KNOWLEDGE REPRESENTATION (Cont.)Today’s Lecture •Review of previous lecture •Limitations of...

ARTIFICIAL

INTELLIGENCE

LECTURE # 04

Artificial Intelligence 2012 Lecture 04 Delivered By Zahid Iqbal 1

Review of Last Lecture


Review

• Reasoning

• Types of Reasoning

• Logic

• Propositional Logic


Today’s Lecture

• Review of previous lecture

• Limitations of Propositional Logic

• Predicate Calculus

• First Order Predicate Logic

• First Order Predicate Logic Constructs

• Inference Rules


Limitations of Propositional Logic

We Can’t describe things in terms of their properties or relationships (very limited expressive power)

Propositional logic is declarative Propositional logic is compositional.

We can’t express rules or generalizations

If the train is late and there are no taxis, john is late for the meeting If trains are late and there are no taxis, anyone traveling by trains is late

for the meeting…


Limitations…

• Propositions can only represent knowledge as complete sentences, e.g.

• a = the ball’s color is blue. • Cannot analyze the internal structure of the sentence.

• No quantifiers are available, e.g. for-all, there-exists • Propositional logic provides no framework for proving statements

such as: • All humans are mortal • All women are humans • Therefore, all women are mortals

• This is a limitation in its representational power.


Predicate Calculus

• Provides a richer modeling language

• We have objects and properties

• We have relationships between objects

• We have quantification – an ability to refer to all or some objects

Retain connectives such as ~ =

Instead of looking at sentences that are of interest merely for their

truth values, predicate calculus is used to represent statements

about specific objects or individuals .

Examples of individuals:

you, this page of lecture, the number 1, Socrates


Predicate Calculus (cont.)

A predicate is that which says something about the subject.

e. g., The book is red.

subject color of the book

represented as:

is-red(book) or simply red(book)

is-red: predicate

book: argument

· A predicate statement takes the value true or false .



• red(book) is true if the book is red, false if it is not, then ~red( book) becomes false.

• Predicate with one argument is called a 1- place predicate.

• A predicate can have more than 1 argument: e. g., color( book, red)

mother( john, mary)

greater- than( 7, 4)

• The number of arguments of a predicate is called its arity.

• book, red, john, mary, 7, 4 … are constants.

• We need variables and quantifiers to express sentences such as

“Everyone likes ice cream”

“Ali has some friends”



• " for all, for every (universal quantifier) • $ there exists (some) (existential quantifier )

Examples:

"X likes(X, ice_ cream)

$Y friends(Y, Ali)

• The quantifier specifies the extension of the variable (the total number of objects it applies, or the range of values it can take).

• . Universal and existential quantifiers allow expressing general rules with variables

• Universal quantification

• All cats are mammal • It is equivalent to the conjunction of all the sentences obtained by

substitution the name of an object for the variable x.



• Functions -- have a fixed number of arguments (arity)

-- return (or evaluate to) objects instead of truth values.

e. g., uncle- of( mary) = john

plus( 4, 3) = 7

• Arguments can be constants, variables, or functions.

e. g., father- of( father- of( john))

• Sometime we use something called a term , which is either a constant, variable, or function expression


Quantification: Existential

• Existential quantification : • an existentially quantified sentence is true in case one of the

disjunct is true

$ x Sister(x, Ibrahem) // $ : There exists

• Equivalent to disjunction: Sister(fatima, ibrahem) V

Sister(zainab, ibrahem) V

Sister(kalsoom, ibrahem)

…

• We can mix existential and universal

quantification.


Predicate Calculus Syntax

• Every atomic sentence is a sentence.

• If s is a sentence, so is ~s.

• If s1 and s2 are sentences, so is s1 s2 ;

.… so is s1 s2 ;

.… so is s1 s2 ;

.… so is s1 = s2 ;

• If X is a variable and s a sentence, then "X s is a sentence.

• .… then $X s is a sentence.

• For example:

"X "Y father( X, Y) mother( X, Y) parent( X, Y)

is a well- formed predicate calculus sentence.


First-order logic

• Simplest form of predicate logic,

• Propositional logic assumes the world contains facts

• First-order logic (like natural language) assumes the world contains • Objects: people, houses, numbers, colors, baseball games, wars,

centuries…

• properties: blue, small, tall, ugly, ...

• Relations: prime, brother of, bigger than, part of, comes between, bogus …

• Functions: father of, best friend, third inning of, one more than, plus, …


FOPC

• The main types of symbols used are • Constant: name specific object

• Predicates: a fact or predicate is divided into 2 parts. • Predicate: the assertion of the proposition

• Argument: the object of the proposition.

