A2A1, then

A1 A2

If A1A2 and A2A1 is true, then

A1 A2 is also true

Eliminating (E:) IfA1 A2 then A1A2,

A2A1

IfA1 A2 is true then A1A2 &

A2A1 are also true

Introducing ~ (I: ~) If from A infer A1 ^ ~A1 is

proved then ~A is proved

If from A (which is true), a

contradiction is proved then truth

of ~A is also proved

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Example

Prove that A ^ (B v C) is deduced from A ^ B

Description Formula Comments

Theorem From A^B infer A^(BvC) To be proved

Hypothesis(given) A ^ B 1

E: ^(1) A 2

E: ^(1) B 3

I: v(3) B v C 4

I: ^(2,4) A ^ (B v C) Proved

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Example: NDS HOMEWORK

Prove the theorem infer *(A B) ^ (B C)+ (A C)

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Axiomatic System

The axiomatic system is based on a set of 3 axioms and onerule of deduction.

In axiomatic system, the proofs of the theorem are often

difficult and require a guess in selection of appropriate axiom.

In this system, only two operators are used to form a formula:

not (~) and implies ().

These can be converted as follows:

A ^ B ~(~A v B) ~(A~B)

A v B ~A B

A B (A B) ^ (B A) ~[(A B) ~ (B A) ]

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3 Axioms & 1Rule

Axiom 1: ( )

Axiom 2: [ ( )] [( ) ( )]

Axiom 3: (~ ~) ( )

Modus Ponen Rule

Hypothesis: , and , Consequent:

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Example

Establish that A C is a deductive consequence

of A B, B C-

Description Formula Comments

Theorem A B, B C- - (A C) ProveHypothesis 1 A B 1

Hypothesis 2 B C 2

Instance of Axiom 1 (B C) *A (B C)+ 3

MP(2,3) [A (B C)+ 4

Instance of Axiom 2 [A (BC)+ *(AB) (AC)+ 5

MP(4,5) (A B) (A C) 6

MP(1,6) (A C) Proved

Semantic Tableau System in

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Semantic Tableau System inPropositional Logic

Semantic tableau is a binary tree which is

constructed by using semantic tableau rules

with a formula as a root.

Both NDS and Axiomatic System uses forward

Chaining Approach.

Semantic tableau methods as well as

Resolution Refutation method uses backward

chaining method.

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Semantic Tableau Rules list 1

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Example: Semantic Tableau

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Example: Semantic Tableau.

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Example: Semantic Tableau.

Resolution Refutation in

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Resolution Refutation inPropositional Logic

Clause is defined as a special formula

containing the boolean operators ~ and v.

Resolution Refutation method is the most

favoured method for developing computer

based systems that can be used to prove

theorems automatically.

In this method, the negation of the goal to beproved is added to the given set of clauses, and

using the resolution principle, it is shown that

there is a refutation in the new set.

Conversion of Formula to set of

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Conversion of Formula to set ofClauses

There are 2 Normal Forms:

Disjunctive Normal Form (DNF)

Conjunctive Normal Form (CNF)

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Conversion of a Formula to its CNF

Eliminate double negation signs by using

~(~A) A

Use De Morgans law to push ~(negation) immediately before the

atomic formula

~(A ^ B) ~A v ~B

~(A v B) ~A ^ ~B

Use Distributive law to get CNF

A v (B ^ C) (A v B) ^ (A v C)

Eliminate the & by using following equivalence laws:

A B ~ A v B

A B (A B) ^ (B A)

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Resolution of Clauses

Two clauses can be resolved by eliminating

complimentary pair of literals, from both.

A new clause is formed by disjunction of the

remaining literals in both the clauses.

The complimentary literals are resolved

together by deleting complimentary literals

and hence a new clause is formed.

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Example of Resolution

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Example of Resolution

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Predicate Logic

It is the extension of Propositional Logic.

Due to the disadvantages of propositionallogic, there is the need of predicate logic.

It will provide more better inferencing and alsoto have common validation mechanism.

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Predicate Logic

The propositional logic, is not powerful enough for all types of

assertions;

Example : The assertion "x > 1", where x is a variable, is not a

proposition because it is neither true nor false unless value of x

is defined.

