14864_7. symbolic reasoning under uncertainty
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SYMBOLIC REASONINGUNDER
UNCERTAINTY
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SYMBOLIC REASONINGUNDER
UNCERTAINTY
Story so far
We have described techniques for reasoning with a complete,
consistent and unchanging model of the world.
But in many problem domains, it is not possible to create such
models.
So here we are going to explore techniques for solving problems
with incomplete and uncertain models.
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SYMBOLIC REASONINGUNDERUNCERTAINTY
Introduction to Non-monotonic Reasoning
The ABC Murder Mystery example
Non monotonic reasoning is one in which the axioms and/or the
rules of inference are extended to make it possible to reason with
incomplete information.
These systems preserve, however, the property that , at any given
moment, a statement is either believed to be true, believed to be
false, or not believed to be either.
Statistical Reasoning : in which the representation is extendedto allow some kind of numeric measure of certainty(rather than
true or false) to be associated with each statement.
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SYMBOLIC REASONINGUNDERUNCERTAINTY
At times we need to maintain many parallel belief spaces, each of
which would correspond to the beliefs of one agent.
Such techniques are complicated by the fact that the belief spaces of
various agents, although not identical, are sufficiently similar that it
is unacceptably in efficient to represent them as completely separateknowledge bases.
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MONOTONICITY
Conventional reasoning systems, such as FOPL are designed to work
with information that has three important properties.
It is complete with respect to domain of interest.
It is consistent.
The only way it can change is that new facts can be added as
they become available.
All this leads to monotonicity.
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If any of these properties is not satisfied, conventional logic based
reasoning systems become inadequate. Non monotonic reasoning
systems, are designed to be able to solve problems in which all of
these properties may be missing
Issues to be addressed
How can the knowledge base be extended to allow inferences to
be made on the basis of lack of knowledge as well as on the
presence of it?
How can the knowledge base be updated properly when a newfact is added to the system(or when the old one is removed)?
How can knowledge be used to help resolve conflicts when there
are several in consistent non monotonic inferences that could be
drawn?
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Logic for monotonic reasoning
Monotonicity is kind of a definition to FOPL, we have to find
some alternative to support non monotonic reasoning.
No single formalization has all the desired properties.
We want to find a formalism that does all of the following things
Define the set of possible worlds that could exist given the
facts that we do have.
Provide a way to say that we prefer to believe in some modelsrather than others.
Provide the basis for a practical implementation of this kind of
reasoning.
Corresponds to our intuitions about how this kind of reasoning
works.
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Default Reasoning
We use non monotonic reasoning to perform, what is commonly
called Default Reasoning.
We want to draw conclusions based on what is most likely to be
true.
Two approaches are
Nonmonotonic Logic
Default Logic Two common kinds of nonmonotonic reasoning that can be
defined in these logics :
Abduction
Inheritance
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Non Monotonic Logic(NML)
It is one in which the language of FOPL is augmented with a
modal operator M, which can be read as is consistent.
Vx,y : Related(x,y) ^ M GetAlong(x,y)WillDefend(x,y)
For all x and y, if x and y are related and if the fact that x gets
along with y is consistent with everything else that is believed,
then conclude that x will defend y.
If we are allowing statements in this form, one important issue
that must be resolved if we want our theory to be even semidecidable, we must decide what is consistent means.
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Default Logic
An alternative logic for performing default-based reasoning is
Reiters Default Logic(DL) in which a new class of inference
rules is introduced. In this approach, we allow inference rules of
this form
A : B
C
The rule should be read as If A is provable and it is
consistent to assume B, then conclude C
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Abduction
Standard logic performs deductions. Given 2 axioms
Vx : A(x) B(x)
A(C)
We conclude B(C) using deduction
Vx : MeaselsSpots(x)
To conclude that if somebody has spots will surely have measels is
incorrect, but it may be the best guess we can make about what isgoing on. Deriving conclusions in this way is this another form of
default reasoning. We call this abductive reasoning.
To accurately define abductive reasoning we may state that Given
2 wffs AB and B, for any expression A & B, if it is consistent to
assume A, do so.
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SYMBOLIC REASONINGUNDER
UNCERTAINTY
Inheritance
A rule for the Baseball Player can be as
Baseball-Player(x) : height(x,6-1) [This is a rule]
height(x,6-1)
Adult-Male(x) : height(x,5-10) [This is a rule]
height(x,5-10)
Adult-Male(x) : --- Baseball-Player(x) height (x,5-10) [ Rule]
height(x,5-10)
Vx: Adult-Male(x) --AB(x,aspect1)height(x,5-10) and so on
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Minimalist Reasoning
We describe methods for saying a very specific and highly useful
class of things that are generally true.
These methods are based on some variant of the idea of a
minimal model.
We will define a model to be minimal if there are no other
models in which fewer things are true.
The idea behind using minimal models as a basis for
nonmonotonic reasoning about the world is the following
There are 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 statements are those that necessarily must be
true in order to maintain the consistency.
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The Closed World Assumption
CWA says that the only objects that satisfy any predicate P are
those that must.
Eg. A companys employee database.
Airline example
Although the CWA is both simple & powerful, it can fail to produce
an appropriate answer for either of the two reasons.
The assumptions are not always true in the world; some parts ofthe world are not realistically closable. - unrevealed facts in
murder case
It is a purely syntactic reasoning process. Thus, the result depend
on the form of assertions that are provided
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Augment a Problem- Solver
How to write a program that solves problems using axioms.
Even uncertain knowledge can be solved using forward &
backward reasoning.
As a result there are 2 approaches to this kind of problem solving
Reason forward from what is known. Treat non monotonically
derivable conclusions the same way monotonically derivable
ones are handled.
Everything is same as monotonic except that here we have
forward chaining rules but with UNLESS clause.
Reason backward to determine whether some expressions P is
true?
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Augment a Problem- Solver
Reason backward to determine whether some expressions P is
true?(or perhaps to find a bindings for its variables that make
it true) Non-monotonic reasoning systems that support this
kind of reasoning may do either or both of the following
Allow default(unless) clauses in backward ruling.
Support a kind of debate in which an attempt is made to
construct arguments both in favor of P and opposed to it.
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