Biểu diễn tri thức nhờ logic vị từ bậc một

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Limitations of Propositional Logic 2

\ Can't directly talk about properties of individuals

or relations between individuals

u E.g., how to represent the fact that John is tall?

\ We have no way to conclude that John is good at

Predicate Logic Overview

\ Extensions and Variations

\ Proof in Predicate Logic

\ Important Concepts and Terms

Delimiters

, ( )

Delimiters , ( )

Wff

P∧ Q → R

Wff P∧ Q → R

\ Constants are used to name existing objects:

u The interpretation identifies the object in the real world

u No constant can name more than one object

u An object can have more than one name or no name at all

BNF Grammar Predicate Logic

<Sentence> → <AtomicSentence>

| (<Sentence> <Connective> <Sentence>)

| <Quantifier> <Variable>, <Sentence>

<Constant> → a, b, c, max, carl, jim, jack

<Variable> → A, B, C, X1 , X2, COUNTER, POSITION

<Function> → father-of, square-position, sqrt, cosine

<Predicate> → P, Q, LARGER, BETWEEN, YOUNGER-THAN

Ambiguities are resolved through precedence or parentheses

First Order Predicate Logics Syntax

term ::= variable

| function_symbol_of_arity_n(t1, …, tn) n>0

| function_symbol_of_arity_0 constantatom ::= predicate_symbol_of_arity_n(t1, …, tn) n>0

| predicate_symbol_of_arity_0 constantliteral ::= atom positive literal

| ¬ atom negative literalwff ::= atom well formed formula (sentence)

| (∀ variable wff) universal formula

| (∃ variable wff) existential formula

Function Symbols

\ Function symbols

u function_name(arg1, arg2, …, argn)

u Identifies the object referred to by a tuple of objects

u May be defined implicitly through other functions, or explicitly

through tables

\ Function names begin with a lowercase letter or are

expressed with a symbol

u F = {f, g, h, …} = F0 ∪ F1 ∪ F2 ∪ …

\ Function arities:

u F0: function symbols of arity 0 (constants): a, b, max, jim

u F1: function symbols of arity 1 (one argument)

u F2: function symbols of arity 2 (two arguments)

Functions Examples

\ A function is used to express complex names

u password(claire) Claire’s password

\ A function may be nested

u Max’s age’s double double(age(max))

u father(mother(max)) Max’s mother’s father

u starship(son(dr_crusher)) Dr_Crusher’s son’s starship

\ A function is never a predicate

u Can’t nest predicates TALL(TALL(max))

\ Function symbols of arity >1

u youngestChild(max, ann) Max and Ann’s youngest child

\ A predicate forms a sentence,

while a function names an individual

\ Ví d các hàm sau đây đ m sau đây đ u là các h ng :

u successor(X, Y), weight(b), successor(b, wight(Z))

\ Nh ng các hàm sau đây không ph m sau đây không ph i là h ng :

u P(X, blue) vì P là v t

u weight (P(b))

vì P(b) không ph i là h ng (v t không làm đ m đ i cho hàm)

Predicates

\ Predicate symbols :

u PREDICATE(arg1, arg2, …, argn)

u A (determinate) property possessed by an object: Shape, Size

u A (determinate) relationship among objects:

Shape relationship, size relationship, positional relationship…

u The number of arguments is called the predicate’s arity

u The order of the arguments is important

\ Predicates have names beginning with an uppercase letter

or are represented by an operator symbol

u P = P0 ∪ P1 ∪ P2 ∪ …

\ Predicate arities:

P0: predicate symbols of arity 0 (constants: proposition) : P, Q, R, …

P1: predicate symbols of arity 1 (one argument)

P2: predicate symbols of arity 2 (two arguments) …

\ Ví d các v t sau đây l sau đây l à các nguyên t :

u P(X, blue), EMPTY, BETWEEN(table, X, sill(window))

