On rough definability and types of approximation of classifications

The study of rough definability of classifications was initiated by Busse [3]. Classifications are of great interest, as in the process of learning from examples, rules are derived from classifications generated by single decisions. It was established by him that not all concepts associated with rough definability of sets can be extended to classifications. Four propositions were established in [3] and used in defining types of classifications. In this article we extend these propositions to obtain necessary and sufficient type theorems from which several results besides the above four results of Busse could be derived. We interpret and illustrate each of these results through examples. It is found that there are 11 possible types of classifications. Out of which only 5 which are basic in nature were considered by Busse without stating any reason for doing so. However, in this paper we shall establish that the other six types reduce to these five considered by him, either directly or transitively. In this paper, we prove a general theorem which provides a complete picture of the types of elements in a classification and introduce an algorithm to generate these classifications, given the number of elements in the classification. In fact, this result establishes the statement of Pawlak [9] that the notions of complement for sets and that for classifications are different. Also, we shall establish some properties of the parameters of measures of uncertainty; the accuracy of approximation and the quality of approximation of classifications.