Symbolic Reasoning under 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.
Symbolic Reasoning under uncertainty
The ABC murder mystery example:
Let Abbott, Babbitt and Cabot be suspects in a murder case. Abbott has an alibi, in
the register of a respected hotel in Albany. Babbitt also has an alibi, for his
brother-in-law testified that Babbitt was visiting him in Brooklyn at the time.
Cabot pleads alibi too, claiming to have been watching a ski meet in the catskills.
We have only his words for that, So we believe
1. The Abbott did not commit the crime
2. The Babbitt did not commit the crime
3. The Abbott or Babbitt or Cabot did.
But presently Cabot documents his alibi-He had the good luck to have been caught
by television in the sidelines at the ski meet. A new belief is thus trust upon us:
4. That Cabot did not.
Symbolic Reasoning under uncertainty
Introduction to Non-monotonic Reasoning
 Non monotonic reasoning: 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 extended to allow
some kind of numeric measure of certainty(rather than true or false) to be
associated with each statement.
Symbolic Reasoning under uncertainty
At times we need to maintain many parallel belief spaces, each of which
would correspond to the beliefs of one agent.
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 (these new facts are consistent with
all the other facts that have already been asserted).
Symbolic Reasoning under uncertainty
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 new fact 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?
Symbolic Reasoning under uncertainty
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
Non-monotonic Logic
Default Logic
Two common kinds of non-monotonic reasoning that can be defined in
these logics :
Abduction
Inheritance
Nonmonotonic Logic
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 semi decidable, we must
decide what “is consistent ” means.
Default Logic
Default Logic
An alternative logic for performing default-based reasoning is Reiter’s
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
Abduction
Abduction
Standard logic performs deductions. Given 2 axioms
Vx : A(x)  B(x)
A(C)
We conclude B(C) using deduction
Vx : Measels (x)Spots(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 is going 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 wff’s
AB and B, for any expression A & B, if it is consistent to assume A, do so.
Inheritance in Default Logic
Inheritance is a basis for inheriting attribute values from a prototype description of a class
To individual entities that belong to the class. Inheritance says that “An object inherits attribute
Values from all the classes which it is member unless doing so leads to contradiction, in which
case a value from a more restricted class has precedence over a value from a broader class”.
Given :
Conclude
But this is blocked by
Now we add:
Revised axiom :
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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 non-monotonic
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.
Two kinds of minimalist reasoning are:
(1) Closed world assumption (CWA).
(2) Circumscription.
The Closed World Assumption
This type of assumption is useful in application where most of the facts are
true and it is, therefore, reasonable to assume that if a proposition cannot be
proven, it is false.
This is known as the closed world assumption with failure as negation. This
means that in a kb if the ground literal p(a) is not provable, then ~p(a) is
assumed to hold true.
CWA says that the only objects that satisfy any predicate P are those that
must. Best for data base but some time fails with knowledge base.
Eg. A company’s employee database.
Airline example
The Closed World Assumption
Consider the following Clause
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 of the 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.- Consider A(Joe) V B(Joe)
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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Circumscription
Suppose that a child knows only two kinds of birds parrot and sparrow .
Formally we can write this as
Predicate Expression
AB Predicated
Adult male(x): ¬Baseball_player(x)^ ¬Midget(x)^ ¬Jockey(x) ^height(x,5-10)
height(x,5-10)
EX-Non monotonic
Example 1:
-Most thing do not fly
-Most birds do fly, unless they are too young or dead or have broken wing.
-Penguin and Ostriches do not fly