On Querying Incomplete Information in Databases under Bag Semantics

Abstract

Querying incomplete data is an important task both in data management, and in many AI applications that use query rewriting to take advantage of relational database technology. Usually one looks for answers that are certain, i.e., true in every possible world represented by an incomplete database. For positive queries -expressed either in positive relational algebra or as unions of conjunctive queriesfinding such answers can be done efficiently when databases and query answers are sets. Real-life databases however use bag, rather than set, semantics. For bags, instead of saying that a tuple is certainly in the answer, we have more detailed information: namely, the range of the numbers of occurrences of the tuple in query answers. We show that the behavior of positive queries is different under bag semantics: finding the minimum number of occurrences can still be done efficiently, but for maximum it becomes intractable. We use these results to investigate approximation schemes for computing certain answers to arbitrary first-order queries that have been proposed for set semantics. One of them cannot be adapted to bags, as it relies on the intractable maxima of occurrences, but another scheme only deals with minima, and we show how to adapt it to bag semantics without losing efficiency.

Fragments of bag relational algebra: Expressiveness and certain answers

“…Then, in Section 5.2, we focus on the problem MIN θ [q] and prove the exact tractability boundary shown in Figure 1. We start by showing that in all of the fragments up to RA + {∩}, the value of min can be computed by naive evaluation of queries, which extends a result in [14]. We then show that outside this fragment the problem is intractable, namely NP-complete for <, coNP-complete for >, and DP-complete for =.…”

“…There is a much bigger variety of relational algebra fragments for bags, but little is known about finding min(ā, q, D) and max(ā, q, D) for queries in those fragments. We know that min is easy to compute for RA + queries and that for full RA the problem is computationally hard: checking whether min(ā, q, D) ≥ n is NP-complete [2,14]. Checking whether max(ā, q, D) ≥ n is NP-complete even for SPC queries [14].…”

“…We know that min is easy to compute for RA + queries and that for full RA the problem is computationally hard: checking whether min(ā, q, D) ≥ n is NP-complete [2,14]. Checking whether max(ā, q, D) ≥ n is NP-complete even for SPC queries [14]. The complexity of actually computing min and max (or, in terms of a decision problem, checking whether min(D, q, D) = n, and likewise for max) is still open.…”

“…In this model databases are populated by constants and nulls, coming from two disjoint and countably infinite sets denoted by Const and Null, respectively. More formally, a k-ary relation is a finite bag of k-ary tuples over Const [14]). Let D be a database, let q be a relational algebra expression of arity n, and letā ∈ Const(D) n be a tuple of constants, we define min(ā, q, D) and max(ā, q, D) as follows: Thus, from now on we assume min(ā, q, D) and max(ā, q, D) as our standard notion of query answers and study their complexity.…”

“…Next, in Section 5.3, we look at MAX θ [q]. It was shown in [14] that for > the problem is NP-complete, even for very simple queries. Here, we complete the picture and settle the case for =, even when q is a query that simply returns a relation from the database.…”