Querying the Guarded Fragment

Abstract

Abstract. Evaluating a Boolean conjunctive query q against a guarded first-order theory ϕ is equivalent to checking whether ϕ ∧ ¬q is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories ϕ and (unions of) conjunctive queries q that (i) ϕ |= q iff ϕ |= fin q, that is, iff q is true in every finite model of ϕ and (ii) determining whether ϕ |= q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded bisimulation invariant PTIME.

Finite model reasoning over existential rules

“…Conversely, the proof of this property for the subsequent three classes has been a very different matter. Complex, yet intriguing, constructions have been devised for linear (Rosati 2006;Bárány et al 2014), guarded (Bárány et al 2014), and more recently for sticky (Gogacz and Marcinkowski 2013). To complete the picture, we have addressed the same problem for shy and get the following positive result, which is the main contribution of the paper.…”

“…The problem of answering a Boolean query q against a logical theory consisting of an extensional database D paired with an ontology Σ is attracting the increasing attention of scientists in various fields of Computer Science, ranging from Artificial Intelligence (Baget et al 2011;Calvanese et al 2013; to Database Theory (Bienvenu et al 2014;Bourhis et al 2016) and Logic (Pérez-Urbina et al 2010;Bárány et al 2014;). This problem, called ontology-based query answering, for short OBQA (Calì et al 2009b), is usually stated as D ∪ Σ |= q, and it is equivalent to checking whether q is satisfied by all models of D ∪ Σ according to the standard approach of first-order logics, yielding an open world semantics.…”

“…This is usually stated as D ∪ Σ |= q if, and only if, D ∪ Σ |= fin q (where |= fin stands for entailment under finite models), as the "only if" direction is always trivially true. And there are contexts, like in databases (Johnson and Klug 1984;Rosati 2006;Bárány et al 2014), in which reasoning with respect to finite models is preferred.…”

Ontology-based data access

“…Note that all ontology languages considered in this paper enjoy finite controllability, meaning that finite relational structures can be replaced with unrestricted ones without changing the certain answers to unions of conjunctive queries [6,7].…”

“…2 To overcome this restriction, we consider the guarded fragment of first-order logic and the unary-negation fragment of first-order logic [6,46]. Both generalize the description logic ALC in different ways.…”