1. Theoretical Foundations Chapter 1 :: Knowledge Presented by Gerald Alva

2. 2 Chapter 1 Outline What is Knowledge? How is it defined? What is a Knowledge Base? How can a Knowledge Base be used? What is Knowledge Specification/Generalization?

3. 3 Knowledge & Classification Knowledge consists of a family of various classification patterns of a domain.

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5. 5 Knowledge Base Equivalence Relations/categories provide a foundation of the Knowledge Base. A Knowledge Base is a relational system where U is finite and R is a family of equivalence relations over U. K = (U,R)

6. 6 Knowledge Base Example Given a set of toy blocks U that can be classified according to Color, Shape and Size. 1.Find the family of Equivalence Relations over the universe U. 2.Define the Knowledge Base based of these Equivalence Relations. 3.Find all the Equivalence Classes for each Equivalence Relation. XColorShapeSize X1RedRoundSmall X2BlueSquareLarge X3RedTriangularSmall X4BlueTriangularSmall X5YellowRoundSmall X6YellowSquareSmall X7RedTriangularLarge X8YellowTriangularLarge

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

8. 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. Equivalence Relation Equivalent Class

9. 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: U toys /C red {x1,x3,x7} U toys /C blue {x2,x4} U toys /C yellow {x5,x6,x8} XColor X1Red X2Blue X3Red X4Blue X5Yellow X6Yellow X7Red X8Yellow

10. 10 Equivalence Classes U toys /R size U toys /C large {x1,x3,x4,x5,x6} U toys /C small {x2,x7,x8} U toys /R color U toys /C red {x1,x3,x7} U toys /C blue {x2,x4} U toys /C yellow {x5,x6,x8} U toys /R shape U toys /C square {x2,x6} U toys /C round {x1,x5} U toys /C triangular {x3,x4,x7,x8} Equivalence classes for each Equivalence Relation:

11. 11 Knowledge Base Concept U toys /C red {x1,x3,x7}  U toys /C tri {x3,x4,x7,x8} = {x3, x7} U toys /C yellow {x5,x6,x8}  U toys /C round {x1,x5} = {x5} The sets {x3,x7} and {x5} are concepts that belong to our knowledge base. They represent red-triangular and yellow-square elements. Concept: Red and Triangular Concept: Yellow and Round Set theory can be used to create concepts associated with the universe.

12. 12 Knowledge Base Concept U toys /C round {x1,x5}  U toys /C blue {x2,x6} =  Concept: Round and Blue 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. 13 Equivalent Knowledge Bases K = (U, P) K = (U, Q) K  K if U/P  U/Q Two knowledge bases are considered equivalent if they have the same set of elementary categories.

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

15. 15 Questions???