Introduction to Artificial Intelligence

Introduction to Artificial Intelligence First-order Logic (Logic, Deduction, Knowledge Representation) Bernhard Beckert

NIVERSITÄT

OBLENZ

ANDAU Winter Term 2004/2005

-L

Why first-order logic? Syntax and semantics of first-order logic Fun with sentences Wumpus world in first-order logic

Propositional logic is declarative : pieces of syntax correspond to facts

Propositional logic is declarative : pieces of syntax correspond to facts Propositional logic allows partial / disjunctive / negated information (unlike most data structures and databases)

B B P

meaning of is derived from meaning of and of _{, ,} ∧ P _{, ,}

B B P

meaning of is derived from meaning of and of _{, ,} ∧ P _{, ,}

2 Meaning in propositional logic is context-independent

(unlike natural language, where meaning depends on context)

B B P

meaning of is derived from meaning of and of _{, ,} ∧ P _{, ,}

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)

Example:

Cannot say “pits cause breezes in adjacent squares” except by writing one sentence for each square

Propositional logic

Assumes that the world contains facts

Propositional logic

Assumes that the world contains facts

First-order logic

Assumes that the world contains

Objects people, houses, numbers, theories, Donald Duck, colors, centuries, . . .

Propositional logic

Assumes that the world contains facts

First-order logic

Assumes that the world contains

Objects people, houses, numbers, theories, Donald Duck, colors, centuries, . . . Relations . . .

red, round, prime, multistoried, brother of, bigger than, part of, has color, occurred after, owns, . . .

Propositional logic

Assumes that the world contains facts

First-order logic

Assumes that the world contains

Objects people, houses, numbers, theories, Donald Duck, colors, centuries, . . . Relations . . .

red, round, prime, multistoried, brother of, bigger than, part of, has color, occurred after, owns, . . .

Functions

, middle of, father of, one more than, beginning of,

. . .

Symbols Constants KingJohn, 2, Koblenz, C, . . .

Predicates Brother, >, =, . . . Functions Sqrt, LeftLegOf , . . . Variables x, y, a, b, . . . Connectives

∧ ∨ ¬ ⇒ ⇔

Quantifiers

∀ ∃

Symbols Constants KingJohn, 2, Koblenz, C, . . .

Predicates Brother, >, =, . . . Functions Sqrt, LeftLegOf , . . . Variables x, y, a, b, . . . Connectives

∧ ∨ ¬ ⇒ ⇔

Quantifiers

∀ ∃

Note

The equality predicate is always in the vocabulary It is written in infix notation

Atomic sentence predicate

(

term

, . . . ,

term ⁿ

) or

term

Atomic sentence predicate

term

constant

, . . . , term ⁿ ) or

term

(

function

(

term

) or

term ⁿ

, . . . ,

term

variable

Example Brother

( KingJohn, RichardTheLionheart )

Example Brother | {z } predicate

( KingJohn

| {z } constant

, RichardTheLionheart

| {z } constant

)

Example Brother , RichardTheLionheart

( KingJohn )

| {z } | {z } | {z }

constant constant

predicate

| {z } | {z }

term term

Example Brother | {z } predicate

( KingJohn

| {z } constant

| {z } term

, RichardTheLionheart

| {z } constant

| {z } term

)

| {z }

atomic sentence

Example

> ( Length(LeftLegOf (Richard)), Length(LeftLegOf (KingJohn)) )

Example

| {z } predicate

( Length

| {z } function

(LeftLegOf

| {z } function

(Richard

| {z } constant

)), Length

| {z } function

(LeftLegOf

| {z } function

(KingJohn

| {z } constant

)) )

Example

> , Length

( Length (LeftLegOf (Richard )) (LeftLegOf (KingJohn )) )

|{z} | {z } | {z } | {z } | {z } | {z } | {z } predicate function function constant function function constant

| {z } | {z } term term

Example

> , Length

( Length (LeftLegOf (Richard )) (LeftLegOf (KingJohn )) )

|{z} | {z } | {z } | {z } | {z } | {z } | {z } predicate function function constant function function constant

| {z } | {z } term term

}

atomic sentence

Built from atomic sentences using connectives

S S S S

¬S ∧ S ∨ S ⇒ S ⇔ S

(as in propositional logic)

Built from atomic sentences using connectives

S S S S

¬S ∧ S ∨ S ⇒ S ⇔ S

(as in propositional logic)

Example Sibling

Sibling

( KingJohn, Richard ) ⇒ ( Richard, KingJohn )

