Symbolic Reasoning under uncertainty

1. 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

2. 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. Symbolic Reasoning under uncertainty

3. 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

4. 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

5. 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

6. 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 Symbolic Reasoning under uncertainty

7. 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. Nonmonotonic Logic

8. 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 Default Logic

9. 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. Abduction

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 ismember unless doing so leads to contradiction, in which case a value from a more restricted class hasprecedence over a value from a broader class”. Given : Conclude But this is blocked by Now we add: Revised axiom : 163

11. 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. Minimalist Reasoning

12. 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

13. Consider the following Clause The Closed World Assumption • 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. 167

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!!!