Artificial IntelligenceLecture No. 10

Dr. Asad Ali Safi

Assistant Professor,Department of Computer Science,

COMSATS Institute of Information Technology (CIIT) Islamabad, Pakistan.

Summary of Previous Lecture

• A knowledge-based agent• The Wumpus World• Syntax , semantics• Entailment• Logic as a KR language

Today’s Lecture

• Logic• Propositional logic• Pros and cons of propositional logic• First-order logic• Syntax of FOL: Basic elements• Atomic/complex sentences• Connections between Quantifiers

4

No independent access to the world • The reasoning agent often gets its knowledge about

the facts of the world as a sequence of logical sentences and must draw conclusions only from them without independent access to the world.

• Thus it is very important that the agent’s reasoning is sound!

LANGUAGES

What is logic?

• We can also think of logic as an “algebra” for manipulating only two values: true (T) and false (F)

• We will cover:– Propositional logic--the simplest kind

Propositional logic• Propositional logic consists of:

– The logical values true and false (T and F)– Propositions: “Sentences,” which

• Are atomic (that is, they must be treated as indivisible units, with no internal structure), and

• Have a single logical value, either true or false

– Operators, both unary and binary; when applied to logical values, yield logical values

• The usual operators are and, or, not, and implies

Propositional logic: Syntax• Propositional logic is the simplest logic – illustrates basic

ideas• The proposition symbols P1, P2 etc are sentences

– If S is a sentence, ¬S is a sentence (negation, not)– If S1 and S2 are sentences, S1 S2 is a sentence (conjunction, ∧

AND)– If S1 and S2 are sentences, S1 S2 is a sentence (disjunction, ∨

OR)– If S1 and S2 are sentences, S1 S2 is a sentence (implication, ⇒

IMPLIES)– If S1 and S2 are sentences, S1 S2 is a sentence ⇔

(biconditional)

Truth tables• Logic, like arithmetic, has operators, which apply to

one, two, or more values (operands)• A truth table lists the results for each possible

arrangement of operands– Order is important: x op y may or may not give the

same result as y op x

• The rows in a truth table list all possible sequences of truth values for n operands, and specify a result for each sequence– Hence, there are 2n rows in a truth table for n

operands

Unary operators• There are four possible unary operators:

Only the last of these (negation) is widely used (and has a symbol,¬ ,for the operation

X Constant T Constant F

Identity ¬X

T T F T F

F T F F T

Useful binary operators• Here are the binary operators that are traditionally used:

Notice in particular that material implication (⇒) only approximately means the same as the English word “implies”

Any other binary operators can be constructed from a combination of these (along with unary not, ¬)

X YANDX ∧ Y

ORX ∨ Y

IMPLIESX ⇒ Y

BICONDITIONAL

X ⇔ YT T T T T T

T F F T F F

F T F T T F

F F F F T T

Logical expressions• All logical expressions can be computed with some combination

of and (∧), or (∨ ), and not (¬) operators• For example, logical implication can be computed this way:

Notice that ¬X ∨ Y is equivalent to X ⇒ Y

X Y ¬X ¬X ∨ Y X ⇒ Y

T T F T T

T F F F F

F T T T T

F F T T T

Pros and cons of propositional logic

Propositional logic is declarative

Propositional logic allows partial/disjunctive/negated information(unlike most data structures and databases)

–– Propositional logic is compositional:

meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2

–– Meaning in propositional logic is context-independent

(unlike natural language, where meaning depends on context)–– Propositional logic has very limited expressive power

– (unlike natural language)– E.g., cannot say "pits cause breezes in adjacent squares“

• except by writing one sentence for each squar

First-order logic• While 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, …

–– Relations: red, round, prime, brother of, bigger than, part of, comes

between, …

– Functions: father of, best friend, one more than, plus, …–“Evil King John ruled England in 1200.”– Object: John, England, 1200; Relation; ruled; Properties: evil, king

Logics in General

• Ontological Commitment: – What exists in the world — TRUTH– PL : facts hold or do not hold.– FOL : objects with relations between them that hold or do

not hold

• Epistemological Commitment: – What an agent believes about facts — BELIEF

Syntax of FOL: Basic elements

• Constant Symbols:– Stand for objects– e.g., KingJohn, 2, UCI,...

