In this chapter, classification reasoning is considered. The concept of good classification test lies in the foundation of this reasoning. Inferring good classification tests from data sets is the inductive phase of reasoning resulted in generating implicative and functional dependencies supporting the deductive phase of reasoning. An algorithm of inferring good classification tests is given with the decomposition of it into subtasks allowing to choose sub-contexts for each obtained dependency and to control sub-contexts during both deductive and inductive phases of classification reasoning.
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The symbolic methods of machine learning work on objects with symbolic, Boolean, integer, and categorical attributes. With this point of view, these methods can be considered as ones of mining conceptual knowledge. We concentrate on the supervised conceptual learning. Now, the theory of conceptual learning does not include classification reasoning as its inalienable component, although precisely this reasoning constitutes an integral part of any mode of reasoning (Mill, 1872; Michalski & Kaufman, 1998; Spencer, 1898; Piaget & Inelder, 1954; Sechenov, 2001). Furthermore, current models of commonsense reasoning do not include classification too (Russel & Norvig, 2010). However, classification, as a process of thinking, performs the following operations (Polia, 1954; Bynum, 1972, Mill, 1872; Quinlan, 1989):
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Generalizing or specifying object descriptions;
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Interpreting logical expressions on a set of all thinkable objects;
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Learning concepts from examples;
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Decision tree construction;
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Extracting hierarchical object classifications from examples;
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Forming knowledge and data contexts adequate to a current situation of reasoning;
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Reducing the domain of searching for a solution of some problem;
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Revealing essential elements of reasoning (objects, attributes, values of attributes etc);
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Revealing the links of object sets and their descriptions with external contexts interrelated with them.
This list can be continued.
We believe that conceptual learning is a special class of methods based on mining and using conceptual knowledge the elements of which are objects, attributes (values of attributes), classifications (partitions of objects into disjoint blocks), and links between them. These links are expressed by the use of implications: “object ↔ class”, “object ↔ property”, “values of attributes ↔ class”, and “subclass ↔ class”.
We understand classification reasoning as a process of thinking based on which the causal connections between objects, their properties and classes of objects are revealed. In fact, this reasoning is critical for the formation of conceptual knowledge or ontology in the contemporary terminology.
Studying the processes of classification within the framework of machine learning and knowledge discovery led to the necessity of reformulating the entire class of symbolic machine learning problems as the problems of finding approximations of a given classification of objects (Naidenova, 1996). This reformulation is based on the concept of a good diagnostic (or classification) test (GDT) for the given classification of objects (Naidenova & Polegaeva, 1986; Naidenova, 2006). A GDT has a dual nature. On the one hand, it is a logical expression in the form of implication or functional dependency; on the other hand, it generates the partition of a set of objects equivalent to a given classification of this set or partition that is nearest to the given classification with respect to the inclusion relation between partitions.