For x > 1 to be a proposition ,

either we substitute a specific number for x ;

or change it to something like

"There is a number x for which x > 1 holds";

or "For every number x, x > 1 holds".

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Predicate logic

Consider example :

All men are mortal.

Socrates is a man.

Then Socrates is mortal ,

These cannot be expressed in propositional logic as a finite and logicallyvalid argument (formula).

We need languages : that allow us to describe properties (predicates)

of objects, or a relationship among objects represented by the

variables .

Predicate logic satisfies the requirements of a language.

Predicate logic is powerful enough for expression and reasoning.

Predicate logic is built upon the ideas of propositional logic.

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Using Predicate Logic

1. Marcus was a man.

2. Marcus was a Pompeian.

3. All Pompeians were Romans.

4. Caesar was a ruler.

5. All Pompeians were either loyal to Caesar or hated him.

6. Every one is loyal to someone.

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

8. Marcus tried to assassinate Caesar.

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Predicate Logic Example

1. Marcus was a man.

man(Marcus)


Pompeian(Marcus)


x: Pompeian(x) Roman(x)


ruler(Caesar)

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inclusive-or

x: Pompeians (x) loyalto(x, Caesar) hate(x, Caesar)

exclusive-or

x: Pompeians (x) (loyalto(x, Caesar) hate(x, Caesar))

(loyalto(x, Caesar) hate(x, Caesar))

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Predicate Logic Example...


x: y: loyalto(x, y)

y: x: loyalto(x, y)

P di t L i E l

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7. People only try to assassinate rulers they

are not loyal to.

x: y: person(x) ruler(y) tryassassinate(x, y) loyalto(x, y)


tryassassinate(Marcus, Caesar)

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Using Predicate Logic

Was Marcus loyal to Caesar?

Using 7 & 8 fact, we can predict

Backward chaining

man(Marcus)

ruler(Caesar)

tryassassinate(Marcus, Caesar)

x: man(x) person(x)loyalto(Marcus, Caesar)

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Resolution

The basic ideas

KB |= KB |=false

() () ()

sound and complete

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Conversion to Clause Form1. Eliminate .

P Q P Q

2. Reduce the scope of each to a single term.

(P Q) P Q

(P Q) P Qx: P x: P

x: p x: P

P P

3. Standardize variables so that each quantifier binds a uniquevariable.

(x: P(x)) (x: Q(x)) (x: P(x)) (y: Q(y))

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Conversion to Clause Form

4. Move all quantifiers to the left without changing their relative order.(x: P(x)) (y: Q(y)) x: y: (P(x) (Q(y))

5. Eliminate (Skolemization).

x: P(x) P(c) Skolem constant

x:y P(x, y) x: P(x, f(x)) Skolem function

6. Drop.

x: P(x) P(x)

7. Convert the formula into a conjunction of disjuncts.

(P Q) R (P R) (Q R)

8. Create a separate clause corresponding to each conjunct.

9. Standardize apart the variables in the set of obtained clauses.

Con ersion to Cla se Form S mmar

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Conversion to Clause Form Summary

1. Eliminate .2. Reduce the scope of each to a single term.

3. Standardize variables so that each quantifier binds a unique variable.

4. Move all quantifiers to the left without changing their relative order.

5. Eliminate (Skolemization).

6. Drop.

7. Convert the formula into a conjunction of disjuncts.

8. Create a separate clause corresponding to each conjunct.

9. Standardize apart the variables in the set of obtained clauses.

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Example1. Marcus was a man.






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


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Example1. Man(Marcus).

2. Pompeian(Marcus).

3. x: Pompeian(x) Roman(x).

4. ruler(Caesar).

5. x: Roman(x) loyalto(x, Caesar) hate(x, Caesar).6. x: y: loyalto(x, y).

7. x: y: person(x) ruler(y) tryassassinate(x, y)

loyalto(x, y).

8. tryassassinate(Marcus, Caesar).

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Example

Prove:hate(Marcus, Caesar)

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More Questions that can be raised

1. When did Marcus die?

2. Whom did Marcus hate?

3. Who tried to assassinate a ruler?

4. What happen in 79 A.D.?.

5. Did Marcus hate everyone?

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