Semantic Acyclicity for Conjunctive Queries: Approximations and Constraints

Abstract

Evaluation of conjunctive queries (CQs) is NP-complete, but becomes tractable for syntactically defined fragments. One of the oldest and most studied such fragments is the class of acyclic CQs. Here we look at the problem of semantic acyclicity, i.e., given a CQ q, is there an acyclic CQ $q'$ that is equivalent to it? This notion is important in CQ evaluation, as semantically acyclic CQs can be evaluated in polynomial time. The notion of semantic acyclicity itself is decidable, with the same complexity as the usual static analysis tasks for CQs, i.e., NP-complete.

Unfortunately, semantic acyclic is not general enough for practical purposes, as only CQs whose core is acyclic belong to this class. In this tutorial we present two approaches that have been developed to make the notion more flexible and take better advantage of the ideas that underlie it. These are computing approximations and making use of semantic information in the form of constraints. For approximations, we look at the case when q is not semantically acyclic and explain how to find and evaluate those acyclic CQs $q'$ that are as “close” as possible to q in terms of containment. As for constraints, they enrich semantic acyclicity since they can be applied on a CQ q to produce an acyclic reformulation of it. We present results that establish the boundary of decidability for semantic acyclicity under usual database constraints such as tuple and equality-generating dependencies, and show their applicability in query evaluation.