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Pertemuan 9Symbolic Reasoning Under Uncertainty

Matakuliah : T0264/Inteligensia Semu

Tahun : Juli 2006

Versi : 2/1

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Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• << TIK-99 >>

• << TIK-99>>

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Outline Materi

• Materi 1

• Materi 2

• Materi 3

• Materi 4

• Materi 5

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7.1 Introduction to Nonmonotonic Reasoning

Penalaran monotonic knowledge bersifat : lengkap (complete), konsisten dan model tidak berubah (monotonic). Tetapi banyak permasalahan tidak dapat dimodelkan secara lengkap dan kosisten.

Oleh karena itu muncul penalaran non-monotonic dengan ciri-ciri :

• Mengandung ketidakpastian• Adanya perubahan pada pengetahuan• Adanya penambahan fakta baru dapat mengubah

konklusi yang sudah terbentuk• Misalka S adalah konlusi dari D, maka S buka konklusi

pada D + fakta baru

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Introduction to Nonmonotonic Reasoning

• Premis 1 : Mekanika adalah pelajaran sulit• Premis 2 : Getaran dan Gelombang adalah pelajaran sulit• Premis 3 : Listrik magnet adalah pelajaran sulit• Premis 4 : Temodinamika dan panas adalah pelajaran sulit

• Kesimpulan : Fisika adalah pelajaran sulit

• Jika ada fakta baru : • Premis 5 : Kalkulus adalah pelajaran sulit

• Sehingga Kesimpulan tidak berlaku lagi.• Jadi penambahan fakta baru mengakibatkan

ketidakkonsistenan, dan keadaan ini disebut penalaran non-monotonic.

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7.2 Logic for Nonmonotonic Reasoning

Models and Interpretations• An interpretation of a set of wff ’s consists of :

- A domain (D)- A function that assigns• to each predicate a relation• to each n-ary function an operator that maps

from Dn into D• to each constant an element of D

• A model of a set of wff ’s is an interpretation that satisfies them.

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Logic for Nonmonotonic Reasoning

The essence of nonmonotonic reasoning is :

1. Default Reasoning : to use nonmonotonic reasoning to perform what is commonly

- Nonmonotonic logic

- Default logic

- Abduction

- Inheritance

2. Minimalist reasoning : these methods are based on some variant of the idea of a minimal model.

- The closed World Assumtion

- Circumscription

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Logic for Nonmonotonic Reasoning

• Models, Wff’s, and Nonmonotonic Reasoning

A

B

C

B

A

C

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7.2.1. Default Reasoning

Nonmonotomic Logic• M is operator “first-order predicate logic”• Example 1 :

x,y : Related(x,y) M GetAlong(x,y) WillDefend(x,y)

• Example 2 :x : Republican(x) M Pacifist(x)

Pacifist(x)x : Quaker(x) M Pacifist(x) Pacifist(x)Republican(Dick)Quaker(Dick)

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Default Reasoning

Nonmonotomic Logic

Rules are wff ’s

A M B B

A M B B

We can derive the expression

M B B

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Default Reasoning

• Default Logic

From :

A : B

C

Such rule should be read as “If A is provable and it is consistent to assume B then conclude C”

Use rules to compute one or more extensions.

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Default Reasoning

Rules are not wff ‘s. For example, given the two rules

A : B A : B

B B

no assertion about A, no conclusion about B can be drawn, since neither inference rule applies.

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Default Reasoning

Abduction

Example :

Given two axioms :

x : measles(x) Spots(x)

Spots(Jill)

conclude

Measles(Jill)

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Default Reasoning

Definition :

Given two wff’s (A B) and (B), for any expresions A and B, if it is consistent to assume A, do so.

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Default Reasoning

Inheritance in Default Logic

Given :

Baseball-Player(x) : height(x,6-1)

height(x,6-1)

x,y,z : height(x,y) height(x,z) y = z

Pitcher(Three-Finger-Brown)

Conclude :

height(Three-Finger-Brown, 6-1)

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Default Reasoning

But this is blocked by

height(Three-Finger-Brown, 5-11)

Now we add :

Adult - Male(x) : height(x,5-10)

height(x,5-10)

But now there are two extensions.

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Default Reasoning

Revised axiom :Adult-Male(x) : Baseball-Player(x) height(x,5-

10)

height(x,5-10)

But this approach becomes unwiedly :

Adult-Male(x) : Baseball-Player(x) Midget(x) Jockey(x) eight(x,5-10)

height(x,5-10)

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Default Reasoning

So we introduce AB predicates :x : Adult-male(x) AB(x,aspect1)

height(x,5-10)x : Baseball-Player(x) AB(x, aspect1)x : Midget(x) AB(x, aspect1)x : Jockey(x) AB(x, aspect1)and single default rule :

: AB(x,y)AB(x,y)

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7.2. Minimalist Reasoning

• These methods are based on some variant of the idea of a minimal model.