\ Còn :

u successor (X, Y, sill (window)

không ph i nguyên t , mà là các hàm

Atomic Sentences

\ A atomic sentence:

u Expresses a claim that is either true or false

u Formed by a single predicate followed by one or more arguments

Well-formed Formula (wff)

\ Any atomic sentence is a wff

\ If A are B are wffs then so are

Equality

\ Equality indicates that two terms refer to the same object

=(A, B)

u A and B are identical

u Usually, written in infix form A = B

u The equality symbol “=“ is an (in-fix) shorthand

u FATHER(jane) = jim, or =(FATHER(jane), jim)

u E.g Jim is Jane’s and John’s father

\ Equality by reference and equality by value

u Sometimes the distinction between referring to the same object and referring to two objects that are identical (indistinguishable) can be important

\ E ≠ E E is not identical to iteself

m t nguyên t hay có d ng (¬G), v i G là m t nguyên t

u Trong m t CTC, tr c ho c sau các ký t n i, ký t phân

cách, các h ng, các bi n, các hàm, các v t , ng i ta có th

đ t tùy ý các d u cách (space hay blank)

u Cu Tý quê Hà T nh : QUÊ(cutý, hàt nh), QUÊ(X, Y)

u Cu Tý xa quê Hà T nh : XAQUÊ(cutý, hàt nh), XAQUÊ(X, Y)

Predicate Logics: some terminology

\ There is a predicate logic for each basis B=(F, P)

of function and predicate symbols

\ Terms formed on basis B are called B-terms:

the set of all B-terms is denoted TB

\ Formulas formed on basis B are called B-formulas:

the set of all B-formulas is denoted WFFB

\ Formulas with all variables bound to a quantifier

are called closed formulas

\ Formulas with no quantifier

are called quantifier free formulas

\ Formulas with no quantifier and no variable

are called ground formulas

\ T nay ta quy c r ng, trong m t CTC :

u M t bi n đ n đ c l ng t hóa s xu t hi n ngay sau ∃ hay ∀

u Ph m vi l ng t hóa c a bi n k t v trí xu t hi n tr đi đi

M t s nh n xét 2

\ Tri th c di n t theo ngôn ng t nhiên hay toán h c

không ph i luôn luôn d dàng chuy n đ n đ i thành các CTC trong lôgic v t b c m t

\ Ch ng h n, đ đ di n t r ng :

«N u hai v t y chang nhau thì chúng có cùng tính ch t»,

ng i ta có th vi t :

(∀P) (∀X) (∀Y) (EQUAL(X, Y) → (P(X) ↔ P(Y)))

\ Nh ng bi u th c trên không ph i là logic v t b c m t

vì có l ng t ∀ áp d ng cho m t ký t v t là P

\ Trong lôgic v t b c m t, s ki n trên đ n trên đ c vi t :

(∀X) (∀Y) (SAME_P(X, Y) → (P(X) ↔ P(Y))), ho c

(∀X) (∀Y) (SAME_P(X, Y) → (HAVE(X, p) ↔ HAVE (Y, p)))

Example: Family Relationships

\ Objects: people

\ Properties: gender, …

u Expressed as unary predicates:

MALE(X), FEMALE(Y) ho c SEX(X, male), SEX(Y, female)

\ Relations: parenthood, brotherhood, marriage

u Expressed through binary predicates

PARENT(X, Y), BROTHER(X, Y), …

\ Functions: motherhood, fatherhood

u Because every person has exactly one mother and one father:

mother(X), father(Y)

u There may also be a relation

mother-of(X, Y), father-of(X, Y)

Atomic Sentences Translation

Examples Atomic Sentences

FATHER(jack, john), MOTHER(jill, john), SISTER(jane, john)

PARENTS(jack, jill, john, jane)

PARENTS(jack, jill, john, jane) ∧ MARRIED(jack, jill)

PARENTS(jack, jill, john, jane) → MARRIED(jack, jill)

OLDER-THAN(jane, john) ∨ OLDER-THAN(john, jane)

OLDER(father-of(john), 30) ∨ OLDER (mother-of(john), 20)

