Inference First Order Logic

(1)

Inference First Order Logic

Chastine Fatichah

Teknik Informatika

Institut Teknologi Sepuluh Nopember

(2)

Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(3)

Sejarah Reasoning

• 450b.c.

Stoics

propositional logic,

inference

• 322b.c.

Aristotle

“syllogisms" (inference rules), quantifiers

• 1565

Cardano

probability theory (propositional logic +

uncertainty)

• 1847

Boole

propositional logic (again)

• 1879

Frege

logic

• 1922

Wittgenstein

proof by truth tables

• 1930

Gӧdel

 complete algorithm for FOL

• 1930

Herbrand

complete algorithm for FOL (reduce to

propositional)

• 1931

Gӧdel

 complete algorithm for arithmetic

(4)

Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(5)

Universal instantiation (UI)

Semua kalimat dengan universal quantifier (



) meng-entail

semua instantiation-nya:

v

α

Subst({v/g}, α)

untuk sembarang variabel v and ground term g



x King(x)



Greedy(x)



Evil(x) meng-entail:

King(John) 

Greedy(John)



Evil(John)

King(Richard) 

Greedy(Richard)



Evil(Richard)

Contoh

(6)

Existential instantiation (EI)

Untuk sembarang kalimat α, variabel v, and konstanta k

yang tidak muncul di knowledge-base:

v

α

Subst({v/k}, α)

x Crown(x) 

OnHead(x,John) meng-entail:

Crown(C

₁

)



OnHead(C

₁

,John)

Dengan syarat C

₁

adalah konstanta baru, disebut konstanta

Skolem

(7)

Existential instantiation (EI)

• Universal instantiation bisa digunakan

berkali-kali

untuk menambahkan kalimat baru.

• KB yang baru logically equivalent dengan yang

lama.

• Existential instantiation cukup digunakan

sekali

untuk menggantikan kalimat existential.

• KB yang baru

tidak

logically equivalent dengan

yang lama, tetapi satisfiable iff KB yang lama

juga satisfiable  inferentially equivalent

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Mengubah ke Propositional Inference

Misalkan KB berisi kalimat-kalimat berikut:

Dan jika kita mengambil semua kemungkinan instantiation dari kalimat

universal, maka kita dapatkan KB sbb:

KB yang baru disebut

propositionalized

: proposition symbolnya adalah:

King(John), Greedy(John), Evil(John), King(Richard), dll.



x King(x)



Greedy(x)



Evil(x)

King(John)

Greedy(John)

Brother(Richard,John)

King(John)



Greedy(John)



Evil(John)

King(Richard)



Greedy(Richard)



Evil(Richard)

King(John)

Greedy(John)

(9)

Mengubah ke Propositional

Inference

• Klaim : Setiap FOL KB dapat di propositional kan

sebagai entailment

• Klaim: Kalimat ground di entail dengan KB baru jika dan

hanya jika entail terhadap original KB

• Idea: propositionalize KB dan query, lalu gunakan

resolution

• Problem: dengan adanya

function

, ground terms menjadi

infinite

.

(10)

Masalah dengan propositionalization

• Propositionalization

menghasilkan banyak kalimat irrelevant.

• Contoh:



x King(x)



Greedy(x)



Evil(x)

King(John)



y Greedy(y)

Brother(Richard,John)

manusia bisa cepat mengerti kalau Evil(John), namun

propositionalization menghasilkan:

King(Richard)



Greedy(Richard)



Evil(Richard)

Greedy(Richard)

yang irrelevant

(11)

Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(12)

Unification

• Inference : Substitution θ sedemikian hingga

King(x)

dan

Greedy(x)

cocok dengan

King(John)

dan

Greedy(y)

θ = {x/John,y/John}

• Unify(α,β) = θ jika αθ = βθ

p

q

θ

Knows(John,x) Knows(John,Jane)

Knows(John,x) Knows(y,OJ)

Knows(John,x) Knows(y,Mother(y))

Knows(John,x) Knows(x,OJ)

(13)

Unification

• Inference : Substitution θ sedemikian hingga

King(x)

dan

Greedy(x)

cocok dengan

King(John)

dan

Greedy(y)

θ = {x/John,y/John}

• Unify(α,β) = θ jika αθ = βθ

p

q

θ

Knows(John,x) Knows(John,Jane)

{x/Jane}}

Knows(John,x) Knows(y,OJ)

Knows(John,x) Knows(y,Mother(y))

Knows(John,x) Knows(x,OJ)

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Unification

• Inference : Substitution θ sedemikian hingga

King(x)

dan

Greedy(x)

cocok dengan

King(John)

dan

Greedy(y)

θ = {x/John,y/John}

• Unify(α,β) = θ jika αθ = βθ

p

q

θ

Knows(John,x) Knows(John,Jane)

{x/Jane}}

Knows(John,x) Knows(y,OJ)

{x/OJ,y/John}}

Knows(John,x) Knows(y,Mother(y))

Knows(John,x) Knows(x,OJ)

(15)

Unification

• Inference : Substitution θ sedemikian hingga

King(x)

dan

Greedy(x)

cocok dengan

King(John)

dan

Greedy(y)

θ = {x/John,y/John}

• Unify(α,β) = θ jika αθ = βθ

p

q

θ

Knows(John,x) Knows(John,Jane)

{x/Jane}}

Knows(John,x) Knows(y,OJ)

{x/OJ,y/John}}

Knows(John,x) Knows(y,Mother(y))

{y/John,x/Mother(John)}}

(16)

