Semantic Optimization of Conjunctive Queries

Abstract

This work deals with the problem of semantic optimization of the central class of conjunctive queries (CQs). Since CQ evaluation is NP-complete, a long line of research has focussed on identifying fragments of CQs that can be efficiently evaluated. One of the most general such restrictions corresponds to generalized hypetreewidth bounded by a fixed constant k ≥ 1; the associated fragment is denoted GHW k. A CQ is semantically in GHW k if it is equivalent to a CQ in GHW k. The problem of checking whether a CQ is semantically in GHW k has been studied in the constraint-free case, and it has been shown to be NP-complete. However, in case the database is subject to constraints such as tuple-generating dependencies (TGDs) that can express, e.g., inclusion dependencies, or equality-generating dependencies (EGDs) that capture, e.g., key dependencies, a CQ may turn out to be semantically in GHW k under the constraints, while not being semantically in GHW k without the constraints. This opens avenues to new query optimization techniques. In this paper, we initiate and develop the theory of semantic optimization of CQs under constraints. More precisely, we study the following natural problem: Given a CQ and a set of constraints, is the query semantically in GHW k , for a fixed k ≥ 1, under the constraints, or, in other words, is the query equivalent to one that belongs to GHW k over all those databases that satisfy the constraints? We show that, contrary to what one might expect, decidability of CQ containment is a necessary but not a sufficient condition for the decidability of the problem in question. In particular, we show that checking whether a CQ is semantically in GHW 1 is undecidable in the presence of full TGDs (i.e., Datalog rules) or EGDs. In view of the above negative results, we focus on the main classes of TGDs for which CQ containment is decidable, and that do not capture the class of full TGDs, i.e., guarded, non-recursive and sticky sets of TGDs, and show that the problem in question is decidable, while its complexity coincides with the complexity of CQ containment. We also consider key dependencies over unary and binary relations, and show that the problem in question is decidable in elementary time. Furthermore, we investigate whether being semantically in GHW k alleviates the cost of query evaluation. Finally, in case a CQ is not semantically in GHW k , we discuss how it can be approximated via a CQ that falls in GHW k in an optimal way. Such approximations might help finding "quick" answers to the input query when exact evaluation is intractable.

The Limits of Efficiency for Open- and Closed-World Query Evaluation Under Guarded TGDs

“…From Theorem 5.1 and Proposition 5.5, we get a 2E T upper bound for deciding uniform UCQ k -equivalence for guarded CQSs, while the lower bound is inherited from [8]. As in Theorem 5.1, k should be greater than the arity of the schema, minus one.…”

“…The fact that (3) implies ( 1) is shown in [8], while (1) implies ( 2) holds by definition. The interesting task is to show that (2) implies (3).…”

The Complexity of Conjunctive Queries with Degree 2

“…(3) In consequence, we show that BCQ over a class of hypergraphs H is tractable if and only if H has bounded generalised hypertree width. We extend the result to classes of queries with bounded semantic generalised hypertree width [4] and to the corresponding counting problem of counting answers of CQs.…”

“…Note that semantic width is also commonly referred to as width modulo homomorphism in the literature since CQ equivalence coincides with homomorphic equivalence of queries. For full details and related definitions see [4].…”

“…Let H core( Q) be the class of all hypergraphs of the cores of the queries in Q. It is known that the semantic generalised hypertree width sem-𝑔ℎ𝑤 (𝑞) of a CQ 𝑞 is precisely ghw(core(𝑞)) [4]. Thus, if Q has unbounded sem-𝑔ℎ𝑤, H core( Q) has unbounded ghw.…”