\ The universal quantifier, represented by the symbol ∀

means “for every” or “for all”

\ The existential quantifier, represented by the symbol ∃

means “there exists”

\ Limitations of predicate logic – most quantifier

Universal Quantifiers ∀ (for all)

Usage of Universal Qualification

\ Universal quantification is frequently used to make

statements like:

u “all humans are mortal”

u “all cats are mammals”

u “all birds can fly” …

\ This can be expressed through sentences like

Existential Quantifier ∃ (there exists)

\ ∃X P(x) states that

u a predicate P holds for some objects in the universe

u The sentence is true if and only if there is at least one true

individual sentence where the variable X is replaced by the

individual objects it can stand for

Usage of Existential Quantification

\ Existential quantification is used to make statements like:

u “some humans are computer scientists”

u “john has a sister who is a computer scientist”

u “some birds can’t fly” …

\ This can be expressed through sentences like

Multiple Quantifiers

\ More complex sentences can be formulated by multiple

variables and by nesting quantifiers

u The order of quantification is important

u Variables must be introduced by quantifiers, and belong to the

innermost quantifier that mention them

\ Examples

∀X, ∀Y PARENT(X, Y) → CHILD(Y, X)

∀X HUMAN(X) ∧ ∃Y MOTHER(Y, X) ⇒ ∀X ∃Y HUMAN(X) ∧ MOTHER(Y, X)

∀X HUMAN(X) ∧ ∃Y LOVES(X, Y) ⇒ ∀X ∃Y HUMAN(X) ∧ LOVES(X, Y)

∃X HUMAN(X) ∧ ∀Y LOVES(X, Y) ⇒ ∀X ∀Y HUMAN(X) ∧ LOVES(X, Y)

∃X HUMAN(X) ∧ ∀Y LOVES(Y, X) ⇒ ∃X ∀Y HUMAN(X) ∧ LOVES(Y, X)

\ They can translate to the form:

V: Variable

\ All statements made with one quantifier can be converted into equivalent statements with the other quantifier by

using negation

∀ is a conjunction over all objects under discourse

∃ is a disjunction over all objects under discourse

\ De Morgan’s rules apply to quantified sentences

\ Strictly speaking, only one quantifier is necessary

u Using both is more convenient

Family Relationships

\ One’s mother is one’s sibling’s mother

∀M, C, D MOTHER(M, C) ∧ SIBLING(C, D) → MOTHER(M, D)

\ Brothers are siblings

∀X, Y BROTHER(X, Y) → SIBLING(X, Y)

\ Sibling is transitive

∀X, Y, Z SIBLING(X, Y) ∧ SIBLING(Y, Z) → SIBLING(X, Z)

\ One’s mother is one’s sibling’s mother

∀M, C MOTHER(M, C) ∧ SIBLING(C, D) → MOTHER(M, D)

\ A first cousin is a child of a parent’s sibling

¬(X=Y) ∧ ∃P PARENT(P, X) ∧ PARENT(P, Y)

Ng i này là m c a ng i kia kh m t ng i là ph n và là thân sinh c a ng i kia

Ng i này là m c a ng i kia kh m t ng i là ph n và là thân sinh c a ng i kia

Ng i này là ch ng c a ng i kia kh ng i này là đàn ông và ng i kia là v c a ng i này

Ng i này là ch ng c a ng i kia kh ng i này là đ đ àn ông và ng i kia là v c a ng i này

Translating English to FOL

\ Every gardener likes the sun

∀X GARDENER(X) → LIKES(X, sun)

\ You can fool some of the people all of the time

∃X ∀T (PERSON(X) ∧ TIME(T)) → CAN-FOOL(X, T)

\ You can fool all of the people some of the time

∀X ∃T (PERSON(X) ∧ TIME(T) → CAN-FOOL(X, T)

\ All purple mushrooms are poisonous

∀X (MUSHROOM(X) ∧ PURPLE(X)) → POISONOUS(X)

