14864_7. Symbolic Reasoning Under Uncertainty

7/30/2019 14864_7. Symbolic Reasoning Under Uncertainty

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SYMBOLIC REASONINGUNDER

UNCERTAINTY


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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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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.

14864_7. Symbolic Reasoning Under Uncertainty

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