• E.g. “Ali likes bananas” in predicate logic will b,

• Likes( Ali, bananas),

• Variables: use for general representation of objects • Likes ( X, Y),

• Formula: combine predicate and quantifiers to represent information.

• Connectives: ^, v, ~,

• Quantifiers: $, "

• Function: father(x) = y: A function that specifies the unique element, that is the father of Ali


Predicates, variables, functions, constants, connectives, quantifiers

Constants: (first letter small)

bLUE a color

sanTRO a car

crow a bird

Variables: (first letter capital)

Represent general class of objects/properties

Dog: an element that is a dog, but unspecified

Color: an unspecified color

FOL: Basic Elements

Function (evaluates to a constant / variable)Maps Sentences to Objects

• Denote a mapping 4m the domain of function to range of function.

Ali is father of Babar father(babar) = ali

Babar is son of Ali son(ali) = babar

• If you write father(baber), the answer should be ali

• For the above functions the arity is 1 (number of arguments to the function)

• We can have functions with arity > 1, e.g. student(amar, mcs)

FOL: Basic Elements

Functions:

1) ali likes akram likes(ali) = akram

2) atif likes abid likes(atif) = abid

3) Constants to Variables likes(X) = Y

{X,Y} have two possible BINDINGS

{X, Y} could be {ali, akram}

Or

{X,Y} could be {atif, abid}

FOL: Basic Elements

Predicate

Maps Sentences to Truth Values (True/False)

Predicate name a relationship b/w zero or more

objects in the world. (arity)

1) Shahid is student student(shahid)

2) Sana is a girl girl(sana)

3) Father of baber is elder than Hamza elder(father(babar),

hamza)

For 1 and 2 arity is 1 and for 3 the arity is 2

FOL: Basic Elements

Predicate

1) Shahid is a good student

student(shahid,good) or good_student(shahid) or

is_good(shahid,student)

2) Sana is a friend of Saima, Sana and Saima both are

girls

friend_of(sana,saima)^girl(sana)^girl(saima)

3) Bill helps Fred

helps(bill,fred)

FOL: Basic Elements

Connectives:

^ and

v or

~ not

Implication

Quantification

To express sentences like:

All persons can see

There is a person who cannot see

Use:

Universal quantifiers " (ALL)

Existential quantifiers $ (There exists)

FOL: Basic Elements

Complex sentences

• Complex sentences are made from atomic sentences using

connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

sibling(ali,hamza) sibling(hamza,ali)

>(1,2) ≤ (1,2) (1 is greater than 2 or less than equal

to 2)

>(1,2) >(1,2) (1 is greater than 2 and is not

greater than equal to 2)

Universal quantification • "<variables> <sentence>

Every girl at UOG is smart:

"X at(X, uog) smart(X)

• Roughly speaking, equivalent to the conjunction of instantiations of P

at(rabia,uog) smart(rabia)

at(safia,uog) smart(safia)

at(amna,uog) smart(amna)

…

• Typically, is the main connective with ".

• Common MISTAKE: ^ is main connective with ".

• "X at(X, uog) smart(X)

• Every girl is at UOG and every girl is smart.

Existential quantification

• $<variables> <sentence>

• Some boys at UOG are smart:

$X at(X,UOG) smart(X)

• Roughly speaking, equivalent to the disjunction of instantiations of P

• at(amir,UOG) smart(amir)

at(bashir,UOG) smart(bashir)

at(asim,UOG) smart(asim)

...

• Typically, is the main connective with $ .

Properties of quantifiers

• Quantifiers can be nested E.g. "X $Y or $X "Y

• "X "Y is the same as "Y "X

• $X $Y is the same as $Y $X

• $X "Y is not the same as "Y $X

• $X "Y loves(X,Y)

“There is a person who loves everyone in the world”

• "Y $X loves(X,Y)

“Everyone in the world is loved by at least one person”

Properties of quantifiers (Contd.)

• Quantifier duality: each can be expressed using the other

• "X likes(X,car) $X likes(X,car)

• $X likes(X,bread) "X likes(X,bread)

• "X likes(X,car) $X likes(X,car)

• $X likes(X,bread) "X likes(X,bread)

FOPC Inference Rules

• The ability .to infer new correct expressions from a set of true assertion is an important feature of the predicate calculus

• Allow the deduction of new sentences from previously given sentences.

If we know that [all humans are mortal] is true, and that [Socrates is a human] is true than we can conclude that, [Socrates is mortal] In FOPC. e. g., "X human( X) mortal( X) human( Socrates) It should logically follow that: mortal( Socrates)


Some rules of inference

• Here are some examples of sound rules of inference • A rule is sound if its conclusion is true whenever the premise is true

• Each can be shown to be sound using a truth table RULE PREMISE CONCLUSION Modus Ponens A, A B B And Introduction A, B A B And Elimination A B A Double Negation A A Unit Resolution A B, B A Resolution A B, B C A C

Some Useful Inference Rules

29

modus ponens (MP)

If P is true and P Q is true

then Q is true

e. g., If we know that “John is an uncle” is true and that “If John is an uncle then John is male ” is true. Then we can conclude that “John is male” is true.

