Artificial Intelligence Lecture 1-2: Representational Methods - - PDF document

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Artificial Intelligence Lecture 1-2: Representational Methods

CS 231: LUMS Lahore

• Dr. M M Awais
• Propositional Logic and Predicate Logic

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Lecture 1-2

Propositional Logic

• Symbols

P,Q,R,S ……..

• Truth Symbols

True (T), False (F)

• Connectives

(and), (or), (Implication), (Equality, equivalence) (not)

Statements (Propositions) could be true/false FACTS are also called Atomic Proposition

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Truth Table

T T T T F F T T F T T F F F T F F T T T F F T T

P Q

P Q Q P

Q P

Are same 4

Lecture 1-2

Possible Sentences

• P Q

Conjuncts

• P Q

Disjuncts

• P Q

P=Premise/Antecedent Q=conclusion/Consequent

• P
• (P Q) = (P Q)

Inter conversion

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Lecture 1-2

Laws of Propositional Expressions

• Demorgan’s Law (P V Q)=(P Q)
• Distributive Law

P V(Q R)= (P Q) (P R)

• Commutative Law P Q= Q P
• Associative Law (P Q) R=P (Q R)
• Contrapositive Law P Q= P Q

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Lecture 1-2

Inferencing

• If simple facts are known to be true one can

find the truth value for the expressions

• Thus INTERPRETATIONS can be done.
• Interpretation is the assignment of truth

values to the sentences

Symbols T/F Mapping

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Expressions for KR

• Fact 1: Ali likes cakes

P

• Fact 2: Ali eats cakes

Q

• PQ : Ali Likes cakes or eats cakes
• PQ : Ali likes cakes and eats ckes
• Q : Ali does not eat cakes
• PQ: If Ali likes cakes then he eats cakes
• PQ:?????

– Above and vice versa

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Formal Logic Formal Logic

• The most widely used formal logic method

is FIRST-ORDER PREDICATE LOGIC Components :

Alphabets Formal language Axioms Inference Rules

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Alphabets Alphabets-

• I

I

Predicates, variables, functions,constants, connectives, quantifiers, and delimiters Constants: (first letter small) bLUE a color sanTRO a car crow a bird Variables: (first letter capital) Dog: an element that is a dog, but unspecified Color: an unspecified color

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Lecture 1-2

Alphabets Alphabets-

• II

II

Function: father(ali) A function that specifies the unique element, that is the father of Ali killer(X) x is a killer

Have arity ‘n=1’ (number of arguments to the function)

Predicate man(shahid) A predicate which gets TRUTH value equal to 1 (or represented by T) when the interpretation is true. Here Shahid is a man so the predicate is true. bigger(ali , father(babar)) Ali is bigger than the father of Babar.

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Alphabets Alphabets-

• III

III

Connectives: ^ and v or ~ not Implication (when applied to representing logic

consider implication sentences as IF-THEN rules)

Quantification All persons can see There is a person who cannot see Universal quantifiers Existential quantifiers

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Examples Examples

My house is a blue, two -story, with red shutters, and is a corner house blue(my-house)^two-story(my-house)^red-shutters(my- house)^corner(my-house) Ali bought a scooter or a car bought(ali , car) v bought(ali , scooter) IF fuel, air and spark are present the fuel will combust present(spark)^present(fuel)^present(air) combustion(fuel)

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Examples Examples

All people need air ∀X[person(X) need_AIR(X)] The owner of the car also owns the boat [owner(X , car) ^ car(X , boat)] Formulate the following expression in the PC: “Ali is a computer science student but not a pilot or a football player”

cs_STUDENT(ali) ∧ (¬ pilot(ali) ∨ ¬ ft_PLAYER(ali) ) 14

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Examples Examples

Restate the sentence in the following way:

• 1. Ali is a computer science (CS) student
• 2. Ali is not a pilot
• 3. Ali is not a football player

cs_student(ali)^ ~pilot(ali)^ ~football_player(ali)

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Examples Examples

Studying fuzzy systems is exciting and applying logic is great fun if you are not going to spend all of your time slaving

• ver the terminal

∀ X(~slave_terminal(X)

[fs_eciting(X)^logic_fun(X)]) Every voter either favors the amendment or despises it

∀X[voter(X) [favor(X , amendment) v

despise(X,amendment)] ^ ~[favor(X , amendment) v despise(X , amendment)] (this part simply endorses the statement, may not be required) 16

Lecture 1-2

Undecidable Predicate

• For which exhaustive testing is required
• Example:
• ∀X likes(zahra, X)
• This sentence is computationally impossible

to calculate

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Example: Robotic Arm

• Represent the initial details of the systems
• Generate sentences of descriptive nature

and or implicative nature

• Modify the facts using new sentences

A B C D

• n(b,a)

On( c,d)

• ntable(b)
• ntable(d)

clear(a) clear(c) hand_empty 18

Lecture 1-2

Definitions

• Logically Follows: X logically follows from a set
• f predicate calculus expressions S if every

interpretation that satisfy S also satisfy X (X F S)

• Satisfy: If S has value T under interpretation I

then I satisfy S (interpretation that makes the sentence true)

• Model: If I satisfies S for all variables then I is a

model of S

• Satisfiable: S is satisfiable iff there exists an

interpretation and variable assignments that satisfy it

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S F X

S: All birds fly S: Sparrow is bird X: Sparrow flies All humans are mortal Shahid is a human Shahis is mortal All birds fly Sparrow is bird Sparrow flies

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Definitions

• Inconsistent: If a set of expressions are not satisfiable
• Valid:If any expression has a value T for all

possible interpertations

• Sound: If an expression logically follows from

another expression then the inferential rule is sound

• Complete: When inferential rule produces every

expression that logically follows a particular expression

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Operations

• Unification: Algorithm for determining the

subitutions needed to make two predicate calculus expressions match

• Skolemization: A method of removing or

replacing existential quantifiers

• Composition: If S and S` are two substitutions

sets, then the composition of S and S` (SS`) is

• btained by applying the elements of S to the

elements of S` and finally adding the results