1. Theoretical FoundationsChapter 1 :: Knowledge Presented by Gerald Alva

2. Chapter 1 Outline

3. Knowledge & Classification

5. Knowledge Base

6. Knowledge Base Example • Given a set of toy blocks U that can be classified according to Color, Shape and Size. • Find the family of Equivalence Relations over the universe U. • Define the Knowledge Base based of these Equivalence Relations. • Find all the Equivalence Classes for each Equivalence Relation.

7. Equivalence Relations Define the Universe: Utoys{x1,x2,x3,x4,x5,x6,x7,x8} The three categories provides the following family of equivalence relations: {Rcolor }{Rshape}{Rsize} Therefore, Knowledge Base can be defined as: K = (U,R) = K = (Utoys, {Rcolor,Rshape,Rsize})

8. Equivalent Classes An Equivalence Class is used to further refine our understanding of an Equivalence Relation. An Equivalence Class is a subset of an Equivalence Relation.

9. Equivalence Classes • Equivalence Classes for Color: • The three possible colors are Red, Blue, and Yellow. Therefore, we have the following sets representing all elements that are red, blue, and yellow: • Utoys/Cred{x1,x3,x7} • Utoys/Cblue{x2,x4} • Utoys/Cyellow{x5,x6,x8}

10. Equivalence Classes Equivalence classes for each Equivalence Relation:

11. Knowledge Base Concept Set theory can be used to create concepts associated with the universe.

12. Knowledge Base Concept • The set {{x1,x5},{x2,x6}} is a concept that does not belong to our knowledge base. • There are no round-blue elements in our Knowledge Base.

13. Equivalent Knowledge Bases

14. Specialization & Generalization Let K = (U,P) and K ¢ = (U,Q) be two knowledge bases. If U/P Ì of U/Q then P is finer than Q. Furthermore, P is a Specializationof Q and Q is a Generalization of P. Specialization :: Provides a more precise representation Generalization :: Provides a more broad representation

15. Questions???