A Model-Theoretic Semantics of Integrity Constraints in Deductive Databases

Conclusion We have developed the sustained model semantics for query answering and integrity checking in deductive databases. Several points have motivated this endeavor. First, it appeared as disturbing that, while most of today's semantics for query answering are three-valued, most semantics in the literature for answering the query if integrity is satisfied or violated are two-valued. After all, the answer "un-known" may make as much sense to the query about integrity as to any other query. Hence, we have developed a three-valued view of integrity which concedes that integrity may be neither violated nor satisfied in certain cases. Second, we found that most semantics for integrity are, in a sense, one-sided, by paying a higher attention to either satisfaction or violation. We have argued that an application-independent semantics for integrity should not be one-sided wrt conditions that ensure satisfaction (or, resp., violation) while being less concerned wrt the opposite. Rather, it should be left to the particular requirements of a given application to relax positions, for taking sides wrt weaker or stronger notions of satisfaction or violation of integrity, if necessary. This led to a refinement of our three-valued view, resulting in a non-biased two-valued semantics. The four possible cases that an integrity constraint is either satisfied by each sustained model, or only weakly satisfied by some sustained model, or that it is neither satisfied nor violated (hence weakly violated), or that it is violated in all sustained models, might even suggest to investigate a four-valued semantics for integrity. Third, there was a need to clarify the relationships between the traditional the-oremhood and consistency views of integrity [SK] and more recent proposals concerning database integrity in abductive logic programming [KM1] [KKT] [TK]. That need arose in the context of ongoing work on using abductive logic programming for knowledge assimilation in deductive databases [De5− 8]. To repeat, the starting point of this paper was a somehow uneasy feeling wrt the subtle diversity of existing proposals for capturing the meaning of integrity constraints. This paper is not the first in this regard. Similar observations have been made in [Re2]. However, the conclusions drawn in [Re2] lead into a direction which is different from ours. Reiter proposes to make the epistemic meta-level character of integrity constraints explicit by using a first-order calculus enhanced with modal operators. As opposed to that, my goal is to see how far model theory together with abductive logic …