Dual Tableaux for Information Logics of Relative Frames

Abstract

In the preceding chapter in Sect. 11.3 we demonstrated that a specification of information relations which would be meaningful for information systems with incomplete information requires an explicit reference to a set of attributes with respect to which the relations are defined. In order to incorporate in a logical formalism the sets of attributes which determine the relations, the notions of a relative relation and a relative frame i.e., the frame whose relations are relative, were introduced in [Orł88], see also [DO02]. More precisely, a relative frame consists of a family, or several families of relations such that the relations within the family are indexed with subsets of a set of parameters, intuitively understood as attributes of an information system. Apart from the ordinary properties of relations such as, for example, symmetry or transitivity, relative relations may have some properties which refer to the family of relations as a whole. These properties are collectively named global properties of relations. For example, one of the typical global property says that a relation indexed with the union of two sets equals intersection of relations indexed with the components of the union. Such an assumption is relevant, among others, when the family consists of equivalence relations.