\ No purple mushroom is poisonous

¬(∃X) PURPLE(X) ∧ MUSHROOM(X) ∧ POISONOUS(X)

or, equivalently,

(∀X) (MUSHROOM(X) ∧ PURPLE(X)) → ¬POISONOUS(X)

Translating English to FOL…

\ There are exactly two purple mushrooms

(∃X)(∃Y) MUSHROOM(X) ∧ PURPLE(X) ∧ MUSHROOM(Y) ∧ PURPLE(Y)

∧ ¬(X=Y) ∧ (∀Z) (MUSHROOM(Z) ∧ PURPLE(Z))

another starting with X and ending with Y

(∀X)(∀Y) ABOVE(X, Y) ↔ (ON(X, Y) ∨ (∃Z) (ON(X, Z) ∧ ABOVE(Z, Y)))

Colonel West Example

\ Assume:

u It is a crime for an American to sell weapons to a hostile nation.The country Nono, an enemy of America, has some missiles, and all its missiles were sold to it by Colonel West, who is an American

u Constants: nono, america, west

u Predicates: CRIMINAL, AMERICAN, WEAPON, HOSTILE, NATION, ENEMY, MISSILE, OWNS, SELLS

\ Then

u It is a crime for an American to sell weapons to a hostile nation

⇒ If any american X sells any weapon Y to any hostile nation Z, then that american X is a criminal

∧ HOSTILE(Z) ∧ SELLS(X, Z, Y) → CRIMINAL(X))

Other Translating

\ Nono’s missiles

u Nono has some missiles

u All of Nono’s missiles were sold to it by West

⇒ If X is a missile owned by Nono then West sold X to Nono

\ The other facts

u An enemy of America is hostile

u West is an American

u Nono is an enemy of America

∃ X (MISSILE(X) ∧ OWNS(nono, X))

∀ X ((MISSILE(X) ∧ OWNS(nono, X)) → SELLS(west, nono, X))

∀ X (ENEMY(X, america) → HOSTILE(X))

AMERICAN(west)

NATION(nono) ∧ NATION(america) ∧ ENEMY(nono, america)

Other Examples

\ Love

u There is a girl who is loved by every boy

⇒ There is a girl X and if Y is a boy then Y loves her

u Every boy loves some girl

u For every boy Y there exists a girl X that he loves

\ Socrates Example

u All men are mortal

u Socrates is a man

u Socrates is mortal

∃X (GIRL(X) ∧ ∀Y(BOY(Y) → LOVES(Y,X)))

∀Y(BOY(Y) → ∃X (GIRL(X) ∧ LOVES(Y,X)))

∀X (MAN(X) → MORTAL(X))

MAN(socrates)

MORTAL(socrates)

Curiosity Example

\ Assume:

Jack owns a dog

Every dog owner is an animal lover

No animal lover kills an animal

Either Jack or Curiosity killed the cat

The cat’s name is Tuna

Then

u Constants: jack, curiosity, tuna.

u Predicates: OWNS, DOG, ANIMALLOVER, KILLS, ANIMAL

Curiosity Example

\ Dog owners

u Jack owns a dog

⇒ Some dog is owed by Jack

u Every dog owner is an animal lover

u No animal lover kills an animal

⇒ An animal lover does not kill an animal

⇒ For any animal lover X and any animal Y,

then Y is not killed by X

∃ X (DOG(X) ∧ OWNS(jack, X))

∀ X ((∃ Y (DOG(Y) ∧ OWNS(X,Y))) → ANIMALLOVER(X))

∀ X, Y ((ANIMALLOVER(X) ∧ ANIMAL(Y)) → ¬KILLS(X,Y))

Curiosity Example

u The cat’s name is Tuna

u Either Jack or Curiosity killed the cat

u All cats are animals

CAT(tuna)

KILLS(jack, tuna) ∨ KILLS(curiosity, tuna)

∀X CAT(X) → ANIMAL(X)

