Counting Database Repairs under Primary Keys Revisited

Abstract

Consistent query answering (CQA) aims to deliver meaningful answers when queries are evaluated over inconsistent databases. Such answers must be certainly true in all repairs, which are consistent databases whose difference from the inconsistent one is somehow minimal. An interesting task in this context is to count the number of repairs that entail the query. This problem has been already studied for conjunctive queries and primary keys; we know that it is #Pcomplete in data complexity under polynomial-time Turing reductions (a.k.a. Cook reductions). However, as it has been already observed in the literature of counting complexity, there are problems that are "hard-to-count-easy-to-decide", which cannot be complete (under reasonable assumptions) for #P under weaker reductions, and, in particular, under standard many-one logspace reductions (a.k.a. parsimonious reductions). For such "hard-to-count-easy-to-decide" problems, a crucial question is whether we can determine their exact complexity by looking for subclasses of #P to which they belong. Ideally, we would like to show that such a problem is complete for a subclass of #P under many-one logspace reductions. The main goal of this work is to perform such a refined analysis for the problem of counting the number of repairs under primary keys that entail the query.

Counting Problems over Incomplete Databases

“…Concerning approximation results, it is known that the problems #Repairs( , Σ) and Prob( ) admit an FPRAS in some important settings. In particular, when is a union of BCQs, this holds for #Repairs( , Σ) when Σ is a set of primary keys [15], and for Prob( ) over BID and tuple-independent probabilistic databases [18]. We observe here that this is reminescent of our Corollary 5.3, which shows that problems of the form #Val( ) have an FPRAS for every union of BCQs.…”

Evaluating top-k queries with inconsistency degrees

“…Usually, there exist several possible repairs of a given database and for such a reason, following the certain answers semantics, consistent answers to a given query are defined as being the intersection of the query answers on all possible repairs of the initial database instance. The problem of consistent query answering has been investigated in several settings depending on: (i) the definition of repairs, where a repair is defined in terms of the sets of inserted and/or deleted tuples [11,10,34,13], or updates [39,32,6] and (ii) the considered integrity constraints and the query language. The latter has been studied for a variety of queries and constraints such as first-order queries and binary universal constraints or single functional dependencies [5], union of conjunctive queries and key and inclusion dependencies [14], first-order queries and denial constraints [16], to mention a few.…”

Benchmarking Approximate Consistent Query Answering

“…The key elements underlying CQA are (1) the notion of repair of an inconsistent database D, that is, a consistent database whose difference with D is somehow minimal, and (2) the notion of query answering based on certain answers, i.e., answers that are entailed by every repair. Here is a simple example taken from [6]:…”

“…The notion of certain answers only says that a candidate answer is entailed by all repairs, or is not entailed by at least one repair. But, as it has been discussed in [6,7], the former is too strict, while the latter is not very useful in a practical context.…”

“…With the aim of obtaining more informative answers, we would like to rely on a notion that is more flexible than certain answers that tells us how likely a candidate answer is to be a consistent answer. This is achieved via the notion of relative frequency, which essentially computes the percentage of repairs that entail a candidate answer [6]. To illustrate this notion, let us consider again Example 1.1.…”