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Incomplete and Nondeterministic Information Systems: Object-Directed Semantics for Descriptor Languages

In the paper we discuss logical approaches to incomplete and/or nondeterministic data. As is well-known, complete and deterministic information systems induce indiscernibility relations and the lower and upper approximations regarded as operators obey the laws of S5 modal system. In the case of incomplete and/or nondeterministic systems, this modal approach yields a few more binary relations on the set of objects (e.g. NIL-structures and NIL-logics). Anyway, in both cases, there are relations which play a dominant role. The main idea of this study is to shift the focus from relations to objects - that is why we use the term object-directed. Following well-established traditions from modal logics, we would like to consider objects with empty or non-single attribute values as a special kind of worlds. In consequence, in the object-directed approach there would be two sorts of objects and (usually) one relation in contrast to the relation-directed approach where we have one sort of objects and a number of relations. In the first part of our paper we shall discuss a global kind of non-normality and show how rough approximations can be linked to weak modal systems. In the second part we shall consider a local kind of non-normality; this time we use a multi-valued modal system Q introduced by A. N. Prior. The key idea offered by the paper is to regard incomplete and/or nondeterministic information systems as a source of non-normal models for (modal) descriptor languages.