Unification

• Inference : Substitution θ sedemikian hingga

King(x)

dan

Greedy(x)

cocok dengan

King(John)

dan

Greedy(y)

θ = {x/John,y/John}

• Unify(α,β) = θ jika αθ = βθ

p

q

θ

Knows(John,x) Knows(John,Jane)

{x/Jane}}

Knows(John,x) Knows(y,OJ)

{x/OJ,y/John}}

Knows(John,x) Knows(y,Mother(y))

{y/John,x/Mother(John)}}

Knows(John,x) Knows(x,OJ)

{fail}

• Standardizing apart

variable menghilangkan overlap, contoh

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Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(18)

Generalized Modus Ponens

(GMP)

p

₁

' is King(John)

p

₁

is King(x)

p

₂

' is Greedy(y)

p

₂

is Greedy(x)

θ is {x/John,y/John}

q is Evil(x)

q θ is Evil(John)

• GMP dengan KB yang berisi

definite clauses:

p

₁



p

₂



…



p

_n



q

• Semua variabel diasumsikan

universally quantified

p

₁

', p

₂

', … , p

_n

', ( p

₁

 p

₂

 …  p

_n

 q)

qθ

dimana p

_i

'θ = p

_i

θ untuk semua i

Inference Rule GMP

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Soundness pada GMP

• Perlu menunjukkan bahwa

p

₁

', …, p

_n

', (p

₁



…



p

_n



q) ╞ qθ

dimana

p

_i

'θ = p

_i

θ

untuk semua i

• Lemma: Untuk setiap definite clause p, terdapat

p ╞ pθ

dengan Universal instantiation (UI)

1. (p

₁



…



p

_n



q) ╞ (p

₁



…



p

_n



q)θ = (p

₁

θ



…



p

_n

θ



qθ)

2. p

₁

', …, p

_n

' ╞ p

₁

'



…



p

_n

' ╞ p

₁

'θ



…



p

_n

'θ

(20)

Contoh Knowledge Base

Buktikan bahwa Colonel West is a criminal !!

“ The law says that it is a crime for an American to sell weapons to

hostile nations. The country Nono, an enemy of America, has

some missiles, and all of its missiles were sold to it by Colonel

West, who is American. “

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Contoh Knowledge Base

• ... it is a crime for an American to sell weapons to hostile nations:

American(x)



Weapon(y)



Sells(x,y,z)



Hostile(z)



Criminal(x)

• Nono … has some missiles:



x Owns(Nono,x)



Missile(x):

Owns(Nono,M

₁

) and Missile(M

₁

)

• … all of its missiles were sold to it by Colonel West

Missile(x)



Owns(Nono,x)



Sells(West,x,Nono)

• Missiles are weapons:

Missile(x)



Weapon(x)

• An enemy of America counts as "hostile“:

Enemy(x,America)



Hostile(x)

• West, who is American …

(22)

Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(23)

Forward chaining FOL

dengan GMP

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(25)

(26)

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Forward chaining properti

• Sound dan complete untuk first-order definite clauses

• Datalog

= first-order definite clauses + tanpa function

Time complexity FC pada Datalog  polynomial

• Tapi pada kasus umum, bisa infinite loop kalau tidak

di-entail.

(28)

Pokok Bahasan

• Mengubah First Order Logic ke Propotional

Logic

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

(29)

Algoritma backward chaining

Backward chaining FOL

dengan GMP

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(31)

(32)

(33)

(34)

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(36)

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(38)

Backward chaining properti

• Depth-first search:

• linear space complexity

• incomplete (infinite loops)

• i

nefficient (repeated subgoals)

(39)

Logic programming: Prolog

Logic programming

Ordinary programming

1. Identify problem

Identify problem

2. Assemble information

Assemble information

3. Tea break

Figure out solution

4. Encode information in KB

Program solution

5. Encode problem instance as facts

Encode problem instance as data

6. Ask queries

Apply program to data

7. Find false facts

Debug procedural errors

(40)

Prolog

• Appending two lists to produce a third:

append([],Y,Y).

append([X|L],Y,[X|Z]) :- append(L,Y,Z).

• query:

append(A,B,[1,2]) ?

• answers:

A=[] B=[1,2]

• A=[1] B=[2]

A=[1,2] B=[]

(41)

Konversi ke CNF

• Everyone who loves all animals is loved by

someone:



x [



y Animal(y)



Loves(x,y)]



[



y Loves(y,x)]

• 1. Eliminasi biconditionals dan implikasi



x [



y



Animal(y)



Loves(x,y)]



[



y Loves(y,x)]

• 2. Pindah  inwards: x p ≡ x p,  x p ≡

x p



x [



y



(



Animal(y)



Loves(x,y))]



[



y Loves(y,x)]

(42)

Konversi ke CNF.

3. Standarisasi variabel:



x [



y Animal(y)





Loves(x,y)]



[



z Loves(z,x)]

4. Skolemize: bentuk umum pada existential instantiation.

Setiap variabel existential diganti dengan

Skolem function

pada universally

quantified variables:



x [Animal(F(x))





Loves(x,F(x))]



Loves(G(x),x)

5. Hapus semua universal quantifiers:

[Animal(F(x))





Loves(x,F(x))]



Loves(G(x),x)

6. Distribusi  over  :

(43)

(44)

Sumber :

1.Slide perkuliahan Stuart Russell's (Berkeley)

http://aima.cs.berkeley.edu/

2.Slide perkuliahan Sistem Cerdas Ruli Manurung (Universitas Indonesia)