Naïve Evaluation of Queries over Incomplete Databases

Abstract

The term naïve evaluation refers to evaluating queries over incomplete databases as if nulls were usual data values, that is, to using the standard database query evaluation engine. Since the semantics of query answering over incomplete databases is that of certain answers, we would like to know when naïve evaluation computes them, that is, when certain answers can be found without inventing new specialized algorithms. For relational databases it is well known that unions of conjunctive queries possess this desirable property, and results on preservation of formulae under homomorphisms tell us that, within relational calculus, this class cannot be extended under the open-world assumption.Our goal here is twofold. First, we develop a general framework that allows us to determine, for a given semantics of incompleteness, classes of queries for which naïve evaluation computes certain answers. Second, we apply this approach to a variety of semantics, showing that for many classes of queries beyond unions of conjunctive queries, naïve evaluation makes perfect sense under assumptions different from open world. Our key observations are: (1) naïve evaluation is equivalent to monotonicity of queries with respect to a semanticsinduced ordering, and (2) for most reasonable semantics of incompleteness, such monotonicity is captured by preservation under various types of homomorphisms. Using these results we find classes of queries for which naïve evaluation works, for example, positive first-order formulae for the closed-world semantics. Even more, we introduce a general relation-based framework for defining semantics of incompleteness, show how it can be used to capture many known semantics and to introduce new ones, and describe classes of first-order queries for which naïve evaluation works under such semantics. ACM Reference Format:Amélie Gheerbrant, Leonid Libkin, and Cristina Sirangelo. 2014. Naïve evaluation of queries over incomplete databases.

Correctness of SQL Queries on Databases with Nulls

“…FACT 1 ( [12,18,25]). For positive relational algebra queries, naïve evaluation computes exactly certain answers with nulls, and thus it has correctness guarantees.…”

Negative Knowledge for Certain Query Answers

“…Using informativeness ordering, one can state when a query answering algorithm behaves rationally: this happens if it produces more informative answers on more informative inputs. For relational databases, these ideas led to new large classes of queries for which certain answers can be computed efficiently [17], and to a new account of many-valued query answers [13], as employed by all standard DBMSs [14].…”

“…To answer the first question, we follow the approach of [17,28] which treats incompleteness at an abstract level applicable to many data models. The key elements of the approach are the notions of complete and incomplete models, the semantics of an incomplete object, which is a set of complete ones it can denote, and a set of formulae representing knowledge about objects.…”

“…The semantic function makes it possible to define informativeness ordering, which says when one object is more informative than another. The restriction of frameworks in [17,28] was that knowledge was positive: if a fact is known about an object, it remains true in more informative ones. Negative knowledge is not such: we can think of it as saying that we do not know some fact about an object; therefore, we do not know that fact about less informative objects.…”

“…A general model We now recall the basic setting of [17,28] that lets us talk about the essential features of incompleteness without recourse to a particular data model. The two basic concepts are objects, and formulae they satisfy.…”