1. 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.

2. 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
The Abbott did not commit the crime
The Babbitt did not commit the crime
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.

3. 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.

4. 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).

5. 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?

6. 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

7. 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.

8. 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

9. 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.

10. 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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11. 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:
Closed world assumption (CWA).
Circumscription.

12. 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

13. 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)

14. 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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15. Circumscription
Suppose that a child knows only two kinds of birds parrot and sparrow . Formally we can write this as
Predicate Expression

16. AB Predicated
Adult male(x): ¬Baseball_player(x)^ ¬Midget(x)^ ¬Jockey(x) ^height(x,5-10)
height(x,5-10)

17. 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

18. Thank You!!!