Matakuliah
Tahun
Versi
: T0264/Inteligensia Semu
: Juli 2006
: 2/1
Pertemuan 9
Symbolic Reasoning Under
Uncertainty
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Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• << TIK-99 >>
• << TIK-99>>
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Outline Materi
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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 :
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Mengandung ketidakpastian
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Adanya perubahan pada pengetahuan
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Adanya penambahan fakta baru dapat mengubah
konklusi yang sudah terbentuk
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Misalka S adalah konlusi dari D, maka S buka konklusi
pada D + fakta baru
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Introduction to Nonmonotonic Reasoning
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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
C
B
A
B
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MBB
A  M B  B
We can derive the expression
MB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 a
minimal 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
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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
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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
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