The idea behind using minimal models as basis for nonmonotonic reasoning about the world is following :

“There many fewer true statements than false ones. If something is true and relevant it makes sense to assume that it has been entered into our knowledge base. Therefore, assume that the only true statement are those that necessarily must be true in order to maintain the consistency of knowledge base”.

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Minimalist Reasoning

The Closed World Assumption (CWA)

The only objects that satisfy any predicate P are those that must.Very useful for databases and AB predicates

Problem :• Some worlds are not closed• The CWA is a purely syntactic reasoning

process

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Minimalist Reasoning

Example 1 :

From :

A(Joe) B(Joe)

we derive :

A(Joe) B(Joe)

A (Joe)

B (Joe)

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Minimalist Reasoning

Example 2 :From :

Single(John)Single(Mary)

We derive :Single(Jane)

From :Married(John)Married(Mary)

We derive :Married(Jane)

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Minimalist Reasoning

Circumscription

Two advantages over CWA :• Operates on whole formulas, not individual

predicates.• Allows some predicates to be marked as closed

and others as open.

Accomplished by adding axioms that force aminimal interpretation on a selected portion of the KB.

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Minimalist Reasoning

Example 1 : x : Adult(x) AB(x,aspect1) Literate(x)

Example 2 :

A(Joe) B(Joe)

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7.3 Implementation Issues

1. How to derive exactly those nonmonotonic conclusions that are relevant to solving the problem.

2. How to update our knowledge incrementally as problem solving progresses.

3. In nonmonotonic reasoning system, it often happens that more than one interpretation of know fact licensed by a available inference rules.

4. In general, these theories are not computationally effective.

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Implementation Issues

• Techniques for doing this can be divided into two classes, determine by their approach to the search control problem :

1. Dept-first : in which we follow a single.

2. Breadth-first : all the possibilities as equally likely

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7.4 Augmenting a Problem Solver

• Problem solving can be done using either forward or backward reasoning.

• Problem solving using uncertain knowledge is no exception.

• Two basic approach to this kind of problem solving :

1. Reason forward from what is known

2. Reason backward to determine whether some expression P is true

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Augmenting a Problem Solver

• Backward Rules Using UNLESS

Suspect(x) Beneficiary(x)

UNLESS Alibi(x)

Alibi(x) SomewhereElse(x)

SomewhereElse(x) RegisteredHotel(x,y) and FarAway(y)

UNLESS ForgetRegister(y)

Alibi(x) Defend(x,y)

UNLESS Lies(y)

SomewhereElse(x) PictureOf(x,y) and FarAway(y)

Contradiction() TRUE

UNLESS x : Suspect(x)

Beneficiary(Aboutt)

Beneficiary(Babbitt)

Beneficiary(Cabot)

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Augmenting a Problem Solver

• Forward Rules Using UNLESS

If : Beneficiary(x), UNLESS Alibi(x) then Suspect(x)

If : SomewhereElse(x), then Alibi(x)

If : RegisteredHotel(x,y) and FarAway(y),

UNLESS ForgetRegister(y),

If : Defends(x,y), UNLESS Lies(y) then Alibi(x)

If : PictureOf(x,y) and FarAway(y),then SomewhereElse(x)

If : TRUE,

UNLESS x : Suspect(x) then Contradiction()

Beneficiary(Abbott)

Beneficiary(Babbitt)

Beneficiary(Cabot)

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7.5 Implementation: Depth First Search

Justification - Based TMSs

• Used in conjunction with a separate problem solver.

• Connect nodes via a network of dependencies.

• Provide an algorithm for labeling nodes with their belief status.

• Search depth-first.

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Implementation: Depth First Search

Justifications

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Implementation: Depth First Search

• A justification is valid if every assertion in the IN-list is believed and none of those in the OUT-list is.

• A justification is nonmonotonic if its OUT-list is not empty, or, recursively, if any assertion in its IN-list has a nonmonotonic justification.

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7.6 Implemetation : Breadth-First Search

• The assumption-based truth maintenance system (ATMS) is an alternarive way of implementing nonmonotonic reasoning.

• In an ATMS, alternative paths are maintained in parallel.

• The ATMS is designed to be used in conjunction with a separate problem solver. The problem solver job is to :

- Create nodes that correspond to assertions- Associate with each such node one or more

justifications- Inform the ATMS of inconsistent contexts

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Implemetation : Breadth-First Search

The role of ATMS system is then to :

• Propagate inconsistencies, then ruling out contexts that include subcontexts that are known to be inconsistent.

• Label each problem solver node with the contexts in which it has a valid justification.

A1 A2 A3 …… An C

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Implemetation : Breadth-First Search

A Context Lattice

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<< CLOSING>>

End of Pertemuan 9

Good Luck