Built from atomic sentences using connectives

S S S S

¬S ∧ S ∨ S ⇒ S ⇔ S

(as in propositional logic)

Example Sibling , Richard Sibling , KingJohn

( KingJohn ) ⇒ ( Richard )

| {z } | {z } | {z } | {z } | {z } | {z } term term term term predicate predicate

Built from atomic sentences using connectives

S S S S

¬S ∧ S ∨ S ⇒ S ⇔ S

(as in propositional logic)

Example Sibling , Richard Sibling , KingJohn

( KingJohn ) ⇒ ( Richard )

| {z } | {z } | {z } | {z } | {z } | {z } term term term term predicate predicate

| {z } | {z }

atomic sentence atomic sentence

Built from atomic sentences using connectives

| {z } term

| {z } atomic sentence

)

| {z } term

, KingJohn

| {z } term

( Richard

Sibling | {z } predicate

⇒

| {z } atomic sentence

)

| {z } term

, Richard

( KingJohn

¬S

Example Sibling | {z } predicate

(as in propositional logic)

⇔ S

2 S

⇒ S

2 S

∨ S

∧ S

}

complex sentence

Models of first-order logic

Sentences are true or false with respect to models, which consist of – a domain (also called universe) – an interpretation

Models of first-order logic

Sentences are true or false with respect to models, which consist of – a domain (also called universe) – an interpretation

Domain

A non-empty (finite or infinite) set of arbitrary elements

Models of first-order logic

Sentences are true or false with respect to models, which consist of – a domain (also called universe) – an interpretation

Domain

A non-empty (finite or infinite) set of arbitrary elements

Interpretation

Assigns to each – constant symbol: a domain element – predicate symbol: a relation on the domain (of appropriate arity) – function symbol: a function on the domain (of appropriate arity)

Definition

An atomic sentence , . . . , n

predicate term term

( )

is true in a certain model (that consists of a domain and an interpretation) iff _n the domain elements that are the interpretations of term , . . . , term

are in the relation that is the interpretation of predicate

Definition

An atomic sentence , . . . , n

predicate term term

( )

is true in a certain model (that consists of a domain and an interpretation) iff _n the domain elements that are the interpretations of term , . . . , term

are in the relation that is the interpretation of predicate The truth value of a complex sentence in a model is computed from the truth-values of its atomic sub-sentences in the same way as in propositional logic

Constants: KingJohn , Richard Predicates:

person

king

crown

Functions:

brother

head

left

leg

Syntax

∀

variables sentence

Syntax variables sentence

∀

Example

“Everyone studying in Koblenz is smart:

∀ (StudiesAt(x, Koblenz) ⇒ Smart(x))

|{z} | {z } sentence variables

Semantics

is true in a model ∀xP iff

for all domain elements in the model:

P is true in the model when x is interpreted by d

Semantics

is true in a model ∀xP iff

for all domain elements in the model:

P is true in the model when x is interpreted by d Intuition

is roughly equivalent to the conjunction of all instances of P ∀x P

Semantics

is true in a model ∀xP iff

for all domain elements in the model:

P is true in the model when x is interpreted by d Intuition

is roughly equivalent to the conjunction of all instances of P ∀x P

Example

equivalent to: ∀ x StudiesAt(x, Koblenz) ⇒ Smart(x)

StudiesAt

(KingJohn, Koblenz) ⇒ Smart(KingJohn) ∧ StudiesAt(Richard, Koblenz) ⇒ Smart(Richard) ∧ StudiesAt(Koblenz, Koblenz) ⇒ Smart(Koblenz) ∧ . . .

Note

is the main connective with ⇒

∀

Common mistake

Using as the main connective with ∧

∀

Note

is the main connective with ⇒

∀

Common mistake

Using as the main connective with ∧

∀

Example Correct: Smart

∀x (StudiesAt(x, Koblenz) ⇒ (x)) “Everyone who studies at Koblenz is smart”

Note

is the main connective with ⇒

∀

Common mistake

Using as the main connective with ∧

∀

Example Correct: Smart

∀x (StudiesAt(x, Koblenz) ⇒ (x)) “Everyone who studies at Koblenz is smart”

Smart Wrong:

∀x (StudiesAt(x, Koblenz) ∧ (x)) “Everyone studies at Koblenz and is smart”, i.e., “Everyone studies at Koblenz and everyone is smart”

Syntax

∃

variables sentence

Syntax variables sentence

∃

Example

“Someone studying in Landau is smart:

