On Data Bases with Incomplete Information

Abstract

Semantic and logical problems arising in an incomplete information data base are investigated. A simple query language is described and its semantics is defined, which refers the queries to the information about reality contained in a data base, rather than to reality itself. This approach, called the internal interpretation, is shown to lead in a natural way to the notions of a topological Booleans algebra and a model logic related to S4, in teh same way as referring queries directly to reality (external interpretation) leads to Boolean algebras and classical logic. An axiom system is given for equivalent (with respect to the internal interpretation) transformation of queries, which is then exploited as a basic tool in a method for computing the internal intepretation for a broad class of queries. An interesting special case of the problem of determining the internal intepretation amounts to deciding whether an assertion about reality (a "yes-no" query) is consistent with the incomplete information about reality contained in a data base. We give a solution to this problem, which relies on the classical combinatorial problem of distinct representatives of subsets.