IF614 – Inteligensia Semu
09 / 01 - 13
Chapter 7
Symbolic Reasoning Under Uncertainty
7.1 Introduction to Nonmonotonic
Reasoning
Problems with Our Current Logic
 Knowledge must be complete.
 Knowledge must be consistent.
 Knowledge base must grow monotonically.
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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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09 / 03 - 13
Models, Wff’s, and Nonmonotonic
Reasoning





 







A
B
C
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09 / 04 - 13
Nonmonotomic 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)
Rules are wff ’s
AMBB
A  M B  B
We can derive the expression
MBB
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09 / 05 - 13
Default Logic
From :
A:B
C
Use rules to
extensions.
compute
one
or
more
Rules are not wff ‘s. For example, given the
two rules
A : B
B
A:B
B
no conclusion about B can be drawn.
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Abduction
Example :
Given
x : measles(x)  Spots(x)
Spots(Jill)
conclude
Measles(Jill)
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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09 / 07 - 13
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)
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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09 / 08 - 13
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)  height(x,5-10)
height(x,5-10)
So we introduce AB predicates :
x : Adult-male(x)  AB(x,aspect1) height(x,510)
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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09 / 09 - 13
The Closed World Assumption
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
Example 1
From :
:
A(Joe) B(Joe)
we derive :
A(Joe)  B(Joe)
A (Joe)
B (Joe)
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Example 2
From :
09 / 010 - 13
:
Single(John)
Single(Mary)
We derive :
Single(Jane)
From :
Married(John)
Married(Mary)
We derive :
Married(Jane)
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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.
Example 1 :
x : Adult(x)  AB(x,aspect1)  Literate(x)
Example 2 :
A(Joe)  B(Joe)
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7.3 Implementation Issues
7.4 Augmenting Problem Solver
7.5 Implementation: Depth First
Search
Justification - Based TMSs
 Used in conjunction
problem solver.
 Connect nodes
dependencies.
via
with
a
a
separate
network
of
 Provide an algorithm for labeling nodes with
their belief status.
 Search depth-first.
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Justifications
Suspect Abbott
supported belief

+
-
Beneficiary Abbott
IN -list
justification
Alibi Abbott
OUT - list
 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 OUTlist is not empty, or, recursively, if any
assertion in its IN-list has a nonmonotonic
justification.
7.6 Implemetation : Breadth-First
Search
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