∀x : ROMAN(X) → LOYALTO(X, caesar) ∨ HATE(X, caesar)

\ Tuy nhiên khi s d ng v i ngh a ho c có lo i tr , có th vi t l i :

∀x : ROMAN(X) → ((LOYALTO(X, caesar) ∧ HATE(X, caesar)) ∧

(¬LOYALTO(X, caesar) ∧ HATE(X, caesar)))

Bi u di n wff’s trong lôgic v t

\ M i ng i đ i đ u trung thành v i m t ng i nào đ o đ ó

∀X, ∃Y LOYALTO(X, Y) ho c ∃Y, ∀X LOYALTO(X, Y)

\ Nhân dân ch mu n gi t nh ng k c m quy n

mà h không trung thành

∀X, ∀Y PERSON(X) ∧ RULER(Y) ∧ TRYASSASSINATE(X, Y) →

¬LOYALTO(X, Y)

M nh đnh đ này t ra m p m (ambiguous) Có ph i ch nh ng k c m

quy n mà nhân dân mu n gi t là nh ng k h không trung thành (theo

nh ng k c m quy n mà h không trung thành ?

\ Marcus mu n gi t Ceresar

u TRYASSASSINATE(marcus, caesar)

\ Câu h i : Marcus có trung thành v i Caesar không ?

vì núi l a phun vào n m 79 A.D.

u Không có ng i nào (không ai) s ng nhi u h n 150 tu i

\ Marcus là m t ng i

MAN(marcus)

\ Marcus là ng i Pompeii

POMPEIAN (marcus)

\ Marcus sinh n m 40 trong công nguyên

(A.D.: Anno Domini)

BORN(marcus, 40)

\ M i ng i đ i đ u (ai c ng ph i) ch t

∀X MAN(X) Ø MORTAL(X)

\ T t c m i ng i dân Pompeii đ i dân Pompeii đ u b ch t

vì núi l a phun vào n m 79 A.D.

∀X ERUPTED(vocalno, 79) ∧ POMPEIAN(X) Ø DIED(X, 79)

Di n gi i (Interpretation)

\ Cho G là m t CTC, m t di n gi i c a G, ký hi u I, đ đ c xác

đ nh t n m b c sau đây c sau đây :

u Ch n mi n di n gi i (Interpretation Domain) là các t p h p khác r ng, ký hi u D ≠ ∅

H nga z

Hàm b c nf(X1, Xn)

Quan h Translation-Interpretation

Ngôn ng

t nhiên

Ngôn nglôgic v tWFF

Tính giá tr c a CTC theo di n gi i (ti p)

\ N u G có d ng (¬G’) :

u I(G) = T n u I(G’) = F trong D

u I(G) = F n u I(G’) = T trong D

Bi n đ i các CTC : lo i b l ng t ∀ và ∃

\ Standardize Variables

in a single sentence (unless you really meant to)

Eg

\ Move all quantifiers left, but keep them in order!

\ Translation into the form:

Q<V> M<V>

with Q: quantifiers, in order ∀s and then ∃s

M: Matrix: wffs including V, <V>: Variable

Bi n đ i các CTC

\ Skolemize: Eliminate Existential Quantifiers

u Existential quantifiers can be eliminated by the introduction

of a new constant that does not appear elsewhere

Bi n đ i các CTC

\ Skolemize: Drop Existential Quantifiers

quantifiers… Consider

becomes by SUBST{Y|h}:

person to their HEART(f(x))

Xây d ng c s lu t cho HCG

\ S ki n 1 :

u Con mèo mà trèo cây cau

H i th m chú chu t đi đâu vt đi đâu v ng nhà

Chú chu t đi cht đi ch đ đ ng xa

Mua m m, mua mu i v gi cha con mèo

u Con mèo mà trèo cây cau : TREO(X, Y)

u H i th m chú chu t đi đâu vt đi đâu v ng nhà :THAM(X, Y), VANGNHA(X)

u Mua m m, mua mu i v gi cha con mèo