Let P = John is an uncle

Q = John is male

Hence if P is true and P Q is true

then Q is true.

• This is known as the modus ponens rule, or the implication elimination rule.

inference rules

• Modus Ponens (MP)

• Modus pones can also be applied to expression containing variables.??????????????????

"X (man(X) mortal( X)

man(waleed)

It should logically follow that:

mortal(waleed)

P, PQ

____________

Q

inference rules

Modus Tolens

• If X is true then Y is true. ( X Y)

Y is false. ~Y

Therefore X is false. ~A

• Example

• If there is smoke, there is fire. There is not fire, so there is no smoke.

• If I am happy, then I smile. I am not smiling, therefore I am not happy.

if P Q is true and Q is false or ~Q is true

then ~P is true

e. g., sick( student) not_ attend_ lecture( student)

~not_ attend_ lecture( student)

produces: ~sick( student)

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Inference Rules

Intelligent( Saira) CSMajor (Saira)

___________________________

Intelligent(Saira) ^ CSMajor( Saira)

P Q

__________

P ^ Q

And-Introduction (AI)

Inference Rules

• And-Elimination (AE)

P ^Q

_______

P

Intelligent(Saira) ^ CSMajor( Saira)

___________________________

Intelligent( Saira)

Inference Rules

• Universal Elimination (UE)

• Any universal quantified variable in a true sentence is replaced

by any appropriate term form the domain, the result is a true

sentence.

"x Takes( x , AI) & Intelligent(x)

_______________________

Takes( Pat , AI ) & Intelligent (Pat)

The substitution has to be done by a Ground Term.

Methods of Inference

• Forward Chaining: Starts with the available data and

uses inference rules to extract more data until a goal is

reached.

• Backward Chaining: Starts with a goal and works

backwards from the goal to the facts to see if there is data

available that will support the goal

Forward Chaining

• From "x[man(x) → mortal(x)], we infer that man(Talha) →

mortal(Talha) using UE.

• From man(Talha) and man(Talha) → mortal(Talha), we

infer mortal(Talha) using Modus Ponens.

• We started with the available data and used inference

rules to extract more data until a goal is reached.

Backward Chaining

• We start with goal.

• man(Talha) → mortal(Talha) using UE.

• To show that mortal(Talha), we have to show man(Talha).

• Man(Talha) is given in KB.

• Therefore, mortal(Talha).

• We Started with a goal and worked backwards from the goal to the facts to see if there is data available that will support the goal

Resolution Rule

• Deduction mechanism we discussed above, using the

four rules of inference may be used in practical systems,

but is not feasible. It uses a lot of inference rules that

introduce a large branch factor in the search for a proof.

• An alternative is approach is called resolution, a strategy

used to determine the truth of an assertion, using only

one resolution rule:


A B

~B C

__________

A C

Resolution-Truth Table


Conjunctive Normal Form (CNF)

• Resolution requires all sentences to be converted into a

special form called conjunctive normal form (CNF). A

statement in conjunctive normal form (CNF) consists of

ANDs of ORs.

• The outermost structure is made up of conjunctions. Inner

units called clauses are made up of disjunctions. The

components of a statement in CNF are clauses and

literals. A clause is the disjunction of many units. The units

that make up a clause are called literals. And a literal is

either a variable or the negation of a variable.


Conversion to CNF

• Eliminate arrows (implications)

A B = ~A B

• 2. Drive in negations using De Morgan’s Laws, which are

given below

~(A B) = ~A ~B

• 3. Distribute OR over AND

A (B C) = (A B) (A C)


Example: CNF

(A B) (C D)

1. ~(A B) (~C D)

2. (~A ~B) (~C D)

3. (~A ~C D) (~b ~C D)


Resolution by Refutation

• Now, we will look at a proof strategy called resolution

refutation. The steps for proving a statement using

resolution refutation are:

1. Write all sentences in CNF

2. Negate the desired conclusion

3. Apply the resolution rule until you derive a contradiction

or cannot apply the rule anymore.

4. If we derive a contradiction, then the conclusion follows

from the given axioms

5. If we cannot apply anymore, then the conclusion cannot

be proved from the given axioms


Resolution-Refutation Example 1


References

• Artificial Intelligence, A modern approach by Russell:

(Chapter # 8,9)

• Artificial Intelligence: Structures and Strategies for

Complex Problem Solving (Chapter # 2)

• Internet


End of Lecture


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