Dependency Structures for Decision Tables

In this paper we study the decision tables from a more general and abstract point of view. We focus our attention on the {em consistency} of decision tables. The key point of our analysis is that we want to extrapolate the best possible degree of knowledge by inconsistent tables that can arise in a natural way starting from situations in which there is no a priori correlation between condition attributes and the decision ones. In order to study inconsistencies, we analyze the membership of the $A$-indiscernibility class of any element to a local positive region. We formalize our approach by means of an operatorial perspective, providing in this way a more abstract mathematical context to the study of decision tables. In this perspective, we are also interested in finding which condition subsets preserve the local positive region. This is the main reason to introduce the notions of {em local positive essentials} and {em local positive reducts}. These attribute subset families, generally, do not satisfy the properties of their corresponding notions for information tables. Hence, in order to extend naturally these results to the decision tables case, we can follow two different approaches: to define a subclass of decision tables in which they hold or to change the nature of the hypergraph induced by the decision discernibility matrix.