Query Evaluation with Null Values: How Complex is Completeness?

TL;DR: This paper reexamine the proof-theoretic approach proposed by Reiter (which as proposed is incomplete) and presents a “natural” interpretation of null values based on various possible null assignments and shows that this approach leads to completeness of query evaluation.

Abstract: The problem of evaluating queries on a relational database which is allowed to contain null values has been extensively studied. In general, most of the approaches to query evaluation in the literature seem to fall into two categories. Those in the first guarantee that answers to queries can be efficiently computed (i.e. in time polynomial in the database size), while being “incomplete” in the sense that they do not compute all “valid” answers to certain queries. The second kind guarantee “completeness” but unfortunately suffer from intractability. In this paper, we reexamine the proof-theoretic approach proposed by Reiter [Re 86] (which as proposed is incomplete) and present a “natural” interpretation of null values based on various possible null assignments. We show that this approach leads to completeness of query evaluation. We bring out the drawback of such an extension by showing that evaluation of even “simple” queries using this approach is co-NP-complete. We then propose an approach based on intuitionistic logic for the problem. The advantages are that query evaluation is now guaranteed to be complete (w.r.t. the new approach) and computable in time polynomial in the database size.