Two Applications of Model-Theoretic Forcing to Lipski’s Data Bases with Incomplete Information 1

This paper concerns two aspects of incomplete information in data bases:

- computations of answers to queries which are externally interpreted in some incomplete first-order structures with dependencies, and

- the proper treatment of modal operators in such structures.

We introduce the notion of an incomplete first-order structure with dependencies which seems to be an adequate model of data bases with incomplete information.

We show that the widely accepted implicit assumption that first-order models (of a data base) have a unique domain, has no first-order consequences if the set of premises (represented in the data base) contains only a finite amount of information explicitly involving the elements of this domain. This observation allows us to evaluate the degree of unsolvability of the problem of answering externally interpreted queries in the incomplete first-order structures.

Moreover, we propose a forcing-based definition of the internal interpretation of modal queries in such structures, and investigate some of the properties of this interpretation.