On Databases with Incomplete Information

Abstract

Semantic and logical problems arising in an incomplete information database are investigated. A simple query language is described, and its semantics, which refers the queries to the information about reality contained in a database, rather than to reality itself, is defined. This approach, called the internal interpretation, is shown to lead in a natural way to the notions of a topological Boolean algebra and a modal logic related to $4 in the 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 interpretation for a broad class of queries. An interesting special case of the problem of determining the internal interpretation amounts to deciding whether an assertion about reality (a "yes-no" query) is consistent with the incomplete information about reality contained in a database. A solution to this problem, which relies on the classical combinatorial problem of distinct representatives of subsets, is given.