∃ (StudiesAt(x, Landau) ∧ Smart(x))

|{z} | {z } sentence variables

Semantics

is true in a model ∃xP iff

there is a domain element in the model such that:

P is true in the model when x is interpreted by d

Semantics

is true in a model ∃xP iff

there is a domain element in the model such that:

P is true in the model when x is interpreted by d Intuition

is roughly equivalent to the disjunction of all instances of P ∃x P

Semantics

is true in a model ∃xP iff

there is a domain element in the model such that:

P is true in the model when x is interpreted by d Intuition

is roughly equivalent to the disjunction of all instances of P ∃x P

Example

equivalent to: ∃ x StudiesAt(x, Landau) ∧ Smart(x)

StudiesAt

(KingJohn, Landau) ∧ Smart(KingJohn) ∨ StudiesAt(Richard, Landau) ∧ Smart(Richard) ∨ StudiesAt(Landau, Landau) ∧ Smart(Landau) ∨ . . .

Note

is the main connective with ∧

∃

Common mistake

Using as the main connective with ⇒

∃

Note

is the main connective with ∧

∃

Common mistake

Using as the main connective with ⇒

∃

Example Correct: Smart

∃x (StudiesAt(x, Landau) ∧ (x)) “There is someone who studies at Landau and is smart”

Note

is the main connective with ∧

∃

Common mistake

Using as the main connective with ⇒

∃

Example Correct: Smart

∃x (StudiesAt(x, Landau) ∧ (x)) “There is someone who studies at Landau and is smart”

Smart Wrong:

∃x (StudiesAt(x, Landau) ⇒ (x)) “There is someone who, if he/she studies at Landau, is smart” This is true if there is anyone not studying at Landau

Quantifiers of same type commute

is the same as ∀x∀y

∀y∀x is the same as

∃x∃y ∃y∃x

Quantifiers of different type do NOT commute

is not the same as ∃x∀y

∀y∃x

Example

∃x∀yLoves(x, y) “There is a person who loves everyone in the world” ∀y∃xLoves(x, y) “Everyone in the world is loved by at least one person” (Both hopefully true but different)

Quantifiers of different type do NOT commute

is not the same as ∃x∀y

∀y∃x

Example

∀x∃yMother(x, y) “Everyone has a mother” (correct) ∃y∀xMother(x, y) “There is a person who is the mother of everyone” (wrong)

Quantifier duality

is the same as ∀xLikes(x, IceCream)

¬∃x¬Likes(x, IceCream) is the same as

∃xLikes(x, Broccoli) ¬∀x¬Likes(x, Broccoli)

“Brothers are siblings” ∀x, y (Brother(x, y) ⇒ Sibling(x, y))

“Brothers are siblings” ∀x, y (Brother(x, y) ⇒ Sibling(x, y)) “Sibling” is symmetric ∀x, y (Sibling(x, y) ⇔ Sibling(y, x))

“Brothers are siblings” ∀x, y (Brother(x, y) ⇒ Sibling(x, y)) “Sibling” is symmetric ∀x, y (Sibling(x, y) ⇔ Sibling(y, x)) “One’s mother is one’s female parent” ∀x, y (Mother(x, y) ⇔ (Female(x) ∧ Parent(x, y))) “A first cousin is a child of a parent’s sibling” ∀x, y (FirstCousin(x, y) ⇔ ∃p, ps (Parent(p, x) ∧ Sibling(ps, p) ∧ Parent(ps, y)))

Semantics term is true under a given interpretation

= term

if and only if

term and term have the same interpretation

Example

Definition of (full) sibling in terms of Parent ∀x, y Sibling(x, y) ⇔ (¬(x = y) ∧

∃m, f (¬(m = f ) ∧

Parent

(m, x) ∧ Parent( f , x) ∧

Parent

(m, y) ∧ Parent( f , y)))

Important notions

– validity – satisfiability – unsatisfiablity – entailment are defined for first-order logic in the same way as for propositional logic

Important notions

– validity – satisfiability – unsatisfiablity – entailment are defined for first-order logic in the same way as for propositional logic

Calculi

There are sound and complete calculi for first-order logic (e.g. resolution) Whenever KB , it is also true that KB

⊢ α |= α Whenever KB , it is also true that KB

|= α ⊢ α But these calculi CANNOT decide validity, entailment, etc.

In propositional logic

Validity, satisfiability, unsatisfiablity are decidable

In first-order logic

The set of valid, and the set of unsatisfiable formulas are enumerable The set of satisfiable formulas is NOT EVEN enumerable

Introduction to Artificial Intelligence