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Integrated Model of Inductive-Deductive Inference Based on Finite Predicates and Implicative Regularities
Abstract
The theory of Boolean functions, especially in respect to representing these functions in the disjunctive or conjunctive normal forms, is extended in this chapter onto the case of finite predicates. Finite predicates are decomposed by that into some binary units, which will correspond to components of Boolean vectors and matrices and are represented as combinations of these units. Further, the main concepts used for solving pattern recognition problems are defined, namely world model, data, and knowledge. The data presenting information about the existence of some objects with definite combinations of properties is considered, as well as the knowledge presenting information about the existence of regular relationships between attributes. These relationships prohibit some combinations of properties. In this way, the knowledge gives the information about the non-existence of objects with some definite (prohibited) combinations of attribute values. A special form of regularity representation, called implicative regularities, is introduced. Any implicative regularity generates an empty interval in the Boolean space of object descriptions, which do not contradict the data. The problem of plausibility evaluation of induced implicative regularities should be solved by that. The pattern recognition problem is solved by two steps. First, regularities are extracted from the database (inductive inference); second, the obtained knowledge is used for the object recognition (deductive inference).
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