• Predicate Symbols– Stand for relations– E.g., Brother(Richard, John), greater_than(3,2)...

• Function Symbols– Stand for functions– E.g., Sqrt(3), LeftLegOf(John),...

Syntax of FOL: Basic elements• Constants KingJohn, 2, UCI,...

• Predicates Brother, >,...

• Functions Sqrt, LeftLegOf,...

• Variables x, y, a, b,...

• Connectives , , , ,

• Equality =

• Quantifiers ,

Relations

• Some relations are properties: they state some fact about a single object: Round(ball),

Prime(7).

• n-ary relations state facts about two or more objects: Married(John,Mary), LargerThan(3,2).

• Some relations are functions: their value is another object: Plus(2,3), Father(Dan).

Atomic sentencesTerm = function (term1,...,termn) or constant or variable• A logical expression that refers to an object

– LeftLegOf(Richard)

• There are 2 kinds of terms:– constant symbols: Table, Computer– function symbols: LeftLeg(Pete), Sqrt(3), Plus(2,3) etc

Atomic sentence = predicate (term1,...,termn) or term1 = term2

• An Atomic sentence is formed from a predicate symbol followed by list of terms. • Examples: LargerThan(2,3) is false. Brother_of(Mary,Pete) is false.

Married(Father(Richard), Mother(John)) could be true or false

• Note: Functions do not state facts and form no sentence: – Brother(Pete) refers to John (his brother) and is neither true nor false.

• Brother_of(Pete,Brother(Pete)) is True. Binary relation Function

Complex Sentences• We make complex sentences with connectives

(just like in propositional logic).( ( ), ) ( ( ))Brother Lef tLeg Richard J ohn Democrat Bush

binary relation

function

property

objects

connectives

More Examples• Brother(Richard, John) Brother(John, Richard)

• King(Richard) King(John)

• King(John) => King(Richard)

• LessThan(Plus(1,2) ,4) GreaterThan(1,2)

(Semantics are the same as in propositional logic)

Variables

• Person(John) is true or false because we give it a single argument ‘John’

• We can be much more flexible if we allow variables which can take on values in a domain. e.g., all persons x, all integers i, etc.– E.g., can state rules like Person(x) => HasHead(x) or Integer(i) => Integer(plus(i,1)

Universal Quantification • means “for all”

• Allows us to make statements about all objects that have certain properties

• Can now state general rules:

x King(x) => Person(x)

x Person(x) => HasHead(x) " i Integer(i) => Integer(plus(i,1))

Note that " x King(x) Person(x) is not correct! This would imply that all objects x are Kings and are People

x King(x) => Person(x) is the correct way to say this

Existential Quantification • x means “there exists an x such that….” (at least one object x)

• Allows us to make statements about some object without naming it

• Examples:

x King(x)

x Lives_in(John, Castle(x))

i Integer(i) GreaterThan(i,0)

Note that is the natural connective to use with

(And => is the natural connective to use with )

Combining Quantifiers

x y Loves(x,y) – For everyone (“all x”) there is someone (“y”) who

loves them

$ y x Loves(x,y) - there is someone (“y”) who loves everyone

Clearer with parentheses: y ( x Loves(x,y) )

Connections between Quantifiers

• Asserting that all x have property P is the same as asserting that does not exist any x that don’t have the property P

x Likes(x, 271 class) x Likes(x, 271 class)

In effect: - is a conjunction over the universe of objects - is a disjunction over the universe of objects Thus, DeMorgan’s rules can be applied

De Morgan’s Law for Quantifiers( )

( )

( )

( )

x P x P

x P x P

x P x P

x P x P

( )

( )

( )

( )

P Q P Q

P Q P Q

P Q P Q

P Q P Q

De Morgan’s Rule Generalized De Morgan’s Rule

Rule is simple: if you bring a negation inside a disjunction or a conjunction,always switch between them (or and, and or).

Summery of Today’s Lecture• logic• Propositional logic• Pros and cons of propositional logic• First-order logic• Syntax of FOL: Basic elements• Atomic/complex sentences• Connections between Quantifiers