DEALING WITHINCONSITENCIES ~ANKIT SHARMA M.Tech 3rd Sem ROLL No. 312
LOGIC-a brief intro• Logic is used to represent textual information in a formal way in order to give the information a precise meaning and to remove ambiguity.• For example, we might want to express that every day when it rains, the streets are wet.• In first-order logic (FOL) we might express the fact that it is raining at a specific day using a predicate rain with an argument representing the date when it is raining, e.g., the fact that it is raining on August, 24th, 2009 might be expressed with the following predicate rain(24082009).• So we conclude: ∀X : rain(X) → streets wet(X) if we know that rain(24082009) because we observed the rain at the given date, then we infer streets wet(24082009), which is not known before. Hence, logical reasoning can be used to derive new facts.
Certainty factor• The Door Bell Problem• The door bell rang at 12 O’clock in midnight.• -was someone at the door?• -did Mohan wake up?• -Proposition 1: atdoor(x) doorbell.• -Proposition 2: doorbell wake(Mohan).
Reasoning about Door Bell Given doorbell can we say at doorbell(x), because atDoor(x) doorbell. Abductive reasoning. But no, the doorbell might start ringing due to other reasons:- -short circuit. -wind. -animals.
Cnt….• Given doorbell, can we say• Wake(Mohan), because doorbell wake(Mohan).• Deductive reasoning.• yes, only if proposition 2 is always true.• However in general Mohan may not wakeup even if the bell rings.
Cnt…• Therefore, we cannot answer either of the questions with certainty.• Proposition 1 is incomplete so modifying it as• Atdoor(x)v shortcircuit v wind…….. Doorbell.• Doesnt help because the list of possible causes on the left is huge(infinite may be).• Proposition is often true but not a tautology.
Any way out?• However, problems like that of the doorbell are very common in real life.• In A.I we often need to reason under such It will rain in December. UNCERTAINITY circumstances.• We solve it by proper modeling of uncertainty• & impreciseness and developing appropriate reasoning techniques. IMPRECISENESS Often rarely sometimes. e.g. boy is very tall.
Non monotonic reasoning• In first order logic, adding new axioms increases the amount of knowledge base. Therefore set of facts and inferences in such systems can grow larger, they cannot be reduced i.e. they increase monotonically.• But Nonmonotonic reasoning means adding new facts to the database will contradict and invalidate the old knowledge.
example• We first state that all birds can fly and that one bird is named Tweety.• ∀X : bird(X) → fly(X)• bird(tweety)• From this two pieces of knowledge we can derive that Tweety can fly, i.e.,• bird(tweety) ∀X : bird(X) → fly(X)• fly(tweety)• If we add another fact like Tom is also a bird bird(Tom), then we can also derive that Tom is also able to fly. Hence, more knowledge allows us to derive more new rules and facts. As a consequence we conclude that classical reasoning (in FOL) is monotonic.• The situation changes if we add new knowledge like penguins cannot fly and that Tweety is a penguin.• ∀X : penguin(X) → ￢fly(X)• penguin(tweety)• In this case we derive a contradiction from which we can derive everything.
Truth maintenance systems• Necessary when changes in the fact-base lead to inconsistency / incorrectness among the facts non-monotonic reasoning• A Truth Maintenance System tries to adjust the Knowledge Base or Fact Base upon changes to keep it consistent and correct.• A TMS uses dependencies among facts to keep track of conclusions and allow revision / retraction of facts and conclusions.
Dependency…….example• Suppose the knowledge base KB contained only the propositions P, P →Q. From this IE would right fully conclude Q and this conclusion to the KB. Later if it was learned that if P was inappropriate, it would be added to the KB resulting in an contradiction. Consequenlty it woluld be necessary to remove P to eliminate the inconsistency. But with the P now removed, Q is no longer a justified belief. It should be removed . This type of job is done by TMS.• Actually the TMS does not discard the conclusions like Q as suggested. That could be wasteful since p became again valid. so again we have to re-derive it .instead TMS maintains a dependency records for all such conclusions. The records determine which se of beliefs are current (which are to be used by IE). Thus Q would be removed from the current belief set and not deleted. INFERENCE TELL ENGINE ASK TMS Architecture of the problem solver with TMS KNOWLEDGE BASE
Structured knowledge PROFESSION(bob,professor) FACULTY(bob,engineering) . . MARRIED(bob,sandy) FATHER-OF(bob,sue,joey) OWNS(bob,house) When the quantity of information becomes large, the maintenance of knowledge becomes difficult. in such cases, some form of knowledge structuring is done.
ASSOCIATIVE NETWORKS• Network representations provides a means of structuring and exhibiting the structure in knowledge.• Network representations give a pictorial presentation of objects , their attributes and their relationships that exist between them and other entities. Can fly• a-kind-of colorbird tweety yellow Has wings fragment of associative n/w.
Understanding Frames• Frames were first introduced by Marvin minsky as a data structure to represent a mental model of a stereotypical situation such as driving a car, attending a meeting or eating in a restaurant.• Knowledge about an object or event is stored together in memory as a unit, then when a new frame is encountered, an appropriate frame is selected from memory for use in reasoning about the situation.
Understanding Frames – Facts Frames are record-like structures that have slots & slot- values for an entity Using frames, the knowledge about an object/event can be stored together in the KB as a unit A slot in a frame specify a characteristic of the entity which the frame represents Contains information as attribute-value pairs, default values etc.
Graphical Representation• Graphs easy to store in a computer• To be of any use must impose a formalism
Conceptual Graphs• Semantic network where each graph represents a single proposition• Concept nodes can be – Concrete (visualisable) such as restaurant, my dog Spot – Abstract (not easily visualisable) such as anger• Edges do not have labels – Instead, conceptual relation nodes – Easy to represent relations between multiple objects